Black Hole Spacetime

The black hole spacetime was first obtained by Karl Schwarzschild in 1916, just within a year of the original publication of Einstein's final gravitational theory. As such it represents some of the truly unique features of Einstein's geometric point of view about gravity. Here we discuss a few of the features of the Schwarzschild spacetime, focussing on those which show how Einstein's theory, taken to its limits, does indeed predict some interesting new features which are strictly absent from the Newtonina conception of gravitation.

We begin with the expression for the interval between spacetime events occurring at two slightly different locations.

[Graphics:schw.version3gr1.gif]

The above tells us how to find the interval between events labeled by points in the spacetime exterior to a black hole's special surface called the event horizon.

The Event Horizon

In Newtonian physics, space is Euclidean and time runs uniformly forward, there being no variation in the properties of space or time throughout the universe. Nevertheless we can still gain a preliminary view of the significance of an event horizon by first looking at the Newtonian 'event horizon'.

Consider an object of mass M, such as the earth or a planet. Suppose one were to pose the following question:

What is the minimum speed with which a rocket ship must have in order to escape the gravitational effect of the attracting object?

The speed which answers the above question is called the escape velocity. It turns out that the speed can be worked out by using the conservation of energy. We only give the result here:

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr3.gif]

This formula tells us that if the mass of the attracting object and its radius are given, then we can find the needed speed. If one puts in the mass of the earth for M and the radius of the earth for R, one finds that the escape velocity for the earth is 11 km/sec. For an initial speed smaller than this the rocket shot straight upward will rise to some maximum height, stop, and then fall back to the earth's surface.

Now we use this escape velocity idea in conjunction with a tenant of the special theory of relativity to arrive at our qualitative estimate for the size of an event horizon. Since we know that the biggest velocity that can be generated is the speed of light c, let's plug it into the above formula. One immediately sees that only at a special value of R
will the condition [Graphics:schw.version3gr4.gif] = c be satisfied. When we algebraically solve for this special value fro R we get

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr5.gif]

This result is a peculiar mix of Newtonian and Special Relativity and is strictly not valid. Nevertheless, it gives us a nice intiutive conception of just what the event horizon represents: the event horizon represents thesphere whic is at the smallest distance from an object of mass M for which one has a chance to escape by projecting straight out.This is the smallest such distance because to start from a distance closer than this would entail having an escape velocity larger than the speed of light...and we know nature does not allow such speeds.

Thus, if one somehow gets closer than the Schwarzschild radius, one is trapped! Not even light can get out!

Thus, the event horizon is a critical surface inside which one is doomed to a short life, soon being pulled to the center where the gravitational force becomes infinitely large.

Also, the surface of the Schwarzschild radius is black since no light emerges from it. Thus the name BLACK HOLE: stuff falls in, but nothing comes out!!

Schwarzschild Event Horizon

The Schwarzschild event horizon turns out to have an effective radius of the same form as we have for the 'Newtonian black hole'...

Note that when the value of the radius becomes the Schwarzschild radius, the
part of the interval between events that describes the contribution from the timelike part becomes zero. Note also that the contribution from the part which describes the spatial distance between nearby points becomes infinite...but the coordinate speed of light, which is defined to be ds^2 =0, becomes

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr7.gif]

One of the first things one should notice about this result is that the coordinate speed of light is different at different locations, unlike the Special Theory in which it is a universal constant. In the Schwarzschild spacetime, the speed is strictly zero at the event horizon and grows toward a value of c only as the point r gets closer to infinitely large values. Concerning the fact that the coordinate speed of light is zero at the Schwarzschild radius, the implication is that photons of light which begin exactly at the horizon, stay at the horizon because their local speed of light is zero. Thus light is doomed to stay at the Schwarzschild radius. Light which starts its outward trek at a r which is slightly greater than the Schwarzschild radius has an effective local speed of light which is very small, indicating that the effects of the geometry are a significant impediment to its outward motion. A final point: the coordinate speed of light we have defined is the one which approaches the 'ordinary' speed of light at infinity. Hence, only for observers at a very large distance from the black hole will this definition be relevant. For observers who are at rest at a given point and who will make measurements in local intertial frames of reference, they will find the local speed of light to be c! It is the observers who are far away who find the variation with position of the coordinate speed. So, there is both the dependence on the concept of speed on the local geometry as well as dependence upon the class of observers which make the measurements. All these different observers' characterization of the events can be related and a consistent picture emerges. It is one of the strengths of Einstein's conceptions that one has this tremendous leeway in which frame one uses and how one characterizes the events in a given frame [which set of numbers one uses to coordinatize the events is up to the observer].

Another interesting thing to consider is the effect of spatial curvature. From the interval between events, if we consider the proper distance between two events at the same temporal point but differing spatial locations r and r+ dr, we see that the spatial interval becomes:

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr10.gif]

Finally, we can get the proper time between events at the same spatial location by putting dr= 0 in the Schwarzschild spacetime metric. The result is:

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr11.gif]

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr12.gif]

Singularity at center of a Black Hole


As we emphasized, it is the curvature at a point of a spacetime which is the real indicator of a permanent gravitational field. For the Schwarzschild spacetime given above, the curvature can be shown to go as

[Graphics:schw.version3gr2.gif][Graphics:schw.version3gr13.gif]

Thus at the Schwarzschild radius the curvature is finite. However, at the origin r = 0, the curvature becomes infinite. This is a clear indication that something strange is going on at such a point, because the very geometrical object which describes the permanency of the gravitational field is unboundedly large at r = 0. Such a point is called a singularity and its existence in a spacetime is bad news. Nothing is well-defined at such a point.


The interesting thing about the Schwarzschild singularity is that it is hidden from an external observer's view by the event horizon! Is nature is hiding these bizarre points from us? If nature makes singularities which possess no event horizon, then the singular point would be visible. Since all known laws of nature would break down there, it is not even clear that we would have a universe which is predictable. This is because with naked singularities sitting around and visible, there is no predicting what would emerge from such points.

The Cosmic Censorship Conjecture

An interesting, but difficult question emerges: When a star collapses under its own weight does it always form a black hole with the attendant event horizon or can the singularity be formed without an event horizon? The conjecture is that black holes only form with event horizons. The problem is that this has not been proved to be true. It has been found to be true in all situations considered so far and is probably true...