Notes on Relativity and Cosmology - Physics Department, UCSB

238 CHAPTER 9. BLACK HOLES
r = R s
r = R s
t
r
oo
r = R s
Here, I have used arrows to indicate the direction in which the time coordinate
t increases on this diagram.
Not only do light rays directed along the horizon remain at r = R s , any light ray
at the horizon which is directed a little bit sideways (and not perfectly straight
outward) cannot even stay at r = R s , but must move to smaller r. The diagram
below illustrates this by showing the horizon as a surface made up of light rays.
If we look at a light cone emitted from a point on this surface, only the light
ray that is moving in the same direction as the rays on the horizon can stay in
the surface. The other light rays all fall behind the surface and end up inside
the black hole (at r < R s ).
Similarly, any object of nonzero mass requires an infinite acceleration (directed
straight outward) to remain at the horizon. With any finite acceleration, the
object falls to smaller values of r. At any value of r less than R s no object
can ever escape from the black hole. This is clear from the above spacetime