Notes on Relativity and Cosmology - Physics Department, UCSB

9.5. BLACK HOLE ASTROPHYSICS AND OBSERVATIONS 261
a 2 s/c 2 = c2 4
(
Rs
r 2 ) 2
(1 − R s /r) −1 . (9.43)
To evaluate the relative acceleration, equation (9.38) tells us to add these
two results together. Clearly, there is a major cancellation and all that we have
left is:
( )
relative acceleration = c 2 Rs
r 3 L. (9.44)
This gives the relative acceleration of two freely falling observers who, at that
moment, are at rest with respect to the static observers. (The free fallers are
also located at radius r and are separated by a radial distance L, which is much
smaller than r.) The formula holds anywhere that the Schwarzschild metric
applies. In particular, anywhere outside a black hole.
Now for the question you have all been waiting for .... what happens at the
horizon (or, perhaps just barely outside)? Well, this is just r = R s . In this case,
equation (9.44) reads
( ) L
relative acceleration = c 2 R 2 . (9.45)
s
The most important thing to notice about this formula is that the answer is
finite. Despite the fact that a static observer at the horizon would need an
infinite acceleration relative to the free fallers, any two free fallers have only a
finite acceleration relative to each other.
The second thing to notice is that, for a big black hole (large R s ), this relative
acceleration is even small. (However, for a small black hole, it can be rather
large.) I’ll let you plug in numbers on your own and see how the results come
out. Have fun!
9.5 Black Hole Astrophysics and Observations
We have now come to understand basic round (Schwarzschild) black holes fairly
well. We have obtained several perspectives on black hole exteriors and interiors
and we have also learned about black hole singularities. However, there are
several issues associated with black holes that we have yet to discuss. Not least
of these is the observational evidence that indicates that black holes actually
exist! This section will be devoted to this evidence and to the physics that
surrounds it.
9.5.1 The observational evidence for black holes
We argued back in section 9.2.1 that big black holes should not be too hard
to make. So, the question arises, are there really such things out there in the