gwyn@BRL@sri-unix (11/18/82)
From: Doug Gwyn <gwyn@BRL>
For purposes of discussion it is useful to state more specifically
what the Schwarzschild solution is.
1) It is a STATIC,
2) spatially SPHERICALLY SYMMETRIC
3) solution to the ORIGINAL General-Relativistic field equations
4) (which implies PURE gravitational field),
5) meeting certain ASYMPTOTIC conditions (such as approximating
the Newtonian point-mass field at large distance).
Any one (or more) of the above conditions is susceptible to
change, generally breaking the Schwarzschild solution (sometimes
leading to another similar possible solution).
One thing to keep in mind is that General Relativity does not
completely determine solutions of its equations in any way
nearly as simple as the classical electromagnetic boundary-value
problems. In particular, space-time topology is a free variable.
One can "patch" together Schwarzschild solutions to obtain a
"wormhole", for example.
To my mind, the weakest area in popular discussion of black holes
is the idea of what is "inside" the Schwarzschild radius. One
should try not to project the asymptotically flat space-time metric
in toward the "interesting" region, because intuition will then be
misleading. The Schwarschild radius is not an "essential" singularity
in the metric, but an apparent one due to coordinate choice; Kruskal
coordinates are more convenient for some questions about this model.
Another common misconception is probably due to the standard
illustration showing a "generating mass" blob hovering centered
in the neck of a funnel, the latter representing "curved space"
and the funnel neck diameter the Schwarzschild radius. Actually
in this picture there should not be a blob outside the curved
surface. According to the classical unified field theorist, the
funnel is not generated by an object; the funnel IS the object.
(This can be seen in standard General Relativity if one writes
T = f(R)
rather than
R = f'(T)
where R and T are the curvature and mass tensors, respectively.)