Notes on Relativity and Cosmology - Physics Department, UCSB

9.3. BEYOND THE HORIZON 247
and the past interior should be much like the future interior. This part of the
spacetime is often called a ‘white hole’ as there is no way that any object can
remain inside: everything must pass outward into one of the exterior regions
through one of the horizons!
r = R s
r = 0
r = R s
r < R s
r > R s
r < R s
r > R s
r = R s
r = 0
r = R s
As we mentioned briefly with regard to the second exterior, the past interior
does not really exist for the common black holes found in nature. Let’s talk
about how this works. So far, we have been studying the pure Schwarzschild
solution. As we have discussed, it is only a valid solution in the region in which
no matter is present. Of course, a little bit of matter will not change the picture
much. However, if the matter is an important part of the story (for example,
if it is matter that causes the black hole to form in the first place), then the
modifications will be more important.
Let us notice that in fact the ‘hole’ (whether white or black) in the above spacetime
diagram has existed since infinitely far in the past. If the Schwarzschild
solution is to be used exactly, the hole (including the wormhole) must have been
created at the beginning of the universe. We expect that most black holes were
not created with the beginning of the universe, but instead formed later when
too much matter came too close together. Recall that a black hole must form
when, for example, too much thin gas gets clumped together.
Once the gas gets into a small enough region (smaller than its Schwarzschild
radius), we have seen that a horizon forms and the gas must shrink to a smaller
size. No finite force (and, in some sense, not even infinite force) can prevent the
gas from shrinking. Now, outside of the gas, the Schwarzschild solution should
be valid. So, let me draw a worldline on our Schwarzschild spacetime diagram
that represents the outside edge of the ball of gas. This breaks the diagram into
two pieces: an outside that correctly describes physics outside the gas, and an
inside that has no direct physical relevance and must be replaced by something
that depends on the details of the matter: