Schwarzschild Spacetime

Here we will review some of the features of the Schwarzschild Spacetime. This spacetime constitutes the black hole and many of the interesting consequences of Einstein's theory of gravity show up in this model spacetime.

As we have emphasized, the goal of the Einstein theory is to give a four-dimensional characterization of the interval between events and to build into the geometric fabric of spacetime those properties of both special relativity and gravity. Now it turns out that Karl Schwarzschild found the interval between events for a single point mass in 1916, just one year after the publication of the theory in its final form in 1915.

The spacetime for a single point mass of mass M sitting at the origin of spherical coordinates is given by the following


The above tells us how to find the interval between events labeled by points which are nearby 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:


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


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 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 to have 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 local speed of light, which is defined to be:[Graphics:schw.version3gr6.gif] becomes


One of the first things to notice about the above result is that the speed of light is not a universal constant, but rather depends on where one is in relation to the massive object. As one makes the r coordinate larger and larger in comparison to the Schwarzschild radius, one sees that the value of the local speed of light gets larger, approaching the Special Relativity speed c when the point is a very large distance from the massive object.

A second thing to notice about this result is that the coordinates one has used to describe the propagation of light in this spacetime are not universally valid. When one tries to use the r coordinate for points closer than the Schwarzschild radius, one sees that the speed is negative. While one might think that this could mean that the light is inward pointing, it turns out that the coordinates used are not valid inside the Schwarzschild radius. One would have to construct another set of coordinates to cover the interior of the black hole event horizon. So, don't try to use these formulae inside the Schwarzschild radius.


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:



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


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! It
seems that 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.