Notes on Relativity and Cosmology - Physics Department, UCSB

9.3. BEYOND THE HORIZON 239
diagram, since to move from the future interior to, say, the right exterior the
object would have to cross the light ray at r = R s , which is not possible.
Note that we could have started with this geometric insight at the horizon and
used it to argue for the existence of the photon sphere: Light aimed sideways
around the black hole escapes when started far away but falls in at the horizon.
Somewhere in the middle must be a transition point where the light neither
escapes nor falls in. Instead, it simply circles the black hole forever at the same
value of r.
9.3 Beyond the Horizon
Of course, the question that everyone would like to answer is “What the heck is
going on inside the black hole?” To understand this, we will turn again to the
Schwarzschild metric. In this section we will explore the issue in quite a bit of
detail and obtain several useful perspectives.
9.3.1 The interior diagram
To make things simple, let’s suppose that all motion takes place in the r, t plane.
This means that dθ = dφ = 0, and we can ignore those parts of the metric. The
relevant pieces are just
ds 2 = −(1 − R s /r)dt 2 +
dr 2
1 − R s /r . (9.16)
Let’s think for a moment about a line of constant r (with dr = 0). For such
a line, ds 2 = −(1 − R s /r)dt 2 . The interesting thing is that, for r < R s , this is
positive. Thus, for r < R s , a line of constant r is spacelike. You will therefore
not be surprised to find that, near the horizon, the lines of constant r are just
like the hyperbolae that are a constant proper time from where the two horizons
meet. Below, I have drawn a spacetime diagram in a reference frame that is in
free fall near the horizon.