Research ArticleField Equations and Radial Solutions in a NoncommutativeSpherically Symmetric Geometry

Aref Yazdani

Department of Physics Faculty of Basic Sciences University of Mazandaran PO Box 47416-95447 Babolsar Iran

Correspondence should be addressed to Aref Yazdani ayazdanistuumzacir

Received 10 June 2014 Revised 6 October 2014 Accepted 19 October 2014 Published 11 November 2014

Academic Editor Luis A Anchordoqui

Copyright copy 2014 Aref Yazdani This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study a noncommutative theory of gravity in the framework of torsional spacetime This theory is based on a Lagrangianobtained by applying the technique of dimensional reduction of noncommutative gauge theory and that the yielded diffeomorphisminvariant field theory can be made equivalent to a teleparallel formulation of gravity Field equations are derived in the frameworkof teleparallel gravity through Weitzenbock geometry We solve these field equations by considering a mass that is distributedspherically symmetrically in a stationary static spacetime in order to obtain a noncommutative line elementThis new line elementinterestingly reaffirms the coherent state theory for a noncommutative Schwarzschild black hole For the first time we derive theNewtonian gravitational force equation in the commutative relativity framework and this result could provide the possibility toinvestigate examples in various topics in quantum and ordinary theories of gravity

1 Introduction

Field equations of gravity and radial solutions have beenpreviously derived in noncommutative geometry [1ndash5] Thegeneralization of quantum field theory by noncommutativitybased on coordinate coherent state formalism also curesthe short distance behavior of point-like structures [6ndash13] In this method the particle mass 119872 instead of beingcompletely localized at a point is dispensed throughout aregion of linear size radic120579 substituting the position Dirac-deltafunction describing point-like structures with a Gaussianfunction and describing smeared structures In other wordswe assume that the energy density of a static sphericallysymmetric particle-like gravitational source cannot be adelta function distribution and will be given by a Gaussiandistribution of minimal width radic120579 as follows

120588120579 (119903) =119872

(4120587120579)32

exp(minus1199032

4120579) (1)

Furthermore noncommutative gauge theory appears instring theory [14ndash18] the boundary theory of an open stringis noncommutative when it ends on D-bran with a constant

B-field or an Abelian gauge field (particularly see [14])Therefore closed string theories are expected to remaincommutative as long as the background is geometric Recentevidence has found a connection between nongeometry andclosed string noncommutativity and even nonassociativity[19ndash21] approaches using dual membrane theories [22] andmatrix models [23 24] arrive at the same conclusion

The ordinary quantumfield theory is unable to present anexact description of exotic effects of the inherent nonlocalityof interactions so we need a model to provide an effectivedescription of many of the nonlocal effects in string theorywithin a simpler setting [25]

The model leads to the gauge theories of gravitationthrough an ordinary class of dimensional reductions of non-commutative electrodynamics on flat space which then canbe made equivalent to a formulation of teleparallel gravitymacroscopically describing general relativity Moreover thismodel is developed by the parallel theories of gravitationgiving a clear understanding of Einsteinrsquos principle of absoluteparallelism It is defined by a nontrivial vierbein field andformed by a linear connection For carrying nonvanishingtorsion this connection is known as Wietzenbock geometryon spacetime

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 349659 9 pageshttpdxdoiorg1011552014349659

2 Advances in High Energy Physics

This model is given appropriately by a noncommutativeLagrangian and introduced by authors in [4 5] Admittedlythis Lagrangian and the relevant explanationswill be the basisof our next general calculations In this paper are going to usethe Greek alphabet (120583 ] 120588 = 0 1 2 3) to denote indicesrelated to spacetime and the first half of the Latin alphabet(119886 119887 119888 = 0 1 2 3) to denote indices related to thetangent space AMinkowski spacetime whose Lorentzmetricis assumed to have the form of 120578119886119887 = diag(minus1 +1 +1 +1)Themiddle letters of the Latin alphabet (119894 119895 119896 = 1 2 3)will bereserved for space indices The noncommutative Lagrangianis expressed as

119866119903 =1205940

1198902det (ℎ

1205901015840

120590) 1205781205831205831015840

times [1

4120578]]1015840

1205781205821205821015840 120582

120583]1205821015840

1205831015840]1015840 minus ]120583]

]10158401205831015840]1015840 +

1

2]1015840120583]

]1205831015840]1015840]

(2)

In the usual way having a Lagrangian which describesgravitation based on noncommutative background is likethose of gauge theories written in terms of contractions ofits field strength here represented by torsion of Weitzenbockconnection Its behavior under a local change of Δ 120583 isthe main invariance property of the particular combinationtorsion tensor fields Here 119890 is Yang-Mills coupling constantnoncommutative scale determines the Planck length and thePlanck scale of 119899-dimensional spacetime is given by

119896 = radic16120587119866119873 = 11989010038161003816100381610038161003816119875119891119886119891119891 (Θ

119860119861)10038161003816100381610038161003816

12119899

(3)

In mass dimension 2 the weight constant 1205940 is

1205940 =10038161003816100381610038161003816119875119891119886119891119891 (Θ

119860119861)10038161003816100381610038161003816

minus1119899

(4)

In the commutative limit it reduces to gravitational constantTherefore Θ

119860119861 is a noncommutative parameter defined as

Θ119860119861

= (120579120583]

120579120583119887

120579120583119887

120579119886119887) 997888rarr 120579

120583]= 120579119886119887

= 0 (5)

By considering the calculation of superpotential and energy-momentum current with respect to noncommutative gaugepotential given by 119861

120583

119886= | det(120579120583

10158401198861015840

)|

12119899

120579]120583

120596119886] the version ofnon-commutative gravitational field equations is produced120596119886] are gauge fields corresponding to the gauging of thetranslation group that is replacing 119877

119899 by the Lie algebra119892 of local gauge transformations with gauge functions andits relation with the non-trivial tetrad field is expressed asℎ120583

119886= 120575120583

119886minus 119890120579

]120583120596119886] and 120575

120583

119886has the perturbative effect in the

trivial holonomic tetrad fields of flat spaceIt is important to note that by applying the ldquodimensional

reduction of gauge theoriesrdquo noncommutative electrody-namics gauge field shown by the noncommutative Yang-Miles theory reduces to the gauge theories of gravitationwhich naturally yields Weitzenbock geometry on the space-timeAlso the induced diffeomorphism invariant field theorycan bemade equivalent to a teleparallel formulation of gravitymacroscopically describing general relativity In Section 2

we show that our Lagrangian can be made equivalent withgeneral relativity In Section 3 we are going to derive thefield equations by utilizing various definitions of teleparal-lel gravity By simplifying and solving the field equationswe obtain the line element in the spherically symmetricspacetime in Section 4 We continue our discussion withinvestigations about the limiting cases of our line element andhorizons of noncommutative Schwarzschild black hole in thismethod Finally we show how the Newtonian gravitationalforce equation can be derived from our line element in thecommutative limit in Section 5

2 Equivalence with General Relativity

In order to continue our discussion to achieve to noncom-mutative field equations we should show how our model canbe coupled with general relativity With respect to the givenrelation of

Γ120588

120583] = Γ120588

120583] + 120588

120583] (6)

for the vanishing curvature of the Weitzenbock connectionwe have

120588

120579120583] = 119877120588

120579120583] + 120588

120579120583] equiv 0 (7)

where

119877120588

120579120583] = 120597120583Γ120588

120579] minus 120597]Γ120588

120579] + Γ120588

120590120583Γ120590

120579] minus Γ120588

120590]Γ120590

120579120583 (8)

is the curvature of the Levi-Civita connection The aboveequations show that whereas in general relativity torsionvanishes in teleparallel gravity it is curvature that vanishesWe rewrite (7) based on their components in order to findthe scaler of

120588

120579120583] therefore we have

120588

120579120583] = (120597120583120588

120579] minus 120597]120588

120579120583+ 120588

120590120583120590

120579] minus Γ120588

120590]120590

120579120583

minus Γ120590

120579120583120588

120590] + Γ120590

120579]120588

120590120583) +

120588

120590]120590

120579120583minus 120588

120590120583120590

120579]

(9)

that is the tensor written in terms of the Weitzenbockconnection only Like the Riemanian curvature tensor it is 2-form assuming values in the Lie algebra of the Lorentz group(see [26 27]) By taking appropriate contractions it is easy toshow that

120588

120579120583] = (120583120588

120579] minus ]120588

120579120583) +

120588

120590]120590

120579120583minus 120588

120590120583120590

120579] (10)

By considering (20) and the following term

minus119877 = equiv1

2Λ120579

120588120588

120579120583]119889119909120583

and 119889119909] (11)

we achieve to the scalar version of (7)

119877 equiv (120583]120588

120588]120583 minus ]120583120588

120583120588

] ) +2

ℎ120597120583 (ℎ

]120583] ) (12)

The Lagrangian of (2) can be written in a simple form of

=1205940

1198902det (ℎ

1205901015840

120590) (120583]120588

120588]120583 minus ]120583120588

120583120588

] ) (13)

Advances in High Energy Physics 3

with a combination of (12) and (13) where takes thefollowing form

=1205940

1198902det (ℎ

1205901015840

120590) (119877 minus

2

ℎ(120597120583 (ℎ

]120583] ))) (14)

By considering (3) and (4) exchanges to

= 119871 minus 120597120583 (ℎ

8120587119866]120583] ) (15)

up to a divergence at the commutative limit therefore theLagrangian of (2) is equivalent to the Lagrangian of generalrelativity

=minus1

16120587119866radicminus119892119877 (16)

is the Einstein-Hilbert Lagrangian of general relativity How-ever this result could be extended with many further termsbut this is enough to derive a valid field equations

3 Noncommutative Field Equations

In this section we are going to present a reformulation ofteleparallel gravity (which is made equivalent to general rela-tivity) Due to the introduced noncommutative Lagrangian(2) we are able to derive the field equations similarly tothe teleparallel method Weitzenbock geometric definitionsand some well-known concepts of general relativity [28ndash30]and teleparallel gravity are required (more explanations aboutthese equations can be found in [29 31 32]) In 4-dimensionthe noncommutative action integral is given by

119878 = int 1198661199031198894119909 (17)

Under an arbitrary variation 120575ℎ120583

119886of the tetrad field the action

variation is written in the following form

120575119878 = int Ξ119886

120583120575ℎ120583

119886ℎ1198894119909 (18)

where

ℎΞ119886

120583=

120575119866119903

120575119861120583

119886

equiv120575119866119903

120575ℎ120583

119886

=120597119866119903

120597ℎ120583

119886

minus 120597120582

120597119866119903

120597120582120597ℎ120583

119886

(19)

is the matter energy-momentum tensor (More definitionsabout this tensor can be found in [33]) Now consider firstan infinitesimal Lorentz transformation as

Λ119887

119886= 120575119887

119886+ 120576119887

119886 (20)

where 120576119887

119886= minus120576119887

119886 Because of such transformation the tetrad

should be changed as

120575ℎ120583

119886= 120576119887

119886ℎ120583

119887 (21)

The requirement of invariance of the action under localLorentz transformation therefore yields

int Ξ119887

119886120576119887

119886ℎ1198894119909 = 0 (22)

Since 120576119887

119886is antisymmetric symmetric of energy-momentum

tensor yields some specific results that can be seen in [28]Consider spacetime coordinates that are transformed asfollows

1199091015840120588

= 119909120588

+ 120577120588 (23)

Whereby we retrieve the tetrad in the form of

120575ℎ120583

119886equiv ℎ1015840120583

119886(119909) minus ℎ

120583

119886(119909) = ℎ

120588

119886120597120588120577120583

minus 120577120588120597120588ℎ120583

119886 (24)

Substituting in (18) we have

120575119878 = int Ξ119886

120583[ℎ119886

120588120597120583120577120588

minus 120577120588120597120588ℎ120583

119886] ℎ1198894119909 (25)

or equivalently

120575119878 = int [Ξ120588

119888120597120588120577119888

+ 120577119888Ξ120588

120583120597120588ℎ120583

119888minus 120577120588120597120588ℎ120583

119886] ℎ1198894119909 (26)

Substitute the identity

120597120588ℎ120583

119886= 119887

119886120588ℎ120583

119887minus Γ120583

120582120588ℎ120582

119886 (27)

where is the spin connection in teleparallel gravity Theimportant property of teleparallel gravity is that its spinconnection is related only to the inertial properties of theframe not to gravitation In fact it is possible to choose anappropriate frame in which it vanishes everywhereWe knowthat the above formula vanishes by (41) (see also [34 35])and making use of the symmetric of the energy-momentumtensor the action variation assumes the form of

120575119878 = int Ξ120588

119888[120597120588120577119888

+ (119888

119887120588minus 119888

119887120588) 120577119887] ℎ1198894119909 (28)

Integrating the first term by parts and neglecting the surfaceterm the invariance of the action yields

int [120597120583 (ℎΞ120583

119886) minus (

119887

119886120583minus 119887

119886120583) (ℎΞ120583

119887)] 120577119886ℎ1198894119909 = 0 (29)

From arbitrariness of 120577119888 under the covariant derivative 120583 it

follows that

120583ℎΞ120583

119886equiv 120597120583 (ℎΞ

120583

119886) minus (

119887

119886120583minus 119887

119886120583) (ℎΞ120583

119887) = 0 (30)

By identity of

120597120588ℎ = ℎΓ]]120588 equiv ℎ (Γ

]120588] minus

]120588]) (31)

the above conservation law becomes

120597120583Ξ120583

119886+ (Γ120583

120588120583minus 120583

120588120583) Ξ120588

119886minus (119887

119886120583minus 119887

119886120583) Ξ120583

119887= 0 (32)

In a purely spacetime form it reads

120583Ξ120583

120582equiv 120597120583Ξ

120583

120582+ (Γ120583

120583120588minus 120583

120588120583) Ξ120588

120582minus (Γ120588

120582120583minus 120588

120582120583) Ξ120583

120588= 0 (33)

4 Advances in High Energy Physics

This is the conservation law of the source of energy-momentum tensor Variation with respect to the noncom-mutative gauge potential 119861

120583

119886yields the noncommutative

teleparallel version of the gravitational field equations

120597120590 (ℎ 119878120583120590

119886) minus 119896ℎ 119869

120583

119886= 119896ℎΞ

120583

119886 (34)

where

ℎ 119878120583120590

119886= ℎℎ120582

119886119878120583120590

120582equiv minus119896

120597

120597 (120597120590ℎ119886120583

)

(35)

which defines the superpotential For the gauge current wehave

ℎ 119869120583

119886= minus

120597

120597119861119886120583

equiv minus120597

120597ℎ119886120583

(36)

Note that the matter energy-momentum tensor which isdefined in this relation appears as the source of torsionsimilarly the energy-momentum tensor appears as the sourceof curvature in general relativity Our computation has led usto the following results

119878120583120590

119886= 2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578

119869120583

119886=

1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888

(37)

for noncommutative superpotential and gauge current Thelagrangian appears again in our equations but notice thatthis term cross its coefficient yields a term purely based onits field strength according to (2) This simplified expressionmaintains equivalence to general relativity We can observethat the gravitational field equations depend on the torsiononly Finally the field equations can be written as

120597120590 (ℎ (2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578))

minus 119896ℎ (1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888) = 119896ℎΞ

120583

119886

(38)

where 119896 = 12059401198902 is a constant These field equations are

similar to teleparallel field equations although it wouldbe distinguished with different field strength

120583120590

119886which is

given by the covariant rotational of noncommutative gaugepotential of 119861

120583

119886 Consider the following equations from the

teleparallel theory (see for instance [32 34ndash36])

119886

120583] = 120597]ℎ119886

120583minus 120597120583ℎ119886

] + 119886

119890]ℎ119890

120583minus 119886

119890120583ℎ119890

] (39)

Γ120588

]120583 = ℎ120588

119886120597120583ℎ119886

] + ℎ120588

119886119886

119887120583ℎ119887

] (40)

120597120583ℎ119886

] minus Γ120588

]120583ℎ119886

120588+ 119886

119887120583ℎ119887

] = 0 (41)

120588

]120583 = Γ120588

120583] minus Γ120588

]120583 (42)

The field equations take the exact following form

120597

120597119909120590(Γ120590

119886120583minus Γ120590

120583119886) minus

120597

120597119909120583Γ120582

119886120582+

120597

120597119909120582Γ120582

119886120583

minus Γ120578

119886120582Γ120582

120583120578+ Γ120578

119886120583Γ120582

120582120578=

1205940

1198902120588 (119903)

120597

120597119909119886

120597

120597119909120583

(43)

which unlike the left hand side of (38) is written purelybased on noncommutative field strength and the abovefield equation is written in terms of Weitzenbock connec-tion only Regarding the equivalency between correspondingLagrangians and the above simplified field equations andapplying (34) we have therefore

119877119886120583 minus1

2ℎ119886120583119877 = 119896Ξ119886120583 (44)

as equivalent to Einsteinrsquos field equations Note that (44) isnot Einsteinrsquos field equations but the teleparallel field equa-tions made equivalent to general relativity And equivalentmodel of teleparallel field equations with general relativity isexpressed in the references in detail (see for instance [32 3435]) We continue our discussion to derive noncommutativeline element by solving these field equations

4 Noncommutative Line Element

Teleparallel versions of the stationary static spherically axis-symmetric and symmetric of the Schwarzschild solutionhave been previously obtained [37 38] Within a frameworkinspired by noncommutative geometry we solve the fieldequations for a distribution of spherically symmetricallymassin a stationary static spacetime like the exterior solution ofSchwarzschild (see also [29])Then it is natural to assume thatthe line element is as follows

1198891199042

= minus119891 (119903) 1198891199052

+ 119892 (119903) 1198891199032

+ ℎ (119903) 1199032

(1198891205792

+ sin21205791198891206012)

(45)

With a new radial coordinate defined as 119903 = 119903radicℎ(119903) the lineelement becomes

1198891199042

= minus119860 (119903) 1198891199052

+ 119861 (119903) 1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (46)

Usually one replaces the functions 119860(119903) and 119861(119903) with expo-nential functions to obtain somewhat simpler expressionsfor the noncommutative tensor components Hence weintroduce the functions 120572(119903) and 120573(119903) by 119890

2120572(119903)= 119860(119903) and

119890120573(119903)

= 119861(119903) to get

1198891199042

= minus1198902120572

1198891199052

+ 1198902120573

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (47)

Tetrad components of the above metric take the followingform

ℎ119886

120583=

[[[

[

minus1198902120572

0 0 0

0 1198902120573 sin 120579 cos120601 119903 cos 120579 cos120601 minus119903 cos 120579 sin120601

0 1198902120573 sin 120579 sin120601 119903 cos 120579 sin120601 119903 sin 120579 cos120601

0 1198902120573 cos 120579 minus119903 sin 120579 0

]]]

]

(48)

Weitzenbock connection Γ120588

120583] has the following expression

Γ120588

120583] = ℎ120588

119886120597]ℎ119886

120583 (49)

Advances in High Energy Physics 5

Now we can calculate the nonvanishing components ofWeitzenbock connection as follows

Γ0

01= minus2120572

1015840 Γ

1

11= 21205731015840 Γ

1

22= minus119903119890

minus120573

Γ1

33= minus119903119890

minus120573sin2120579 Γ2

12=

119890120572

119903= Γ3

13

Γ2

21=

1

119903= Γ3

31 Γ

2

33= minus sin 120579 cos 120579

Γ3

23= Γ3

32= cot 120579

(50)

By replacing these components in (43) the noncommutativetensors of (51)ndash(53) for the left-hand side of the field equationswill produce the following expression

119873 =1

1199032(minus4119890minus2120573

+ 1 minus 120595120579) minus2

1199031205731015840119890minus2120573

=1205940

1198902120588 (119903) 120575 (51)

119873119903119903 =1

1199032(2119890minus2120573

+ 1 minus 120595120579) +2

1199031205721015840119890minus2120573

=1205940

1198902120588 (119903) 120575119903119903 (52)

119873120579120579

= 119873120601120601

=1

119903119890minus2120573

(11990312057210158401015840

+ 11990312057210158402

minus 11990312057210158401205731015840+ 1205721015840minus 1205731015840minus 1)

+ 12057210158402

119890minus2120573

=1205940

1198902120588 (119903) 120575

120579120579=

1205940

1198902120588 (119903) 120575

120601120601

(53)

Adding (51) and (52) we get simply

1

1199032(120595120579 minus 119890

minus2120573+ 119890minus2120573

(1205721015840minus 1205731015840) + 1) =

1205940

1198902120588 (119903) (54)

where 120572(119903) = 120573(119903) It should also be noted that by recalling(47) we can consider the limiting case for our solutionassuming (120572

1015840minus 1205731015840) = 119896 where 119896 is a constant and by

considering the time coordinate we can shift this constantto an arbitrary value It is possible therefore without loss ofgenerality to choose 119896 = 0 It does not contradict (47) toset 1205721015840

= 1205731015840 According to this analysis the equation 119873 =

(12059401198902)120588(119903)120575 can be written as

minus1

119903

119889

119889119903[119903 (119890minus2120573

minus 120595120579 minus 1)] =1205940

1198902120588 (119903) (55)

For a perfect fluid in thermodynamic equilibrium the stress-energy tensor takes on a particularly simple form

Ξ120583]

= (120588 + 119875) 119906120583119906]

+ 119901119892120583]

(56)

where the pressure 119875 can be neglected due to the distributionof mass and the gravitational effects consequently only oneterm will remain in the above formula as follows

Ξ119886120583

= 120588 (119903)119889119909119886

119889119905

119889119909120583

119889119905 (57)

or

Ξ119886120583

= 120588 (119903) 120575119886120583

(58)

Therefore for spherically symmetric distribution ofmass thatdepends on r-coordinate we can write

119898 (119903) = int

119903

0

41205871199032120588 (119903) 119889119903 (59)

Note that the 120588(119903) is defined by (1) Indeed we introducethe same energy density indicated in the noncommutativeperturbation theory [39]

119898 (119903) = 119872120579 (119903) =2119872

radic120587120574 (

3

2

1199032

4120579) (60)

Equation (55) can be integrated to find

119890minus2120573

= 1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579 (61)

where 120595120579 is a function that carries the tetrad field factor andwill be defined later by (66) and (67) Now by considering

119890minus2120573

= minus1

ℎ11

= ℎ00 (62)

the noncommutative line element for a spherically symmetricmatter distribution is therefore

1198891199042

= minus (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579) 119889119905

2

+ (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579)

minus1

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012)

(63)

The constant field of 12059401198902 in terms of (3) (4) and (5) can be

retrieved as

1205940

1198902=

1003816100381610038161003816100381612057912058311988710038161003816100381610038161003816

16120587119866119873

(64)

where 119866119873 is the Newtonian constant and |120579120583119887

| is determinedby 120579120583119887

= 12057921

= minus12057912

equiv 120579 Where 120579 is a real antisymmetricand constant tensor therefore the above equation can besimplified to yield

1205940

1198902=

120579

16120587119866119873

(65)

New line element (63) in particular depends on 120595120579 andnaturally 120595120579 has its origin on the quantum fluctuations ofthe noncommutative background geometry and originallycomes from the field equations The presented solution forour field equations produces naturally some additional termsin comparison with the solution of noncommutative versionof general relativity (naturally because it has some additionalterms in its components) These terms appear in the new lineelement because 120595120579 relates to the noncommutative torsionalspacetime and algebraic properties in spherically symmetricsolution of the tetrad fields We have therefore 120595120579 in thefollowing simplified equation

120595120579 = 120576119903120579120601

120576119903120579120601

ℎ119903

119903119890minus120573

(66)

Definition 120576119903120579120601

120576119903120579120601

= minus6ℎ2 is applied here (see also

[40 41]) According to this definition and (48) and (50)

6 Advances in High Energy Physics

Through simplification we find the following form of 120595120579 wefind the following form of 120595120579

120595120579 cong sum

119896=2119899

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

minus sum

119896=2119899+1

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

(67)

Note that 120595120579 is considered with the lower bound of sum Ifwe want to consider at least the second order of 120579 (which isproposed by [13]) for 120595120579 then it is natural to assume 119899 = 1Therefore two states for our line element will be producedthe imperfect state and the perfect state Let us now considerthe perfect stateThere is a proof for this state in terms of sometheorems in mathematics that allows us to introduce our lineelement as an appropriate description for a noncommutativespacetime Combination of these theoremswith regard to ourresults is given by [42ndash47]

Theorem 1 Let 119871 be a perfect field Recall that a polynomial119891(119909) isin 119871[119909] is called additive if 119891(119909 + 119910) = 119891(119909) + 119891(119910)

identically It is easy to see that a polynomial is additive if andonly if it is of the form

119891 (119909) = 1 minus 1198860119909 + 1198862

11199092

minus sdot sdot sdot plusmn 119886119899

119899119909119899

= sum

119899=0

1198862119899

1198991199092119899

minus sum

119899=0

1198862119899+1

1198991199092119899+1

(68)

The set of additive polynomials forms a noncommutative fieldin which (119891 ∘ 119892)(119909) = 119891(119892(119909)) This field is generated byscalar multiplications 119909 997891rarr 119886119909 for 119886 isin 119871 and 119909119894 isin 119891(119909) doesnot commute with the 119909119895120598119891(119909) Note that 119886 can be a constantfield and it has given as asymp 12059404120587119890

2 here (see [42ndash47] and thereferences cited therein) It is clear that components of119891(119909) canbe exactly replaced with components of ℎ00

Regarding other investigations into descriptions of non-commutative spacetime we should expand our discussioninto a comparisonmethodwith other line elements presentedfor noncommutative spacetime Reference [13] suggests thefollowing line element for noncommutative Schwarzschildspacetime

1198891199042

= minus (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)) 119889119905

2

+ (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579))

minus1

1198891199032

+ 1199032119889Ω2

(69)

where 119889Ω2

= 1198891205792

+ sin21205791198891206012 and 120574(32 119903

24120579) is the lower

incomplete gamma function

120574 (3

2

1199032

4120579) equiv int

11990324120579

0

119889119905radic119905119890minus119905

(70)

We note that nonvanishing radial pressure is a consequenceof the quantum fluctuation of the spacetimemanifold leadingto an inward gravitational pull and preventing the mattercollapsing into a point According to the line element (63)

in a neighborhood of the origin at 119903 le 120579 the energy densitydistribution of a static symmetric and noncommutative fuzzyspacetime is described by (1) which replaces the Dirac 120575

distribution by a smeared Gaussian profile Meanwhile inthe imperfect state our line element can be made equivalentto the line element of (69) and it is expected to happenwhen 120595120579 vanishes Assuredly it is due to vanishing ofthe tetrad components ℎ

119886

120583in (48) or even Wietzenbock

connections in (50) It means that in absence of torsionalspacetime the coordinate coherent state will be produced inthe noncommutative field theory It is completely reasonablesince coherent state theory is derived in the noncommutativeframework of the general relativity and the torsion is notdefined in the general relativity This equivalency is shownwith the following relation

1 minus119872

2119903radic120587120574 (

3

2

1199032

4120579) cong 119892

coherent state00

= 1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)

(71)

According to this proof the solution of the presented non-commutative field equations in the imperfect state of itselfresults in the exact solution of noncommutative generalrelativity field equations through coordinate coherent state ofour line element

41 Schwarzschild Black Hole Horizons In this paper wehave not extended our discussion into black holes butour introduced equations can be the basis of a subject onnoncommutative black holes Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black holewould be done by the horizon equation minusℎ119903

119867

= ℎ11

(119903119867) = 0Answers to this equation are illustrated in Figures 1 and 2Figure 1 shows the behavior of ℎ00 versus the horizon radiiwhen 120595120579 vanishes It is clear that 120595120579 vanishing approximatelyresults in 11989200 of (69) Figure 2 shows the behavior of ℎ00 atthe same conditions when we have 120595120579 As we can see fromthese figures there is a different behavior in the perfect statein comparison with the imperfect state near the horizon radiiwhich is due to the nature of torsional spacetime Howeverthe same behaviors have been indicated in the origin and thehigher bound of 119903

5 Force Equation in Commutative Limit

In teleparallel gravity the Newtonian force equation isobtained by assuming the class of frames in which theteleparallel spin connection vanishes and the gravitationalfield is stationary and weak [40 41 48 49] In our modelthe Newtonian gravitational force equation directly derivesfrom torsion components by 120595120579 in its commutative limitWhen we write the expansion of new line element in thenoncommutative limit we have

ℎ00 = 1 minus1205940

41198902

119898 (119903)

119903+ (

1205940

41198902

119898 (119903)

119903)

2

minus (1205940

41198902

119898 (119903)

119903)

3

(72)

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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AstronomyAdvances in

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Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

2 Advances in High Energy Physics

This model is given appropriately by a noncommutativeLagrangian and introduced by authors in [4 5] Admittedlythis Lagrangian and the relevant explanationswill be the basisof our next general calculations In this paper are going to usethe Greek alphabet (120583 ] 120588 = 0 1 2 3) to denote indicesrelated to spacetime and the first half of the Latin alphabet(119886 119887 119888 = 0 1 2 3) to denote indices related to thetangent space AMinkowski spacetime whose Lorentzmetricis assumed to have the form of 120578119886119887 = diag(minus1 +1 +1 +1)Themiddle letters of the Latin alphabet (119894 119895 119896 = 1 2 3)will bereserved for space indices The noncommutative Lagrangianis expressed as

119866119903 =1205940

1198902det (ℎ

1205901015840

120590) 1205781205831205831015840

times [1

4120578]]1015840

1205781205821205821015840 120582

120583]1205821015840

1205831015840]1015840 minus ]120583]

]10158401205831015840]1015840 +

1

2]1015840120583]

]1205831015840]1015840]

(2)

In the usual way having a Lagrangian which describesgravitation based on noncommutative background is likethose of gauge theories written in terms of contractions ofits field strength here represented by torsion of Weitzenbockconnection Its behavior under a local change of Δ 120583 isthe main invariance property of the particular combinationtorsion tensor fields Here 119890 is Yang-Mills coupling constantnoncommutative scale determines the Planck length and thePlanck scale of 119899-dimensional spacetime is given by

119896 = radic16120587119866119873 = 11989010038161003816100381610038161003816119875119891119886119891119891 (Θ

119860119861)10038161003816100381610038161003816

12119899

(3)

In mass dimension 2 the weight constant 1205940 is

1205940 =10038161003816100381610038161003816119875119891119886119891119891 (Θ

119860119861)10038161003816100381610038161003816

minus1119899

(4)

In the commutative limit it reduces to gravitational constantTherefore Θ

119860119861 is a noncommutative parameter defined as

Θ119860119861

= (120579120583]

120579120583119887

120579120583119887

120579119886119887) 997888rarr 120579

120583]= 120579119886119887

= 0 (5)

By considering the calculation of superpotential and energy-momentum current with respect to noncommutative gaugepotential given by 119861

120583

119886= | det(120579120583

10158401198861015840

)|

12119899

120579]120583

120596119886] the version ofnon-commutative gravitational field equations is produced120596119886] are gauge fields corresponding to the gauging of thetranslation group that is replacing 119877

119899 by the Lie algebra119892 of local gauge transformations with gauge functions andits relation with the non-trivial tetrad field is expressed asℎ120583

119886= 120575120583

119886minus 119890120579

]120583120596119886] and 120575

120583

119886has the perturbative effect in the

trivial holonomic tetrad fields of flat spaceIt is important to note that by applying the ldquodimensional

reduction of gauge theoriesrdquo noncommutative electrody-namics gauge field shown by the noncommutative Yang-Miles theory reduces to the gauge theories of gravitationwhich naturally yields Weitzenbock geometry on the space-timeAlso the induced diffeomorphism invariant field theorycan bemade equivalent to a teleparallel formulation of gravitymacroscopically describing general relativity In Section 2

we show that our Lagrangian can be made equivalent withgeneral relativity In Section 3 we are going to derive thefield equations by utilizing various definitions of teleparal-lel gravity By simplifying and solving the field equationswe obtain the line element in the spherically symmetricspacetime in Section 4 We continue our discussion withinvestigations about the limiting cases of our line element andhorizons of noncommutative Schwarzschild black hole in thismethod Finally we show how the Newtonian gravitationalforce equation can be derived from our line element in thecommutative limit in Section 5

2 Equivalence with General Relativity

In order to continue our discussion to achieve to noncom-mutative field equations we should show how our model canbe coupled with general relativity With respect to the givenrelation of

Γ120588

120583] = Γ120588

120583] + 120588

120583] (6)

for the vanishing curvature of the Weitzenbock connectionwe have

120588

120579120583] = 119877120588

120579120583] + 120588

120579120583] equiv 0 (7)

where

119877120588

120579120583] = 120597120583Γ120588

120579] minus 120597]Γ120588

120579] + Γ120588

120590120583Γ120590

120579] minus Γ120588

120590]Γ120590

120579120583 (8)

is the curvature of the Levi-Civita connection The aboveequations show that whereas in general relativity torsionvanishes in teleparallel gravity it is curvature that vanishesWe rewrite (7) based on their components in order to findthe scaler of

120588

120579120583] therefore we have

120588

120579120583] = (120597120583120588

120579] minus 120597]120588

120579120583+ 120588

120590120583120590

120579] minus Γ120588

120590]120590

120579120583

minus Γ120590

120579120583120588

120590] + Γ120590

120579]120588

120590120583) +

120588

120590]120590

120579120583minus 120588

120590120583120590

120579]

(9)

that is the tensor written in terms of the Weitzenbockconnection only Like the Riemanian curvature tensor it is 2-form assuming values in the Lie algebra of the Lorentz group(see [26 27]) By taking appropriate contractions it is easy toshow that

120588

120579120583] = (120583120588

120579] minus ]120588

120579120583) +

120588

120590]120590

120579120583minus 120588

120590120583120590

120579] (10)

By considering (20) and the following term

minus119877 = equiv1

2Λ120579

120588120588

120579120583]119889119909120583

and 119889119909] (11)

we achieve to the scalar version of (7)

119877 equiv (120583]120588

120588]120583 minus ]120583120588

120583120588

] ) +2

ℎ120597120583 (ℎ

]120583] ) (12)

The Lagrangian of (2) can be written in a simple form of

=1205940

1198902det (ℎ

1205901015840

120590) (120583]120588

120588]120583 minus ]120583120588

120583120588

] ) (13)

Advances in High Energy Physics 3

with a combination of (12) and (13) where takes thefollowing form

=1205940

1198902det (ℎ

1205901015840

120590) (119877 minus

2

ℎ(120597120583 (ℎ

]120583] ))) (14)

By considering (3) and (4) exchanges to

= 119871 minus 120597120583 (ℎ

8120587119866]120583] ) (15)

up to a divergence at the commutative limit therefore theLagrangian of (2) is equivalent to the Lagrangian of generalrelativity

=minus1

16120587119866radicminus119892119877 (16)

is the Einstein-Hilbert Lagrangian of general relativity How-ever this result could be extended with many further termsbut this is enough to derive a valid field equations

3 Noncommutative Field Equations

In this section we are going to present a reformulation ofteleparallel gravity (which is made equivalent to general rela-tivity) Due to the introduced noncommutative Lagrangian(2) we are able to derive the field equations similarly tothe teleparallel method Weitzenbock geometric definitionsand some well-known concepts of general relativity [28ndash30]and teleparallel gravity are required (more explanations aboutthese equations can be found in [29 31 32]) In 4-dimensionthe noncommutative action integral is given by

119878 = int 1198661199031198894119909 (17)

Under an arbitrary variation 120575ℎ120583

119886of the tetrad field the action

variation is written in the following form

120575119878 = int Ξ119886

120583120575ℎ120583

119886ℎ1198894119909 (18)

where

ℎΞ119886

120583=

120575119866119903

120575119861120583

119886

equiv120575119866119903

120575ℎ120583

119886

=120597119866119903

120597ℎ120583

119886

minus 120597120582

120597119866119903

120597120582120597ℎ120583

119886

(19)

is the matter energy-momentum tensor (More definitionsabout this tensor can be found in [33]) Now consider firstan infinitesimal Lorentz transformation as

Λ119887

119886= 120575119887

119886+ 120576119887

119886 (20)

where 120576119887

119886= minus120576119887

119886 Because of such transformation the tetrad

should be changed as

120575ℎ120583

119886= 120576119887

119886ℎ120583

119887 (21)

The requirement of invariance of the action under localLorentz transformation therefore yields

int Ξ119887

119886120576119887

119886ℎ1198894119909 = 0 (22)

Since 120576119887

119886is antisymmetric symmetric of energy-momentum

tensor yields some specific results that can be seen in [28]Consider spacetime coordinates that are transformed asfollows

1199091015840120588

= 119909120588

+ 120577120588 (23)

Whereby we retrieve the tetrad in the form of

120575ℎ120583

119886equiv ℎ1015840120583

119886(119909) minus ℎ

120583

119886(119909) = ℎ

120588

119886120597120588120577120583

minus 120577120588120597120588ℎ120583

119886 (24)

Substituting in (18) we have

120575119878 = int Ξ119886

120583[ℎ119886

120588120597120583120577120588

minus 120577120588120597120588ℎ120583

119886] ℎ1198894119909 (25)

or equivalently

120575119878 = int [Ξ120588

119888120597120588120577119888

+ 120577119888Ξ120588

120583120597120588ℎ120583

119888minus 120577120588120597120588ℎ120583

119886] ℎ1198894119909 (26)

Substitute the identity

120597120588ℎ120583

119886= 119887

119886120588ℎ120583

119887minus Γ120583

120582120588ℎ120582

119886 (27)

where is the spin connection in teleparallel gravity Theimportant property of teleparallel gravity is that its spinconnection is related only to the inertial properties of theframe not to gravitation In fact it is possible to choose anappropriate frame in which it vanishes everywhereWe knowthat the above formula vanishes by (41) (see also [34 35])and making use of the symmetric of the energy-momentumtensor the action variation assumes the form of

120575119878 = int Ξ120588

119888[120597120588120577119888

+ (119888

119887120588minus 119888

119887120588) 120577119887] ℎ1198894119909 (28)

Integrating the first term by parts and neglecting the surfaceterm the invariance of the action yields

int [120597120583 (ℎΞ120583

119886) minus (

119887

119886120583minus 119887

119886120583) (ℎΞ120583

119887)] 120577119886ℎ1198894119909 = 0 (29)

From arbitrariness of 120577119888 under the covariant derivative 120583 it

follows that

120583ℎΞ120583

119886equiv 120597120583 (ℎΞ

120583

119886) minus (

119887

119886120583minus 119887

119886120583) (ℎΞ120583

119887) = 0 (30)

By identity of

120597120588ℎ = ℎΓ]]120588 equiv ℎ (Γ

]120588] minus

]120588]) (31)

the above conservation law becomes

120597120583Ξ120583

119886+ (Γ120583

120588120583minus 120583

120588120583) Ξ120588

119886minus (119887

119886120583minus 119887

119886120583) Ξ120583

119887= 0 (32)

In a purely spacetime form it reads

120583Ξ120583

120582equiv 120597120583Ξ

120583

120582+ (Γ120583

120583120588minus 120583

120588120583) Ξ120588

120582minus (Γ120588

120582120583minus 120588

120582120583) Ξ120583

120588= 0 (33)

4 Advances in High Energy Physics

This is the conservation law of the source of energy-momentum tensor Variation with respect to the noncom-mutative gauge potential 119861

120583

119886yields the noncommutative

teleparallel version of the gravitational field equations

120597120590 (ℎ 119878120583120590

119886) minus 119896ℎ 119869

120583

119886= 119896ℎΞ

120583

119886 (34)

where

ℎ 119878120583120590

119886= ℎℎ120582

119886119878120583120590

120582equiv minus119896

120597

120597 (120597120590ℎ119886120583

)

(35)

which defines the superpotential For the gauge current wehave

ℎ 119869120583

119886= minus

120597

120597119861119886120583

equiv minus120597

120597ℎ119886120583

(36)

Note that the matter energy-momentum tensor which isdefined in this relation appears as the source of torsionsimilarly the energy-momentum tensor appears as the sourceof curvature in general relativity Our computation has led usto the following results

119878120583120590

119886= 2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578

119869120583

119886=

1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888

(37)

for noncommutative superpotential and gauge current Thelagrangian appears again in our equations but notice thatthis term cross its coefficient yields a term purely based onits field strength according to (2) This simplified expressionmaintains equivalence to general relativity We can observethat the gravitational field equations depend on the torsiononly Finally the field equations can be written as

120597120590 (ℎ (2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578))

minus 119896ℎ (1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888) = 119896ℎΞ

120583

119886

(38)

where 119896 = 12059401198902 is a constant These field equations are

similar to teleparallel field equations although it wouldbe distinguished with different field strength

120583120590

119886which is

given by the covariant rotational of noncommutative gaugepotential of 119861

120583

119886 Consider the following equations from the

teleparallel theory (see for instance [32 34ndash36])

119886

120583] = 120597]ℎ119886

120583minus 120597120583ℎ119886

] + 119886

119890]ℎ119890

120583minus 119886

119890120583ℎ119890

] (39)

Γ120588

]120583 = ℎ120588

119886120597120583ℎ119886

] + ℎ120588

119886119886

119887120583ℎ119887

] (40)

120597120583ℎ119886

] minus Γ120588

]120583ℎ119886

120588+ 119886

119887120583ℎ119887

] = 0 (41)

120588

]120583 = Γ120588

120583] minus Γ120588

]120583 (42)

The field equations take the exact following form

120597

120597119909120590(Γ120590

119886120583minus Γ120590

120583119886) minus

120597

120597119909120583Γ120582

119886120582+

120597

120597119909120582Γ120582

119886120583

minus Γ120578

119886120582Γ120582

120583120578+ Γ120578

119886120583Γ120582

120582120578=

1205940

1198902120588 (119903)

120597

120597119909119886

120597

120597119909120583

(43)

which unlike the left hand side of (38) is written purelybased on noncommutative field strength and the abovefield equation is written in terms of Weitzenbock connec-tion only Regarding the equivalency between correspondingLagrangians and the above simplified field equations andapplying (34) we have therefore

119877119886120583 minus1

2ℎ119886120583119877 = 119896Ξ119886120583 (44)

as equivalent to Einsteinrsquos field equations Note that (44) isnot Einsteinrsquos field equations but the teleparallel field equa-tions made equivalent to general relativity And equivalentmodel of teleparallel field equations with general relativity isexpressed in the references in detail (see for instance [32 3435]) We continue our discussion to derive noncommutativeline element by solving these field equations

4 Noncommutative Line Element

Teleparallel versions of the stationary static spherically axis-symmetric and symmetric of the Schwarzschild solutionhave been previously obtained [37 38] Within a frameworkinspired by noncommutative geometry we solve the fieldequations for a distribution of spherically symmetricallymassin a stationary static spacetime like the exterior solution ofSchwarzschild (see also [29])Then it is natural to assume thatthe line element is as follows

1198891199042

= minus119891 (119903) 1198891199052

+ 119892 (119903) 1198891199032

+ ℎ (119903) 1199032

(1198891205792

+ sin21205791198891206012)

(45)

With a new radial coordinate defined as 119903 = 119903radicℎ(119903) the lineelement becomes

1198891199042

= minus119860 (119903) 1198891199052

+ 119861 (119903) 1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (46)

Usually one replaces the functions 119860(119903) and 119861(119903) with expo-nential functions to obtain somewhat simpler expressionsfor the noncommutative tensor components Hence weintroduce the functions 120572(119903) and 120573(119903) by 119890

2120572(119903)= 119860(119903) and

119890120573(119903)

= 119861(119903) to get

1198891199042

= minus1198902120572

1198891199052

+ 1198902120573

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (47)

Tetrad components of the above metric take the followingform

ℎ119886

120583=

[[[

[

minus1198902120572

0 0 0

0 1198902120573 sin 120579 cos120601 119903 cos 120579 cos120601 minus119903 cos 120579 sin120601

0 1198902120573 sin 120579 sin120601 119903 cos 120579 sin120601 119903 sin 120579 cos120601

0 1198902120573 cos 120579 minus119903 sin 120579 0

]]]

]

(48)

Weitzenbock connection Γ120588

120583] has the following expression

Γ120588

120583] = ℎ120588

119886120597]ℎ119886

120583 (49)

Advances in High Energy Physics 5

Now we can calculate the nonvanishing components ofWeitzenbock connection as follows

Γ0

01= minus2120572

1015840 Γ

1

11= 21205731015840 Γ

1

22= minus119903119890

minus120573

Γ1

33= minus119903119890

minus120573sin2120579 Γ2

12=

119890120572

119903= Γ3

13

Γ2

21=

1

119903= Γ3

31 Γ

2

33= minus sin 120579 cos 120579

Γ3

23= Γ3

32= cot 120579

(50)

By replacing these components in (43) the noncommutativetensors of (51)ndash(53) for the left-hand side of the field equationswill produce the following expression

119873 =1

1199032(minus4119890minus2120573

+ 1 minus 120595120579) minus2

1199031205731015840119890minus2120573

=1205940

1198902120588 (119903) 120575 (51)

119873119903119903 =1

1199032(2119890minus2120573

+ 1 minus 120595120579) +2

1199031205721015840119890minus2120573

=1205940

1198902120588 (119903) 120575119903119903 (52)

119873120579120579

= 119873120601120601

=1

119903119890minus2120573

(11990312057210158401015840

+ 11990312057210158402

minus 11990312057210158401205731015840+ 1205721015840minus 1205731015840minus 1)

+ 12057210158402

119890minus2120573

=1205940

1198902120588 (119903) 120575

120579120579=

1205940

1198902120588 (119903) 120575

120601120601

(53)

Adding (51) and (52) we get simply

1

1199032(120595120579 minus 119890

minus2120573+ 119890minus2120573

(1205721015840minus 1205731015840) + 1) =

1205940

1198902120588 (119903) (54)

where 120572(119903) = 120573(119903) It should also be noted that by recalling(47) we can consider the limiting case for our solutionassuming (120572

1015840minus 1205731015840) = 119896 where 119896 is a constant and by

considering the time coordinate we can shift this constantto an arbitrary value It is possible therefore without loss ofgenerality to choose 119896 = 0 It does not contradict (47) toset 1205721015840

= 1205731015840 According to this analysis the equation 119873 =

(12059401198902)120588(119903)120575 can be written as

minus1

119903

119889

119889119903[119903 (119890minus2120573

minus 120595120579 minus 1)] =1205940

1198902120588 (119903) (55)

For a perfect fluid in thermodynamic equilibrium the stress-energy tensor takes on a particularly simple form

Ξ120583]

= (120588 + 119875) 119906120583119906]

+ 119901119892120583]

(56)

where the pressure 119875 can be neglected due to the distributionof mass and the gravitational effects consequently only oneterm will remain in the above formula as follows

Ξ119886120583

= 120588 (119903)119889119909119886

119889119905

119889119909120583

119889119905 (57)

or

Ξ119886120583

= 120588 (119903) 120575119886120583

(58)

Therefore for spherically symmetric distribution ofmass thatdepends on r-coordinate we can write

119898 (119903) = int

119903

0

41205871199032120588 (119903) 119889119903 (59)

Note that the 120588(119903) is defined by (1) Indeed we introducethe same energy density indicated in the noncommutativeperturbation theory [39]

119898 (119903) = 119872120579 (119903) =2119872

radic120587120574 (

3

2

1199032

4120579) (60)

Equation (55) can be integrated to find

119890minus2120573

= 1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579 (61)

where 120595120579 is a function that carries the tetrad field factor andwill be defined later by (66) and (67) Now by considering

119890minus2120573

= minus1

ℎ11

= ℎ00 (62)

the noncommutative line element for a spherically symmetricmatter distribution is therefore

1198891199042

= minus (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579) 119889119905

2

+ (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579)

minus1

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012)

(63)

The constant field of 12059401198902 in terms of (3) (4) and (5) can be

retrieved as

1205940

1198902=

1003816100381610038161003816100381612057912058311988710038161003816100381610038161003816

16120587119866119873

(64)

where 119866119873 is the Newtonian constant and |120579120583119887

| is determinedby 120579120583119887

= 12057921

= minus12057912

equiv 120579 Where 120579 is a real antisymmetricand constant tensor therefore the above equation can besimplified to yield

1205940

1198902=

120579

16120587119866119873

(65)

New line element (63) in particular depends on 120595120579 andnaturally 120595120579 has its origin on the quantum fluctuations ofthe noncommutative background geometry and originallycomes from the field equations The presented solution forour field equations produces naturally some additional termsin comparison with the solution of noncommutative versionof general relativity (naturally because it has some additionalterms in its components) These terms appear in the new lineelement because 120595120579 relates to the noncommutative torsionalspacetime and algebraic properties in spherically symmetricsolution of the tetrad fields We have therefore 120595120579 in thefollowing simplified equation

120595120579 = 120576119903120579120601

120576119903120579120601

ℎ119903

119903119890minus120573

(66)

Definition 120576119903120579120601

120576119903120579120601

= minus6ℎ2 is applied here (see also

[40 41]) According to this definition and (48) and (50)

6 Advances in High Energy Physics

Through simplification we find the following form of 120595120579 wefind the following form of 120595120579

120595120579 cong sum

119896=2119899

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

minus sum

119896=2119899+1

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

(67)

Note that 120595120579 is considered with the lower bound of sum Ifwe want to consider at least the second order of 120579 (which isproposed by [13]) for 120595120579 then it is natural to assume 119899 = 1Therefore two states for our line element will be producedthe imperfect state and the perfect state Let us now considerthe perfect stateThere is a proof for this state in terms of sometheorems in mathematics that allows us to introduce our lineelement as an appropriate description for a noncommutativespacetime Combination of these theoremswith regard to ourresults is given by [42ndash47]

Theorem 1 Let 119871 be a perfect field Recall that a polynomial119891(119909) isin 119871[119909] is called additive if 119891(119909 + 119910) = 119891(119909) + 119891(119910)

identically It is easy to see that a polynomial is additive if andonly if it is of the form

119891 (119909) = 1 minus 1198860119909 + 1198862

11199092

minus sdot sdot sdot plusmn 119886119899

119899119909119899

= sum

119899=0

1198862119899

1198991199092119899

minus sum

119899=0

1198862119899+1

1198991199092119899+1

(68)

The set of additive polynomials forms a noncommutative fieldin which (119891 ∘ 119892)(119909) = 119891(119892(119909)) This field is generated byscalar multiplications 119909 997891rarr 119886119909 for 119886 isin 119871 and 119909119894 isin 119891(119909) doesnot commute with the 119909119895120598119891(119909) Note that 119886 can be a constantfield and it has given as asymp 12059404120587119890

2 here (see [42ndash47] and thereferences cited therein) It is clear that components of119891(119909) canbe exactly replaced with components of ℎ00

Regarding other investigations into descriptions of non-commutative spacetime we should expand our discussioninto a comparisonmethodwith other line elements presentedfor noncommutative spacetime Reference [13] suggests thefollowing line element for noncommutative Schwarzschildspacetime

1198891199042

= minus (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)) 119889119905

2

+ (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579))

minus1

1198891199032

+ 1199032119889Ω2

(69)

where 119889Ω2

= 1198891205792

+ sin21205791198891206012 and 120574(32 119903

24120579) is the lower

incomplete gamma function

120574 (3

2

1199032

4120579) equiv int

11990324120579

0

119889119905radic119905119890minus119905

(70)

We note that nonvanishing radial pressure is a consequenceof the quantum fluctuation of the spacetimemanifold leadingto an inward gravitational pull and preventing the mattercollapsing into a point According to the line element (63)

in a neighborhood of the origin at 119903 le 120579 the energy densitydistribution of a static symmetric and noncommutative fuzzyspacetime is described by (1) which replaces the Dirac 120575

distribution by a smeared Gaussian profile Meanwhile inthe imperfect state our line element can be made equivalentto the line element of (69) and it is expected to happenwhen 120595120579 vanishes Assuredly it is due to vanishing ofthe tetrad components ℎ

119886

120583in (48) or even Wietzenbock

connections in (50) It means that in absence of torsionalspacetime the coordinate coherent state will be produced inthe noncommutative field theory It is completely reasonablesince coherent state theory is derived in the noncommutativeframework of the general relativity and the torsion is notdefined in the general relativity This equivalency is shownwith the following relation

1 minus119872

2119903radic120587120574 (

3

2

1199032

4120579) cong 119892

coherent state00

= 1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)

(71)

According to this proof the solution of the presented non-commutative field equations in the imperfect state of itselfresults in the exact solution of noncommutative generalrelativity field equations through coordinate coherent state ofour line element

41 Schwarzschild Black Hole Horizons In this paper wehave not extended our discussion into black holes butour introduced equations can be the basis of a subject onnoncommutative black holes Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black holewould be done by the horizon equation minusℎ119903

119867

= ℎ11

(119903119867) = 0Answers to this equation are illustrated in Figures 1 and 2Figure 1 shows the behavior of ℎ00 versus the horizon radiiwhen 120595120579 vanishes It is clear that 120595120579 vanishing approximatelyresults in 11989200 of (69) Figure 2 shows the behavior of ℎ00 atthe same conditions when we have 120595120579 As we can see fromthese figures there is a different behavior in the perfect statein comparison with the imperfect state near the horizon radiiwhich is due to the nature of torsional spacetime Howeverthe same behaviors have been indicated in the origin and thehigher bound of 119903

5 Force Equation in Commutative Limit

In teleparallel gravity the Newtonian force equation isobtained by assuming the class of frames in which theteleparallel spin connection vanishes and the gravitationalfield is stationary and weak [40 41 48 49] In our modelthe Newtonian gravitational force equation directly derivesfrom torsion components by 120595120579 in its commutative limitWhen we write the expansion of new line element in thenoncommutative limit we have

ℎ00 = 1 minus1205940

41198902

119898 (119903)

119903+ (

1205940

41198902

119898 (119903)

119903)

2

minus (1205940

41198902

119898 (119903)

119903)

3

(72)

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 3

with a combination of (12) and (13) where takes thefollowing form

=1205940

1198902det (ℎ

1205901015840

120590) (119877 minus

2

ℎ(120597120583 (ℎ

]120583] ))) (14)

By considering (3) and (4) exchanges to

= 119871 minus 120597120583 (ℎ

8120587119866]120583] ) (15)

up to a divergence at the commutative limit therefore theLagrangian of (2) is equivalent to the Lagrangian of generalrelativity

=minus1

16120587119866radicminus119892119877 (16)

is the Einstein-Hilbert Lagrangian of general relativity How-ever this result could be extended with many further termsbut this is enough to derive a valid field equations

3 Noncommutative Field Equations

In this section we are going to present a reformulation ofteleparallel gravity (which is made equivalent to general rela-tivity) Due to the introduced noncommutative Lagrangian(2) we are able to derive the field equations similarly tothe teleparallel method Weitzenbock geometric definitionsand some well-known concepts of general relativity [28ndash30]and teleparallel gravity are required (more explanations aboutthese equations can be found in [29 31 32]) In 4-dimensionthe noncommutative action integral is given by

119878 = int 1198661199031198894119909 (17)

Under an arbitrary variation 120575ℎ120583

119886of the tetrad field the action

variation is written in the following form

120575119878 = int Ξ119886

120583120575ℎ120583

119886ℎ1198894119909 (18)

where

ℎΞ119886

120583=

120575119866119903

120575119861120583

119886

equiv120575119866119903

120575ℎ120583

119886

=120597119866119903

120597ℎ120583

119886

minus 120597120582

120597119866119903

120597120582120597ℎ120583

119886

(19)

is the matter energy-momentum tensor (More definitionsabout this tensor can be found in [33]) Now consider firstan infinitesimal Lorentz transformation as

Λ119887

119886= 120575119887

119886+ 120576119887

119886 (20)

where 120576119887

119886= minus120576119887

119886 Because of such transformation the tetrad

should be changed as

120575ℎ120583

119886= 120576119887

119886ℎ120583

119887 (21)

The requirement of invariance of the action under localLorentz transformation therefore yields

int Ξ119887

119886120576119887

119886ℎ1198894119909 = 0 (22)

Since 120576119887

119886is antisymmetric symmetric of energy-momentum

tensor yields some specific results that can be seen in [28]Consider spacetime coordinates that are transformed asfollows

1199091015840120588

= 119909120588

+ 120577120588 (23)

Whereby we retrieve the tetrad in the form of

120575ℎ120583

119886equiv ℎ1015840120583

119886(119909) minus ℎ

120583

119886(119909) = ℎ

120588

119886120597120588120577120583

minus 120577120588120597120588ℎ120583

119886 (24)

Substituting in (18) we have

120575119878 = int Ξ119886

120583[ℎ119886

120588120597120583120577120588

minus 120577120588120597120588ℎ120583

119886] ℎ1198894119909 (25)

or equivalently

120575119878 = int [Ξ120588

119888120597120588120577119888

+ 120577119888Ξ120588

120583120597120588ℎ120583

119888minus 120577120588120597120588ℎ120583

119886] ℎ1198894119909 (26)

Substitute the identity

120597120588ℎ120583

119886= 119887

119886120588ℎ120583

119887minus Γ120583

120582120588ℎ120582

119886 (27)

where is the spin connection in teleparallel gravity Theimportant property of teleparallel gravity is that its spinconnection is related only to the inertial properties of theframe not to gravitation In fact it is possible to choose anappropriate frame in which it vanishes everywhereWe knowthat the above formula vanishes by (41) (see also [34 35])and making use of the symmetric of the energy-momentumtensor the action variation assumes the form of

120575119878 = int Ξ120588

119888[120597120588120577119888

+ (119888

119887120588minus 119888

119887120588) 120577119887] ℎ1198894119909 (28)

Integrating the first term by parts and neglecting the surfaceterm the invariance of the action yields

int [120597120583 (ℎΞ120583

119886) minus (

119887

119886120583minus 119887

119886120583) (ℎΞ120583

119887)] 120577119886ℎ1198894119909 = 0 (29)

From arbitrariness of 120577119888 under the covariant derivative 120583 it

follows that

120583ℎΞ120583

119886equiv 120597120583 (ℎΞ

120583

119886) minus (

119887

119886120583minus 119887

119886120583) (ℎΞ120583

119887) = 0 (30)

By identity of

120597120588ℎ = ℎΓ]]120588 equiv ℎ (Γ

]120588] minus

]120588]) (31)

the above conservation law becomes

120597120583Ξ120583

119886+ (Γ120583

120588120583minus 120583

120588120583) Ξ120588

119886minus (119887

119886120583minus 119887

119886120583) Ξ120583

119887= 0 (32)

In a purely spacetime form it reads

120583Ξ120583

120582equiv 120597120583Ξ

120583

120582+ (Γ120583

120583120588minus 120583

120588120583) Ξ120588

120582minus (Γ120588

120582120583minus 120588

120582120583) Ξ120583

120588= 0 (33)

4 Advances in High Energy Physics

This is the conservation law of the source of energy-momentum tensor Variation with respect to the noncom-mutative gauge potential 119861

120583

119886yields the noncommutative

teleparallel version of the gravitational field equations

120597120590 (ℎ 119878120583120590

119886) minus 119896ℎ 119869

120583

119886= 119896ℎΞ

120583

119886 (34)

where

ℎ 119878120583120590

119886= ℎℎ120582

119886119878120583120590

120582equiv minus119896

120597

120597 (120597120590ℎ119886120583

)

(35)

which defines the superpotential For the gauge current wehave

ℎ 119869120583

119886= minus

120597

120597119861119886120583

equiv minus120597

120597ℎ119886120583

(36)

Note that the matter energy-momentum tensor which isdefined in this relation appears as the source of torsionsimilarly the energy-momentum tensor appears as the sourceof curvature in general relativity Our computation has led usto the following results

119878120583120590

119886= 2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578

119869120583

119886=

1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888

(37)

for noncommutative superpotential and gauge current Thelagrangian appears again in our equations but notice thatthis term cross its coefficient yields a term purely based onits field strength according to (2) This simplified expressionmaintains equivalence to general relativity We can observethat the gravitational field equations depend on the torsiononly Finally the field equations can be written as

120597120590 (ℎ (2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578))

minus 119896ℎ (1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888) = 119896ℎΞ

120583

119886

(38)

where 119896 = 12059401198902 is a constant These field equations are

similar to teleparallel field equations although it wouldbe distinguished with different field strength

120583120590

119886which is

given by the covariant rotational of noncommutative gaugepotential of 119861

120583

119886 Consider the following equations from the

teleparallel theory (see for instance [32 34ndash36])

119886

120583] = 120597]ℎ119886

120583minus 120597120583ℎ119886

] + 119886

119890]ℎ119890

120583minus 119886

119890120583ℎ119890

] (39)

Γ120588

]120583 = ℎ120588

119886120597120583ℎ119886

] + ℎ120588

119886119886

119887120583ℎ119887

] (40)

120597120583ℎ119886

] minus Γ120588

]120583ℎ119886

120588+ 119886

119887120583ℎ119887

] = 0 (41)

120588

]120583 = Γ120588

120583] minus Γ120588

]120583 (42)

The field equations take the exact following form

120597

120597119909120590(Γ120590

119886120583minus Γ120590

120583119886) minus

120597

120597119909120583Γ120582

119886120582+

120597

120597119909120582Γ120582

119886120583

minus Γ120578

119886120582Γ120582

120583120578+ Γ120578

119886120583Γ120582

120582120578=

1205940

1198902120588 (119903)

120597

120597119909119886

120597

120597119909120583

(43)

which unlike the left hand side of (38) is written purelybased on noncommutative field strength and the abovefield equation is written in terms of Weitzenbock connec-tion only Regarding the equivalency between correspondingLagrangians and the above simplified field equations andapplying (34) we have therefore

119877119886120583 minus1

2ℎ119886120583119877 = 119896Ξ119886120583 (44)

as equivalent to Einsteinrsquos field equations Note that (44) isnot Einsteinrsquos field equations but the teleparallel field equa-tions made equivalent to general relativity And equivalentmodel of teleparallel field equations with general relativity isexpressed in the references in detail (see for instance [32 3435]) We continue our discussion to derive noncommutativeline element by solving these field equations

4 Noncommutative Line Element

Teleparallel versions of the stationary static spherically axis-symmetric and symmetric of the Schwarzschild solutionhave been previously obtained [37 38] Within a frameworkinspired by noncommutative geometry we solve the fieldequations for a distribution of spherically symmetricallymassin a stationary static spacetime like the exterior solution ofSchwarzschild (see also [29])Then it is natural to assume thatthe line element is as follows

1198891199042

= minus119891 (119903) 1198891199052

+ 119892 (119903) 1198891199032

+ ℎ (119903) 1199032

(1198891205792

+ sin21205791198891206012)

(45)

With a new radial coordinate defined as 119903 = 119903radicℎ(119903) the lineelement becomes

1198891199042

= minus119860 (119903) 1198891199052

+ 119861 (119903) 1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (46)

Usually one replaces the functions 119860(119903) and 119861(119903) with expo-nential functions to obtain somewhat simpler expressionsfor the noncommutative tensor components Hence weintroduce the functions 120572(119903) and 120573(119903) by 119890

2120572(119903)= 119860(119903) and

119890120573(119903)

= 119861(119903) to get

1198891199042

= minus1198902120572

1198891199052

+ 1198902120573

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (47)

Tetrad components of the above metric take the followingform

ℎ119886

120583=

[[[

[

minus1198902120572

0 0 0

0 1198902120573 sin 120579 cos120601 119903 cos 120579 cos120601 minus119903 cos 120579 sin120601

0 1198902120573 sin 120579 sin120601 119903 cos 120579 sin120601 119903 sin 120579 cos120601

0 1198902120573 cos 120579 minus119903 sin 120579 0

]]]

]

(48)

Weitzenbock connection Γ120588

120583] has the following expression

Γ120588

120583] = ℎ120588

119886120597]ℎ119886

120583 (49)

Advances in High Energy Physics 5

Now we can calculate the nonvanishing components ofWeitzenbock connection as follows

Γ0

01= minus2120572

1015840 Γ

1

11= 21205731015840 Γ

1

22= minus119903119890

minus120573

Γ1

33= minus119903119890

minus120573sin2120579 Γ2

12=

119890120572

119903= Γ3

13

Γ2

21=

1

119903= Γ3

31 Γ

2

33= minus sin 120579 cos 120579

Γ3

23= Γ3

32= cot 120579

(50)

By replacing these components in (43) the noncommutativetensors of (51)ndash(53) for the left-hand side of the field equationswill produce the following expression

119873 =1

1199032(minus4119890minus2120573

+ 1 minus 120595120579) minus2

1199031205731015840119890minus2120573

=1205940

1198902120588 (119903) 120575 (51)

119873119903119903 =1

1199032(2119890minus2120573

+ 1 minus 120595120579) +2

1199031205721015840119890minus2120573

=1205940

1198902120588 (119903) 120575119903119903 (52)

119873120579120579

= 119873120601120601

=1

119903119890minus2120573

(11990312057210158401015840

+ 11990312057210158402

minus 11990312057210158401205731015840+ 1205721015840minus 1205731015840minus 1)

+ 12057210158402

119890minus2120573

=1205940

1198902120588 (119903) 120575

120579120579=

1205940

1198902120588 (119903) 120575

120601120601

(53)

Adding (51) and (52) we get simply

1

1199032(120595120579 minus 119890

minus2120573+ 119890minus2120573

(1205721015840minus 1205731015840) + 1) =

1205940

1198902120588 (119903) (54)

where 120572(119903) = 120573(119903) It should also be noted that by recalling(47) we can consider the limiting case for our solutionassuming (120572

1015840minus 1205731015840) = 119896 where 119896 is a constant and by

considering the time coordinate we can shift this constantto an arbitrary value It is possible therefore without loss ofgenerality to choose 119896 = 0 It does not contradict (47) toset 1205721015840

= 1205731015840 According to this analysis the equation 119873 =

(12059401198902)120588(119903)120575 can be written as

minus1

119903

119889

119889119903[119903 (119890minus2120573

minus 120595120579 minus 1)] =1205940

1198902120588 (119903) (55)

For a perfect fluid in thermodynamic equilibrium the stress-energy tensor takes on a particularly simple form

Ξ120583]

= (120588 + 119875) 119906120583119906]

+ 119901119892120583]

(56)

where the pressure 119875 can be neglected due to the distributionof mass and the gravitational effects consequently only oneterm will remain in the above formula as follows

Ξ119886120583

= 120588 (119903)119889119909119886

119889119905

119889119909120583

119889119905 (57)

or

Ξ119886120583

= 120588 (119903) 120575119886120583

(58)

Therefore for spherically symmetric distribution ofmass thatdepends on r-coordinate we can write

119898 (119903) = int

119903

0

41205871199032120588 (119903) 119889119903 (59)

Note that the 120588(119903) is defined by (1) Indeed we introducethe same energy density indicated in the noncommutativeperturbation theory [39]

119898 (119903) = 119872120579 (119903) =2119872

radic120587120574 (

3

2

1199032

4120579) (60)

Equation (55) can be integrated to find

119890minus2120573

= 1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579 (61)

where 120595120579 is a function that carries the tetrad field factor andwill be defined later by (66) and (67) Now by considering

119890minus2120573

= minus1

ℎ11

= ℎ00 (62)

the noncommutative line element for a spherically symmetricmatter distribution is therefore

1198891199042

= minus (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579) 119889119905

2

+ (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579)

minus1

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012)

(63)

The constant field of 12059401198902 in terms of (3) (4) and (5) can be

retrieved as

1205940

1198902=

1003816100381610038161003816100381612057912058311988710038161003816100381610038161003816

16120587119866119873

(64)

where 119866119873 is the Newtonian constant and |120579120583119887

| is determinedby 120579120583119887

= 12057921

= minus12057912

equiv 120579 Where 120579 is a real antisymmetricand constant tensor therefore the above equation can besimplified to yield

1205940

1198902=

120579

16120587119866119873

(65)

New line element (63) in particular depends on 120595120579 andnaturally 120595120579 has its origin on the quantum fluctuations ofthe noncommutative background geometry and originallycomes from the field equations The presented solution forour field equations produces naturally some additional termsin comparison with the solution of noncommutative versionof general relativity (naturally because it has some additionalterms in its components) These terms appear in the new lineelement because 120595120579 relates to the noncommutative torsionalspacetime and algebraic properties in spherically symmetricsolution of the tetrad fields We have therefore 120595120579 in thefollowing simplified equation

120595120579 = 120576119903120579120601

120576119903120579120601

ℎ119903

119903119890minus120573

(66)

Definition 120576119903120579120601

120576119903120579120601

= minus6ℎ2 is applied here (see also

[40 41]) According to this definition and (48) and (50)

6 Advances in High Energy Physics

Through simplification we find the following form of 120595120579 wefind the following form of 120595120579

120595120579 cong sum

119896=2119899

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

minus sum

119896=2119899+1

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

(67)

Note that 120595120579 is considered with the lower bound of sum Ifwe want to consider at least the second order of 120579 (which isproposed by [13]) for 120595120579 then it is natural to assume 119899 = 1Therefore two states for our line element will be producedthe imperfect state and the perfect state Let us now considerthe perfect stateThere is a proof for this state in terms of sometheorems in mathematics that allows us to introduce our lineelement as an appropriate description for a noncommutativespacetime Combination of these theoremswith regard to ourresults is given by [42ndash47]

Theorem 1 Let 119871 be a perfect field Recall that a polynomial119891(119909) isin 119871[119909] is called additive if 119891(119909 + 119910) = 119891(119909) + 119891(119910)

identically It is easy to see that a polynomial is additive if andonly if it is of the form

119891 (119909) = 1 minus 1198860119909 + 1198862

11199092

minus sdot sdot sdot plusmn 119886119899

119899119909119899

= sum

119899=0

1198862119899

1198991199092119899

minus sum

119899=0

1198862119899+1

1198991199092119899+1

(68)

The set of additive polynomials forms a noncommutative fieldin which (119891 ∘ 119892)(119909) = 119891(119892(119909)) This field is generated byscalar multiplications 119909 997891rarr 119886119909 for 119886 isin 119871 and 119909119894 isin 119891(119909) doesnot commute with the 119909119895120598119891(119909) Note that 119886 can be a constantfield and it has given as asymp 12059404120587119890

2 here (see [42ndash47] and thereferences cited therein) It is clear that components of119891(119909) canbe exactly replaced with components of ℎ00

Regarding other investigations into descriptions of non-commutative spacetime we should expand our discussioninto a comparisonmethodwith other line elements presentedfor noncommutative spacetime Reference [13] suggests thefollowing line element for noncommutative Schwarzschildspacetime

1198891199042

= minus (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)) 119889119905

2

+ (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579))

minus1

1198891199032

+ 1199032119889Ω2

(69)

where 119889Ω2

= 1198891205792

+ sin21205791198891206012 and 120574(32 119903

24120579) is the lower

incomplete gamma function

120574 (3

2

1199032

4120579) equiv int

11990324120579

0

119889119905radic119905119890minus119905

(70)

We note that nonvanishing radial pressure is a consequenceof the quantum fluctuation of the spacetimemanifold leadingto an inward gravitational pull and preventing the mattercollapsing into a point According to the line element (63)

in a neighborhood of the origin at 119903 le 120579 the energy densitydistribution of a static symmetric and noncommutative fuzzyspacetime is described by (1) which replaces the Dirac 120575

distribution by a smeared Gaussian profile Meanwhile inthe imperfect state our line element can be made equivalentto the line element of (69) and it is expected to happenwhen 120595120579 vanishes Assuredly it is due to vanishing ofthe tetrad components ℎ

119886

120583in (48) or even Wietzenbock

connections in (50) It means that in absence of torsionalspacetime the coordinate coherent state will be produced inthe noncommutative field theory It is completely reasonablesince coherent state theory is derived in the noncommutativeframework of the general relativity and the torsion is notdefined in the general relativity This equivalency is shownwith the following relation

1 minus119872

2119903radic120587120574 (

3

2

1199032

4120579) cong 119892

coherent state00

= 1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)

(71)

According to this proof the solution of the presented non-commutative field equations in the imperfect state of itselfresults in the exact solution of noncommutative generalrelativity field equations through coordinate coherent state ofour line element

41 Schwarzschild Black Hole Horizons In this paper wehave not extended our discussion into black holes butour introduced equations can be the basis of a subject onnoncommutative black holes Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black holewould be done by the horizon equation minusℎ119903

119867

= ℎ11

(119903119867) = 0Answers to this equation are illustrated in Figures 1 and 2Figure 1 shows the behavior of ℎ00 versus the horizon radiiwhen 120595120579 vanishes It is clear that 120595120579 vanishing approximatelyresults in 11989200 of (69) Figure 2 shows the behavior of ℎ00 atthe same conditions when we have 120595120579 As we can see fromthese figures there is a different behavior in the perfect statein comparison with the imperfect state near the horizon radiiwhich is due to the nature of torsional spacetime Howeverthe same behaviors have been indicated in the origin and thehigher bound of 119903

5 Force Equation in Commutative Limit

In teleparallel gravity the Newtonian force equation isobtained by assuming the class of frames in which theteleparallel spin connection vanishes and the gravitationalfield is stationary and weak [40 41 48 49] In our modelthe Newtonian gravitational force equation directly derivesfrom torsion components by 120595120579 in its commutative limitWhen we write the expansion of new line element in thenoncommutative limit we have

ℎ00 = 1 minus1205940

41198902

119898 (119903)

119903+ (

1205940

41198902

119898 (119903)

119903)

2

minus (1205940

41198902

119898 (119903)

119903)

3

(72)

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

This is the conservation law of the source of energy-momentum tensor Variation with respect to the noncom-mutative gauge potential 119861

120583

119886yields the noncommutative

teleparallel version of the gravitational field equations

120597120590 (ℎ 119878120583120590

119886) minus 119896ℎ 119869

120583

119886= 119896ℎΞ

120583

119886 (34)

where

ℎ 119878120583120590

119886= ℎℎ120582

119886119878120583120590

120582equiv minus119896

120597

120597 (120597120590ℎ119886120583

)

(35)

which defines the superpotential For the gauge current wehave

ℎ 119869120583

119886= minus

120597

120597119861119886120583

equiv minus120597

120597ℎ119886120583

(36)

Note that the matter energy-momentum tensor which isdefined in this relation appears as the source of torsionsimilarly the energy-momentum tensor appears as the sourceof curvature in general relativity Our computation has led usto the following results

119878120583120590

119886= 2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578

119869120583

119886=

1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888

(37)

for noncommutative superpotential and gauge current Thelagrangian appears again in our equations but notice thatthis term cross its coefficient yields a term purely based onits field strength according to (2) This simplified expressionmaintains equivalence to general relativity We can observethat the gravitational field equations depend on the torsiononly Finally the field equations can be written as

120597120590 (ℎ (2120583120590

119886minus 120590120583

119886minus ℎ120590

119886120578120583

120578+ ℎ120583

119886120578120590

120578))

minus 119896ℎ (1

119896ℎ120582

119886119878]120583119888

119888

]120582 minusℎ120583

119886

ℎ +

1

119896119888

119886120590119878120583120590

119888) = 119896ℎΞ

120583

119886

(38)

where 119896 = 12059401198902 is a constant These field equations are

similar to teleparallel field equations although it wouldbe distinguished with different field strength

120583120590

119886which is

given by the covariant rotational of noncommutative gaugepotential of 119861

120583

119886 Consider the following equations from the

teleparallel theory (see for instance [32 34ndash36])

119886

120583] = 120597]ℎ119886

120583minus 120597120583ℎ119886

] + 119886

119890]ℎ119890

120583minus 119886

119890120583ℎ119890

] (39)

Γ120588

]120583 = ℎ120588

119886120597120583ℎ119886

] + ℎ120588

119886119886

119887120583ℎ119887

] (40)

120597120583ℎ119886

] minus Γ120588

]120583ℎ119886

120588+ 119886

119887120583ℎ119887

] = 0 (41)

120588

]120583 = Γ120588

120583] minus Γ120588

]120583 (42)

The field equations take the exact following form

120597

120597119909120590(Γ120590

119886120583minus Γ120590

120583119886) minus

120597

120597119909120583Γ120582

119886120582+

120597

120597119909120582Γ120582

119886120583

minus Γ120578

119886120582Γ120582

120583120578+ Γ120578

119886120583Γ120582

120582120578=

1205940

1198902120588 (119903)

120597

120597119909119886

120597

120597119909120583

(43)

which unlike the left hand side of (38) is written purelybased on noncommutative field strength and the abovefield equation is written in terms of Weitzenbock connec-tion only Regarding the equivalency between correspondingLagrangians and the above simplified field equations andapplying (34) we have therefore

119877119886120583 minus1

2ℎ119886120583119877 = 119896Ξ119886120583 (44)

as equivalent to Einsteinrsquos field equations Note that (44) isnot Einsteinrsquos field equations but the teleparallel field equa-tions made equivalent to general relativity And equivalentmodel of teleparallel field equations with general relativity isexpressed in the references in detail (see for instance [32 3435]) We continue our discussion to derive noncommutativeline element by solving these field equations

4 Noncommutative Line Element

Teleparallel versions of the stationary static spherically axis-symmetric and symmetric of the Schwarzschild solutionhave been previously obtained [37 38] Within a frameworkinspired by noncommutative geometry we solve the fieldequations for a distribution of spherically symmetricallymassin a stationary static spacetime like the exterior solution ofSchwarzschild (see also [29])Then it is natural to assume thatthe line element is as follows

1198891199042

= minus119891 (119903) 1198891199052

+ 119892 (119903) 1198891199032

+ ℎ (119903) 1199032

(1198891205792

+ sin21205791198891206012)

(45)

With a new radial coordinate defined as 119903 = 119903radicℎ(119903) the lineelement becomes

1198891199042

= minus119860 (119903) 1198891199052

+ 119861 (119903) 1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (46)

Usually one replaces the functions 119860(119903) and 119861(119903) with expo-nential functions to obtain somewhat simpler expressionsfor the noncommutative tensor components Hence weintroduce the functions 120572(119903) and 120573(119903) by 119890

2120572(119903)= 119860(119903) and

119890120573(119903)

= 119861(119903) to get

1198891199042

= minus1198902120572

1198891199052

+ 1198902120573

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012) (47)

Tetrad components of the above metric take the followingform

ℎ119886

120583=

[[[

[

minus1198902120572

0 0 0

0 1198902120573 sin 120579 cos120601 119903 cos 120579 cos120601 minus119903 cos 120579 sin120601

0 1198902120573 sin 120579 sin120601 119903 cos 120579 sin120601 119903 sin 120579 cos120601

0 1198902120573 cos 120579 minus119903 sin 120579 0

]]]

]

(48)

Weitzenbock connection Γ120588

120583] has the following expression

Γ120588

120583] = ℎ120588

119886120597]ℎ119886

120583 (49)

Advances in High Energy Physics 5

Now we can calculate the nonvanishing components ofWeitzenbock connection as follows

Γ0

01= minus2120572

1015840 Γ

1

11= 21205731015840 Γ

1

22= minus119903119890

minus120573

Γ1

33= minus119903119890

minus120573sin2120579 Γ2

12=

119890120572

119903= Γ3

13

Γ2

21=

1

119903= Γ3

31 Γ

2

33= minus sin 120579 cos 120579

Γ3

23= Γ3

32= cot 120579

(50)

By replacing these components in (43) the noncommutativetensors of (51)ndash(53) for the left-hand side of the field equationswill produce the following expression

119873 =1

1199032(minus4119890minus2120573

+ 1 minus 120595120579) minus2

1199031205731015840119890minus2120573

=1205940

1198902120588 (119903) 120575 (51)

119873119903119903 =1

1199032(2119890minus2120573

+ 1 minus 120595120579) +2

1199031205721015840119890minus2120573

=1205940

1198902120588 (119903) 120575119903119903 (52)

119873120579120579

= 119873120601120601

=1

119903119890minus2120573

(11990312057210158401015840

+ 11990312057210158402

minus 11990312057210158401205731015840+ 1205721015840minus 1205731015840minus 1)

+ 12057210158402

119890minus2120573

=1205940

1198902120588 (119903) 120575

120579120579=

1205940

1198902120588 (119903) 120575

120601120601

(53)

Adding (51) and (52) we get simply

1

1199032(120595120579 minus 119890

minus2120573+ 119890minus2120573

(1205721015840minus 1205731015840) + 1) =

1205940

1198902120588 (119903) (54)

where 120572(119903) = 120573(119903) It should also be noted that by recalling(47) we can consider the limiting case for our solutionassuming (120572

1015840minus 1205731015840) = 119896 where 119896 is a constant and by

considering the time coordinate we can shift this constantto an arbitrary value It is possible therefore without loss ofgenerality to choose 119896 = 0 It does not contradict (47) toset 1205721015840

= 1205731015840 According to this analysis the equation 119873 =

(12059401198902)120588(119903)120575 can be written as

minus1

119903

119889

119889119903[119903 (119890minus2120573

minus 120595120579 minus 1)] =1205940

1198902120588 (119903) (55)

For a perfect fluid in thermodynamic equilibrium the stress-energy tensor takes on a particularly simple form

Ξ120583]

= (120588 + 119875) 119906120583119906]

+ 119901119892120583]

(56)

where the pressure 119875 can be neglected due to the distributionof mass and the gravitational effects consequently only oneterm will remain in the above formula as follows

Ξ119886120583

= 120588 (119903)119889119909119886

119889119905

119889119909120583

119889119905 (57)

or

Ξ119886120583

= 120588 (119903) 120575119886120583

(58)

Therefore for spherically symmetric distribution ofmass thatdepends on r-coordinate we can write

119898 (119903) = int

119903

0

41205871199032120588 (119903) 119889119903 (59)

Note that the 120588(119903) is defined by (1) Indeed we introducethe same energy density indicated in the noncommutativeperturbation theory [39]

119898 (119903) = 119872120579 (119903) =2119872

radic120587120574 (

3

2

1199032

4120579) (60)

Equation (55) can be integrated to find

119890minus2120573

= 1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579 (61)

where 120595120579 is a function that carries the tetrad field factor andwill be defined later by (66) and (67) Now by considering

119890minus2120573

= minus1

ℎ11

= ℎ00 (62)

the noncommutative line element for a spherically symmetricmatter distribution is therefore

1198891199042

= minus (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579) 119889119905

2

+ (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579)

minus1

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012)

(63)

The constant field of 12059401198902 in terms of (3) (4) and (5) can be

retrieved as

1205940

1198902=

1003816100381610038161003816100381612057912058311988710038161003816100381610038161003816

16120587119866119873

(64)

where 119866119873 is the Newtonian constant and |120579120583119887

| is determinedby 120579120583119887

= 12057921

= minus12057912

equiv 120579 Where 120579 is a real antisymmetricand constant tensor therefore the above equation can besimplified to yield

1205940

1198902=

120579

16120587119866119873

(65)

New line element (63) in particular depends on 120595120579 andnaturally 120595120579 has its origin on the quantum fluctuations ofthe noncommutative background geometry and originallycomes from the field equations The presented solution forour field equations produces naturally some additional termsin comparison with the solution of noncommutative versionof general relativity (naturally because it has some additionalterms in its components) These terms appear in the new lineelement because 120595120579 relates to the noncommutative torsionalspacetime and algebraic properties in spherically symmetricsolution of the tetrad fields We have therefore 120595120579 in thefollowing simplified equation

120595120579 = 120576119903120579120601

120576119903120579120601

ℎ119903

119903119890minus120573

(66)

Definition 120576119903120579120601

120576119903120579120601

= minus6ℎ2 is applied here (see also

[40 41]) According to this definition and (48) and (50)

6 Advances in High Energy Physics

Through simplification we find the following form of 120595120579 wefind the following form of 120595120579

120595120579 cong sum

119896=2119899

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

minus sum

119896=2119899+1

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

(67)

Note that 120595120579 is considered with the lower bound of sum Ifwe want to consider at least the second order of 120579 (which isproposed by [13]) for 120595120579 then it is natural to assume 119899 = 1Therefore two states for our line element will be producedthe imperfect state and the perfect state Let us now considerthe perfect stateThere is a proof for this state in terms of sometheorems in mathematics that allows us to introduce our lineelement as an appropriate description for a noncommutativespacetime Combination of these theoremswith regard to ourresults is given by [42ndash47]

Theorem 1 Let 119871 be a perfect field Recall that a polynomial119891(119909) isin 119871[119909] is called additive if 119891(119909 + 119910) = 119891(119909) + 119891(119910)

identically It is easy to see that a polynomial is additive if andonly if it is of the form

119891 (119909) = 1 minus 1198860119909 + 1198862

11199092

minus sdot sdot sdot plusmn 119886119899

119899119909119899

= sum

119899=0

1198862119899

1198991199092119899

minus sum

119899=0

1198862119899+1

1198991199092119899+1

(68)

The set of additive polynomials forms a noncommutative fieldin which (119891 ∘ 119892)(119909) = 119891(119892(119909)) This field is generated byscalar multiplications 119909 997891rarr 119886119909 for 119886 isin 119871 and 119909119894 isin 119891(119909) doesnot commute with the 119909119895120598119891(119909) Note that 119886 can be a constantfield and it has given as asymp 12059404120587119890

2 here (see [42ndash47] and thereferences cited therein) It is clear that components of119891(119909) canbe exactly replaced with components of ℎ00

Regarding other investigations into descriptions of non-commutative spacetime we should expand our discussioninto a comparisonmethodwith other line elements presentedfor noncommutative spacetime Reference [13] suggests thefollowing line element for noncommutative Schwarzschildspacetime

1198891199042

= minus (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)) 119889119905

2

+ (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579))

minus1

1198891199032

+ 1199032119889Ω2

(69)

where 119889Ω2

= 1198891205792

+ sin21205791198891206012 and 120574(32 119903

24120579) is the lower

incomplete gamma function

120574 (3

2

1199032

4120579) equiv int

11990324120579

0

119889119905radic119905119890minus119905

(70)

We note that nonvanishing radial pressure is a consequenceof the quantum fluctuation of the spacetimemanifold leadingto an inward gravitational pull and preventing the mattercollapsing into a point According to the line element (63)

in a neighborhood of the origin at 119903 le 120579 the energy densitydistribution of a static symmetric and noncommutative fuzzyspacetime is described by (1) which replaces the Dirac 120575

distribution by a smeared Gaussian profile Meanwhile inthe imperfect state our line element can be made equivalentto the line element of (69) and it is expected to happenwhen 120595120579 vanishes Assuredly it is due to vanishing ofthe tetrad components ℎ

119886

120583in (48) or even Wietzenbock

connections in (50) It means that in absence of torsionalspacetime the coordinate coherent state will be produced inthe noncommutative field theory It is completely reasonablesince coherent state theory is derived in the noncommutativeframework of the general relativity and the torsion is notdefined in the general relativity This equivalency is shownwith the following relation

1 minus119872

2119903radic120587120574 (

3

2

1199032

4120579) cong 119892

coherent state00

= 1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)

(71)

According to this proof the solution of the presented non-commutative field equations in the imperfect state of itselfresults in the exact solution of noncommutative generalrelativity field equations through coordinate coherent state ofour line element

41 Schwarzschild Black Hole Horizons In this paper wehave not extended our discussion into black holes butour introduced equations can be the basis of a subject onnoncommutative black holes Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black holewould be done by the horizon equation minusℎ119903

119867

= ℎ11

(119903119867) = 0Answers to this equation are illustrated in Figures 1 and 2Figure 1 shows the behavior of ℎ00 versus the horizon radiiwhen 120595120579 vanishes It is clear that 120595120579 vanishing approximatelyresults in 11989200 of (69) Figure 2 shows the behavior of ℎ00 atthe same conditions when we have 120595120579 As we can see fromthese figures there is a different behavior in the perfect statein comparison with the imperfect state near the horizon radiiwhich is due to the nature of torsional spacetime Howeverthe same behaviors have been indicated in the origin and thehigher bound of 119903

5 Force Equation in Commutative Limit

In teleparallel gravity the Newtonian force equation isobtained by assuming the class of frames in which theteleparallel spin connection vanishes and the gravitationalfield is stationary and weak [40 41 48 49] In our modelthe Newtonian gravitational force equation directly derivesfrom torsion components by 120595120579 in its commutative limitWhen we write the expansion of new line element in thenoncommutative limit we have

ℎ00 = 1 minus1205940

41198902

119898 (119903)

119903+ (

1205940

41198902

119898 (119903)

119903)

2

minus (1205940

41198902

119898 (119903)

119903)

3

(72)

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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FluidsJournal of

Atomic and Molecular Physics

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Advances in Condensed Matter Physics

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AstronomyAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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Statistical MechanicsInternational Journal of

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Soft MatterJournal of

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ThermodynamicsJournal of

Advances in High Energy Physics 5

Now we can calculate the nonvanishing components ofWeitzenbock connection as follows

Γ0

01= minus2120572

1015840 Γ

1

11= 21205731015840 Γ

1

22= minus119903119890

minus120573

Γ1

33= minus119903119890

minus120573sin2120579 Γ2

12=

119890120572

119903= Γ3

13

Γ2

21=

1

119903= Γ3

31 Γ

2

33= minus sin 120579 cos 120579

Γ3

23= Γ3

32= cot 120579

(50)

By replacing these components in (43) the noncommutativetensors of (51)ndash(53) for the left-hand side of the field equationswill produce the following expression

119873 =1

1199032(minus4119890minus2120573

+ 1 minus 120595120579) minus2

1199031205731015840119890minus2120573

=1205940

1198902120588 (119903) 120575 (51)

119873119903119903 =1

1199032(2119890minus2120573

+ 1 minus 120595120579) +2

1199031205721015840119890minus2120573

=1205940

1198902120588 (119903) 120575119903119903 (52)

119873120579120579

= 119873120601120601

=1

119903119890minus2120573

(11990312057210158401015840

+ 11990312057210158402

minus 11990312057210158401205731015840+ 1205721015840minus 1205731015840minus 1)

+ 12057210158402

119890minus2120573

=1205940

1198902120588 (119903) 120575

120579120579=

1205940

1198902120588 (119903) 120575

120601120601

(53)

Adding (51) and (52) we get simply

1

1199032(120595120579 minus 119890

minus2120573+ 119890minus2120573

(1205721015840minus 1205731015840) + 1) =

1205940

1198902120588 (119903) (54)

where 120572(119903) = 120573(119903) It should also be noted that by recalling(47) we can consider the limiting case for our solutionassuming (120572

1015840minus 1205731015840) = 119896 where 119896 is a constant and by

considering the time coordinate we can shift this constantto an arbitrary value It is possible therefore without loss ofgenerality to choose 119896 = 0 It does not contradict (47) toset 1205721015840

= 1205731015840 According to this analysis the equation 119873 =

(12059401198902)120588(119903)120575 can be written as

minus1

119903

119889

119889119903[119903 (119890minus2120573

minus 120595120579 minus 1)] =1205940

1198902120588 (119903) (55)

For a perfect fluid in thermodynamic equilibrium the stress-energy tensor takes on a particularly simple form

Ξ120583]

= (120588 + 119875) 119906120583119906]

+ 119901119892120583]

(56)

where the pressure 119875 can be neglected due to the distributionof mass and the gravitational effects consequently only oneterm will remain in the above formula as follows

Ξ119886120583

= 120588 (119903)119889119909119886

119889119905

119889119909120583

119889119905 (57)

or

Ξ119886120583

= 120588 (119903) 120575119886120583

(58)

Therefore for spherically symmetric distribution ofmass thatdepends on r-coordinate we can write

119898 (119903) = int

119903

0

41205871199032120588 (119903) 119889119903 (59)

Note that the 120588(119903) is defined by (1) Indeed we introducethe same energy density indicated in the noncommutativeperturbation theory [39]

119898 (119903) = 119872120579 (119903) =2119872

radic120587120574 (

3

2

1199032

4120579) (60)

Equation (55) can be integrated to find

119890minus2120573

= 1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579 (61)

where 120595120579 is a function that carries the tetrad field factor andwill be defined later by (66) and (67) Now by considering

119890minus2120573

= minus1

ℎ11

= ℎ00 (62)

the noncommutative line element for a spherically symmetricmatter distribution is therefore

1198891199042

= minus (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579) 119889119905

2

+ (1 minus1205940

41205871198902

119898 (119903)

119903+ 120595120579)

minus1

1198891199032

+ 1199032

(1198891205792

+ sin21205791198891206012)

(63)

The constant field of 12059401198902 in terms of (3) (4) and (5) can be

retrieved as

1205940

1198902=

1003816100381610038161003816100381612057912058311988710038161003816100381610038161003816

16120587119866119873

(64)

where 119866119873 is the Newtonian constant and |120579120583119887

| is determinedby 120579120583119887

= 12057921

= minus12057912

equiv 120579 Where 120579 is a real antisymmetricand constant tensor therefore the above equation can besimplified to yield

1205940

1198902=

120579

16120587119866119873

(65)

New line element (63) in particular depends on 120595120579 andnaturally 120595120579 has its origin on the quantum fluctuations ofthe noncommutative background geometry and originallycomes from the field equations The presented solution forour field equations produces naturally some additional termsin comparison with the solution of noncommutative versionof general relativity (naturally because it has some additionalterms in its components) These terms appear in the new lineelement because 120595120579 relates to the noncommutative torsionalspacetime and algebraic properties in spherically symmetricsolution of the tetrad fields We have therefore 120595120579 in thefollowing simplified equation

120595120579 = 120576119903120579120601

120576119903120579120601

ℎ119903

119903119890minus120573

(66)

Definition 120576119903120579120601

120576119903120579120601

= minus6ℎ2 is applied here (see also

[40 41]) According to this definition and (48) and (50)

6 Advances in High Energy Physics

Through simplification we find the following form of 120595120579 wefind the following form of 120595120579

120595120579 cong sum

119896=2119899

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

minus sum

119896=2119899+1

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

(67)

Note that 120595120579 is considered with the lower bound of sum Ifwe want to consider at least the second order of 120579 (which isproposed by [13]) for 120595120579 then it is natural to assume 119899 = 1Therefore two states for our line element will be producedthe imperfect state and the perfect state Let us now considerthe perfect stateThere is a proof for this state in terms of sometheorems in mathematics that allows us to introduce our lineelement as an appropriate description for a noncommutativespacetime Combination of these theoremswith regard to ourresults is given by [42ndash47]

Theorem 1 Let 119871 be a perfect field Recall that a polynomial119891(119909) isin 119871[119909] is called additive if 119891(119909 + 119910) = 119891(119909) + 119891(119910)

identically It is easy to see that a polynomial is additive if andonly if it is of the form

119891 (119909) = 1 minus 1198860119909 + 1198862

11199092

minus sdot sdot sdot plusmn 119886119899

119899119909119899

= sum

119899=0

1198862119899

1198991199092119899

minus sum

119899=0

1198862119899+1

1198991199092119899+1

(68)

The set of additive polynomials forms a noncommutative fieldin which (119891 ∘ 119892)(119909) = 119891(119892(119909)) This field is generated byscalar multiplications 119909 997891rarr 119886119909 for 119886 isin 119871 and 119909119894 isin 119891(119909) doesnot commute with the 119909119895120598119891(119909) Note that 119886 can be a constantfield and it has given as asymp 12059404120587119890

2 here (see [42ndash47] and thereferences cited therein) It is clear that components of119891(119909) canbe exactly replaced with components of ℎ00

Regarding other investigations into descriptions of non-commutative spacetime we should expand our discussioninto a comparisonmethodwith other line elements presentedfor noncommutative spacetime Reference [13] suggests thefollowing line element for noncommutative Schwarzschildspacetime

1198891199042

= minus (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)) 119889119905

2

+ (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579))

minus1

1198891199032

+ 1199032119889Ω2

(69)

where 119889Ω2

= 1198891205792

+ sin21205791198891206012 and 120574(32 119903

24120579) is the lower

incomplete gamma function

120574 (3

2

1199032

4120579) equiv int

11990324120579

0

119889119905radic119905119890minus119905

(70)

We note that nonvanishing radial pressure is a consequenceof the quantum fluctuation of the spacetimemanifold leadingto an inward gravitational pull and preventing the mattercollapsing into a point According to the line element (63)

in a neighborhood of the origin at 119903 le 120579 the energy densitydistribution of a static symmetric and noncommutative fuzzyspacetime is described by (1) which replaces the Dirac 120575

distribution by a smeared Gaussian profile Meanwhile inthe imperfect state our line element can be made equivalentto the line element of (69) and it is expected to happenwhen 120595120579 vanishes Assuredly it is due to vanishing ofthe tetrad components ℎ

119886

120583in (48) or even Wietzenbock

connections in (50) It means that in absence of torsionalspacetime the coordinate coherent state will be produced inthe noncommutative field theory It is completely reasonablesince coherent state theory is derived in the noncommutativeframework of the general relativity and the torsion is notdefined in the general relativity This equivalency is shownwith the following relation

1 minus119872

2119903radic120587120574 (

3

2

1199032

4120579) cong 119892

coherent state00

= 1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)

(71)

According to this proof the solution of the presented non-commutative field equations in the imperfect state of itselfresults in the exact solution of noncommutative generalrelativity field equations through coordinate coherent state ofour line element

41 Schwarzschild Black Hole Horizons In this paper wehave not extended our discussion into black holes butour introduced equations can be the basis of a subject onnoncommutative black holes Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black holewould be done by the horizon equation minusℎ119903

119867

= ℎ11

(119903119867) = 0Answers to this equation are illustrated in Figures 1 and 2Figure 1 shows the behavior of ℎ00 versus the horizon radiiwhen 120595120579 vanishes It is clear that 120595120579 vanishing approximatelyresults in 11989200 of (69) Figure 2 shows the behavior of ℎ00 atthe same conditions when we have 120595120579 As we can see fromthese figures there is a different behavior in the perfect statein comparison with the imperfect state near the horizon radiiwhich is due to the nature of torsional spacetime Howeverthe same behaviors have been indicated in the origin and thehigher bound of 119903

5 Force Equation in Commutative Limit

In teleparallel gravity the Newtonian force equation isobtained by assuming the class of frames in which theteleparallel spin connection vanishes and the gravitationalfield is stationary and weak [40 41 48 49] In our modelthe Newtonian gravitational force equation directly derivesfrom torsion components by 120595120579 in its commutative limitWhen we write the expansion of new line element in thenoncommutative limit we have

ℎ00 = 1 minus1205940

41198902

119898 (119903)

119903+ (

1205940

41198902

119898 (119903)

119903)

2

minus (1205940

41198902

119898 (119903)

119903)

3

(72)

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

Through simplification we find the following form of 120595120579 wefind the following form of 120595120579

120595120579 cong sum

119896=2119899

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

minus sum

119896=2119899+1

sum

119899=1

(1205940

41205871198902

119898 (119903)

119903)

119896

(67)

Note that 120595120579 is considered with the lower bound of sum Ifwe want to consider at least the second order of 120579 (which isproposed by [13]) for 120595120579 then it is natural to assume 119899 = 1Therefore two states for our line element will be producedthe imperfect state and the perfect state Let us now considerthe perfect stateThere is a proof for this state in terms of sometheorems in mathematics that allows us to introduce our lineelement as an appropriate description for a noncommutativespacetime Combination of these theoremswith regard to ourresults is given by [42ndash47]

Theorem 1 Let 119871 be a perfect field Recall that a polynomial119891(119909) isin 119871[119909] is called additive if 119891(119909 + 119910) = 119891(119909) + 119891(119910)

identically It is easy to see that a polynomial is additive if andonly if it is of the form

119891 (119909) = 1 minus 1198860119909 + 1198862

11199092

minus sdot sdot sdot plusmn 119886119899

119899119909119899

= sum

119899=0

1198862119899

1198991199092119899

minus sum

119899=0

1198862119899+1

1198991199092119899+1

(68)

The set of additive polynomials forms a noncommutative fieldin which (119891 ∘ 119892)(119909) = 119891(119892(119909)) This field is generated byscalar multiplications 119909 997891rarr 119886119909 for 119886 isin 119871 and 119909119894 isin 119891(119909) doesnot commute with the 119909119895120598119891(119909) Note that 119886 can be a constantfield and it has given as asymp 12059404120587119890

2 here (see [42ndash47] and thereferences cited therein) It is clear that components of119891(119909) canbe exactly replaced with components of ℎ00

Regarding other investigations into descriptions of non-commutative spacetime we should expand our discussioninto a comparisonmethodwith other line elements presentedfor noncommutative spacetime Reference [13] suggests thefollowing line element for noncommutative Schwarzschildspacetime

1198891199042

= minus (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)) 119889119905

2

+ (1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579))

minus1

1198891199032

+ 1199032119889Ω2

(69)

where 119889Ω2

= 1198891205792

+ sin21205791198891206012 and 120574(32 119903

24120579) is the lower

incomplete gamma function

120574 (3

2

1199032

4120579) equiv int

11990324120579

0

119889119905radic119905119890minus119905

(70)

We note that nonvanishing radial pressure is a consequenceof the quantum fluctuation of the spacetimemanifold leadingto an inward gravitational pull and preventing the mattercollapsing into a point According to the line element (63)

in a neighborhood of the origin at 119903 le 120579 the energy densitydistribution of a static symmetric and noncommutative fuzzyspacetime is described by (1) which replaces the Dirac 120575

distribution by a smeared Gaussian profile Meanwhile inthe imperfect state our line element can be made equivalentto the line element of (69) and it is expected to happenwhen 120595120579 vanishes Assuredly it is due to vanishing ofthe tetrad components ℎ

119886

120583in (48) or even Wietzenbock

connections in (50) It means that in absence of torsionalspacetime the coordinate coherent state will be produced inthe noncommutative field theory It is completely reasonablesince coherent state theory is derived in the noncommutativeframework of the general relativity and the torsion is notdefined in the general relativity This equivalency is shownwith the following relation

1 minus119872

2119903radic120587120574 (

3

2

1199032

4120579) cong 119892

coherent state00

= 1 minus4119872

119903radic120587120574 (

3

2

1199032

4120579)

(71)

According to this proof the solution of the presented non-commutative field equations in the imperfect state of itselfresults in the exact solution of noncommutative generalrelativity field equations through coordinate coherent state ofour line element

41 Schwarzschild Black Hole Horizons In this paper wehave not extended our discussion into black holes butour introduced equations can be the basis of a subject onnoncommutative black holes Indeed the calculation of eventhorizons of a noncommutative Schwarzschild black holewould be done by the horizon equation minusℎ119903

119867

= ℎ11

(119903119867) = 0Answers to this equation are illustrated in Figures 1 and 2Figure 1 shows the behavior of ℎ00 versus the horizon radiiwhen 120595120579 vanishes It is clear that 120595120579 vanishing approximatelyresults in 11989200 of (69) Figure 2 shows the behavior of ℎ00 atthe same conditions when we have 120595120579 As we can see fromthese figures there is a different behavior in the perfect statein comparison with the imperfect state near the horizon radiiwhich is due to the nature of torsional spacetime Howeverthe same behaviors have been indicated in the origin and thehigher bound of 119903

5 Force Equation in Commutative Limit

In teleparallel gravity the Newtonian force equation isobtained by assuming the class of frames in which theteleparallel spin connection vanishes and the gravitationalfield is stationary and weak [40 41 48 49] In our modelthe Newtonian gravitational force equation directly derivesfrom torsion components by 120595120579 in its commutative limitWhen we write the expansion of new line element in thenoncommutative limit we have

ℎ00 = 1 minus1205940

41198902

119898 (119903)

119903+ (

1205940

41198902

119898 (119903)

119903)

2

minus (1205940

41198902

119898 (119903)

119903)

3

(72)

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

1 2 3 4 5 6 7 8 9 10

r

1

08

06

04

02

0

minus02

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

Figure 1 The imperfect state in a noncommutative sphericallysymmetric geometry The function of ℎ00 versus 119903radic120579 for variousvalues of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579

(without horizon) the middle curve corresponds to 119872 = 1198720

asymp

190radic120579 (with one horizon at 119903119867 = 1199030 asymp 49radic120579) and finally thelowest curve corresponds to 119872 = 302radic120579 (two horizons at 119903119867 =

119903minus asymp 270radic120579 and 119903119867 = 119903+ asymp 720radic120579)

1 2 3 4 5 6 7 8 9 10

r

h00

h00 forM gt M0

h00 forM = M0

h00 forM lt M0

1

08

06

04

02

0

minus02

minus04

minus06

Figure 2 The perfect state in a noncommutative spherically sym-metric geometry The function of ℎ

00versus 119903radic120579 for various values

of 119872radic120579 The upper curve corresponds to 119872 = 100radic120579 (withouthorizon) the middle one corresponds to 119872 = 1198720 asymp 190radic120579 (withone horizon at 119903119867 = 1199030 asymp 49radic120579) and finally the lowest curvecorresponds to 119872 = 302radic120579 (two horizons at 119903119867 = 119903minus asymp 390radic120579

and 119903119867 = 119903+ asymp 580radic120579)

or equivalently

ℎ00 = 1 minus 119860119898 (119903)

119903+ 119861

119898 (119903)2

1199032minus 119862

119898 (119903)3

1199033 (73)

Due to noncommutative effects 119903 in the denominator van-ishes but in the limit case when it goes to the commutativelimit it is modified to the commutative 11989200 of Schwarzschildsolution in addition to a force equation much similar tothe Newtonian gravitational force equation Note that theinduced gravitational constant of (3) vanishes in the com-mutative limit and agrees with that found in [50] usingthe supergravity dual of noncommutative Yang-Mills theoryin four dimensions Newton was the first to consider in

his Principia an extended expression of his law of gravityincluding an inverse-cube term of the form

119865 = 11986611989811198982

1199032+ 119861

11989811198982

1199033 119861 is a constant (74)

He attempts to explain Moonrsquos apsidal motion by the aboverelation In the commutative limit our metric can be definedin the form of

ℎcommutative00

cong 1 minus2119872

119903+

41198722

1199032minus

81198723

1199033 (75)

where 119898(119903) is given by (60) and in the commutative limit ithas the form of

lim120579rarr0

119898 (119903) = 2119872 (76)

By considering the following terms in (74)

(i) relativistic limits 119866 = 1

(ii) set the 1198981 = 1198982 = 2119872 119861 = minus2119872

for our line element we can set

ℎcommutative00

= (119892commutative Schwarszchild solution00

+ 119865 (119903)Newton

)

(77)

As we can see from (63) and (67) (expansion of newline element) the ℎ00 has two parts torsional and nontor-sional parts the above relation states that in the limit ofcommutativity torsional parts reduce to force equation of119865(119903) and nontorsional part yields the 11989200 of commutativeSchwarzschild solution

Einsteinrsquos theory of general relativity attributes gravita-tion to curved spacetime instead of being due to a forcepropagated between bodies Energy and momentum distortspacetime in their vicinity and other particles move in trajec-tories determined by the geometry of spacetime Thereforedescriptions of the motions of light and mass are consistentwith all available observations Meanwhile according to thegeneral relativityrsquos definition the gravitational force is afictitious force due to the curvature of spacetime becausethe gravitational acceleration of a body in free fall is due toits world line being a geodesic of spacetime [51] Howeverthrough a weak equivalence principle that is assumed initiallyin teleparallel gravity [52ndash54] our results are reasonable andwe can conclude that the presented solution in its commutativelimit attributes the gravitation to a force propagated betweenbodies and the curved spacetime or sum of torsion andcurvature This result is similar to Einstein-Cartan theory ofgravity [55]

Moreover the different behaviour of Schwarzschild blackhole horizon which is absent in the previous method is dueto force of heavy pulling from the black hole in terms of thisintroduced force As can be seen in Figure 2 the intensity ofthis force has a direct relation withmass119872 so the heavier theblack hole is the stronger force it has near its horizon

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

6 Conclusion

In this letter we have utilized a noncommutative Lagrangianwhich gives us possibilities to use teleparallel gravity toderive field equations Solution of these field equations inthe spherically symmetric geometry yields a new noncom-mutative line element In the limit cases when the torsionvanishes we have obtained an interesting result in absenceof torsional spacetime the version of coordinate coherent statein noncommutative field theory will be produced IncidentallyFigures 1 and 2 show other limit cases in our solution at thelarge distances and different range of masses

As we expressed before there are conceptual differencesin general relativity curvature is used to geometrize thegravitational interaction geometry replaces the concept offorce and the trajectories are determined not by forceequations but by geodesics Teleparallel gravity on the otherhand attributes gravitation to torsion Torsion howeveraccounts for gravitation not by geometrizing the interactionbut by acting as a force [40 41] This is a definition used inteleparallel gravity whereas our model does not exactly coin-cidewith teleparallel gravity (in the limit case only) thereforeit is natural to have more complex results especially thatthe definition of the existing force in torsion for gravitationalinteractions is approved clearly in the limit of commutativity inour model Absolutely attributing the gravitation to the forceequation in the relativity framework which is shown directlythrough the commutative limit of our line element can beutilized in the various branches of physics

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author would like to thankMr Arya Bandari for effectiveproofreading of this paper

References

[1] M R Douglas and N A Nekrasov ldquoNoncommutative fieldtheoryrdquo Reviews of Modern Physics vol 73 no 4 pp 977ndash10292001

[2] W Kalau and M Walze ldquoGravity non-commutative geometryand theWodzicki residuerdquo Journal of Geometry and Physics vol16 no 4 pp 327ndash344 1995

[3] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[4] R J Szabo ldquoQuantum field theory on noncommutative spacesrdquoPhysics Reports vol 378 no 4 pp 207ndash299 2003

[5] E Langmann and R J Szabo ldquoTeleparallel gravity and dimen-sional reductions of noncommutative gauge theoryrdquo PhysicalReview D vol 64 no 10 Article ID 104019 15 pages 2001

[6] X Calmet and A Kobakhidze ldquoNoncommutative generalrelativityrdquo Physical Review D vol 72 no 4 Article ID 0450102005

[7] X Calmet and A Kobakhidze ldquoSecond order noncommutativecorrections to gravityrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 74 no 4 Article ID 0477022006

[8] A H Chamseddine ldquoDeforming Einsteinrsquos gravityrdquo PhysicsLetters B vol 504 no 1-2 pp 33ndash37 2001

[9] P Aschieri C Blohmann M Dimitrijevic F Meyer P Schuppand J Wess ldquoA gravity theory on noncommutative spacesrdquoClassical and Quantum Gravity vol 22 no 17 p 3511 2005

[10] A Smailagic and E Spallucci ldquoFeynman path integral on thenoncommutative planerdquo Journal of Physics A vol 36 p L4672003

[11] A Smailagic and E Spallucci ldquoUV divergence-free QFT onnoncommutative planerdquo Journal of Physics AMathematical andGeneral vol 36 no 39 pp L517ndashL521 2003

[12] A Smailagic and E Spallucci ldquoLorentz invariance unitarity andUV-finiteness of QFT on noncommutative spacetimerdquo Journalof Physics AMathematical andGeneral vol 37 no 28 pp 7169ndash7178 2004

[13] P Nicolini A Smailagic and E Spallucci ldquoNoncommutativegeometry inspired Schwarzschild black holerdquo Physics Letters Bvol 632 no 4 pp 547ndash551 2006

[14] C-S Chu and P-MHo ldquoNon-commutative open string andD-branerdquo Nuclear Physics B vol 550 no 1-2 pp 151ndash168 1999

[15] N Seiberg and E Witten ldquoString theory and noncommutativegeometryrdquoThe Journal of High Energy Physics vol 9 article 321999

[16] F Ardalan H Arfaei and M Sheikh-Jabbari ldquoNoncommuta-tive geometry from strings and branesrdquo Journal of High EnergyPhysics vol 2 p 16 1999

[17] F Ardalan H Arfaei and M M Sheikh-Jabbari ldquoDiracquantization of open strings and noncommutativity in branesrdquoNuclear Physics B vol 576 no 1ndash3 pp 578ndash596 2000

[18] C Sochichiu ldquoM[any] vacua of IIBrdquo Journal of High EnergyPhysics vol 2000 no 5 article 026 12 pages 2000

[19] R Blumenhagen and E Plauschinn ldquoNonassociative gravityin string theoryrdquo Journal of Physics A Mathematical andTheoretical vol 44 no 1 Article ID 015401 19 pages 2011

[20] D Lust ldquoT-duality and closed string non-commutative (dou-bled) geometryrdquo Journal of High Energy Physics vol 2010 article84 2010

[21] C Condeescu I Florakis and D Lust ldquoAsymmetric orbifoldsnon-geometric fluxes and non-commutativity in closed stringtheoryrdquo Journal of High Energy Physics vol 2012 no 4 p 1212012

[22] C Samann and R J Szabo ldquoGroupoids loop spaces andquantization of 2-plectic manifoldsrdquo Reviews in MathematicalPhysics vol 25 no 3 Article ID 1330005 72 pages 2013

[23] A Konechny and A Schwarz ldquoIntroduction to M(atrix) theoryand noncommutative geometryrdquo Physics Reports vol 360 no5-6 pp 353ndash465 2002

[24] D Mylonas P Schupp and R J Szabo ldquoMembrane sigma-models and quantization of non-geometric flux backgroundsrdquoJournal of High Energy Physics vol 2012 article 12 2012

[25] A Chatzistavrakidis and L Jonke ldquoMatrix theory origins ofnon-geometric fluxesrdquo Journal of High Energy Physics vol 2013no 2 article 040 2013

[26] P Nicolini ldquoVacuum energy momentum tensor in (2+1)NC scalar field theoryrdquo In press httparxivorgabshep-th0401204

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

[27] A Gruppuso ldquoNewtonrsquos law in an effective non-commutativespace-timerdquo Journal of Physics A vol 38 no 9 pp 2039ndash20422005

[28] S Ansoldi P Nicolini A Smailagic and E Spallucci ldquoNon-commutative geometry inspired charged black holesrdquo PhysicsLetters B vol 645 no 2-3 pp 261ndash266 2007

[29] D McMahon Relativity Demystified Tata McGraw-Hill Educa-tion 2006

[30] Oslash Groslashn and S Hervik Einsteinrsquos General Theory of RelativityWith Modern Applications in Cosmology Springer 2007

[31] SWeinbergGravitation and Cosmology vol 126 JohnWiley ampSons New York NY USA 1972

[32] V C deAndrade L C T Guillen and J G Pereira ldquoTeleparallelgravity an overviewrdquo in Proceedings of the 9th Marcel Gross-mann Meeting pp 1022ndash1023 July 2000

[33] D I Olive Lectures on Gauge Theories and Lie AlgebrasUniversity of Virginia Charlottesville Va USA 1982 notestaken by G Bhattacharya and N Turok

[34] J W Maluf M V Veiga and J F da Rocha-Neto ldquoRegularizedexpression for the gravitational energy-momentum in telepar-allel gravity and the principle of equivalencerdquoGeneral Relativityand Gravitation vol 39 no 3 pp 227ndash240 2007

[35] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[36] J G Pereira ldquoTeleparallelism a new insight into gravityrdquohttparxivorgabs13026983

[37] M Sharif and M J Amir ldquoTeleparallel Killing vectors of theEinstein universerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 23 no 13pp 963ndash969 2008

[38] J G Pereira T Vargas and C M Zhang ldquoAxial-vector torsionand the teleparallel Kerr spacetimerdquo Classical and QuantumGravity vol 18 no 5 pp 833ndash841 2001

[39] K Nozari and A Yazdani ldquoThe energy distribution of anoncommutative ReissnermdashNordstrom black holerdquo ChinesePhysics Letters vol 30 no 9 Article ID 090401 2013

[40] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction vol 173 Springer New York NY USA 2013

[41] V C de Andrade and J G Pereira ldquoGravitational Lorentz forceand the description of the gravitational interactionrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 56no 8 pp 4689ndash4695 1997

[42] D Goss ldquoThe adjoint of the Carlitz module and Fermatrsquos lasttheoremrdquo Finite Fields and Their Applications vol 1 no 2 pp165ndash188 1995

[43] D S Thakur ldquoHypergeometric functions for function fieldsrdquoFinite Fields and Their Applications vol 1 no 2 pp 219ndash2311995

[44] D Goss Basic Structures of Function Field Arithmetic SpringerNew York NY USA 1997

[45] K S Kedlaya ldquoThe algebraic closure of the power seriesfield in positive characteristicrdquo Proceedings of the AmericanMathematical Society vol 129 no 12 pp 3461ndash3470 2001

[46] H Hironaka ldquoResolution of singularities of an algebraic varietyover a field of characteristic zero IIrdquoAnnals ofMathematics vol79 pp 205ndash326 1964

[47] B Poonen ldquoFractional power series and pairings on Drinfeldmodulesrdquo Journal of the American Mathematical Society vol 9no 3 pp 783ndash812 1996

[48] G Zet ldquoSchwarzschild solution on a space-time with torsionrdquohttparxivorgabsgr-qc0308078

[49] H I Arcos and J G Pereira ldquoTorsion gravity a reappraisalrdquoInternational Journal of Modern Physics D vol 13 no 10 pp2193ndash2240 2004

[50] N Ishibashi S Iso H Kawai and Y Kitazawa ldquoString scale innoncommutative Yang-Millsrdquo Nuclear Physics B vol 583 no1-2 pp 159ndash181 2000

[51] httpenwikipediaorgwikiNewton27s law of universalgravitation

[52] G J Olmo ldquoViolation of the equivalence principle in modifiedtheories of gravityrdquoPhysical Review Letters vol 98 no 6 ArticleID 061101 2007

[53] R Aldrovandi J G Pereira and K H Vu ldquoGravitation withoutthe equivalence principlerdquo General Relativity and Gravitationvol 36 no 1 pp 101ndash110 2004

[54] C Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 no 3 pp 925ndash935 1961

[55] A Trautman ldquoEinstein-Cartan theoryrdquo httparxivorgabsgr-qc0606062

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of