Notes on Relativity and Cosmology - Physics Department, UCSB

268 CHAPTER 9. BLACK HOLES
9.6 Black Hole Odds and Ends
Black holes are an enormous subject area and there are many parts of the
story that we have not yet discussed. For most of these there is simply not
enough time to address them in a one semester course. However, a few topics
merit special mention either because of their common appearance in the popular
media 9 , because they will be useful in our discussion of cosmology in chapter 10,
or because of their intrinsic interest. We will therefore be addressing Hawking
radiation, Penrose diagrams, and more complicated types of black holes (more
complicated than Schwarzschild black holes) in turn.
9.6.1 A very few words about Hawking Radiation
Strictly speaking, Hawking Radiation is not a part of this course because it
does not fall within the framework of general relativity. However, since someone
always asks about it, I feel the need to make a few very brief comments on the
subject. Believe me, this is only the very tip of the iceberg!
Here’s the story: do you recall that, when we discussed the black hole singularity,
we said that what really happens there will not be described by general
relativity? We mentioned that physicists expect a new and even more fundamental
understanding of physics to be important there, and that the subject
is called “quantum gravity.” We also mentioned that very little is understood
about quantum gravity at the present time.
Well, there is one thing that we think we do understand about quantum effects
in gravity. This is something that happens outside the black hole and therefore
far from the singularity. In this setting, the effects of quantum mechanics in the
gravitational field itself are extremely small. So small that we believe that we
can do calculations by simply splicing together our understanding of quantum
mechanics (which governs the behavior of photons, electrons, and such things)
and our understanding of gravity. In effect, use quantum mechanics together
with the equivalence principle to do calculations.
Stephen Hawking did such a calculation back in the early 1970’s. What he found
came as a real surprise. Consider a black hole by itself, without an accretion disk
or any other sort of obvious matter nearby. It turns out that the region around
the black hole is not completely dark! Instead, it glows like a hot object, albeit
at a very low temperature. The resulting thermal radiation is called Hawking
radiation.
Now, I first want to say that this is an incredibly tiny effect. For a solar mass
black hole the associated temperature is only 10 −5 Kelvin, that is, 10 −5 degrees
above absolute zero. Large black holes are even colder, as the temperature is
proportional to M −2 , where M is the black hole mass. So black holes are very,
very cold. In particular, empty space has a temperature of about 3K due to
9 So that you have likely heard of such things already or will encounter them in the future.
I would like you to have at least some exposure to the real story about these things.