[Relativity FAQ] - [Copyright]

updated 02-FEB-1995 by MM

original by Matt McIrvin

In 1916, when general relativity was new, Karl Schwarzschild worked
out a useful solution to the Einstein equation describing the evolution of
spacetime geometry. This solution, a possible shape of spacetime, would
describe the effects of gravity *outside* a spherically symmetric,
uncharged, nonrotating object (and would serve approximately to describe
even slowly rotating objects like the Earth or Sun). It worked in much the
same way that you can treat the Earth as a point mass for purposes of
Newtonian gravity if all you want to do is describe gravity *outside* the
Earth's surface.

What such a solution really looks like is a "metric," which is a kind of generalization of the Pythagorean formula that gives the length of a line segment in the plane. The metric is a formula that may be used to obtain the "length" of a curve in spacetime. In the case of a curve corresponding to the motion of an object as time passes (a "timelike worldline,") the "length" computed by the metric is actually the elapsed time experienced by an object with that motion. The actual formula depends on the coordinates chosen in which to express things, but it may be transformed into various coordinate systems without affecting anything physical, like the spacetime curvature. Schwarzschild expressed his metric in terms of coordinates which, at large distances from the object, resembled spherical coordinates with an extra coordinate t for time. Another coordinate, called r, functioned as a radial coordinate at large distances; out there it just gave the distance to the massive object.

Now, at small radii, the solution began to act strangely. There
was a "singularity" at the center, r=0, where the curvature of spacetime
was infinite. Surrounding that was a region where the "radial" direction
of decreasing r was actually a direction in *time* rather than in space.
Anything in that region, including light, would be obligated to fall toward
the singularity, to be crushed as tidal forces diverged. This was separated
from the rest of the universe by a place where Schwarzschild's coordinates
blew up, though nothing was wrong with the curvature of spacetime there.
(This was called the Schwarzschild radius. Later, other coordinate systems
were discovered in which the blow-up didn't happen; it was an artifact of
the coordinates, a little like the problem of defining the longitude of the
North Pole. The physically important thing about the Schwarzschild radius
was not the coordinate problem, but the fact that within it the direction
into the hole became a direction in time.)

Nobody really worried about this at the time, because there was no known object that was dense enough for that inner region to actually be outside it, so for all known cases, this odd part of the solution would not apply. Arthur Stanley Eddington considered the possibility of a dying star collapsing to such a density, but rejected it as aesthetically unpleasant and proposed that some new physics must intervene. In 1939, Oppenheimer and Snyder finally took seriously the possibility that stars a few times more massive than the sun might be doomed to collapse to such a state at the end of their lives.

Once the star gets smaller than the place where Schwarzschild's coordinates fail (called the Schwarzschild radius for an uncharged, nonrotating object, or the event horizon) there's no way it can avoid collapsing further. It has to collapse all the way to a singularity for the same reason that you can't keep from moving into the future! Nothing else that goes into that region afterward can avoid it either, at least in this simple case. The event horizon is a point of no return.

In 1971 John Archibald Wheeler named such a thing a black hole, since light could not escape from it. Astronomers have many candidate objects they think are probably black holes, on the basis of several kinds of evidence (typically they are dark objects whose large mass can be deduced from their gravitational effects on other objects, and which sometimes emit X-rays, presumably from infalling matter). But the properties of black holes I'll talk about here are entirely theoretical. They're based on general relativity, which is a theory that seems supported by available evidence.