Research Article Modified Newton s Law of Gravitation due...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 126528 7 pageshttpdxdoiorg1011552013126528

Research ArticleModified Newtonrsquos Law of Gravitation due toMinimal Length in Quantum Gravity

Ahmed Farag Ali1 and A Tawfik23

1 Department of Physics Faculty of Sciences Benha University Benha 13518 Egypt2 Egyptian Center for Theoretical Physics (ECTP) MTI University Cairo Egypt3 Research Center for Einstein Physics Free University of Berlin 14195 Berlin Germany

Correspondence should be addressed to Ahmed Farag Ali ahmedalifscbuedueg

Received 30 November 2012 Accepted 15 January 2013

Academic Editor George Siopsis

Copyright copy 2013 A Farag Ali and A Tawfik This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A recent theory about the origin of the gravity suggests that the gravity is originally an entropic force In this work we discuss theeffects of generalized uncertainty principle (GUP) which is proposed by some approaches to quantum gravity such as string theoryblack hole physics and doubly special relativity theories (DSR) on the area law of the entropyThis leads to aradicarea-type correctionto the area law of entropy which implies that the number of bits N is modified Therefore we obtain a modified Newtonrsquos lawof gravitation Surprisingly this modification agrees with different sign with the prediction of Randall-Sundrum II model whichcontains one uncompactified extra dimension Furthermore such modification may have observable consequences at length scalesmuch larger than the Planck scale

1 Introduction

The earliest idea about the connection between gravitationand the existence of a fundamental length was proposed in[1] In the last two decades the existence of a minimal lengthis one of the most interesting predictions of some approachesrelated to quantum gravity such as string theory black holephysics and noncommutative geometry [2ndash9] The existenceof a minimal length is considered as a consequence of thestring theory because strings obviously cannot interact atdistances smaller than the string size Furthermore the blackhole physics suggests that the uncertainty relation should bemodified near the Planck energy scale due to the fact that thephotons emitted from the black hole suffer from two majorerrors the first one is the error by Heisenbergrsquos classical anal-ysis and the second one is because the black hole mass variesduring the emission process and the radius of the horizonchanges accordingly [2ndash8] An interesting measure gedankenexperiment was proposed in [10] involving microblack holesat the Planck scale of spacetime which leads to the GUPThisindependent model depends on Heisenbergrsquos principle andSchwarzschild radius Recently it was found that polymer

quantization suggests the existence of minimal length in asimilarway to string theory and black hole physics [11]There-fore all these different approaches suggest that the standarduncertainty relation in quantum mechanics is modified toyieldgeneralized uncertainty principle (GUP) [2ndash8 11] Inlight of this such modifications can play an essential role asthe quantum gravitational corrections which would open aninterestingwindow for quantumgravity phenomenology [12ndash26]

In a one-dimensional chain as the Ising model whenassuming that every single spin is positioned at a distance119889 apart from the two neighborhoods the macroscopic stateof such a chain is defined by 119889 The entire chain would havevarious configurations so that if 119889 rarr 119897 the chain has muchless configurations than if 119889 ≪ 119897 where 119897 is the chainrsquoslength Statistically the entropy is given by the number ofmicroscopic states 119878 = 119896

119861lnΩ Due to the second law of

thermodynamics such a system tends to approach a state ofmaximal entropy so that the chain in the macroscopic state119889 tends to go to a 119889 state with a much higher entropy Theforce that causes such a statistical tendency is defined asthe entropic force In light of this the entropic force is a

2 Advances in High Energy Physics

phenomenologicalmechanism deriving a system to approachmaximum entropy that is increasing the number of micro-scopic states that will be inhered in the systemrsquos phase spaceThere are various examples on the entropic force for examplepolymer molecules and the elasticity of rubber bands

Recently Verlinde proposed that the gravity is not a fun-damental force and can be considered as an entropic force[27] The earliest idea about gravity that is regarded as a non-fundamental interaction has been introduced by Sakharov[28ndash31] where the spacetime background is assumed toemerge as a mean field approximation of underlying micro-scopic degrees of freedom Similar behavior is observed inhydrodynamics [32] It is found that the entropy of black holeis related to the horizonrsquos area at the black holersquos horizonwhile the temperature is related to the surface gravity Bothentropy and temperature are assumed to be related to themass of the black hole [33ndash36]Thus the connection betweenthermodynamics and geometry leads to Einsteinrsquos equa-tions of gravitational field from relations connecting heatentropy and temperature [37] Einsteinrsquos equations connectenergy-momentum tensor with space geometry Advocat-ing the gravity as nonfundamental interaction leads to theassumption that gravity would be explained as an entropicforce caused by changes in the information associated withthe positions of material bodies [27] When combining theentropic force with the Unruh temperature the second lawof Newton is obtained But when combining it with the holo-graphic principle and using the equipartition law of energyNewtonrsquos law of gravitation is obtained It was investigated in[38] modification of the entropic force due to corrections tothe area law of entropywhich is derived fromquantum effectsand extra dimensions

Apart from the controversial debate on the origin of grav-ity [39 40] we investigate the impact of GUP on the entropicforce and derive essential quantities including potentialmodification to Newtonrsquos law of gravity

There were some studies for the effect of some versions ofGUPonNewtonrsquos law of gravity in [41ndash44] Also noncommu-tative geometry which is considered as a completely planckscale effect has been studied to derive the modified Newtonrsquoslaw of gravity [45ndash48] All these approaches for studying thePlanck scale effects on the Newtonrsquos law of gravity are basedon the following schememodified theory of gravity rarr mod-ified black hole entropy rarr modified holographic surfaceentropy rarr Newtonrsquos law corrections We followed the samescheme in our paper using the new version of GUP proposedin [49ndash51] and we a got new corrections in our currentwork which are distinct from the previous studiesMoreoverwe compared our results with Randall-Sundrum model ofextra dimension which also predicts the modification ofNewtonrsquos law of gravity at the Planck scale [52 53] wherewe think there may be some connection between generalizeduncertainty principle and extra dimension theories becausethey predict similar physics at least for the case of Newtonrsquoslaw of gravity which may be considered as a distinct resultfrom the previous studies

The present paper is organized as follows Section 2reviews briefly the generalized uncertainty principle that wasproposed in [49ndash51] Section 3 is devoted to review the

entropic force and gravitational interaction [27] The effectof utilizing GUP impact on the entropic force is introducedin Section 4 In Section 5 we estimate the GUP correctionto Newtonrsquos law of gravitation The conclusions are given inSection 6

2 The Generalized Uncertainty Principle

It is conjectured that the standard commutation relations atshort distances would be modified A new form of GUP wasproposed [49ndash51] and found consistent with the doubly spe-cial relativity (DSR) theories the string theory and the blackhole physics It predicts a maximum observable momentumand a minimal measurable length With satisfying Jacobiidentity GUP is found to ensure the relations [119909

119894 119909119895] = 0 =

[119901119894 119901119895]

[119909119894 119901119895] = 119894ℎ [120575

119894119895minus 120572(119901120575

119894119895+119901119894119901119895

119901) + 1205722(1199012120575119894119895+ 3119901119894119901119895)]

(1)

where 120572 = 1205720119872119901119888 = 120572

0ℓ119901ℎ and 119872

1199011198882 stand for Planck

energy119872119901and ℓ119901are Planck mass and length respectively

1205720sets on the upper and lower bounds to 120572For a particle having an energy scale comparable to

Planckrsquos one the physical momentum would be a subject ofmodification [49ndash51]

119901119894= 1199010119894(1 minus 120572119901

0+ 212057221199012

0) (2)

where 119909119894= 1199090119894and 119901

0119895satisfy the canonical commutation

relations [1199090119894 1199010119895] = 119894ℎ120575

119894119895 Here 119901

0119894can be interpreted as

the momentum at low energies and 119901119894as that at high energies

and the variable 1199010is the value of the momentum at the low

energy scalesThis newly proposedGUP suggests that the space is quan-

tized into fundamental units whichmay be the Planck lengthThe quantization of space has been shown within the contextof loop quantum gravity in [54]

In a series of earlier papers the effects of GUP wereinvestigated on atomic condensed matter preheating phaseof the universe systems black holes at LHC [55ndash58] theweak equivalence principle (WEP) and the Liouville theorem(LT) in statistical mechanics [59] It was found that theGUP can potentially explain the small observed violationsof the WEP in neutron interferometry experiments [60ndash62]and also predicts a modified invariant phase space which isrelevant to the Liouville theorem It was derived in [55] thefirst bound for 120572

0is about sim1017 which would approximately

gives 120572 sim 10minus2 GeVminus1 The other bound of 1205720is sim1010 This

bound means that 120572 sim 10minus9 GeVminus1 As discussed in [63]the exact bound on 120572 can be obtained by comparison withobservations and experiments It seems that the gamma raysburst would allow us to set an upper value for the GUP-charactering parameter 120572 which we would like to report onin the future

Recently it has been suggested in [64] that the GUPimplications can bemeasured directly in quantum optics lab-oratories which definitely confirm the theoretical predictions

Advances in High Energy Physics 3

given in [19 55] Definitely this is considered as a milestonein the road of quantum gravity phenomenology

In Section 3 we briefly review the assumptions that thegravitational force would originate from an entropic nature

3 Gravity as an Entropic Force

Recently Verlinde [27] has utilized Sakharovrsquos proposal [28ndash31] that the gravity would not be considered a fundamentalforce Concretely it was suggested that the gravitationalforce might originate from an entropic nature As discussedin the introduction this assumption is based on the rela-tion between the gravitation and thermodynamics [33ndash36]According to thermodynamics and holographic principleVerlindersquos approach results in Newtonrsquos law Moreover theFriedmann equations can also be derived [65] At tempera-ture 119879 the entropic force 119865 of a gravitational system is givenas

119865Δ119909 = 119879Δ119878 (3)

where Δ119878 is the change in the entropy so that at a displace-ment Δ119909 each particle carries its own portion of entropychange From the correspondence between the entropychangeΔ119878 and the information about the boundary of the sys-tem and using Bekensteinrsquos argument [33ndash36] it is assumedthat Δ119878 = 2120587119896

119861 where Δ119909 = ℎ119898 and 119896

119861is the Boltzmann

constant

Δ119878 = 2120587119896119861

119898119888

ℎΔ119909 (4)

where 119898 is the mass of the elementary component 119888 is thespeed of light and ℎ is the Planck constant respectively

Turning to the holographic principle which assumes thatfor any closed surface without worrying about its geometryinside all physics can be represented by the degrees offreedom on this surface itself This implies that the quantumgravity can be described by a topological quantum fieldtheory for which all physical degrees of freedom can beprojected onto the boundary [66]The information about theholographic system is given by 119873 bits forming an ideal gasIt is conjectured that 119873 is proportional to the entropy of theholographic screen

119873 =4119878

119896119861

(5)

then according to Bekensteinrsquos entropy-area relation [33ndash36]

119878 =1198961198611198883

4119866ℎ119860 (6)

Therefore one gets

119873 =1198601198883

119866ℎ=412058711990321198883

119866ℎ (7)

where 119903 is the radius of the gravitational systemand the area ofthe holographic screen 119860 = 41205871199032 is implemented in derivingthis equation It is assumed that each bit emerges outwards

from the holographic screen that is one dimension There-fore each bit carries an energy equal to 119896

1198611198792 so by using the

equipartition rule to calculate the energy of the system onegets

119864 =1

2119873119896119861119879 =

212058711988831199032

119866ℎ119896119861119879 = 119872119888

2 (8)

By substituting (3) and (4) into (8) we get

119865 = 119866119872119898

1199032 (9)

making it clear thatNewtonrsquos law of gravitation can be derivedfrom first principles

In Section 4 we study the effect of GUP approach on theentropic force and hence on the Newtonrsquos law of gravitation

4 GUP Impact on theBlack Hole Thermodynamics

Taking into consideration the GUP approach [49ndash51] andbecause black holes are considered as good laboratories forthe clear connection between thermodynamics and gravitythe black hole thermodynamics will be analyzed in thissection Furthermore how the entropy would be affected willbe investigated as well In Hawkingrsquos radiation the emittedparticles are mostly photons and standard model (SM)particles From kinetic theory of gases let us assume thatgatherings or clouds of points in the velocity space are equallyspread in all directions There is no reason that particleswould prefer to be moving in a certain direction Then thethree moments are simply equal

1199011asymp 1199012asymp 1199013 (10)

leading to

1199012=3

sum119894=1

119901119894119901119894asymp 31199012

119894

⟨1199012

119894⟩ asymp

1

3⟨1199012⟩

(11)

In order to find a relation between ⟨1199012⟩ and Δ1199012 we assumethat the black hole behaves like a black body while it emitsphotons Therefore from Wienrsquos law the temperature corre-sponding to the peak emission is given by

119888 ⟨119901⟩ = 282119879119867 (12)

We should keep in mind that the numerical factor 282should be modified by the grey-body factors which arise dueto the spacetime curvature around the black hole but forsimplicity we are just ignoring this modification

FromHawkingrsquos uncertainty proposed by Scardigli in [14]and Adler et al [67] Hawkingrsquos temperature reads

119879119867=1

120587119888Δ119901 =

1

282119888 ⟨119901⟩ (13)


With the relation ⟨1199012⟩ = Δ1199012 + ⟨119901⟩2 we get

⟨119901⟩ = 2821

120587Δ119901 = radic120583Δ119901

⟨1199012⟩ = (1 + 120583) Δ119901

2

(14)

where 120583 = (282120587)2 Again the parameter 120583 is modifiedif we consider the grey-body factors which arise due to thespacetime curvature around the black hole

In order to have a corresponding inequality for (1) we canutilize the arguments given in [68 69] Then

Δ119909Δ119901 geℎ

2[1 minus 120572 ⟨119901⟩ minus 120572⟨

1199012119894

119901⟩ + 120572

2⟨1199012⟩ + 3120572

2⟨1199012

119894⟩]

(15)

It is apparent that implementing the arguments given in (11)and (14) in the inequality given in (15) leads to

Δ119909Δ119901 geℎ

2[1 minus 120572

0ℓ119901(4

3)radic120583

Δ119901

ℎ+ 2 (1 + 120583) 120572

2

0ℓ2

119901

Δ1199012

ℎ2]

(16)

The resulting inequality (16) is the only one that follows from(1) Solving it as a quadratic equation in Δ119901 results in

Δ119901

ℎge2Δ119909 + 120572

0ℓ119901((43)radic120583)

4 (1 + 120583) 12057220ℓ2119901

times(1 minus radic1 minus8 (1 + 120583) 1205722

0ℓ2119901

(2Δ119909 + 1205720ℓ119901 (43)radic120583)

2)

(17)

The negatively signed solution is considered as the one thatrefers to the standard uncertainty relation as ℓ

119901Δ119909 rarr 0

Using the Taylor expansion we obviously find that

Δ119901 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (18)

Because the energy change reads Δ119864 asymp 119888Δ119901 and accord-ing to Scardigli in [14] and Adler et al [67] one can define theuncertainty in the energyΔ119864 as the energy carried away fromthe black hole through the emitted photon In the followingwe implement the procedure introduced in [70 71]Weutilizethe GUP approach [49ndash51] together with the assumptionsgiven in (11) and (14)Then using natural units that ℎ = 119888 = 1

Δ119864 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (19)

For a black hole absorbing a quantumparticlewith energy119864 and size119877 the area of the black hole is supposed to increaseby the amount [33]

Δ119860 ge 8120587 ℓ2

119901119864119877 (20)

The quantum particle itself implies the existence of finitebound given by

Δ119860min ge 8120587ℓ2

119901119864Δ119909 (21)

Using (19) in the inequality (21) we obtain

Δ119860min ge 8120587ℓ2

119901[1 minus

2

31205720ℓ119901radic120583

1

Δ119909] (22)

According to the argument given in [70 71] the length scaleis chosen to be the inverse surface gravity

Δ119909 = 2119903119904 (23)

where 119903119904is the Schwarzschild radius This argument implies

that

(Δ119909)2sim119860

120587 (24)

Substituting (24) into (22) we got

Δ119860min = 120582ℓ2

119901[1 minus

2

31205720ℓ119901radic120583120587

119860] (25)

where parameter 120582 will be fixed later According to [33ndash36] the black holersquos entropy is conjectured to depend onthe horizonrsquos area From the information theory [72] it hasbeen found that the minimal increase of entropy should beindependent of the area It is just one ldquobitrdquo of informationwhich is 119887 = ln(2)

119889119878

119889119860=Δ119878minΔ119860min

=119887

120582ℓ2119901[1 minus (23) 1205720ℓ119901radic120583120587119860]

(26)

where 119887 is a parameter By expanding the last expression inorders of 120572 and then integrating it we get the entropy

119878 =119887

120582ℓ2119901

[119860 +4

31205720ℓ119901radic120583120587119860] (27)

Using the Hawking-Bekenstein assumption which relatesentropy with the area the value of constants 119887120582 = 14 sothat

119878 =119860

4ℓ2119901

+2

31205720radic120587120583

119860

4ℓ2119901

(28)

Although it was found in [73] that the power-law cor-rections to Bekenstein-Hawking area entropy are ruled outbased on arguments from the Boltzmann-Einstein formulait was found that the power-law corrections may explain theobserved cosmic acceleration today [74]

We conclude that the entropy is directly related to thearea and gets a correction when applying GUP approachThetemperature of the black hole is

119879 =120581

8120587

119889119860

119889119878=120581

8120587[1 minus

2

31205720ℓ119901radic120583120587

119860] (29)

So far we conclude that not only the temperature is propor-tional to the surface gravity but also it depends on the blackholersquos area


5 Modified Newtonrsquos Law ofGravitation due to GUP

In this section we study the implications of the correctionscalculated for the entropy in (28) and calculate how thenumber of bits of (5) would be modified which assume newcorrections to Newtonrsquos law of gravitation Using the cor-rected entropy given in (28) we find that the number of bitsshould also be corrected as follows

1198731015840=4119878

119896119861

=119860

ℓ2119901

+4

31205720radic120583120587

119860

ℓ2119901

(30)

By substituting (30) into (8) and using (3) we get

119864 = 1198651198882(1199032

119898119866+120572radic120583119903

3119898119866) (31)

It is apparent that (31) implies a modification in Newtonrsquos lawof gravitation

119865 = 119866119872119898

1199032(1 minus

120572radic120583

3119903) (32)

This equation states that the modification in Newtonrsquos lawof gravity seems to agree with the predictions of Randall-Sundrum II model [52] which contains one uncompactifiedextra dimension and length scale Λ

119877 The only difference is

the signThemodification in Newtonrsquos gravitational potentialon brane [53] is given as

119881119877119878=

minus119866119898119872

119903(1 +

4Λ119877

3120587119903) 119903 ≪ Λ

119877

minus119866119898119872

119903(1 +

2Λ119877

31199032) 119903 ≫ Λ

119877

(33)

where 119903 and Λ119877are the radius and the characteristic length

scale respectively It is clear that the gravitational potentialis modified at short distance We notice that our result (32)agrees with different sign with (33) when 119903 ≪ Λ

119877 This result

would say that 120572 sim Λ119877which would help set a new upper

bound on the value of the parameter 120572 This means that theproposed GUP approach [49 51] is apparently able to predictthe same physics as Randall-Sundrum II model The latterassumes the existence of one extra dimension compactifiedon a circle whose upper and lower halves are identified Ifthe extra dimensions are accessible only to gravity and notto the standard model field the bound on their size can befixed by an experimental test of Newtonrsquos law of gravitationwhich has only been led down to sim4 millimeter This wasthe result about ten years ago [75] In recent gravitationalexperiments it is found that the Newtonian gravitationalforce the 11199032-law seems to be maintained up to sim013ndash016mm [76 77] However it is unknown whether this lawis violated or not at sub-120583m range Further implications ofthis modifications have been discussed in [78] which couldbe the same for the GUP modification which is calculated inthis paperThis similarity between the GUP implications andextra dimensions implications would assume new bounds onthe GUP parameter 120572 with respect to the extra dimensionlength scale Λ

119877

6 Conclusions

In this paper we tackle the consequences of the quantumgravity on the entropic force approach which assumes a neworigin of the gravitational force We found that the quantumgravity corrections lead to a modification in the area law ofthe entropy which leads to a modification in the numberof bits 119873 According to Verlindersquos theory of entropic forceNewtonrsquos law of gravitation would acquire new quantumgravity corrections due to the modified number of bits Themodification in Newtonrsquos Law of gravitation surprisinglyagrees with the corrections predicted by Randall-Sundrum IImodel with different sign This would open a new naturallyarising question in our proposed research if the GUP andextra dimensions theories would predict the same physicsWe hope to report on this in the future

Acknowledgments

The research of A F Ali is supported by Benha UniversityThe research of AT has been partly supported by theGerman-Egyptian Scientific Projects (GESP ID 1378) A F Ali and AN Tawfik would like to thank Professor Antonino Zichichifor his kind invitation to attend the International School ofSubnuclear Physics 2012 at the ldquoEttore Majorana Foundationand Centre for Scientific Culturerdquo in Erice Italy where thepresent work was started The authors gratefully thank theanonymous referee for useful comments and suggestionswhich helped to improve the paper

References

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[2] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[3] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 pp 145ndash166 1995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 25Article ID 095006 2008

[9] G Amelino-Camelia G Gubitosi and FMercati ldquoDiscretenessof area in noncommutative spacerdquo Physics Letters B vol 676 no4-5 pp 180ndash183 2009

[10] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999


[11] G M Hossain V Husain and S S Seahra ldquoBackground-independent quantization and the uncertainty principlerdquo Clas-sical and Quantum Gravity vol 27 no 16 Article ID 165013 8pages 2010

[12] K Nozari and B Fazlpour ldquoGeneralized uncertainty principlemodified dispersion relations and the early universe thermody-namicsrdquo General Relativity and Gravitation vol 38 no 11 pp1661ndash1679 2006

[13] R J Adler and D I Santiago ldquoOn gravity and the uncertaintyprinciplerdquoModern Physics Letters A vol 14 no 20 article 13711999

[14] F Scardigli ldquoSome heuristic semi-classical derivations of thePlanck length the Hawking effect and the Unruh effectrdquoNuovoCimento B vol 110 no 9 pp 1029ndash1034 1995

[15] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[16] SHossenfelder ldquoInterpretation of quantumfield theories with aminimal length scalerdquo Physical Review D vol 73 no 10 ArticleID 105013 9 pages 2006

[17] C Bambi ldquoA revision of the generalized uncertainty principlerdquoClassical and Quantum Gravity vol 25 no 10 Article ID105003 9 pages 2008

[18] J Y Bang and M S Berger ldquoQuantum mechanics and thegeneralized uncertainty principlerdquo Physical Review D vol 74no 12 Article ID 125012 8 pages 2006

[19] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 no 22 Article ID221301 4 pages 2008

[20] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysics vol 87 no 3 pp 233ndash240 2009

[21] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 article 2 2013

[22] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A vol 30 no 6 pp 2093ndash2101 1997

[23] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 no 44 pp 7691ndash7696 1999

[24] M Sprenger P Nicolini and M Bleicher ldquoPhysics on smallestscalesmdashan introduction to minimal length phenomenologyrdquoEuropean Journal of Physics vol 33 no 4 pp 853ndash862 2012

[25] M Sprenger P Nicolini andM Bleicher ldquoNeutrino oscillationsas a novel probe for a minimal lengthrdquo Classical and QuantumGravity vol 28 no 23 Article ID 235019 2011

[26] J Mureika P Nicolini and E Spallucci ldquoCould any black holesbe produced at the LHCrdquo Physical Review D vol 85 no 10Article ID 106007 8 pages 2012

[27] E P Verlinde ldquoOn the origin of gravity and the laws of NewtonrdquoJournal of High Energy Physics vol 2011 article 29 2011

[28] A D Sakharov ldquoVacuum quantum fluctuations in curved spaceand the theory of gravitationrdquo Soviet Physics vol 12 no 1 pp1040ndash1041 1968

[29] A D Sakharov ldquoVacuum quantum fluctuations in curved spaceand the theory of gravitationrdquo Doklady Akademii Nauk SSSRvol 177 no 1 pp 70ndash71 1967

[30] A D Sakharov ldquoVacuum quantum fluctuations in curved spaceand the theory of gravitationrdquo Soviet Physics Uspekhi vol 34 no5 article 394 1991

[31] A D Sakharov ldquoVacuum quantum fluctuations in curvedspace and the theory of gravitationrdquo General Relativity andGravitation vol 32 no 2 article 365 2000

[32] M Visser ldquoSakharovrsquos induced gravity a modern perspectiverdquoModern Physics Letters A vol 17 no 15ndash17 pp 977ndash991 2002

[33] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review Dvol 7 pp 2333ndash2346 1973

[34] J M Bardeen B Carter and S W Hawking ldquoThe four lawsof black hole mechanicsrdquo Communications in MathematicalPhysics vol 31 no 2 pp 161ndash170 1973

[35] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975

[36] S W Hawking ldquoErratum particle creation by black holesCommunications in Mathematical Physics 43 3 199ndash2201975rdquo Communications in Mathematical Physics vol 46 no 2article 206 1976

[37] T Jacobson ldquoThermodynamics of spacetime the Einsteinequation of staterdquo Physical Review Letters vol 75 no 7 pp1260ndash1263 1995

[38] Y Zhang Y Gong and Z -H Zhu ldquoModified gravity emergingfrom thermodynamics and holographic principlerdquo Interna-tional Journal of Modern Physics D vol 20 no 8 pp 1505ndash15192011

[39] A Kobakhidze ldquoGravity is not an entropic forcerdquo PhysicalReview D vol 83 no 2 Article ID 021502 3 pages 2011

[40] S Hossenfelder ldquoComments on and comments on commentson Verlindersquos paper lsquoon the origin of gravity and the laws ofNewtonrsquordquo httparxivorgabs10031015

[41] K Nozari P Pedram and M Molkara ldquoMinimal lengthmaximal momentum and the entropic force lawrdquo InternationalJournal of Theoretical Physics vol 51 no 4 pp 1268ndash1275 2012

[42] P Nicolini ldquoNonlocal and generalized uncertainty principleblack holesrdquo httparxivorgabs12022102

[43] S Ghosh ldquoPlanck scale effect in the entropic force lawrdquoModernPhysics Letters A httparxivorgabs10030285

[44] M A Santos and I V Vancea ldquoEntropic law of force emergentgravity and the uncertainty principlerdquoModern Physics Letters Avol 27 no 2 Article ID 1250012 11 pages 2012

[45] P Nicolini ldquoEntropic force noncommutative gravity andungravityrdquo Physical Review D vol 82 no 4 Article ID 0440308 pages 2010

[46] C Bastos O Bertolami N C Dias and J N Prata ldquoEntropicgravity phase-space noncommutativity and the equivalenceprinciplerdquoClassical andQuantumGravity vol 28 no 12 ArticleID 125007 8 pages 2011

[47] K Nozari and S Akhshabi ldquoNoncommutative geometryinspired entropic inflationrdquo Physics Letters B vol 700 no 2 pp91ndash96 2011

[48] S H Mehdipour A Keshavarz and A Keshavarz ldquoEntropicforce approach in a noncommutative charged black hole and theequivalence principlerdquo Europhysics Letters vol 98 no 1 ArticleID 10002 2012

[49] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009

[50] A F Ali S Das andE CVagenas ldquoThe generalized uncertaintyprinciple and quantum gravity phenomenologyrdquo in 12th MarcelGrossmann Meeting on General Relativity (MG 12) pp 2407ndash2409 Paris France July 2009

[51] S Das E C Vagenas and A F Ali ldquoDiscreteness of space fromGUP II relativistic wave equationsrdquo Physics Letters B vol 690no 4 pp 407ndash412 2010


[52] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

[53] P Callin and F Ravndal ldquoHigher order corrections to theNewtonian potential in the Randall-Sundrum modelrdquo PhysicalReview D vol 70 no 10 Article ID 104009 12 pages 2004

[54] TThiemann ldquoA length operator for canonical quantumgravityrdquoJournal of Mathematical Physics vol 39 no 6 pp 3372ndash33921998

[55] A F Ali S Das and E C Vagenas ldquoProposal for testingquantum gravity in the labrdquo Physical Review D vol 84 no 4Article ID 044013 10 pages 2011

[56] W Chemissany S Das A F Ali and E C Vagenas ldquoEffectof the Generalized Uncertainty Principle on post-inflationpreheatingrdquo Journal of Cosmology and Astroparticle Physics vol2011 article 017 2011

[57] A F Ali ldquoNo existence of black holes at LHC due to minimallength in quantum gravityrdquo Journal of High Energy Physics vol2012 article 67 2012

[58] A F Ali H Nafie and M Shalaby ldquoMinimal length maximalenergy and black-hole remnantsrdquo Europhysics Letters vol 100no 2 Article ID 20004 2012

[59] A F Ali ldquoMinimal length in quantumgravity equivalence prin-ciple and holographic entropy boundrdquo Classical and QuantumGravity vol 28 no 6 Article ID 065013 2011

[60] R Collela AW Overhauser and S AWerner ldquoObservation ofgravitationally induced quantum interferencerdquo Physical ReviewLetters vol 34 no 23 pp 1472ndash1474 1975

[61] K C Littrell B E Allman and S AWerner ldquoTwo-wavelength-difference measurement of gravitationally induced quantuminterference phasesrdquo Physical Review A vol 56 no 3 pp 1767ndash1780 1997

[62] A Camacho and A Camacho-Galvan ldquoTest of some funda-mental principles in physics via quantum interference withneutrons and photonsrdquo Reports on Progress in Physics vol 70pp 1ndash56 2007 httparxivorgabs08101325

[63] A Tawfik H Magdy and A Farag Ali ldquoLorentz invariance vio-lation and generalized uncertainty principlerdquo httparxivorgabs12055998

[64] I Pikovski M R Vanner M Aspelmeyer et al ldquoProbingPlanck-scale physics with quantum opticsrdquo Nature Physics vol8 no 5 pp 393ndash397 2012

[65] R-G Cai L-M Cao and N Ohta ldquoFriedmann equations fromentropic forcerdquo Physical Review D vol 81 no 6 Article ID061501 4 pages 2010

[66] G trsquoHooft ldquoDimensional reduction in quantum gravityrdquo httparxivorgabsgr-qc9310026

[67] R J Adler P Chen and D I Santiago ldquoThe generalized uncer-tainty principle and black hole remnantsrdquoGeneral Relativity andGravitation vol 33 no 12 pp 2101ndash2108 2001

[68] M Cavaglia S Das and R Maartens ldquoWill we observe blackholes at the LHCrdquo Classical and Quantum Gravity vol 20 no15 pp L205ndashL212 2003

[69] M Cavaglia and S Das ldquoHow classical are TeV-scale blackholesrdquo Classical and Quantum Gravity vol 21 no 19 pp 4511ndash4522 2004

[70] A J M Medved and E C Vagenas ldquoWhen conceptualworlds collide the generalized uncertainty principle and theBekenstein-Hawking entropyrdquo Physical Review D vol 70 no12 Article ID 124021 5 pages 2004

[71] BMajumder ldquoBlack hole entropy and themodified uncertaintyprinciple a heuristic analysisrdquo Physics Letters B vol 703 no 4pp 402ndash405 2011

[72] C Adami ldquoThe physics of informationrdquo httparxivorgabsquant-ph0405005

[73] S Hod ldquoHigh-order corrections to the entropy and area ofquantum black holesrdquo Classical and Quantum Gravity vol 21no 14 pp L97ndashL100 2004

[74] PWang ldquoHorizon entropy inmodified gravityrdquo Physical ReviewD vol 72 no 2 Article ID 024030 4 pages 2005

[75] C D Hoyle U Schmidt B R Heckel et al ldquoSubmillimeter testsof the gravitational inverse square law a search for ldquolargerdquo extradimensionsrdquo Physical Review Letters vol 86 no 8 pp 1418ndash1421 2001

[76] S-Q Yang B-F Zhan Q-L Wang et al ldquoTest of the gravita-tional inverse square law at millimeter rangesrdquo Physical ReviewLetters vol 108 no 8 Article ID 081101 5 pages 2012

[77] CDHoyle D J Kapner B RHeckel et al ldquoSubmillimeter testsof the gravitational inverse-square lawrdquo Physical Review D vol70 no 4 Article ID 042004 31 pages 2004

[78] F Buisseret B Silvestre-Brac and V Mathieu ldquoModifiedNewtonrsquos law braneworlds and the gravitational quantumwellrdquoClassical andQuantumGravity vol 24 no 4 pp 855ndash865 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


phenomenologicalmechanism deriving a system to approachmaximum entropy that is increasing the number of micro-scopic states that will be inhered in the systemrsquos phase spaceThere are various examples on the entropic force for examplepolymer molecules and the elasticity of rubber bands

Recently Verlinde proposed that the gravity is not a fun-damental force and can be considered as an entropic force[27] The earliest idea about gravity that is regarded as a non-fundamental interaction has been introduced by Sakharov[28ndash31] where the spacetime background is assumed toemerge as a mean field approximation of underlying micro-scopic degrees of freedom Similar behavior is observed inhydrodynamics [32] It is found that the entropy of black holeis related to the horizonrsquos area at the black holersquos horizonwhile the temperature is related to the surface gravity Bothentropy and temperature are assumed to be related to themass of the black hole [33ndash36]Thus the connection betweenthermodynamics and geometry leads to Einsteinrsquos equa-tions of gravitational field from relations connecting heatentropy and temperature [37] Einsteinrsquos equations connectenergy-momentum tensor with space geometry Advocat-ing the gravity as nonfundamental interaction leads to theassumption that gravity would be explained as an entropicforce caused by changes in the information associated withthe positions of material bodies [27] When combining theentropic force with the Unruh temperature the second lawof Newton is obtained But when combining it with the holo-graphic principle and using the equipartition law of energyNewtonrsquos law of gravitation is obtained It was investigated in[38] modification of the entropic force due to corrections tothe area law of entropywhich is derived fromquantum effectsand extra dimensions

Apart from the controversial debate on the origin of grav-ity [39 40] we investigate the impact of GUP on the entropicforce and derive essential quantities including potentialmodification to Newtonrsquos law of gravity

There were some studies for the effect of some versions ofGUPonNewtonrsquos law of gravity in [41ndash44] Also noncommu-tative geometry which is considered as a completely planckscale effect has been studied to derive the modified Newtonrsquoslaw of gravity [45ndash48] All these approaches for studying thePlanck scale effects on the Newtonrsquos law of gravity are basedon the following schememodified theory of gravity rarr mod-ified black hole entropy rarr modified holographic surfaceentropy rarr Newtonrsquos law corrections We followed the samescheme in our paper using the new version of GUP proposedin [49ndash51] and we a got new corrections in our currentwork which are distinct from the previous studiesMoreoverwe compared our results with Randall-Sundrum model ofextra dimension which also predicts the modification ofNewtonrsquos law of gravity at the Planck scale [52 53] wherewe think there may be some connection between generalizeduncertainty principle and extra dimension theories becausethey predict similar physics at least for the case of Newtonrsquoslaw of gravity which may be considered as a distinct resultfrom the previous studies

The present paper is organized as follows Section 2reviews briefly the generalized uncertainty principle that wasproposed in [49ndash51] Section 3 is devoted to review the

entropic force and gravitational interaction [27] The effectof utilizing GUP impact on the entropic force is introducedin Section 4 In Section 5 we estimate the GUP correctionto Newtonrsquos law of gravitation The conclusions are given inSection 6

2 The Generalized Uncertainty Principle

It is conjectured that the standard commutation relations atshort distances would be modified A new form of GUP wasproposed [49ndash51] and found consistent with the doubly spe-cial relativity (DSR) theories the string theory and the blackhole physics It predicts a maximum observable momentumand a minimal measurable length With satisfying Jacobiidentity GUP is found to ensure the relations [119909

119894 119909119895] = 0 =

[119901119894 119901119895]

[119909119894 119901119895] = 119894ℎ [120575

119894119895minus 120572(119901120575

119894119895+119901119894119901119895

119901) + 1205722(1199012120575119894119895+ 3119901119894119901119895)]

(1)

where 120572 = 1205720119872119901119888 = 120572

0ℓ119901ℎ and 119872

1199011198882 stand for Planck

energy119872119901and ℓ119901are Planck mass and length respectively

1205720sets on the upper and lower bounds to 120572For a particle having an energy scale comparable to

Planckrsquos one the physical momentum would be a subject ofmodification [49ndash51]

119901119894= 1199010119894(1 minus 120572119901

0+ 212057221199012

0) (2)

where 119909119894= 1199090119894and 119901

0119895satisfy the canonical commutation

relations [1199090119894 1199010119895] = 119894ℎ120575

119894119895 Here 119901

0119894can be interpreted as

the momentum at low energies and 119901119894as that at high energies

and the variable 1199010is the value of the momentum at the low

energy scalesThis newly proposedGUP suggests that the space is quan-

tized into fundamental units whichmay be the Planck lengthThe quantization of space has been shown within the contextof loop quantum gravity in [54]

In a series of earlier papers the effects of GUP wereinvestigated on atomic condensed matter preheating phaseof the universe systems black holes at LHC [55ndash58] theweak equivalence principle (WEP) and the Liouville theorem(LT) in statistical mechanics [59] It was found that theGUP can potentially explain the small observed violationsof the WEP in neutron interferometry experiments [60ndash62]and also predicts a modified invariant phase space which isrelevant to the Liouville theorem It was derived in [55] thefirst bound for 120572

0is about sim1017 which would approximately

gives 120572 sim 10minus2 GeVminus1 The other bound of 1205720is sim1010 This

bound means that 120572 sim 10minus9 GeVminus1 As discussed in [63]the exact bound on 120572 can be obtained by comparison withobservations and experiments It seems that the gamma raysburst would allow us to set an upper value for the GUP-charactering parameter 120572 which we would like to report onin the future

Recently it has been suggested in [64] that the GUPimplications can bemeasured directly in quantum optics lab-oratories which definitely confirm the theoretical predictions






119865Δ119909 = 119879Δ119878 (3)


119861 where Δ119909 = ℎ119898 and 119896


constant

Δ119878 = 2120587119896119861

119898119888

ℎΔ119909 (4)



119873 =4119878

119896119861

(5)


119878 =1198961198611198883

4119866ℎ119860 (6)

Therefore one gets

119873 =1198601198883

119866ℎ=412058711990321198883

119866ℎ (7)





119864 =1

2119873119896119861119879 =

212058711988831199032

119866ℎ119896119861119879 = 119872119888

2 (8)


119865 = 119866119872119898

1199032 (9)





1199011asymp 1199012asymp 1199013 (10)

leading to

1199012=3

sum119894=1

119901119894119901119894asymp 31199012

119894

⟨1199012

119894⟩ asymp

1

3⟨1199012⟩

(11)


119888 ⟨119901⟩ = 282119879119867 (12)



119879119867=1

120587119888Δ119901 =

1

282119888 ⟨119901⟩ (13)



⟨119901⟩ = 2821

120587Δ119901 = radic120583Δ119901

⟨1199012⟩ = (1 + 120583) Δ119901

2

(14)



Δ119909Δ119901 geℎ

2[1 minus 120572 ⟨119901⟩ minus 120572⟨

1199012119894

119901⟩ + 120572

2⟨1199012⟩ + 3120572

2⟨1199012

119894⟩]

(15)


Δ119909Δ119901 geℎ

2[1 minus 120572

0ℓ119901(4

3)radic120583

Δ119901

ℎ+ 2 (1 + 120583) 120572

2

0ℓ2

119901

Δ1199012

ℎ2]

(16)


Δ119901

ℎge2Δ119909 + 120572

0ℓ119901((43)radic120583)

4 (1 + 120583) 12057220ℓ2119901


0ℓ2119901

(2Δ119909 + 1205720ℓ119901 (43)radic120583)

2)

(17)


119901Δ119909 rarr 0


Δ119901 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (18)


Δ119864 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (19)


Δ119860 ge 8120587 ℓ2

119901119864119877 (20)


Δ119860min ge 8120587ℓ2

119901119864Δ119909 (21)


Δ119860min ge 8120587ℓ2

119901[1 minus

2

31205720ℓ119901radic120583

1

Δ119909] (22)


Δ119909 = 2119903119904 (23)


that

(Δ119909)2sim119860

120587 (24)


Δ119860min = 120582ℓ2

119901[1 minus

2

31205720ℓ119901radic120583120587

119860] (25)


119889119878

119889119860=Δ119878minΔ119860min

=119887

120582ℓ2119901[1 minus (23) 1205720ℓ119901radic120583120587119860]

(26)


119878 =119887

120582ℓ2119901

[119860 +4

31205720ℓ119901radic120583120587119860] (27)


119878 =119860

4ℓ2119901

+2

31205720radic120587120583

119860

4ℓ2119901

(28)



119879 =120581

8120587

119889119860

119889119878=120581

8120587[1 minus

2

31205720ℓ119901radic120583120587

119860] (29)





1198731015840=4119878

119896119861

=119860

ℓ2119901

+4

31205720radic120583120587

119860

ℓ2119901

(30)


119864 = 1198651198882(1199032

119898119866+120572radic120583119903

3119898119866) (31)


119865 = 119866119872119898

1199032(1 minus

120572radic120583

3119903) (32)




119881119877119878=

minus119866119898119872

119903(1 +

4Λ119877

3120587119903) 119903 ≪ Λ

119877

minus119866119898119872

119903(1 +

2Λ119877

31199032) 119903 ≫ Λ

119877

(33)



119877 This result



119877

6 Conclusions


Acknowledgments


References






















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119865Δ119909 = 119879Δ119878 (3)


119861 where Δ119909 = ℎ119898 and 119896


constant

Δ119878 = 2120587119896119861

119898119888

ℎΔ119909 (4)



119873 =4119878

119896119861

(5)


119878 =1198961198611198883

4119866ℎ119860 (6)

Therefore one gets

119873 =1198601198883

119866ℎ=412058711990321198883

119866ℎ (7)





119864 =1

2119873119896119861119879 =

212058711988831199032

119866ℎ119896119861119879 = 119872119888

2 (8)


119865 = 119866119872119898

1199032 (9)





1199011asymp 1199012asymp 1199013 (10)

leading to

1199012=3

sum119894=1

119901119894119901119894asymp 31199012

119894

⟨1199012

119894⟩ asymp

1

3⟨1199012⟩

(11)


119888 ⟨119901⟩ = 282119879119867 (12)



119879119867=1

120587119888Δ119901 =

1

282119888 ⟨119901⟩ (13)



⟨119901⟩ = 2821

120587Δ119901 = radic120583Δ119901

⟨1199012⟩ = (1 + 120583) Δ119901

2

(14)



Δ119909Δ119901 geℎ

2[1 minus 120572 ⟨119901⟩ minus 120572⟨

1199012119894

119901⟩ + 120572

2⟨1199012⟩ + 3120572

2⟨1199012

119894⟩]

(15)


Δ119909Δ119901 geℎ

2[1 minus 120572

0ℓ119901(4

3)radic120583

Δ119901

ℎ+ 2 (1 + 120583) 120572

2

0ℓ2

119901

Δ1199012

ℎ2]

(16)


Δ119901

ℎge2Δ119909 + 120572

0ℓ119901((43)radic120583)

4 (1 + 120583) 12057220ℓ2119901


0ℓ2119901

(2Δ119909 + 1205720ℓ119901 (43)radic120583)

2)

(17)


119901Δ119909 rarr 0


Δ119901 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (18)


Δ119864 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (19)


Δ119860 ge 8120587 ℓ2

119901119864119877 (20)


Δ119860min ge 8120587ℓ2

119901119864Δ119909 (21)


Δ119860min ge 8120587ℓ2

119901[1 minus

2

31205720ℓ119901radic120583

1

Δ119909] (22)


Δ119909 = 2119903119904 (23)


that

(Δ119909)2sim119860

120587 (24)


Δ119860min = 120582ℓ2

119901[1 minus

2

31205720ℓ119901radic120583120587

119860] (25)


119889119878

119889119860=Δ119878minΔ119860min

=119887

120582ℓ2119901[1 minus (23) 1205720ℓ119901radic120583120587119860]

(26)


119878 =119887

120582ℓ2119901

[119860 +4

31205720ℓ119901radic120583120587119860] (27)


119878 =119860

4ℓ2119901

+2

31205720radic120587120583

119860

4ℓ2119901

(28)



119879 =120581

8120587

119889119860

119889119878=120581

8120587[1 minus

2

31205720ℓ119901radic120583120587

119860] (29)





1198731015840=4119878

119896119861

=119860

ℓ2119901

+4

31205720radic120583120587

119860

ℓ2119901

(30)


119864 = 1198651198882(1199032

119898119866+120572radic120583119903

3119898119866) (31)


119865 = 119866119872119898

1199032(1 minus

120572radic120583

3119903) (32)




119881119877119878=

minus119866119898119872

119903(1 +

4Λ119877

3120587119903) 119903 ≪ Λ

119877

minus119866119898119872

119903(1 +

2Λ119877

31199032) 119903 ≫ Λ

119877

(33)



119877 This result



119877

6 Conclusions


Acknowledgments


References






















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





⟨119901⟩ = 2821

120587Δ119901 = radic120583Δ119901

⟨1199012⟩ = (1 + 120583) Δ119901

2

(14)



Δ119909Δ119901 geℎ

2[1 minus 120572 ⟨119901⟩ minus 120572⟨

1199012119894

119901⟩ + 120572

2⟨1199012⟩ + 3120572

2⟨1199012

119894⟩]

(15)


Δ119909Δ119901 geℎ

2[1 minus 120572

0ℓ119901(4

3)radic120583

Δ119901

ℎ+ 2 (1 + 120583) 120572

2

0ℓ2

119901

Δ1199012

ℎ2]

(16)


Δ119901

ℎge2Δ119909 + 120572

0ℓ119901((43)radic120583)

4 (1 + 120583) 12057220ℓ2119901


0ℓ2119901

(2Δ119909 + 1205720ℓ119901 (43)radic120583)

2)

(17)


119901Δ119909 rarr 0


Δ119901 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (18)


Δ119864 ge1

Δ119909(1 minus

2

31205720ℓ119901radic120583

1

Δ119909) (19)


Δ119860 ge 8120587 ℓ2

119901119864119877 (20)


Δ119860min ge 8120587ℓ2

119901119864Δ119909 (21)


Δ119860min ge 8120587ℓ2

119901[1 minus

2

31205720ℓ119901radic120583

1

Δ119909] (22)


Δ119909 = 2119903119904 (23)


that

(Δ119909)2sim119860

120587 (24)


Δ119860min = 120582ℓ2

119901[1 minus

2

31205720ℓ119901radic120583120587

119860] (25)


119889119878

119889119860=Δ119878minΔ119860min

=119887

120582ℓ2119901[1 minus (23) 1205720ℓ119901radic120583120587119860]

(26)


119878 =119887

120582ℓ2119901

[119860 +4

31205720ℓ119901radic120583120587119860] (27)


119878 =119860

4ℓ2119901

+2

31205720radic120587120583

119860

4ℓ2119901

(28)



119879 =120581

8120587

119889119860

119889119878=120581

8120587[1 minus

2

31205720ℓ119901radic120583120587

119860] (29)





1198731015840=4119878

119896119861

=119860

ℓ2119901

+4

31205720radic120583120587

119860

ℓ2119901

(30)


119864 = 1198651198882(1199032

119898119866+120572radic120583119903

3119898119866) (31)


119865 = 119866119872119898

1199032(1 minus

120572radic120583

3119903) (32)




119881119877119878=

minus119866119898119872

119903(1 +

4Λ119877

3120587119903) 119903 ≪ Λ

119877

minus119866119898119872

119903(1 +

2Λ119877

31199032) 119903 ≫ Λ

119877

(33)



119877 This result



119877

6 Conclusions


Acknowledgments


References






















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1198731015840=4119878

119896119861

=119860

ℓ2119901

+4

31205720radic120583120587

119860

ℓ2119901

(30)


119864 = 1198651198882(1199032

119898119866+120572radic120583119903

3119898119866) (31)


119865 = 119866119872119898

1199032(1 minus

120572radic120583

3119903) (32)




119881119877119878=

minus119866119898119872

119903(1 +

4Λ119877

3120587119903) 119903 ≪ Λ

119877

minus119866119898119872

119903(1 +

2Λ119877

31199032) 119903 ≫ Λ

119877

(33)



119877 This result



119877

6 Conclusions


Acknowledgments


References






















































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Modified Newton s Law of Gravitation due...

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Transcript of Research Article Modified Newton s Law of Gravitation due...