Orbits in Homogeneous Time Varying Spherical Spacetime

Volume 10 (2014) PROGRESS IN PHYSICS Issue 1 (January)

Chifu E. Ndikilar Physics Department, Federal University Dutse, P.M.B 7156, Dutse, Jigawa State, Nigeria. E-mail: [email protected] The solution to Einstein’s gravitational field equations exterior to time varying distributions of mass within regions of spherical geometry is used to study the behaviour of test particles and photons in the vicinity of the mass distribution. Equations of motion are derived and an expression for deflection of light in this gravitational field is obtained. The expresion obtained differs from that in Schwarzschild’s field by a multiplicative time dependent factor. The concept of gravitational lens in this gravitational field is also studied.

1 Introduction Assuming that the orbits remain permanently in the equato- rial plane (as in Newtonian Theory), then θ = π and the La- In [1], the covariant metric tensor exterior to a homogeneous 2 time varying distribution of mass within regions of spherical grangian reduces to  ( ) ( ) ( )  1 geometry is defined as:  2 2 2 2 1  dt dr dϕ  [ ] L = −g00 − g11 − g33  (7) 2 c dτ dτ dτ g = − 1 + f (t, r) (1) 00 c2 or more explicitly as [ ]−   1 ( ) ( )− 1 2  1  2 g = 1 + f (t, r) (2) 1  2 2 2 2 2 2 11 2 L =  1 + f (t, r) t˙ − 1 + f (t, r) r˙ − r ϕ˙  (8) c c c2 c2 g = 2 22 r (3) where the dot denotes differentiation with respect to proper 2 2 time (τ). g33 = r sin θ (4) Now, using the Euler-Lagrange equations and considering where f (t, r) is a function dependent on the mass distribution the fact that in a gravitational field is a conservative field, it within the sphere that experiences radial displacement. Ein- can be shown that the law of conservation of energy in this stein’s gravitational field equations were constructed in [1] field is given as and an approximate expression for the analytical solution of ( ) 2 the lone field equation was obtained as + f t, r t˙ = d constant 1 2 ( ) ( ) (9) ( ) c k r f (t, r) ≈ − exp iω t − (5) or more explicitly as r c [ ( )] 2GM r where k = GM0 with G being the universal gravitational con- 1 − exp iω t − t˙ = d (10) rc2 c stant and M0 the total mass of the spherical body. ω is the angular frequency of the radial displacement of mass within which differs from that in Schwarzschild’s field by the expo- the sphere. nential factor that describes the radial displacement of mass In this article, we use this solution of Einstein’s field equa- with time. tions to study the behaviour of light in the vicinity of a time It can also be shown that the law of conservation of angu- varying spherical mass distribution. lar momemtum in this gravitational field is given as

2 Orbits in Time Varying Spherical Spacetime r2ϕ˙ = h (constant) (11) In order to study the motion of planets and light rays in a which is the same as that in Schwarzschild’s field. homogeneous time varying spherical spacetime, there is need Let L = ε, and equation (8) becomes to derive the geodesic equations [2]. The Lagrangian (L) for ( ) ( )   −1  this gravitational field can be defined using the metric tensor 2 1  2  ε2= 1+ f (t, r) t˙2−  1+ f (t, r) r˙2−r2ϕ˙2 . (12) as: c2 c2 c2 Substituting equation (10) in (12) yields  ( ) ( ) ( ) ( )  1 [ ( )] 1  dt 2 dr 2 dθ 2 dϕ 2 2 ( )   1 2 2 2 2 2 1 2 2 2 L= −g00 −g11 −g22 −g33  (6) r˙ +r ϕ˙ 1+ f (t, r) +ε f (t, r)= c d −ε . (13) c dτ dτ dτ dτ 2 c2 2

52 Chifu E.N. Orbits in Homogeneous Time Varying Spherical Spacetime Issue 1 (January) PROGRESS IN PHYSICS Volume 10 (2014)

This is the Newtonian energy equation with a modification to and that obtained in Schwarzschild’s field is the ϕ˙2 term. It is similar to that obtained in Schwarzschild’s d2u k 3k field except for the time dependent radial diplacement. Also, + u = + u2. (21) ϕ2 2 2 using equation (11), it can be shown that d h c dr dϕ dr h dr Apart from the first and second terms of equation (19) that r˙ = = ϕ˙ = . (14) are similar to Newton’s equation and that in Schwarzschild’s dϕ dτ dϕ r2 dϕ field, the other terms have terms dependent on the time rate ϕ = 1 of rotation of the mass content within the sphere [3]. Now, let u( ) r(ϕ) then Solution of the Newtonian equation (20) yields the well du known conics r˙ = −h . (15) 1 dϕ u = (1 + e cos θ) (22) 0 l Substituting equation (5) and (15) into equation (13) yields 2 = h where l GM and e is the eccentricity of the orbit. Attempt- ing an approximate solution for equation (19) by substituting ( ) [ ( )] du 2 2k 1 the Newtonian solution into the quadratic term in u on the + u2 1 − u exp iω t − + dϕ c2 uc right hand side and neglecting the term in u, a particular inte- ( ) gral u satisfies equation (19) such that 2ε2k 1 c2 ( ) 1 u exp iω t − = d2 − ε2 (16) [ ] h2 uc h2 d2u 3 1 1 + u = k (1 + e cos θ)2 − exp iωt. (23) dϕ2 1 l2c2 h2 It is worthnoting that integrating equation (16) directly leads ff to elliptical integrals which are ackward to handle; thus di er- Now suppose u1 takes the form: entiating yields the following second order differential equa- = + ϕ ϕ + ϕ tion u1 A B sin C cos 2 (24) [ ( )] d2u 2k 1 where A, B and C are constants, then it can be shown that + u 1 − u exp iω t − − dϕ2 c2 uc ( ) ( ) ( ) = k 3 − 1 − 1 ω 2k 1 1 u1 exp i t u2 1 − exp iω t − + c2 l2 l h2 c2 u uc ( ) ( ) ( ) keϕ 3 1 ke2 2kε2 1 1 + − sin 2ϕ exp iωt + cos 2ϕ. (25) 1 + exp iω t − = 0. (17) 2c2 l2 2l l2c2 h2 u2 uc Then the approximate solution for u can be given as This equation has additional terms not found in Schwarz- schild’s field. u = u0 + u1 (26) 3 Timelike Orbits and Precession or ε = ( ) For timelike orbits 1 and equation (17) becomes 1 k 3 1 1 u = ( + e θ) + − − iωt [ ( )] 1 cos 2 2 2 exp 2 l c l l h d u + − 2k ω − 1 − ( ) u 1 u exp i t keϕ 3 1 ke2 dϕ2 c2 uc + − sin 2ϕ exp iωt + cos 2ϕ. (27) ( ) ( ) 2 2 l 2 2 2k 1 1 2c l 2 l c u2 1 − exp iω t − + c2 u uc ( ) ( ) Hence, this approximate solution introduces corrections to u0 2k 1 1 and hence depicts that the orbits of massive objects is only 1 + exp iω t − = 0. (18) h2 u2 uc approximately elliptical and also accounts for the perihelion precession of planetary orbits in this gravitational field. Now as a first approximation, suppose uc ≫ 1 and k ≪ h2u2 then equation (8) reduces to 4 The Bending of Light [ ] ε = d2u 3 1 1 For null geodesics, 0 and equation (17) yields + u = k u2 + u − exp iωt. (19) ϕ2 2 2 2 [ ( )] d c c h d2u 3k 1 + u = exp iω t − u2 dϕ2 c2 uc The Newtonian equation for a spherical mass is [ ( )] k 1 d2u k + exp iω t − u. (28) + u = (20) c2 uc dϕ2 h2

Chifu E.N. Orbits in Homogeneous Time Varying Spherical Spacetime 53 Volume 10 (2014) PROGRESS IN PHYSICS Issue 1 (January)

In the limit of Special Relativity, equation (28) reduces to

d2u + u = 0. (29) dϕ2 The general solution of equation (29) is given as 1 u = sin (ϕ − ϕ ) (30) b 0 where b is the closest approach to the origin (or impact parameter). This is the equation of a straight line as ϕ goes from ϕ0 to ϕ0 + π. The straight line motion of light is the same as that predicted by Newtonian theory. Now, solving the General Relativity problem (equation 28) by taking the general solution (u) to be a pertubation of the Newtonian solution, and setting ϕ0 = 0, then Fig. 1: Diagram showing the total deﬂection of light. u = u0 + u1 (31) = 1 ϕ where u0 b sin . Thus, u1 satisﬁes the equation ( ) d2u 3k b 1 + u = sin2 ϕ exp iω t − dϕ2 1 b2c2 c sin ϕ ( ) k b + sin ϕ exp iω t − . (32) bc2 c sin ϕ

Now, by considering a particular integral of the form

2 u1 = A + B sin ϕ (33) and substituting in equation (32), it can be shown that ( ) ( ) 2k 1 b u = 1 − sin2 ϕ exp iω t − (34) 1 b2c2 2 c sin ϕ and thus Fig. 2: Einstein’s Ring. ( ) ( ) 1 2k 1 b u = sin ϕ+ 1 − sin2 ϕ exp iω t − . (35) b b2c2 2 c sin ϕ The total deﬂection of light (σ) is given as

Now, consider the deflection of a light ray from a star σ = ψ1 + ψ2 which just grazes the time varying homogeneous spherical or mass (such as the Sun approximately); as in Fig. 1, then as [ ( ) ( )] r → ∞, u → 0, so 2k b b σ = exp iω t + + exp iω t + . (39) ( ) ( ) bc2 cψ cψ 1 2k 1 b 1 2 0 = sin ϕ+ 1 − sin2 ϕ exp iω t − . (36) b b2c2 2 c sin ϕ Thus, the introduction of varying mass distribution with time introduces an exponential term in the deflection of light equa- ϕ = −ψ ϕ = ψ + π ϕ ≪ At the asymptotes, 1 and 2 and taking 1 tion not found in static homogeneous spherical gravitational then equation (36) reduces to fields. ( ) Now, as an example of the bending of light, let us consider 1 2k b 0 = ψ + exp iω t + (37) a gravitational lens. 1 2 2 ψ b b c c 1 Consider a quasar directly behind a galaxy in our line of and ( ) sight as shown in Fig. 2. 1 2k b The distance of closest approach to the time varying = ψ + iω t + . 0 2 2 2 exp (38) σ b b c cψ2 spherical mass distribution corresponds to an angle ( ) given

54 Chifu E.N. Orbits in Homogeneous Time Varying Spherical Spacetime Issue 1 (January) PROGRESS IN PHYSICS Volume 10 (2014)

as equation (39). From Fig. 2, considering that both α and β are small, it can be deduced that α b b σ = + β = + (40) 2 d D and substituting equation (39) yields the impact parameter as

{ ( )[ ( ) ( )]} 1 2k Dd b b 2 b = iω t + + iω t + . 2 exp exp c D + d cψ1 cψ2 Hence, the image of the quasar appears as a ring which subtends an angle α = 2b d or

{ [ ( ) ( )]} 1 2 Dd b b 2 α = exp iω t + + exp iω t + . c d(D + d) cψ1 cψ2

5 Conclusion The results obtained in this study has paved the way for the theoretical study of homogeneous spherical mass distributions in which the mass content is varying with time. This will introduce correction terms found in Schwarzschild’s static ﬁeld. It is hoped that using this approach experimentally and astrophysically more satisfactory expressions and values will be obtained for gravitational phenomena in the universe.

Submitted on December 30, 2013 / Accepted on January 25, 2014

References 1. Chifu E.N. and Howusu S.X.K. Solution of Einstein’s Geometrical Gravitational Field Equations Exterior to Astrophysically Real or Hy- pothetical Time Varying Distributions of Mass within Regions of Spherical Geometry. Progress in Physics, 2009, v. 3, 45–48. 2. Chifu E.N., Usman A. and Meludu O.C. Orbits in Homogeneous Oblate Spheroidal Gravitational Space-Time. Progress in Physics, 2009, v. 3, 49–53. 3. Weinberg S. Gravitation and Cosmology. J. Wiley, New York, 1972.

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