Research Article Gravitational Fields of Conical Mass...
5
Hindawi Publishing Corporation Journal of Gravity Volume 2013, Article ID 546913, 4 pages http://dx.doi.org/10.1155/2013/546913 Research Article Gravitational Fields of Conical Mass Distributions Chifu Ebenezer Ndikilar Physics Department, Faculty of Science, Federal University Dutse, P.M.B. 7156, Dutse, Jigawa State 720222, Nigeria Correspondence should be addressed to Chifu Ebenezer Ndikilar; [email protected] Received 20 February 2013; Revised 11 March 2013; Accepted 19 March 2013 Academic Editor: Maddalena Mantovani Copyright © 2013 Chifu Ebenezer Ndikilar. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e gravitational field of conical mass distributions is formulated using the general theory of relativity. e gravitational metric tensor is constructed and applied to the motion of test particles and photons in this gravitational field. e expression for gravitational time dilation is found to have the same form as that in spherical, oblate spheroidal, and prolate spheroidal gravitational fields and hence confirms an earlier assertion that this gravitational phenomena is invariant in form with various mass distributions. It is shown using the pure radial equation of motion that as a test particle moves closer to the conical mass distribution along the radial direction, its radial speed decreases. 1. Introduction In recent articles [1–4], we introduced an approach of studying gravitational fields of various mass distributions as extensions of Schwarzschild’s method. Of interest in this article is the gravitational field of conical mass distributions placed in empty space. Sputnik III, the third Soviet satellite launched on May 15, 1958 has a conical shape. is study is aimed at studying the behaviour of test particles and photons in the vicinity of conically shaped objects placed in empty space such as Sputnik III. 2. Gravitational Metric Tensors It is well known [5, 6] that the general relativistic metric tensor for flat space-time (empty space without mass) is invariant (invariance of the line element) and can be obtained in any orthogonal curvilinear coordinate (, , V, ) by the transformation (, , , ) → (, , V, ). It is worth noting that the metric tensor obtained through this standard proce- dure (i) satisfies Einstein’s field equations a priori and (ii) yields the expected equations of motion for test particles and photons in flat space-time. Using these crucial facts, we now realize that choosing the particular orthogonal curvilinear coordinate (, V, ) corresponding to the geometry of the body facilitates the formulation of boundary conditions on the universal grav- itational scalar potential which is always expected to be a part of the metric tensor. us, transforming Schwarzschild’s metric into the particular orthogonal curvilinear coordinate using the invariance of the line element subject to the fact that the arbitrary function in the metric tensor transforms as (, , , ) → (, , V, ) yields the metric tensor for the mass distribution of the orthogonal curvilinear coordinate. us, the arbitrary function is determined by the mass or pressure distribution and hence possess symmetries imposed by the latter a priori. In approximate gravitational fields, the arbitrary function is equal to the gravitational scalar potential. us, to obtain the metric tensor for a homogeneous conical mass distribution placed in empty space in conical coordinates: (1) coordinates are transformed from spherical polar (, , ) to conical (, V, ) on the right hand side of Schwarzschild’s world line element; (2) the arbitrary function () in Schwarzschild’s field is transformed to () in the field of a homogeneous conical mass.
Transcript of Research Article Gravitational Fields of Conical Mass...
Hindawi Publishing CorporationJournal of GravityVolume 2013 Article ID 546913 4 pageshttpdxdoiorg1011552013546913
Research ArticleGravitational Fields of Conical Mass Distributions
Chifu Ebenezer Ndikilar
Physics Department Faculty of Science Federal University Dutse PMB 7156 Dutse Jigawa State 720222 Nigeria
Correspondence should be addressed to Chifu Ebenezer Ndikilar ebenechifuyahoocom
Received 20 February 2013 Revised 11 March 2013 Accepted 19 March 2013
Academic Editor Maddalena Mantovani
Copyright copy 2013 Chifu Ebenezer Ndikilar 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
The gravitational field of conical mass distributions is formulated using the general theory of relativity The gravitational metrictensor is constructed and applied to the motion of test particles and photons in this gravitational field The expression forgravitational time dilation is found to have the same form as that in spherical oblate spheroidal and prolate spheroidal gravitationalfields and hence confirms an earlier assertion that this gravitational phenomena is invariant in formwith variousmass distributionsIt is shown using the pure radial equation of motion that as a test particle moves closer to the conical mass distribution along theradial direction its radial speed decreases
1 Introduction
In recent articles [1ndash4] we introduced an approach ofstudying gravitational fields of various mass distributionsas extensions of Schwarzschildrsquos method Of interest in thisarticle is the gravitational field of conical mass distributionsplaced in empty space Sputnik III the third Soviet satellitelaunched on May 15 1958 has a conical shape This study isaimed at studying the behaviour of test particles and photonsin the vicinity of conically shaped objects placed in emptyspace such as Sputnik III
2 Gravitational Metric Tensors
It is well known [5 6] that the general relativistic metrictensor for flat space-time (empty space without mass) isinvariant (invariance of the line element) and can be obtainedin any orthogonal curvilinear coordinate (119905 119906 V 119908) by thetransformation (119905 119909 119910 119911) rarr (119905 119906 V 119908) It is worth notingthat the metric tensor obtained through this standard proce-dure
(i) satisfies Einsteinrsquos field equations a priori and(ii) yields the expected equations of motion for test
particles and photons in flat space-timeUsing these crucial facts we now realize that choosing
the particular orthogonal curvilinear coordinate (119906 V 119908)
corresponding to the geometry of the body facilitates theformulation of boundary conditions on the universal grav-itational scalar potential which is always expected to be apart of the metric tensor Thus transforming Schwarzschildrsquosmetric into the particular orthogonal curvilinear coordinateusing the invariance of the line element subject to the factthat the arbitrary function in the metric tensor transforms as119891(119905 119903 120579 120601) rarr 119891(119905 119906 V 119908) yields the metric tensor for themass distribution of the orthogonal curvilinear coordinateThus the arbitrary function 119891 is determined by the mass orpressure distribution and hence possess symmetries imposedby the latter a priori In approximate gravitational fieldsthe arbitrary function is equal to the gravitational scalarpotential
Thus to obtain the metric tensor for a homogeneousconical mass distribution placed in empty space in conicalcoordinates
(1) coordinates are transformed from spherical polar(119903 120579 120601) to conical (119903 V 119908) on the right hand side ofSchwarzschildrsquos world line element
(2) the arbitrary function 119891(119903) in Schwarzschildrsquos field istransformed to 119891
1015840(119903) in the field of a homogeneousconical mass
2 Journal of Gravity
Now the spherical polar coordinates (119903 120579 120601) are relatedto the Cartesian coordinates (119909 119910 119911) as follows
119903 = (1199092+ 1199102+ 1199112)12
120579 = cosminus1( 119911
(1199092 + 1199102 + 1199112)12
)
120601 = tanminus1 (119910
119909)
(1)
Also the Cartesian coordinates are related to the conicalcoordinates (119903 V 119908) as follows
119909 =1
120572119903(1205722minus V2)12
(1205722+ 1199082)12
119910 =1
120573119903(1205732+ V2)12
(1205732minus 1199082)12
119911 =1
120572120573119903V119908
(2)
where 1205722 + 1205732 = 1Expressing the spherical coordinates in terms of the
conical coordinates we obtain
120579 (119903 V 119908) = cosminus1 (V119908120572120573
) (3)
120601 (119903 V 119908) = tanminus1 [
[
120573(1205722 minus V2)12
(1205722 + 1199082)12
120572(1205732 + V2)12
(1205732 minus 1199082)12
]
]
(4)
Now consider a spherically symmetric mass distributionwith invariant world line element (Schwarzschildrsquos line ele-ment) in the exterior region given [1] as
11988821198891205912= 1198882(1 +
2119891 (119903)
1198882)1198891199052minus (1 +
2119891 (119903)
1198882)
minus1
1198891199032
minus 11990321198891205792minus 1199032sin21205791198891206012
(5)
where 119891(119903) generally is an arbitrary function determinedby the distribution of mass or pressure and possess all thesymmetries of the distribution In approximate gravitationalfields 119891(119903) is equivalent to the gravitational scalar potentialfunction exterior to a massive symmetric sphere
Let the spherically symmetric mass distribution be trans-formed by deformation into a conical mass distribution insuch a way that it retains its density and total mass Thus thegeneral relativistic invariant world line element exterior to aspherical symmetric body is tensorially equivalent to that ofa conical mass distribution and related by the transformationfrom spherical to conical coordinatesTherefore to obtain theinvariant line element exterior to conical mass distributionswe transform coordinates on the right hand side of (5) fromspherical to conical coordinates and 119891(119903) rarr 119891
1015840(119903) thearbitrary function determined by the conical mass
From (3) we have
119889120579 =120597120579
120597119903119889119903 +
120597120579
120597V119889V +
120597120579
120597119908119889119908 (6)
where
120597120579
120597V= minus
119908
(12057221205732 minus V21199082)12
120597120579
120597119908= minus
V
(12057221205732 minus V21199082)12
120597120579
120597119903= 0
(7)
Also from (4) we have
119889120601 =120597120601
120597119903119889119903 +
120597120601
120597V119889V +
120597120601
120597119908119889119908 (8)
where
120597120601
120597V= minus
V31199082(1205722 + 1199082)12
(1205722 minus V2)minus12
1205723120573(1205732 + V2)32
(1205732 minus 1199082)12
120597120601
120597119908=V21199083(1205722 minus V2)
12
(1205722 + 1199082)minus12
1205723120573(1205732 minus 1199082)32
(1205732 + V2)12
120597120601
120597119903= 0
(9)
Substituting (3) (6) and (8) into (5) the nonzero compo-nents of the covariant metric tensor exterior to conical massdistributions are obtained as
11989200
= (1 +2
11988821198911015840
(119903))
11989211
= minus(1 +2
11988821198911015840
(119903))minus1
11989222
= minus(120572120573119903119908)
2
(12057221205732 minus V21199082)2
minus1199032V2 (1205722 + 1199082) (1205732 minus 1199082)
(12057221205732 minus V21199082) (1205722 + V2) (1205732 minus V2)
11989223
equiv 11989232
= minus(120572120573119903)
2
V119908
(12057221205732 minus V21199082)2
+1199032V119908
(12057221205732 minus V21199082) (1205722 + V2) (1205732 minus V2)
11989233
= minus(120572120573119903119908)
2
(12057221205732 minus V21199082)2
minus11990321199082 (1205722 minus V2) (1205732 minus V2)
(12057221205732 minus V21199082) (1205722 + 1199082) (1205732 minus 1199082)
(10)
Journal of Gravity 3
The nonzero components of the contravariant metric tensorfor this gravitational field are obtained using tensor analysisas
11989200
=1
11989200
11989211
=1
11989211
11989222
=11989233
1198922211989233minus (11989223)2
11989223
equiv 11989232
=11989223
1198922211989233minus (11989223)2
11989233
=11989222
1198922211989233minus (11989223)2
(11)
Thus the metric tensor in this gravitational field unlikeSchwarzschildrsquos metric has six nonzero components Thecontravariant metric components 11989200 and 11989211 are simply thereciprocal of their covariant counterparts The remainingnonzero components of the contravariant metric have acommon denominator
3 Motion of Test Particles
The affine connection coefficients of the gravitational fieldare obtained using tensor analysis and the motion of testparticles in the vicinity of a conical mass distribution isstudied using
1198892119909120583
1198891205912+ Γ120583
]120582(120597119909
]
120597120591)(
120597119909120582
120597120591) = 0 (12)
Setting 120583 = 0 in (12) we obtain the time equation of motionas
119905 +2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
119889119903119905 119903 = 0 (13)
where the dot represents differentiation with respect toproper time 120591 Solving (13) yields the expression for gravita-tional time dilation in this gravitational field as
119905 = [1 +2
11988821198911015840
(119903)]minus1
(14)
It has the same form as that obtained in the vicinity ofhomogeneous spherical [1] prolate spheroidal [3] and oblatespheroidal [2 4] mass distributionsThus only motion alongthe radial direction affects gravitational time dilation
Similarly setting 120583 = 1 in (12) yields the radial equationof motion as
119903 minus [1 +2
11988821198911015840
(119903)]1198891198911015840 (119903)
1198891199031199052
+2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
1198891199031199032minus V2[1 +
2
11988821198911015840
(119903)]
times (120572120573119908)
2
119903
(120572120573)2
minus (V119908)2
+V2119903 (1205722 + 1199082) (1205732 minus 1199082)
(1205722 + V2) (1205732 minus V2) [(1205722 + 1199082) (1205732 minus 1199082)] = 0
(15)
To obtain a first approximation to the solution of (15) pureradial motion can be studied For pure radial motion 119905 = 0V = 0 and (15) reduces to
119903 +2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
1198891199031199032= 0 (16)
Solving the pure radial equation yields solution as
119903 = 119860[1 +2
11988821198911015840
(119903)]minus1
(17)
where 119860 is a constant of motion This shows that 119903 variesinversely with 119891
1015840(119903) Thus an increase in the gravitational
potential retards motion in the pure radial direction andhence as a test particle moves closer to the conical mass dis-tribution along the radial direction its radial speed decreasesAn astronomical expedition can be undertaken in the nearfuture at the vicinity of Sputnik III to confirm this interestingresults
Also setting 120583 = 2 and 120583 = 3 the respective V and 119908
equations of motion are obtained as
V + 2Γ2
12119903V + 2Γ
2
13119903 + Γ
2
22V2+ 2Γ2
23V + Γ
2
332= 0
+ 2Γ3
12119903V + 2Γ
3
13119903 + 2Γ
3
23V + Γ
3
332= 0
(18)
4 Orbits in the Vicinity of a Conical MassDistribution
The Langrangian for any gravitational field is defined [7] as
119871 equiv1
119888(minus119892120572120573
119889119909120572
119889120591
119889119909120573
119889120591)
12
(19)
which can be written explicitly in this gravitational field as
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
minus 11989222(1198891199092
119889120591)
2
minus11989233(1198891199093
119889120591)
2
minus 211989223(1198891199092
119889120591)(
1198891199093
119889120591)]
(20)
4 Journal of Gravity
Now consider spatial motion along the radial direction onlythen the Langrangian reduces to
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
] (21)
or more explicitly as
119871 = minus119888 (1 +2
11988821198911015840
(119903)) 1199052
+1
119888(1 +
2
11988821198911015840
(119903))minus1
1199032 (22)
Transforming the radial coordinate 119903 to the angular coordi-nate 120596 using 119903 = 119903(120596) and defining 119906(120596) = 1119903(120596) we have
119903 = minus1
1 + 1199062119889119906
119889120596 (23)
Substituting (23) into (22) and using the fact that 119871 = 1 fortime-like orbits and 119871 = 0 for null orbits we obtain
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596=(1 + 1199062)
2
119888
119889119891 (120596)
119889119906 (24)
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596= 0 (25)
as the respective planetary equation of motion and photonequation of motion in the vicinity of a conical mass distri-bution placed in empty space Equation (24) can be used tostudy the motion of orbiting particles moving in the vicinityof conical masses
5 Remarks and Conclusion
The results obtained in this paper have paved the way forthe study of gravitational fields of conical mass distributionslike Sputnik III The immediate consequences of the resultsobtained in this paper are as follows
(1) Einsteinrsquos geometrical field equations can be con-structed using our metric tensor The striking factabout these field equations is that they have only oneunknown 119891
1015840(119903) which is determined by the mass orpressure distribution Thus the field equations canbe easily solved and explicit expressions obtained for1198911015840(119903)
(2) The planetary equation of motion and the photonequation of motion can be solved to describe themotion of test particles and photons in the vicinityof conical masses These equations are opened up forfurther research work and astrophysical interpreta-tion
(3) In approximate gravitational fields the arbitrary func-tion 119891
1015840(119903) can be conveniently equated to the grav-
itational scalar potential exterior to a conical massThus if the complete expression of 1198911015840(119903) is obtainedfrom the field equations it will depict a hither tounknown generalization of the gravitational scalarpotential function for this field
(4) Other gravitational phenomena such as gravitationalspectral shift could be studied in the vicinity of conicalmasses
References
[1] E N Chifu and S X K Howusu ldquoGravitational radiation andpropagation field equation exterior to astrophysically real orhypothetical time varying distributions of mass within regionsof spherical geometryrdquo Physics Essays vol 22 no 1 pp 73ndash772009
[2] E N Chifu A Usman and O C Meludu ldquoOrbits in homo-geneous oblate spheroidal gravitational space timerdquo Progress inPhysics vol 3 pp 49ndash53 2009
[3] E N Chifu A Usman and O C Meludu ldquoExact analyticalsolutions of Einsteinrsquos gravitational field equations in statichomogeneous prolate spheroidal space-timerdquoAfrican Journal ofMathematical Physics vol 9 pp 25ndash42 2010
[4] E N Chifu ldquoGravitational fields exterior to homogeneousspheroidal massesrdquo Abraham Zelmanov Journal of GeneralRelativity Gravitation and Cosmology vol 5 pp 31ndash67 2012
[5] PG Bergmann Introduction to theTheory of Relativity PrenticeHall New Delhi India 1987
[6] S Weinberg Gravitation and Cosmology John Wiley amp SonsNew York NY USA 1972
[7] K S D Peter ldquoAn introduction to tensors and relativityrdquo CapeTown pp 51ndash110 2000
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2 Journal of Gravity
Now the spherical polar coordinates (119903 120579 120601) are relatedto the Cartesian coordinates (119909 119910 119911) as follows
119903 = (1199092+ 1199102+ 1199112)12
120579 = cosminus1( 119911
(1199092 + 1199102 + 1199112)12
)
120601 = tanminus1 (119910
119909)
(1)
Also the Cartesian coordinates are related to the conicalcoordinates (119903 V 119908) as follows
119909 =1
120572119903(1205722minus V2)12
(1205722+ 1199082)12
119910 =1
120573119903(1205732+ V2)12
(1205732minus 1199082)12
119911 =1
120572120573119903V119908
(2)
where 1205722 + 1205732 = 1Expressing the spherical coordinates in terms of the
conical coordinates we obtain
120579 (119903 V 119908) = cosminus1 (V119908120572120573
) (3)
120601 (119903 V 119908) = tanminus1 [
[
120573(1205722 minus V2)12
(1205722 + 1199082)12
120572(1205732 + V2)12
(1205732 minus 1199082)12
]
]
(4)
Now consider a spherically symmetric mass distributionwith invariant world line element (Schwarzschildrsquos line ele-ment) in the exterior region given [1] as
11988821198891205912= 1198882(1 +
2119891 (119903)
1198882)1198891199052minus (1 +
2119891 (119903)
1198882)
minus1
1198891199032
minus 11990321198891205792minus 1199032sin21205791198891206012
(5)
where 119891(119903) generally is an arbitrary function determinedby the distribution of mass or pressure and possess all thesymmetries of the distribution In approximate gravitationalfields 119891(119903) is equivalent to the gravitational scalar potentialfunction exterior to a massive symmetric sphere
Let the spherically symmetric mass distribution be trans-formed by deformation into a conical mass distribution insuch a way that it retains its density and total mass Thus thegeneral relativistic invariant world line element exterior to aspherical symmetric body is tensorially equivalent to that ofa conical mass distribution and related by the transformationfrom spherical to conical coordinatesTherefore to obtain theinvariant line element exterior to conical mass distributionswe transform coordinates on the right hand side of (5) fromspherical to conical coordinates and 119891(119903) rarr 119891
1015840(119903) thearbitrary function determined by the conical mass
From (3) we have
119889120579 =120597120579
120597119903119889119903 +
120597120579
120597V119889V +
120597120579
120597119908119889119908 (6)
where
120597120579
120597V= minus
119908
(12057221205732 minus V21199082)12
120597120579
120597119908= minus
V
(12057221205732 minus V21199082)12
120597120579
120597119903= 0
(7)
Also from (4) we have
119889120601 =120597120601
120597119903119889119903 +
120597120601
120597V119889V +
120597120601
120597119908119889119908 (8)
where
120597120601
120597V= minus
V31199082(1205722 + 1199082)12
(1205722 minus V2)minus12
1205723120573(1205732 + V2)32
(1205732 minus 1199082)12
120597120601
120597119908=V21199083(1205722 minus V2)
12
(1205722 + 1199082)minus12
1205723120573(1205732 minus 1199082)32
(1205732 + V2)12
120597120601
120597119903= 0
(9)
Substituting (3) (6) and (8) into (5) the nonzero compo-nents of the covariant metric tensor exterior to conical massdistributions are obtained as
11989200
= (1 +2
11988821198911015840
(119903))
11989211
= minus(1 +2
11988821198911015840
(119903))minus1
11989222
= minus(120572120573119903119908)
2
(12057221205732 minus V21199082)2
minus1199032V2 (1205722 + 1199082) (1205732 minus 1199082)
(12057221205732 minus V21199082) (1205722 + V2) (1205732 minus V2)
11989223
equiv 11989232
= minus(120572120573119903)
2
V119908
(12057221205732 minus V21199082)2
+1199032V119908
(12057221205732 minus V21199082) (1205722 + V2) (1205732 minus V2)
11989233
= minus(120572120573119903119908)
2
(12057221205732 minus V21199082)2
minus11990321199082 (1205722 minus V2) (1205732 minus V2)
(12057221205732 minus V21199082) (1205722 + 1199082) (1205732 minus 1199082)
(10)
Journal of Gravity 3
The nonzero components of the contravariant metric tensorfor this gravitational field are obtained using tensor analysisas
11989200
=1
11989200
11989211
=1
11989211
11989222
=11989233
1198922211989233minus (11989223)2
11989223
equiv 11989232
=11989223
1198922211989233minus (11989223)2
11989233
=11989222
1198922211989233minus (11989223)2
(11)
Thus the metric tensor in this gravitational field unlikeSchwarzschildrsquos metric has six nonzero components Thecontravariant metric components 11989200 and 11989211 are simply thereciprocal of their covariant counterparts The remainingnonzero components of the contravariant metric have acommon denominator
3 Motion of Test Particles
The affine connection coefficients of the gravitational fieldare obtained using tensor analysis and the motion of testparticles in the vicinity of a conical mass distribution isstudied using
1198892119909120583
1198891205912+ Γ120583
]120582(120597119909
]
120597120591)(
120597119909120582
120597120591) = 0 (12)
Setting 120583 = 0 in (12) we obtain the time equation of motionas
119905 +2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
119889119903119905 119903 = 0 (13)
where the dot represents differentiation with respect toproper time 120591 Solving (13) yields the expression for gravita-tional time dilation in this gravitational field as
119905 = [1 +2
11988821198911015840
(119903)]minus1
(14)
It has the same form as that obtained in the vicinity ofhomogeneous spherical [1] prolate spheroidal [3] and oblatespheroidal [2 4] mass distributionsThus only motion alongthe radial direction affects gravitational time dilation
Similarly setting 120583 = 1 in (12) yields the radial equationof motion as
119903 minus [1 +2
11988821198911015840
(119903)]1198891198911015840 (119903)
1198891199031199052
+2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
1198891199031199032minus V2[1 +
2
11988821198911015840
(119903)]
times (120572120573119908)
2
119903
(120572120573)2
minus (V119908)2
+V2119903 (1205722 + 1199082) (1205732 minus 1199082)
(1205722 + V2) (1205732 minus V2) [(1205722 + 1199082) (1205732 minus 1199082)] = 0
(15)
To obtain a first approximation to the solution of (15) pureradial motion can be studied For pure radial motion 119905 = 0V = 0 and (15) reduces to
119903 +2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
1198891199031199032= 0 (16)
Solving the pure radial equation yields solution as
119903 = 119860[1 +2
11988821198911015840
(119903)]minus1
(17)
where 119860 is a constant of motion This shows that 119903 variesinversely with 119891
1015840(119903) Thus an increase in the gravitational
potential retards motion in the pure radial direction andhence as a test particle moves closer to the conical mass dis-tribution along the radial direction its radial speed decreasesAn astronomical expedition can be undertaken in the nearfuture at the vicinity of Sputnik III to confirm this interestingresults
Also setting 120583 = 2 and 120583 = 3 the respective V and 119908
equations of motion are obtained as
V + 2Γ2
12119903V + 2Γ
2
13119903 + Γ
2
22V2+ 2Γ2
23V + Γ
2
332= 0
+ 2Γ3
12119903V + 2Γ
3
13119903 + 2Γ
3
23V + Γ
3
332= 0
(18)
4 Orbits in the Vicinity of a Conical MassDistribution
The Langrangian for any gravitational field is defined [7] as
119871 equiv1
119888(minus119892120572120573
119889119909120572
119889120591
119889119909120573
119889120591)
12
(19)
which can be written explicitly in this gravitational field as
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
minus 11989222(1198891199092
119889120591)
2
minus11989233(1198891199093
119889120591)
2
minus 211989223(1198891199092
119889120591)(
1198891199093
119889120591)]
(20)
4 Journal of Gravity
Now consider spatial motion along the radial direction onlythen the Langrangian reduces to
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
] (21)
or more explicitly as
119871 = minus119888 (1 +2
11988821198911015840
(119903)) 1199052
+1
119888(1 +
2
11988821198911015840
(119903))minus1
1199032 (22)
Transforming the radial coordinate 119903 to the angular coordi-nate 120596 using 119903 = 119903(120596) and defining 119906(120596) = 1119903(120596) we have
119903 = minus1
1 + 1199062119889119906
119889120596 (23)
Substituting (23) into (22) and using the fact that 119871 = 1 fortime-like orbits and 119871 = 0 for null orbits we obtain
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596=(1 + 1199062)
2
119888
119889119891 (120596)
119889119906 (24)
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596= 0 (25)
as the respective planetary equation of motion and photonequation of motion in the vicinity of a conical mass distri-bution placed in empty space Equation (24) can be used tostudy the motion of orbiting particles moving in the vicinityof conical masses
5 Remarks and Conclusion
The results obtained in this paper have paved the way forthe study of gravitational fields of conical mass distributionslike Sputnik III The immediate consequences of the resultsobtained in this paper are as follows
(1) Einsteinrsquos geometrical field equations can be con-structed using our metric tensor The striking factabout these field equations is that they have only oneunknown 119891
1015840(119903) which is determined by the mass orpressure distribution Thus the field equations canbe easily solved and explicit expressions obtained for1198911015840(119903)
(2) The planetary equation of motion and the photonequation of motion can be solved to describe themotion of test particles and photons in the vicinityof conical masses These equations are opened up forfurther research work and astrophysical interpreta-tion
(3) In approximate gravitational fields the arbitrary func-tion 119891
1015840(119903) can be conveniently equated to the grav-
itational scalar potential exterior to a conical massThus if the complete expression of 1198911015840(119903) is obtainedfrom the field equations it will depict a hither tounknown generalization of the gravitational scalarpotential function for this field
(4) Other gravitational phenomena such as gravitationalspectral shift could be studied in the vicinity of conicalmasses
References
[1] E N Chifu and S X K Howusu ldquoGravitational radiation andpropagation field equation exterior to astrophysically real orhypothetical time varying distributions of mass within regionsof spherical geometryrdquo Physics Essays vol 22 no 1 pp 73ndash772009
[2] E N Chifu A Usman and O C Meludu ldquoOrbits in homo-geneous oblate spheroidal gravitational space timerdquo Progress inPhysics vol 3 pp 49ndash53 2009
[3] E N Chifu A Usman and O C Meludu ldquoExact analyticalsolutions of Einsteinrsquos gravitational field equations in statichomogeneous prolate spheroidal space-timerdquoAfrican Journal ofMathematical Physics vol 9 pp 25ndash42 2010
[4] E N Chifu ldquoGravitational fields exterior to homogeneousspheroidal massesrdquo Abraham Zelmanov Journal of GeneralRelativity Gravitation and Cosmology vol 5 pp 31ndash67 2012
[5] PG Bergmann Introduction to theTheory of Relativity PrenticeHall New Delhi India 1987
[6] S Weinberg Gravitation and Cosmology John Wiley amp SonsNew York NY USA 1972
[7] K S D Peter ldquoAn introduction to tensors and relativityrdquo CapeTown pp 51ndash110 2000
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
Journal of Gravity 3
The nonzero components of the contravariant metric tensorfor this gravitational field are obtained using tensor analysisas
11989200
=1
11989200
11989211
=1
11989211
11989222
=11989233
1198922211989233minus (11989223)2
11989223
equiv 11989232
=11989223
1198922211989233minus (11989223)2
11989233
=11989222
1198922211989233minus (11989223)2
(11)
Thus the metric tensor in this gravitational field unlikeSchwarzschildrsquos metric has six nonzero components Thecontravariant metric components 11989200 and 11989211 are simply thereciprocal of their covariant counterparts The remainingnonzero components of the contravariant metric have acommon denominator
3 Motion of Test Particles
The affine connection coefficients of the gravitational fieldare obtained using tensor analysis and the motion of testparticles in the vicinity of a conical mass distribution isstudied using
1198892119909120583
1198891205912+ Γ120583
]120582(120597119909
]
120597120591)(
120597119909120582
120597120591) = 0 (12)
Setting 120583 = 0 in (12) we obtain the time equation of motionas
119905 +2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
119889119903119905 119903 = 0 (13)
where the dot represents differentiation with respect toproper time 120591 Solving (13) yields the expression for gravita-tional time dilation in this gravitational field as
119905 = [1 +2
11988821198911015840
(119903)]minus1
(14)
It has the same form as that obtained in the vicinity ofhomogeneous spherical [1] prolate spheroidal [3] and oblatespheroidal [2 4] mass distributionsThus only motion alongthe radial direction affects gravitational time dilation
Similarly setting 120583 = 1 in (12) yields the radial equationof motion as
119903 minus [1 +2
11988821198911015840
(119903)]1198891198911015840 (119903)
1198891199031199052
+2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
1198891199031199032minus V2[1 +
2
11988821198911015840
(119903)]
times (120572120573119908)
2
119903
(120572120573)2
minus (V119908)2
+V2119903 (1205722 + 1199082) (1205732 minus 1199082)
(1205722 + V2) (1205732 minus V2) [(1205722 + 1199082) (1205732 minus 1199082)] = 0
(15)
To obtain a first approximation to the solution of (15) pureradial motion can be studied For pure radial motion 119905 = 0V = 0 and (15) reduces to
119903 +2
1198882[1 +
2
11988821198911015840
(119903)]minus1 1198891198911015840 (119903)
1198891199031199032= 0 (16)
Solving the pure radial equation yields solution as
119903 = 119860[1 +2
11988821198911015840
(119903)]minus1
(17)
where 119860 is a constant of motion This shows that 119903 variesinversely with 119891
1015840(119903) Thus an increase in the gravitational
potential retards motion in the pure radial direction andhence as a test particle moves closer to the conical mass dis-tribution along the radial direction its radial speed decreasesAn astronomical expedition can be undertaken in the nearfuture at the vicinity of Sputnik III to confirm this interestingresults
Also setting 120583 = 2 and 120583 = 3 the respective V and 119908
equations of motion are obtained as
V + 2Γ2
12119903V + 2Γ
2
13119903 + Γ
2
22V2+ 2Γ2
23V + Γ
2
332= 0
+ 2Γ3
12119903V + 2Γ
3
13119903 + 2Γ
3
23V + Γ
3
332= 0
(18)
4 Orbits in the Vicinity of a Conical MassDistribution
The Langrangian for any gravitational field is defined [7] as
119871 equiv1
119888(minus119892120572120573
119889119909120572
119889120591
119889119909120573
119889120591)
12
(19)
which can be written explicitly in this gravitational field as
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
minus 11989222(1198891199092
119889120591)
2
minus11989233(1198891199093
119889120591)
2
minus 211989223(1198891199092
119889120591)(
1198891199093
119889120591)]
(20)
4 Journal of Gravity
Now consider spatial motion along the radial direction onlythen the Langrangian reduces to
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
] (21)
or more explicitly as
119871 = minus119888 (1 +2
11988821198911015840
(119903)) 1199052
+1
119888(1 +
2
11988821198911015840
(119903))minus1
1199032 (22)
Transforming the radial coordinate 119903 to the angular coordi-nate 120596 using 119903 = 119903(120596) and defining 119906(120596) = 1119903(120596) we have
119903 = minus1
1 + 1199062119889119906
119889120596 (23)
Substituting (23) into (22) and using the fact that 119871 = 1 fortime-like orbits and 119871 = 0 for null orbits we obtain
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596=(1 + 1199062)
2
119888
119889119891 (120596)
119889119906 (24)
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596= 0 (25)
as the respective planetary equation of motion and photonequation of motion in the vicinity of a conical mass distri-bution placed in empty space Equation (24) can be used tostudy the motion of orbiting particles moving in the vicinityof conical masses
5 Remarks and Conclusion
The results obtained in this paper have paved the way forthe study of gravitational fields of conical mass distributionslike Sputnik III The immediate consequences of the resultsobtained in this paper are as follows
(1) Einsteinrsquos geometrical field equations can be con-structed using our metric tensor The striking factabout these field equations is that they have only oneunknown 119891
1015840(119903) which is determined by the mass orpressure distribution Thus the field equations canbe easily solved and explicit expressions obtained for1198911015840(119903)
(2) The planetary equation of motion and the photonequation of motion can be solved to describe themotion of test particles and photons in the vicinityof conical masses These equations are opened up forfurther research work and astrophysical interpreta-tion
(3) In approximate gravitational fields the arbitrary func-tion 119891
1015840(119903) can be conveniently equated to the grav-
itational scalar potential exterior to a conical massThus if the complete expression of 1198911015840(119903) is obtainedfrom the field equations it will depict a hither tounknown generalization of the gravitational scalarpotential function for this field
(4) Other gravitational phenomena such as gravitationalspectral shift could be studied in the vicinity of conicalmasses
References
[1] E N Chifu and S X K Howusu ldquoGravitational radiation andpropagation field equation exterior to astrophysically real orhypothetical time varying distributions of mass within regionsof spherical geometryrdquo Physics Essays vol 22 no 1 pp 73ndash772009
[2] E N Chifu A Usman and O C Meludu ldquoOrbits in homo-geneous oblate spheroidal gravitational space timerdquo Progress inPhysics vol 3 pp 49ndash53 2009
[3] E N Chifu A Usman and O C Meludu ldquoExact analyticalsolutions of Einsteinrsquos gravitational field equations in statichomogeneous prolate spheroidal space-timerdquoAfrican Journal ofMathematical Physics vol 9 pp 25ndash42 2010
[4] E N Chifu ldquoGravitational fields exterior to homogeneousspheroidal massesrdquo Abraham Zelmanov Journal of GeneralRelativity Gravitation and Cosmology vol 5 pp 31ndash67 2012
[5] PG Bergmann Introduction to theTheory of Relativity PrenticeHall New Delhi India 1987
[6] S Weinberg Gravitation and Cosmology John Wiley amp SonsNew York NY USA 1972
[7] K S D Peter ldquoAn introduction to tensors and relativityrdquo CapeTown pp 51ndash110 2000
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
4 Journal of Gravity
Now consider spatial motion along the radial direction onlythen the Langrangian reduces to
119871 =1
119888[minus11989200(1198891199090
119889120591)
2
minus 11989211(1198891199091
119889120591)
2
] (21)
or more explicitly as
119871 = minus119888 (1 +2
11988821198911015840
(119903)) 1199052
+1
119888(1 +
2
11988821198911015840
(119903))minus1
1199032 (22)
Transforming the radial coordinate 119903 to the angular coordi-nate 120596 using 119903 = 119903(120596) and defining 119906(120596) = 1119903(120596) we have
119903 = minus1
1 + 1199062119889119906
119889120596 (23)
Substituting (23) into (22) and using the fact that 119871 = 1 fortime-like orbits and 119871 = 0 for null orbits we obtain
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596=(1 + 1199062)
2
119888
119889119891 (120596)
119889119906 (24)
1198892119906
1198891205962minus
2119906
1 + 1199062119889119906
119889120596= 0 (25)
as the respective planetary equation of motion and photonequation of motion in the vicinity of a conical mass distri-bution placed in empty space Equation (24) can be used tostudy the motion of orbiting particles moving in the vicinityof conical masses
5 Remarks and Conclusion
The results obtained in this paper have paved the way forthe study of gravitational fields of conical mass distributionslike Sputnik III The immediate consequences of the resultsobtained in this paper are as follows
(1) Einsteinrsquos geometrical field equations can be con-structed using our metric tensor The striking factabout these field equations is that they have only oneunknown 119891
1015840(119903) which is determined by the mass orpressure distribution Thus the field equations canbe easily solved and explicit expressions obtained for1198911015840(119903)
(2) The planetary equation of motion and the photonequation of motion can be solved to describe themotion of test particles and photons in the vicinityof conical masses These equations are opened up forfurther research work and astrophysical interpreta-tion
(3) In approximate gravitational fields the arbitrary func-tion 119891
1015840(119903) can be conveniently equated to the grav-
itational scalar potential exterior to a conical massThus if the complete expression of 1198911015840(119903) is obtainedfrom the field equations it will depict a hither tounknown generalization of the gravitational scalarpotential function for this field
(4) Other gravitational phenomena such as gravitationalspectral shift could be studied in the vicinity of conicalmasses
References
[1] E N Chifu and S X K Howusu ldquoGravitational radiation andpropagation field equation exterior to astrophysically real orhypothetical time varying distributions of mass within regionsof spherical geometryrdquo Physics Essays vol 22 no 1 pp 73ndash772009
[2] E N Chifu A Usman and O C Meludu ldquoOrbits in homo-geneous oblate spheroidal gravitational space timerdquo Progress inPhysics vol 3 pp 49ndash53 2009
[3] E N Chifu A Usman and O C Meludu ldquoExact analyticalsolutions of Einsteinrsquos gravitational field equations in statichomogeneous prolate spheroidal space-timerdquoAfrican Journal ofMathematical Physics vol 9 pp 25ndash42 2010
[4] E N Chifu ldquoGravitational fields exterior to homogeneousspheroidal massesrdquo Abraham Zelmanov Journal of GeneralRelativity Gravitation and Cosmology vol 5 pp 31ndash67 2012
[5] PG Bergmann Introduction to theTheory of Relativity PrenticeHall New Delhi India 1987
[6] S Weinberg Gravitation and Cosmology John Wiley amp SonsNew York NY USA 1972
[7] K S D Peter ldquoAn introduction to tensors and relativityrdquo CapeTown pp 51ndash110 2000
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
