Notes on Relativity and Cosmology - Physics Department, UCSB

240 CHAPTER 9. BLACK HOLES
r < R s
r = R s
t
r = R s t
r = R s
r > R s
The coordinate t increases along these lines, in the direction indicated by the
arrows. This means that the t-direction is actually spacelike inside the black
hole. The point here is not that something screwy is going on with time inside
a black hole. Instead, it is merely that using the Schwarzschild metric in the
way that we have written it we have done something ‘silly’ and labelled a space
direction t. The problem is in our notation, not the spacetime geometry.
Let us fix this by changing notation when we are in this upper region. We
introduce t ′ = r and r ′ = t. The metric then takes the form
ds 2 = −(1 − R s /t ′ )dr ′2 +
dt ′2
1 − R s /t ′ . (9.17)
You might wonder if the Schwarzschild metric is still valid in a region where the
t direction is spacelike. It turns out that it is. Unfortunately, we were not able
to discuss the Einstein equations in detail. If we had done so, however, then
we could check this by directly plugging the Schwarzschild metric into equation
(8.15) just as we would to check that the Schwarzschild metric is a solution
outside the horizon.
Finally, notice that the lines above look just the like lines we drew to describe
the boost symmetry of Minkowski space associated with the change of reference
frames. In the same way, these lines represent a symmetry of the black hole
spacetime. After all, the lines represent the direction of increasing t = r ′ . But,
the Schwarzschild metric is completely independent of t = r ′ – it depends only
on r = t ′ ! So, sliding events along these lines and increasing their value of
t = r ′ does not change the spacetime in any way. Outside of the horizon, this
operation moves events in time. As a result, the fact that it is a symmetry
says that the black hole’s gravitational field is not changing in time. However,
inside the horizon, the operation moves events in a spacelike direction. Roughly
speaking, we can interpret the fact that this is a symmetry as saying that the
black hole spacetime is the same at every place inside.
However, the metric does depend on r = t ′ , so the interior is dynamical.