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Tetrads in General Relativity

VIII. Schwarzschild solution

Schwarzschild Metric

   In January 1916, only a month after publication of Albert Einstein’s general theory of relativity, Karl Schwarzschild published an exact solution of the vacuum Einstein equations describing the gravitational field outside a spherical mass. Hypothesizing that the metric for this case must be static and spherically symmetric, Schwarzschild obtained the line element that reads, in present day notation:

8.1

\[  d{s^2} = {e^{2\Phi }}d{t^2} - {e^{2\lambda }}d{r^2} - {r^2}(d{\theta ^2} + {\sin ^2}\theta d{\phi ^2}) \]

The scalar parameters $\Phi ,\lambda $ are determined by the Einstein equations; see Section XI. The metric in the above form was independently arrived at by Johannes Droste, a PhD student of Lorentz, a few months later in 1916.

   From the Einstein equations, which are differential equations for the metric tensor ${g_{\mu \nu }}$, Schwarzschild obtained the metric components:

8.2

\[  {g_{tt}} = {e^{2\Phi }} = 1 - \frac{C}{r} = 1 - \frac{{2M}}{r}{\quad}{g_{rr}} = - {e^{2\lambda }} = - {e^{ - 2\Phi }}  \]

The constant $C$ is an integration constant, which is set to $C = 2M$ to be consistent with the Newtonian gravitational potential $\Phi = - M/r$ in the weak gravity limit ${e^{2\Phi }} \simeq 1 + 2\Phi $.

   The Schwarzschild metric is an idealized solution under the assumption that the mass $M$, creating the gravitational field, is a point at the origin of the coordinates. It is a useful approximation for describing slowly rotating astronomical objects such as stars and planets, including Earth and the Sun. It can also be visualized as all that remains of a star that underwent collapse in the past. But most famously nowadays is the recognition that the Schwarzschild solution is a model for the most basic type of black hole.