Schwarzschild solution in R-spacetime Tosaporn Angsachon 1 , Sergey Manida 2 , Department of High energy and Elementary particle physics, Faculty of Physics, Saint-Petersburg State University, Ulyanovskaya str. Petrodvorets 198504, Russia. Email: 1
[email protected], 2
[email protected]
arXiv:1301.4198v1 [gr-qc] 17 Jan 2013
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Abstract Here we construct new solution for the Einstein equations – some analog of the Schwarzschild metric in anti–de Sitter–Beltrami space in the c → ∞ limit (R-space). In this case we derive an adiabatic invariant for finite movement of the massive point particle and separate variables Hamilton–Jacobi equation. Quasi orbital motion is analyzed and its radius time dependence is obtained.
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Schwarzschild metric in Anti de-Sitter-Beltrami spacetime
One of the important solution of Einstein’s Field Equations - this is the Schwarzschild solution, which describes a gravitational field outer the massive spherical symmetric body. If spacetime at the large distance from the source of this field has negative constant curvature R (Anti de-Sitter spacetime), then the corresponding metric in a spherical coordinate system x0 , r, θ, ϕ is written out as [1]: ds2 =
dr2 2M r2 − r2 dΩ2 , dx20 − 1− 2 − R r 2M r2 1− 2 − R r
(1)
where dΩ2 = dθ2 + sin2 θdϕ2 . This is the Schwarzschild Anti de-Sitter metric (SAdS). Now we change the metric (1) to the Beltrami coordinates. To do that we introduce coordinate transformations [2]: x0
x0 ⇒ R arcsin R
r
1+
x20 R2
, r⇒r
r
x2 r2 1 + 02 − 2 R R
.
(2)
Putting these transformations (2) in metric (1), we get
ds2 =
dx2 (xdx)2 2M h 2 − − dx0 − h2 R 2 h4 rh40
2 rx0 dx0 dr − R2 h20 . 2M rh3 rh 1 − h20
2M
(3)
For abbreviation of formulas we define: dx2 ≡ dx20 − d~r2 , xdx ≡ x0 dx0 − ~rd~r, h2 ≡ 1 + x2 /R2 , h20 ≡ 1 + x20 /R2 . A formula (3) is the Schwarzschild-Anti de-Sitter metric in the Beltrami coordinates (SAdSB). Now this metric is changed into the limit c → ∞ and we define M c ≡ g, so the metric (3) will be R2 (tdr − rdt)2 2gt R2 r 2 R4 2 − 2 2 dΩ2 . dt2 − (4) ds = 2 4 1 − 2gt c t rR c t c 2 t4 1 − rR This limiting expression for the metric SAdSB is what we call the Schwarzschild solution in R-spacetime. We can write out an action for classical test particle with mass m : S = −mc
Z
ds = −mR
2
Z
v 2gt dt u u − u1 − 2 t t rR
where r˙ = dr/dt ϕ˙ = dΩ/dt. 2
(tr˙ − r)2 r2 t2 ϕ˙ 2 − , 2gt R2 R2 1 − rR
(5)
Lagrangian of this system has the form v 2gt R2 u u L = −m 2 u1 − − t t rR
r2 t2 ϕ˙ 2 (tr˙ − r)2 − . 2gt R2 R2 1 − rR
(6)
The investigation of the properties of this Lagrangian will be held in next part.
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The limit of classical mechanics in R-spacetime
To get the classical mechanics limit we consider a spacetime region gt/R ≪ r ≪ R, r˙ ≪ R/t and expand Lagrangian (6) in the order of the small parameters r/R, rt/R ˙ gt/(rR) : L ≃ −m
R2 1 − 1 2gt + 2 t 2 rR
(tr˙ − r)2 r2 t2 ϕ˙ 2 + 2 2gt R R2 1 − rR
R2 mgR m(tr˙ − r)2 mr2 ϕ˙ 2 + + + . (7) 2 2 t rt 2t 2 The equations of motion do not change under the addition of some total derivative to the Lagrangian, and (7) will be written as mr2 ϕ˙ 2 mgR d R2 r2 mr˙ 2 , (8) + + + − L= 2 2 rt dt t 2t = −m
and this total derivative will be neglected. Now we transform to the Hamiltonian formalism and introduce radial momentum ∂L = mr˙ ∂ r˙
(9)
∂L = mr2 ϕ˙ ≡ Mϕ , ∂ ϕ˙
(10)
pr = and angular momentum pϕ =
which is an integral of motion. The Hamiltonian function will be constructed with the help of the Legendre transformation p2ϕ p2 mgR H = pr r˙ + pϕ ϕ˙ − L = r + − . (11) 2 2m 2mr rt The last term in (11) can be considered as the gravitational potential V =−
mG(t) r
with slowly changing gravitational constant G(t) = 3
gR . t
Hamiltonian (11) contains a parameter which depends on time, therefore we use the method of adiabatic invariant[4]. The stationary Hamiltonian function of our system is represented as H0 =
p2ϕ mG0 p2r − + = E. 2m 2mr2 r
(12)
The Hamilton-Jacobi equation is derived in a simple form 2 2 1 ∂S 1 mG0 ∂S + − = E0 , 2m ∂r 2mr2 ∂ϕ r
(13)
where S is a completed action for considered system. We will find it in the formula S = Sr (r) + Mϕ ϕ − E0 t.
(14)
We put (14) in the equation (13) and get r Mϕ2 2m2 G0 ∂Sr (r) = 2mE0 + − 2 = pr . ∂r r r
(15)
So the adiabatically invariants are 1 Ir = 2π
I
pr dr = −Mϕ + G0
s
m3 , 2|E0 |
(16)
I 1 pϕ ϕ = M ϕ 2π From the sum of (16) and (17) follows that energy depends on time Iϕ =
E=−
3
(17)
m3 g 2 R 2 . 2(Ir + Iϕ )2 t2
(18)
The Schwarzschild metric in R-spacetime and the motion on the quasi-circular orbits
In this part we consider the motion of a test point particle with mass m in the Schwarzschild metric in R-spacetime (4). To do that we use the Hamilton-Jacobi method. In General Relativity Theory a Hamilton-Jacobi equation can be represented as [5] g µν
∂S ∂S = m2 c2 ∂xµ ∂xν
So the Hamilton-Jacobi equation for a metric in R-spacetime is expressed as
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2 t2 ∂S ∂S 1 1 tr − t + 2 R2 r 2gt ∂r R R2 2gt ∂t 1− 1− rR rR
2 2gt ∂S 1− − rR ∂r
t2 − 2 2 R r
4
"
∂S ∂θ
2
1 + sin2 θ
∂S ∂ϕ
2 #
= m2 . (19)
π the Hamilton-Jacobi equation (19) is simplified 2 2 2 2 t2 ∂S 2gt 1 1 ∂S tr ∂S − t r 1 − − + R2 2gt ∂r R2 R2 rR ∂r 2gt ∂t 1 − 1− rR rR 2 t2 ∂S − 2 2 2 = m2 . (20) R r sin θ ∂ϕ
For the motion on the surface θ =
To solve this equation we use the method of separation of variables and represent the full action as a sum of functions of different variables r + Mϕ ϕ. (21) S = S1 (t) + S2 t When we put (21) in the equation (20) we can find S1 = and
−A t
v u u Z u A2 S2 = dxu t 2 R
Mϕ2 2 R2 + m 1 x2 , − xg xg 2 1− 1− x x where x ≡ r/t, xg ≡ 2g/R, A is a constant of variable separation. Finally the action is written out as v u Mϕ2 u 2 R2 + Z m u A2 −A 1 x2 + M ϕ. S= (22) + dxu 2 − ϕ t 2 x g x t R g 1− 1− x x Two equations of motion are derived from the derivative (22) of two constants A and Mϕ :
and
∂S = const ∂A
(23)
∂S = const. ∂Mϕ
(24)
From an equation (23) we can find the dependence r(t) Z dx A R v = , ! u t m 2 2 M A x xg u g ϕ t 1− − 1+ 2 2 2 1− x m2 m R x x
and an equation of trajectory can be found from the equation (24) Z Mϕ . ϕ = dx v ! u 2 2 uA M xg ϕ 1− + m2 R 2 x2 t 2 − R x2 x 5
(25)
We consider now a motion of the test particle on the almost circular orbit. To derive this we rewrite equation (25) in differential form −R p 2 1 dx 2 xg dt = At2 A − U (x), 1− x where v ! u u Mϕ2 xg 2t U (x) = mR 1+ 2 2 2 1− x m R x which play role of the effective potential. A motion on a circular orbit occurs under following conditions A = U,
dU = 0. dx
(26)
A solutions of the equation (26) are 1 Mϕ2 gt r± = 2 m2 g 2 R
1±
s
3m2 R2 x2g 1− Mϕ2
!
.
(27)
Easy to see that r+ corresponds to a stable “circular” orbit and r− corresponds to a nonstable orbit. Multiplier gt/R in the formula (27) has a sense of the Schwarzschild radius. In the limit r ≫ gt/R radicand in (27) is nearly one and radius become Mϕ2 gt r+ ≃ 2 2 . m g R If we put t = T + τ , where T is the Age of the Universe, then radius of the circular orbit will depend on τ : τ . (28) r+ (τ ) = r+ (0) 1 + T For example we consider the annual change of the radius of Moon orbit. Moon orbit radius rm ≃ 4 · 108 m, Age of the Univers T ≃ 5 · 1017 c, duration of the year τ ≃ 2, 5 · 108 c. From the equation (28) we get the annual increase in the radius of the orbit ∆r ≃ 2 mm/year. This increase is consistent with the observations (34 mm/year), which are explained by the tidal force and decreasing of the Earth angular momentum.
References [1] F. Kottler. Uber die physikalischen Grundlagen der Einsteinschen Gravitationstheorie. Ann. Phys. (Germany), 56 : pp. 401462, 1918. [2] S. N. Manida, Advanced physics course. Mechanics (in Russian), SPb, 99 pages, 2007. [3] S. N. Manida, TheorMathPhys, 2011, 169:2, pp. 323336 . [4] L. D. Landau, E. M. Lifshitz, Mechanics, Butterworth-Heinemann, 1976. [5] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, 1980.
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