Dr. Gyula Bene Department for Theoretical Physics, Loránd Eötvös University Pázmány Péter sétány 1/A, 1117 Budapest 9. week
Spherically symmetric gravitational field
Static gravitational field. Gravitational redshift. Synchronized reference frames. Gravitational field of massive objects.
Spherically symmetric gravitational field. Schwarzschild metric. Motion in a spherically symmetric gravitational field.
Event horizon. Gravitational collapse.
Gravitational field of massive objects.
Spherically symmetric gravitational field. Schwarzschild metric.
Spherical symmetry: space-time metric is the same at points located the same distance from the center.
Most general Spherically symmetric expression for the interval:
\[\begin{align}
ds^2=h(r,t)dr^2+k(r,t)(\sin^2\Theta d\varphi^2 +d\Theta^2)+l(r,t)dt^2+a(r,t)dr dt
\end{align}\]
Transformations of type
\(r=f_1(r',t'),\; t=f_2(r',t')\)
preserve spherical symmetry. One may achieve that \(a(r,t)=0\) and \(k(r,t)=-r^2\).
Thus, not hurting generality of the treatment, one may write
\[\begin{align}
ds^2={\rm e}^\nu c^2 dt^2-r^2(d\Theta^2+\sin^2\Theta d\varphi^2)-{\rm e}^\lambda dr^2
\end{align}\]
With notation
\(x^0=ct,\;x^1=r,\;x^2=\Theta,\;x^3=\varphi\)
nonzero components of the metric tensor read
\[\begin{align}
g_{00}={\rm e}^\nu,\;g_{11}=-{\rm e}^\lambda,\;g_{22}=-r^2,\;g_{33}=-r^2\sin^2\Theta\;,
\end{align}\]
while components of the contravariant metric tensor are
\[\begin{align}
g^{00}={\rm e}^{-\nu},\;g^{11}=-{\rm e}^{-\lambda},\;g^{22}=-r^{-2},\;g^{33}=-r^{-2}\sin^{-2}\Theta\;.
\end{align}\]
Nonzero Christoffel symbols:
\[\begin{align}
\Gamma^1_{11}=\frac{\lambda'}{2},\;\Gamma^0_{10}=\Gamma^0_{01}=\frac{\nu'}{2},\;
\Gamma^2_{33}=-\sin \Theta \cos \Theta
\end{align}\]
\[\begin{align}
\Gamma^0_{11}=\frac{\dot \lambda}{2}{\rm e}^{\lambda-\nu},\;\Gamma^1_{22}=-r{\rm e}^{-\lambda},\;
\Gamma^1_{00}=\frac{\nu'}{2}{\rm e}^{\nu-\lambda}
\end{align}\]
\[\begin{align}
\Gamma^2_{12}=\Gamma^2_{21}=\Gamma^3_{13}=\Gamma^3_{31}=\frac{1}{r},\;\Gamma^3_{23}=\Gamma^3_{32}=\coth \Theta,\;
\Gamma^0_{00}=\frac{\dot \nu}{2}
\end{align}\]
\[\begin{align}
\Gamma^1_{10}=\Gamma^1_{01}=\frac{\dot \lambda}{2},\;
\Gamma^1_{33}=-r\sin^2 \Theta {\rm e}^{-\lambda}
\end{align}\]
(here
\(\nu'=\partial \nu/ \partial r\)
and
\(\dot \nu =\partial\nu / \partial t\)
)
Einstein's equations in case of spherical symmetry:
\[\begin{align}
-{\rm e}^{-\lambda}\left(\frac{\nu'}{r}+\frac{1}{r^2}\right)+\frac{1}{r^2}=\frac{8\pi k}{c^4}T^1_1
\end{align}\]
\[\begin{align}
-\frac{1}{2}{\rm e}^{-\lambda}\left(\nu''+\frac{\nu'^2}{2}+\frac{\nu'-\lambda'}{r}-\frac{\nu'\lambda'}{2}\right)
+\frac{1}{2}{\rm e}^{-\nu}\left(\ddot \lambda +\frac{\dot \lambda^2}{2}-\frac{\dot \nu \dot \lambda}{2}\right)
=\frac{8\pi k}{c^4}T^2_2=\frac{8\pi k}{c^4}T^3_3
\end{align}\]
\[\begin{align}
-{\rm e}^{-\lambda}\left(\frac{1}{r^2}-\frac{\lambda'}{r}\right)+\frac{1}{r^2}=\frac{8\pi k}{c^4}T^0_0
\end{align}\]
\[\begin{align}
-{\rm e}^{-\lambda}\frac{\dot \lambda}{r}
=\frac{8\pi k}{c^4}T^1_0
\end{align}\]
In vacuum, outside of a spherically symmetric mass distribution there are only three independent equations:
\[\begin{align}
{\rm e}^{-\lambda}\left(\frac{\nu'}{r}+\frac{1}{r^2}\right)-\frac{1}{r^2}=0
\end{align}\]
\[\begin{align}
{\rm e}^{-\lambda}\left(\frac{\lambda'}{r}-\frac{1}{r^2}\right)+\frac{1}{r^2}=0
\end{align}\]
\[\begin{align}
\dot \lambda=0
\end{align}\]
Solution of the vacuum equations:
\(\lambda\) does not depend on time.
\(\lambda+\nu=F(t)\), where \(F(t)\) may be set to zero by suitable transform of time, \(t=f(t')\).
Finally
\[\begin{align}
{\rm e}^{-\lambda}={\rm e}^{\nu}=1+\frac{\rm const.}{r}
\end{align}\]
At large distances (weak field) we have
\(g_{00}=1+\frac{2\varphi}{c^2}=1-\frac{2k M }{c^2 r}\), hence
\[\begin{align}
{\rm const.}=-r_g=-\frac{2k M }{c^2}
\end{align}\]
Approximation at large distances:
\[\begin{align}
ds^2=ds_0^2-\frac{2k M }{c^2 r}\left(dr^2+c^2dt^2\right)
\end{align}\]
Eintein's equations in the presence of a spherical mass distribution imply
\[\begin{align}
{\rm e}^{-\lambda}=1-\frac{8\pi k}{c^4 r}\int_0^a T^0_0 r^2 dr=1-\frac{2k M }{c^2 r}
\end{align}\]
\(\rightarrow\)
\[\begin{align}
M=\frac{4\pi}{c^2}\int_0^a T^0_0 r^2 dr
\end{align}\]
(gravitational mass defect)
Motion in spherically symmetric gravitational field}
Lagrangian of a point mass:
\[\begin{align}
L=-mc\frac{ds}{dt}=-mc^2\sqrt{1-\frac{r_g}{r}-\frac{\dot{r}^2}{c^2\left(1-\frac{r_g}{r}\right)}-\frac{r^2\dot{\Theta}^2}{c^2}-\frac{r^2\sin^2\Theta \dot{\varphi}^2}{c^2}}
\end{align}\]
\(m\) stands for the mass of the particle.
Suppose that axis \(z\) is parallel with the angular momentum. Then motion takes place in the
\(xy\) plane, hence \(\Theta=\frac{\pi}{2}\), \(\dot{\Theta}=0\). Then
\[\begin{align}
L=-mc^2\sqrt{1-\frac{r_g}{r}-\frac{\dot{r}^2}{c^2\left(1-\frac{r_g}{r}\right)}-\frac{r^2 \dot{\varphi}^2}{c^2}}
\end{align}\]
Conserved quantities:
Angular momentum:
\[\begin{align}
J=\frac{\partial L}{\partial \dot{\varphi}}=\frac{m r^2 \dot{\varphi}}{\sqrt{1-\frac{r_g}{r}-\frac{\dot{r}^2}{c^2\left(1-\frac{r_g}{r}\right)}-\frac{r^2 \dot{\varphi}^2}{c^2}}}
\end{align}\]
This implies
\[\begin{align}
\frac{r^2}{c^2}\dot{\varphi}^2=\frac{\frac{J^2}{m^2c^2r^2}}{1+\frac{J^2}{m^2c^2r^2}}\left(1-\frac{r_g}{r}-\frac{\dot{r}^2}{c^2\left(1-\frac{r_g}{r}\right)}\right)
\end{align}\]
Hence we get
\[\begin{align}
J=\frac{m r^2 \dot{\varphi}\sqrt{1+\frac{J^2}{m^2c^2r^2}}}{\sqrt{1-\frac{r_g}{r}-\frac{\dot{r}^2}{c^2\left(1-\frac{r_g}{r}\right)}}}
\end{align}\]
Energy:
\[\begin{align}
E=\dot{r}\frac{\partial L}{\partial \dot{r}}+\dot{\varphi}\frac{\partial L}{\partial \dot{\varphi}}-L=\frac{m c^2 \left(1-\frac{r_g}{r}\right)\sqrt{1+\frac{J^2}{m^2c^2r^2}}}{\sqrt{1-\frac{r_g}{r}-\frac{\dot{r}^2}{c^2\left(1-\frac{r_g}{r}\right)}}}
\end{align}\]
Time dependence and orbit:
\[\begin{align}
\dot{\varphi}&=\frac{Jc^2}{E}\frac{1}{r^2}\left(1-\frac{r_g}{r}\right)\\
\dot{r}&=c\left(1-\frac{r_g}{r}\right)\sqrt{1-\left(\frac{mc^2}{E}\right)^2\left(1-\frac{r_g}{r}\right)\left(1+\frac{J^2}{m^2c^2r^2}\right)}
\end{align}\]
These imply:
\[\begin{align}
ct&=\frac{E}{mc^2}\int
\frac{dr}{\left(1-\frac{r_g}{r}\right)\sqrt{\left(\frac{E}{mc^2}\right)^2-\left(1+\frac{J^2}{m^2c^2r^2}\right)\left(1-\frac{r_g}{r}\right)}} \\&=\int
\frac{dr}{\left(1-\frac{r_g}{r}\right)\sqrt{1-\left(\left(\frac{mc^2}{E}\right)^2+\left(\frac{Jc}{E}\right)^2\frac{1}{r^2}\right)\left(1-\frac{r_g}{r}\right)}}\\
\varphi&=\int \frac{J\;dr}{r^2\sqrt{\left(\frac{E}{c}\right)^2-\left(m^2c^2+\frac{J^2}{r^2}\right)\left(1-\frac{r_g}{r}\right)}} \\&=\int \frac{dr}{r^2\sqrt{\left(\frac{E}{Jc}\right)^2-\left(\left(\frac{mc}{J}\right)^2+\frac{1}{r^2}\right)\left(1-\frac{r_g}{r}\right)}}
\end{align}\]
In case of light rays \(m=0\), \(\frac{Jc}{E}=\rho\), where \(\rho\) stands for the impact parameter (a light ray would pass at distance \(\rho\) near the center in the absence of gravity). Thus
\[\begin{align}
ct&=\int
\frac{dr}{\left(1-\frac{r_g}{r}\right)\sqrt{1-\frac{\rho^2}{r^2}\left(1-\frac{r_g}{r}\right)}}\\
\varphi&=\int
\frac{dr}{r^2\sqrt{\frac{1}{\rho^2}-\frac{1}{r^2}\left(1-\frac{r_g}{r}\right)}}
\end{align}\]
Gravitational collapse
Singularity of Schwarzschild's metric at \(r=r_g\) does not imply singularity of space-time. It means only that for
\(r\le\;r_g\) a rigid coordinate frame
\(r,\;\Theta,\;\varphi\)
cannot be accomplished with real physical objects.
Coordinate transform:
\[\begin{align}
c\tau=\pm c t\pm \int \frac{f(r)dr}{1-\frac{r_g}{r}},\;R=ct+\int \frac{dr}{\left(1-\frac{r_g}{r}\right)f(r)}
\end{align}\]
Interval in new coordinates:
\[\begin{align}
ds^2=\frac{1-\frac{r_g}{r}}{1-f^2}\left(c^2d\tau^2-f^2dR^2\right)-r^2(d\Theta^2+\sin^2\Theta d\varphi^2)
\end{align}\]
By choosing
\(f(r)=\sqrt{r_g/r}\), singularity disappears, and synchronized coordinate system is obtained:
\[\begin{align}
R-c\tau=\int \frac{(1-f^2)dr}{\left(1-\frac{r_g}{r}\right)f}=\frac{2}{3}\frac{r^{3/2}}{r_g^{1/2}}
\end{align}\]
\[\begin{align}
r=\left[\frac{3}{2}(R-c\tau)\right]^{2/3}r_g^{1/3}
\end{align}\]
\[\begin{align}
ds^2=c^2d\tau^2-\frac{dR^2}{\left[\frac{3}{2r_g}(R-c\tau)\right]^{2/3}}
-\left[\frac{3}{2}(R-c\tau)\right]^{4/3}r_g^{2/3}(d\Theta^2+\sin^2\Theta d\varphi^2)
\end{align}\]
Schwarzschild's sphere (\(r=r_g\)):
\[\begin{align}
\frac{3}{2}(R-c\tau)=r_g
\end{align}\]
Radial motion as observed from a distant outside observer:
\[\begin{align}J=0\;,\quad E_0=mc^2\sqrt{1-\frac{r_g}{r_0}}\end{align}\]
\[\begin{align}
c(t-t_0)=\sqrt{1-\frac{r_g}{r_0}}\int_{r}^{r_0}\frac{dr}{\left(1-\frac{r_g}{r}\right)\sqrt{\frac{r_g}{r}-\frac{r_g}{r_0}}}
\end{align}\]
An object falling radially needs infinite time to reach Schwarzschild's radius (event horizon):
\[\begin{align}
r-r_g=const.\times {\rm e}^{-\frac{ct}{r_g}}
\end{align}\]
Measured in proper time of the falling object, it crosses the event horizon in a finite time and falls in finite time into the center:
\[\begin{align}
\tau-\tau_0=\frac{1}{c}\int\left(\frac{r_g}{r}-\frac{r_g}{r_0}\right)^{-1/2}dr
\end{align}\]
bene@arpad.elte.hu
