Volume 10 (2014) PROGRESS IN PHYSICS Issue 1 (January) Orbits in Homogeneous Time Varying Spherical Spacetime 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 distribu- tions 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 2 ! ! ! 3 1 geometry is defined as: 6 2 2 27 2 1 6 dt dr dϕ 7 " # L = 46−g00 − g11 − g33 57 (7) 2 c dτ dτ dτ g = − 1 + f (t; r) (1) 00 c2 or more explicitly as " #− 2 3 1 ! !− 1 2 6 1 7 2 g = 1 + f (t; r) (2) 1 6 2 2 2 2 2 27 11 2 L = 46 1 + f (t; r) t˙ − 1 + f (t; r) r˙ − r ϕ˙ 57 (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 i! t − (5) or more explicitly as r c " ( )# 2GM r where k = GM0 with G being the universal gravitational con- 1 − exp i! t − t˙ = d (10) rc2 c stant and M0 the total mass of the spherical body. ! 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 = ", and equation (8) becomes to derive the geodesic equations [2]. The Lagrangian (L) for ! 2 ! 3 6 −1 7 this gravitational field can be defined using the metric tensor 2 1 6 2 7 "2= 1+ f (t; r) t˙2− 46 1+ f (t; r) r˙2−r2ϕ˙257 : (12) as: c2 c2 c2 Substituting equation (10) in (12) yields 2 ! ! ! ! 3 1 " !# 1 6 dt 2 dr 2 dθ 2 dϕ 27 2 ( ) 6 7 1 2 2 2 2 2 1 2 2 2 L= 46−g00 −g11 −g22 −g33 57 (6) r˙ +r ϕ˙ 1+ f (t; r) +" f (t; r)= c d −" : (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 i! t − + dϕ c2 uc right hand side and neglecting the term in u, a particular inte- ! gral u satisfies equation (19) such that 2"2k 1 c2 ( ) 1 u exp i! t − = d2 − "2 (16) " # h2 uc h2 d2u 3 1 1 + u = k (1 + e cos θ)2 − exp i!t: (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 i! t − − dϕ2 c2 uc ! ! ! = k 3 − 1 − 1 ! 2k 1 1 u1 exp i t u2 1 − exp i! t − + c2 l2 l h2 c2 u uc ! ! ! keϕ 3 1 ke2 2k"2 1 1 + − sin 2ϕ exp i!t + cos 2ϕ. (25) 1 + exp i! 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 " = ! For timelike orbits 1 and equation (17) becomes 1 k 3 1 1 u = ( + e θ) + − − i!t " !# 1 cos 2 2 2 exp 2 l c l l h d u + − 2k ! − 1 − ! u 1 u exp i t keϕ 3 1 ke2 dϕ2 c2 uc + − sin 2ϕ exp i!t + cos 2ϕ. (27) ! ! 2 2 l 2 2 2k 1 1 2c l 2 l c u2 1 − exp i! 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 i! 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 " # " = d2u 3 1 1 For null geodesics, 0 and equation (17) yields + u = k u2 + u − exp i!t: (19) ϕ2 2 2 2 " !# d c c h d2u 3k 1 + u = exp i! t − u2 dϕ2 c2 uc The Newtonian equation for a spherical mass is " !# k 1 d2u k + exp i! 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 pa- rameter). 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 i! t − dϕ2 1 b2c2 c sin ϕ ! k b + sin ϕ exp i! 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 i! 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 i! 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 = 1 + 2 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 i! t + + exp i! t + : (39) ! ! bc2 c c 1 2k 1 b 1 2 0 = sin ϕ+ 1 − sin2 ϕ exp i! 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 = − ϕ = + 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.