Schwarzschild solution

This solution is the ``corner stone'' of GR. It is very simple (at least looks like), was found almost immediately after the GR had been formulated, was used for the first (classical) experimental tests of GR and is mentioned almost in every paper concerning gravity (up to the most recent ones).

It has 4 symmetries:

1 translation	( $t \rightarrow t+const$; ``time-independence''),
3 rotations	(keeping $t$ and $r$ unchanged; ``spherical symmetry'').

It is most conveniently written in the spherical coordinates, $x^{\mu}=(t,r,\theta,\phi)$(or, more correctly, ``schwarzschild coordinates'', because time coordinate is included), in which it takes the well-known form

$$ds^2 = -(1-\frac{2M}{r})dt^2+\frac{dr^2}{(1-\displaystyle\frac{2M}{r})}+r^2d\Omega^2.$$

It is interpreted as a gravitational field produced by some spherically symmetrical configuration of matter of total mass $M$. In solving the equations this $M$ arises as a constant of integration. It is given the meaning of mass because in the region where gravitational field is weak (i.e. where metric is approximately that of Minkowski space-time) the $dtdt$-component of metric is related to the Newtonian gravitational potential and we have:

$$-g_{00} = 1-\frac{2MG}{rc^2} \approx 1 + \frac{2\phi}{c^2} \Rightarrow \phi(r) = - \frac{GM}{r},$$

as it should be for the field produced by mass $M$ (I am using $G=c=1$ everywhere else).

Because the Newton's gravitational law is recovered, it is not a surprise that investigation of motion along geodesics in Schwarzschild space-time gives the known planetary motion in the ``0'th order''. But more than that, the next order corrections were found to provide the missing in Newtonian mechanics $43^{\prime\prime}$ for Mercury perihelion shift (the observed shift is $574^{\prime\prime}$ per century, from which $531^{\prime\prime}$ come from interaction with other planets in the solar system; the discrepancy was obvious, but could not be explained for a long time) and to predict (correctly) the bending of light by the Sun. Not bad for such a simple-looking metric!

The metric behaves badly at $r=2M$, but it is only a coordinate singularity. Indeed, curvature tensor for this solution is such that $R_{\mu\nu\alpha\beta} \not= 0$, but $R=0$, $RR=0$ and only $RK = \displaystyle\frac{48M^2}{r^6}$. Thus the only ``place'' where curvature becomes singular is at $r=0$. People also explicitly found transformations to other coordinates (e.g. to the Kruskal-Szekeres coordinates), which make the metric manifestly non-singular at the $r=2M$ hypersurface. But it should be noted, that the only way to get rid of a coordinate singularity is to make a singular coordinate transformation.

And still this hypersurface plays very important role. It is what is now called the ``event horizon'', because of its ``one-way-membrane'' property: all time-like geodesics (along which material objects can move) may cross it only once, coming from exterior ($r>2M$), so that anything which passes through it has no chance to come back to ``this world''. Thus, this solution describes a ``black hole'': not even light (and therefore any information) can get out from ``under the horizon''; the only way to observe it is via some indirect gravitational effects. And this name is also correct from a different perspective: all this ``BH physics'' is really a ``hole'' in our understanding of Nature.

It is not simple to understand physically ``interior'' of the BH (the region in space-time, corresponding to $r<2M$ in Schwarzschild coordinates). The most important point to notice is that there $t$ is no longer a ``coordinate time'' and $r$ is no longer a ``radial coordinate'' ($dtdt$ and $drdr$ components of the metric flip their signs which in a sense flips the meaning of t and r). The symmetries are still present in the interior, but $t$-independence cannot be interpreted as ``staticity'' and ``spherical symmetry'' is not really a spherical symmetry. Finally, the curvature singularity at $r=0$ is called a ``central singularity'', but it is not a singularity ``at the origin'' (see [1, p.154] for clarification of this point).