UDC 530.12

PHYSICS

K. P. STANYUKOVICH

EQUATION OF MOTION IN AN INTERNAL CENTRALLY SYMMETRIC FIELD IN GENERAL RELATIVITY

(Presented by Academician Ya. B. Zel’dovich, 27 XII 1967)

Let us consider a convenient form of the equations of motion in an internal centrally symmetric field in general relativity. As we shall see, the equations take their simplest form if we take \(r\) and the second coordinate \(m\), which plays the role of a Lagrangian coordinate, as the independent variables.

Let us take the metric of the field in the usual form

\[ -ds^2=-c^2dt^2 e^\nu+dr^2 e^\lambda+r^2(d\theta^2+\sin^2\theta\,d\varphi^2). \]

Then the basic equations obtained from the conservation laws,

\[ T^k_{i;k}=[(p+\varepsilon)u_i u^k+\delta_i^k p]_{;k}=[(p+\varepsilon)u_i u^k]_{;k}+p_{,i}=0 \]

will have the form \((^{1})\)

\[ \frac{1}{c^2\theta^2}[Au_t+uu_r] -\frac{\omega^2}{c^2}\left[(\ln v)_r+\frac{Au}{c^2}(\ln v)_t\right] = \frac{1}{2u}[A\lambda_t+u\lambda_r] +\frac{\theta^2T\sigma_r}{W}; \tag{1} \]

\[ -[A(\ln v)_t+u(\ln v)_r] +\frac{1}{\theta^2}\left(u_r+\frac{Au}{c^2}u_t\right) +\frac{2u}{r} = \frac{u}{2}\left(\lambda_r+\frac{Au}{c^2}\lambda_t\right); \tag{2} \]

\[ A\sigma_t+u\sigma_r=0. \tag{3} \]

Here \(A=e^{(\lambda-\nu)/2}\); \(u=A\,dr/dt\); \(\theta^2=1-u^2/c^2\); \(W=(p+\varepsilon)v\), \(u\) is the 3-velocity; \(p\) is the pressure; \(v\) is the specific volume; \(\varepsilon=\rho c^2\) is the energy density; \(W\) is the heat content; \(\omega^2/c^2=-(\partial\ln W/\partial\ln v)_\sigma\); \(\omega\) is the sound velocity; \(\sigma\) is the entropy; \(T\) is the temperature.

The field equations, obtained from the equations \(R_i^k-\frac12\delta_i^kR=\chi T_i^k\), have the form

\[ (re^{-\lambda})_r = 1-\frac{\chi r^2}{\theta^2}\left(\varepsilon+p\frac{u^2}{c^2}\right); \tag{4} \]

\[ Ar(e^{-\lambda})_t = \frac{\chi ur^2}{\theta^2}(\varepsilon+p); \tag{5} \]

\[ (1+rv_r)e^{-\lambda} = 1+\frac{\chi r^2}{\theta^2}\left(p+\varepsilon\frac{u^2}{c^2}\right); \tag{6} \]

\[ \frac12 e^{-\lambda} \left[ \left(v_{rr}+\frac12 v_r^2+\frac12(v-\lambda)_r-\frac{v_r\lambda_r}{2}\right) -\frac{A}{c^2}(A\lambda_t)_t \right] = \chi p. \tag{7} \]

There must be 5 independent equations in all. They determine \(p\), \(u\), \(\sigma\), \(\lambda\), and \(v\); at the same time, the equation of state of the medium \(p=p(\sigma,v)\) must be specified, and the thermodynamic equation \(d(\varepsilon v)=T\,d\sigma-p\,dv\) and the identity \(\partial(p,v)/\partial(T,\sigma)=1\) must be used.

After certain transformations, we write 2 independent equations of the system (4)–(7) in the form

\[ A\lambda_t+u\left(\lambda_r+(e^\lambda-1)/r+\chi pr e^\lambda\right)=0, \tag{8} \]

\[ A(1+u^2/c^2)\lambda_t+u(\lambda+v)_r=0. \tag{9} \]

Considering the system of equations (1), (2), (3), (8), we see that these equations do not contain \(v\). Consequently, it is necessary to investigate a system of only 4 equations.

Let us pass from the independent variables \((t;\ r)\) to the variables \((m;\ r)\), where

\[ \varkappa\left(\frac{\partial m}{\partial r}\right)_t = \varkappa\,\frac{r^2}{\theta^2}\left(\varepsilon+p\frac{u^2}{c^2}\right) = 1-(re^{-\lambda})_r, \tag{10} \]

whence

\[ m=\int_0^r \frac{r^2}{\theta^2}\left(\varepsilon+p\frac{u^2}{c^2}\right)dr = \frac{r}{\varkappa}(1-e^{-\lambda}). \tag{11} \]

Equation (18) in the variables \((m,r)\) takes the form

\[ \left(A-u\frac{\partial t}{\partial m}\right)\frac{\partial\lambda}{\partial m} + u\frac{\partial t}{\partial m} \left( \frac{\partial\lambda}{\partial r} + \frac{e^\lambda-1}{r} + \varkappa p r e^\lambda \right)=0. \tag{12} \]

Equation (10) takes the form

\[ \frac{\partial t}{\partial m}\, \frac{r^2}{\theta^2} \left(\varepsilon+p\frac{u^2}{c^2}\right) + \frac{\partial t}{\partial m}=0. \tag{13} \]

Equation (11), which we write in the form \(e^{-\lambda}=1-\varkappa m/r\), determines
\(\partial e^{-\lambda}/\partial m=-\varkappa/r\),
\(\partial e^{-\lambda}/\partial r=\varkappa m/r^2\), or

\[ \frac{\partial\lambda}{\partial m} = \frac{\varkappa}{r}e^\lambda, \qquad \frac{\partial\lambda}{\partial r} = -\frac{\varkappa m}{r^2}e^\lambda = -\frac{m}{r}\frac{\partial\lambda}{\partial m}, \tag{14} \]

therefore (12) passes into the equation

\[ A-u\frac{\partial t}{\partial r} + u\frac{\partial t}{\partial m}pr^2 =0. \tag{15} \]

Hence, from (13) we shall have

\[ u\,\partial t/\partial m = -A\theta^2/r^2(p+\varepsilon), \qquad u\,\partial t/\partial r = A(\varepsilon+pu^2/c^2)/(p+\varepsilon). \tag{16} \]

Equations (1), (2), (3) take, in the coordinates \((m,r)\), the form

\[ \frac{u}{c^2\theta^2}(u_r-pr^2u_m) - \frac{\omega^2}{c^4}\left[(\ln v)_r+\varepsilon r^2(\ln v)_m\right] = \]

\[ = \frac{1}{2}(\lambda_r-m^2\lambda_m) + \frac{T}{W} \left[ \theta^2\sigma_r + r^2\left(\varepsilon+p\frac{u^2}{c^2}\right)\sigma_m \right], \]

\[ -u\left[(\ln v)_r-pr^2(\ln v)_m\right] + \frac{1}{\theta^2}\left[u_r+\varepsilon r^2u_m\right] + \frac{2u}{r} = \frac{u}{2}(\lambda_r+\varepsilon r^2\lambda_m), \]

\[ \sigma_r-pr^2\sigma_m=0. \]

Transforming the equations, using (14), we shall have

\[ \frac{1}{2c^2\theta^2}\left[u_r^2-pr^2u_m^2\right] - \frac{\omega^2}{c^2}\left[(\ln v)_r+\varepsilon r^2(\ln v)_m\right] + \]

\[ + \frac{\varkappa m/r+pr^2}{2(r-\varkappa m)} = \frac{r^2T\sigma_m}{v} = \frac{T\sigma_r}{pv}; \tag{17} \]

\[ -\left[(\ln v)_r-pr^2(\ln v)_m\right] + \frac{1}{2\theta^2} \left[(\ln u^2)_r+\varepsilon r^2(\ln u^2)_m\right] + \]

\[ +\frac{2}{r} + \frac{\varkappa m/r-\varepsilon r^2}{2(r-\varkappa m)} =0. \tag{18} \]

As a result we have arrived at a system of only 3 quasilinear equations of the first order.

For isentropic processes, when \(\sigma=\mathrm{const}\), the problem reduces to a system of 2 equations.

From equation (16), eliminating \(A\), we shall have

\[ \frac{r^2}{\theta^2} \left(\varepsilon+p\frac{u^2}{c^2}\right) \frac{\partial t}{\partial m} + \frac{\partial t}{\partial r} =0; \tag{19} \]

knowing \(\varepsilon=\varepsilon(r,m)\), \(u=u(r,m)\) (for a given equation of state), one can (formally) find \(t=t(r,m)\), which as a result (again formally) makes it possible to determine \(u=u(t,r)\), \(\varepsilon=\varepsilon(t,r)\), and, finally, from any

from equation (16) we find

\[ A=e^{(\lambda-\nu)/2}=-\frac{r^2}{\theta^2}(\varepsilon+p)u\,\frac{\partial t}{\partial m}; \quad \text{since} \quad e^\lambda=\frac{1}{1-\chi m/r}, \quad \text{then} \]

\[ e^{\nu/2}= \frac{1}{\sqrt{1-\dfrac{\chi m}{r}}}\, \frac{A^2}{r^2 u\,\dfrac{\partial t}{\partial m}(p+\varepsilon)}, \tag{20} \]

which completely solves the self-consistent problem of finding \(u,\varepsilon,\lambda,\nu\) for centrally symmetric motions.

Let us now write the characteristic equations of the system of equations (17), (18). Along the lines

\[ m'+pr^2=0,\qquad \sigma'=0; \tag{21} \]

here, for example, \(\sigma'=d\sigma/dr,\ u'=du/dr,\ m'=dm/dr,\ (\ln\nu)'=d\ln\nu/dr\), etc. Further, expanding the corresponding determinants, we find that along the lines

\[ (\varepsilon r^2-m')=\pm \frac{u}{\omega}(pr^2+m') \tag{22} \]

the relations

\[ (pr^2+m')\left[ \left(\frac{\omega}{c}\pm\frac{u}{c}\right) \left(\frac{\omega}{c}(\ln\nu)'\mp\frac{u'}{c\theta^2}\right) \right]\mp \]

\[ \mp\frac{2u\omega}{c^2r} -\frac{\chi}{2(r-\chi m)} \left[ \frac{m}{r}\left(1\pm\frac{u\omega}{c^2}\right) +r^2\left(p\mp\frac{u\omega}{c^2}\varepsilon\right) \right] = \frac{\sigma'Tr^2}{\nu}. \tag{23} \]

In the case \(p=0\), the basic equations (17) and (18) are greatly simplified and are integrated directly (Tolman’s problem \(({}^2)\)).

The system of equations (1), (2), (3), and (8) also has a solution for the ultrarelativistic approximation, when

\[ u/c=1-2\Delta,\qquad \Delta\ll 1. \]

In this case, if terms of order \(\Delta^2\) are neglected, the equations take the form

\[ \frac{1}{2\Delta}\,[A\Delta_\tau+\Delta_r] +\frac{\omega_0^2}{c^2} \left[A(\ln\nu)_\tau+(\ln\nu)_r+\frac{1}{2}[A\lambda_\tau+\lambda_r]\right]=0, \]

\[ [A(\ln\nu)_\tau+(\ln\nu)_r] +\frac{1}{2\Delta}(A\Delta_\tau+\Delta_r) -\frac{2}{r} +\frac{1}{2}[A\lambda_\tau+\lambda_r]=0, \tag{24} \]

\[ A\sigma_\tau+\sigma_r=0,\qquad A\lambda_\tau+\lambda_r+\frac{e^\lambda-1}{r}+\chi pr e^\lambda=0, \]

where \(\tau=ct,\ \omega_0^2/c^2=k-1\).

It is more convenient to write the first two equations of this system in the form

\[ A(\ln\nu)_\tau+(\ln\nu)_r=\frac{2}{(2-k)r}, \]

\[ \frac{1}{2\Delta}[A\Delta_\tau+\Delta_r] +\frac{2(k-1)}{(2-k)r} = \frac{1}{2}\left[\frac{e^\lambda-1}{r}+\chi pr e^\lambda\right]. \tag{25} \]

Let us pass to the independent variables \((\nu,r)\); then the last two equations of system (24) and the second equation (25), after eliminating the quantity

\[ A-\frac{\partial t}{\partial r} = \frac{2\nu}{(2-k)r}\,\frac{\partial t}{\partial \nu} \tag{26} \]

(the first equation (25)), take the form

\[ \frac{2\nu}{2-k}\frac{\partial\sigma}{\partial\nu} +r\frac{\partial\sigma}{\partial r}=0,\qquad \frac{2\nu}{2-k}\frac{\partial\lambda}{\partial\nu} +r\frac{\partial\lambda}{\partial r} +e^\lambda-1+\chi pr^2 e^\lambda=0, \]

\[ \frac{1}{2\Delta} \left[ \frac{2\nu}{2-k}\frac{\partial\Delta}{\partial\nu} +r\frac{\partial\Delta}{\partial r} \right] +\frac{2(k-1)}{2-k} = \frac{1}{2}\left[e^\lambda-1+\chi pr^2e^\lambda\right]. \tag{27} \]

We now transform equation (5) to the coordinates \((\nu,r)\); as a result we shall have

\[ \frac{\partial\tau}{\partial\nu} \left[ r\frac{\partial\lambda}{\partial r} +e^\lambda-1 -\frac{\chi r^2}{4\Delta}(\varepsilon+p)e^\lambda \right] - r\frac{\partial\tau}{\partial r}\frac{\partial\lambda}{\partial\nu}=0. \tag{28} \]

The solution of the resulting system of equations is carried out as follows. From the first equation of system (27) we obtain

\[ \sigma=F_1(r,\nu^{-(2-k)/2})=\mathrm{const}\cdot p\nu^k . \tag{29} \]

Next we solve the second equation (29) and determine

\[ (1-e^{-\lambda})=\frac{1}{r}F_2(r\nu^{-(2-k)/2})+\frac{2-k}{5k-6}\varkappa r^2p . \tag{30} \]

Further, from the last equation (27) we find

\[ \Delta=e^{-\lambda}F_3(r\nu^{-(2-k)/2})r^{-4(k-1)/(2-k)} . \tag{31} \]

From equation (28) we find \(\tau=\tau(r,\nu)\), and, finally, equation (26) determines \(v=v(r,\nu)\), since

\[ A=e^{(\lambda-\nu)/2}=\frac{\partial \tau}{\partial r} +\frac{\partial \tau}{\partial \nu}\frac{2v}{(2-k)r}, \quad\text{whence}\quad e^{-\nu/2}=e^{-\lambda/2}\frac{1}{r} \left[ r\frac{\partial \tau}{\partial r} +\frac{\partial \tau}{\partial \nu}\frac{2v}{(2-k)} \right]. \]

Detailed calculations here are meaningful only when solving particular problems with already specified functions \(F_1,F_2,F_3\). As a result we obtain exact asymptotic solutions depending on 5 arbitrary functions.

In conclusion, let us make limiting transitions. For \(\varkappa=0\) we shall have the equations

\[ \frac{1}{2c^2\theta^2}[u_r^2-pr^2u_m^2] -\frac{\omega^2}{c^2}\{(\ln \nu)_r+\varepsilon r^2(\ln \nu)_m\} =\frac{r^2T\sigma_m}{\nu} =\frac{T\sigma_r}{p\nu}; \tag{32} \]

\[ -[(\ln \nu)_r-pr^2(\ln \nu)_m] +\frac{1}{2\theta^2u^2}[u_r^2+\varepsilon r^2u_m^2] +\frac{2}{r}=0, \tag{33} \]

which corresponds to special relativity. For \(u/c\ll 1\) we obtain the limiting transition to Newton’s theory. Indeed, since in this case \(\theta=1\), \(\nu\rho=1\), \(u\,\partial t/\partial r=1\), \(u\,\partial t/\partial M_0=-1/4\pi\rho r^2\), we shall have\(^3\)

\[ uu_r+\omega^2[(\ln \rho)_r+4\pi\rho r^2(\ln \rho)_{M_0}] +GM_0/r=4\pi r^2\rho T\sigma_{M_0}, \tag{34} \]

\[ u(\ln \rho)_r+u_r+4\pi\rho r^2u_{M_0}+2u/r=0, \quad \sigma_r=0. \tag{35} \]

The equations introduced by us are the simplest for analyzing spherically symmetric motions in the general theory of relativity.

Received
25 XII 1967

CITED LITERATURE

\(^1\) S. M. Kolesnikov, K. P. Stanyukovich, PMM, 29, no. 4 (1965). \(^2\) R. C. Tolman, Relativity, Thermodynamics and Cosmology, Oxford, 1934. \(^3\) F. A. Baum, S. A. Kaplan, K. P. Stanyukovich, Introduction to Cosmic Gas Dynamics, ch. VIII, § 23, Moscow, 1958.