The Schwarzschild Metric

Schwarzschild solved the Einstein equations under the assumption of spherical symmetry in 1915, two years after their publication. This in itself is a good indication that the equations of General Relativity are a good deal more complicated than Electromagnetism.

The most obvious spherically symmetric problem is that of a point mass. The mass curves space-time and thus affects the particles moving nearby. The metric tensor in Schwarzschild (spherical coordinates becomes

\bgroup\color{black}$\displaystyle g_{\mu\nu}=\begin{pmatrix}-\left(1 - \frac{r_...
...r_s}{r} \right)} &0&0\cr 0&0&r^2&0\cr0&0&0&r^2\sin^2\theta\end{pmatrix} $\egroup

and the space-time interval in spherical coordinates in the Schwarzschild solution is.
\bgroup\color{black}$ \displaystyle {ds}^{2} =-c^2{d\tau}^2
= -\left(1 - \frac{r...
...\frac{r_s}{r}} + r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)$\egroup
where \bgroup\color{black}$ \tau$\egroup is the proper time (time measured by a clock moving with the particle) in seconds, \bgroup\color{black}$ t$\egroup is the coordinate time (measured by a stationary clock at infinity) in seconds, \bgroup\color{black}$ r$\egroup is the radial coordinate (circumference of a circle centered on the star divided by \bgroup\color{black}$ 2\pi$\egroup) in meters, \bgroup\color{black}$ \theta$\egroup is the polar angle, \bgroup\color{black}$ \varphi$\egroup is the azimuthal angle, and \bgroup\color{black}$ r_s$\egroup is the Schwarzschild radius (in meters) of the massive body, which is related to its mass \bgroup\color{black}$ M$\egroup by.
\bgroup\color{black}$ \displaystyle r_{s} = \frac{2GM}{c^{2}} $\egroup

This goes to the normal flat Minkowski space-time interval (in spherical coordinates) for \bgroup\color{black}$ r\rightarrow\infty$\egroup or for zero mass \bgroup\color{black}$ M$\egroup.

The Schwarzschild radius for normal planets and stars is much smaller than the actual size of the object so the Schwarzschild solution is only valid outside the object. For black holes, the Schwarzschild radius is the horizon inside of which nothing can escape the black hole.

Jim Branson 2012-10-21