154
10 Black Holes and Gravitational Collapse
Fig. 10.7 The supermassive black hole in Messier 87 as imaged by the EHT
There are two elementary ways to heuristically obtain the Hawking formula for
the temperature of a black hole, one using the uncertainty principle, and one using the
second law of thermodynamics. The quantum field theory derivation used originally
by Hawking is beyond our present scope so we will use the uncertainty principle
derivation even though it is heuristic and crude (Hawking 1974; Adler 2001, 2006).
The derivation based on thermodynamics is discussed in Exercise 10.15 (Ohanian
1994).
Our derivation uses the uncertainty principle combined with some qualitative
concepts from quantum field theory, so it is better motivated and more convincing
than dimensional analysis alone. In field theory we find that the vacuum is not at
all empty but is filled with virtual particles interacting as symbolized by Feynman
diagrams, one of which is shown in Fig. 10.8: an electron and positron and photon
materialize out of nothing and have a fleeting existence before they recombine and
vanish.
Fig. 10.8 On the left is a vacuum bubble diagram of quantum electrodynamics. The particles have
only a brief existence before they are forced by energy conservation to recombine and vanish. On
the right the photon may escape since the black hole can provide energy to the system
10 Black Holes and Gravitational Collapse
Fig. 10.7 The supermassive black hole in Messier 87 as imaged by the EHT
There are two elementary ways to heuristically obtain the Hawking formula for
the temperature of a black hole, one using the uncertainty principle, and one using the
second law of thermodynamics. The quantum field theory derivation used originally
by Hawking is beyond our present scope so we will use the uncertainty principle
derivation even though it is heuristic and crude (Hawking 1974; Adler 2001, 2006).
The derivation based on thermodynamics is discussed in Exercise 10.15 (Ohanian
1994).
Our derivation uses the uncertainty principle combined with some qualitative
concepts from quantum field theory, so it is better motivated and more convincing
than dimensional analysis alone. In field theory we find that the vacuum is not at
all empty but is filled with virtual particles interacting as symbolized by Feynman
diagrams, one of which is shown in Fig. 10.8: an electron and positron and photon
materialize out of nothing and have a fleeting existence before they recombine and
vanish.
Fig. 10.8 On the left is a vacuum bubble diagram of quantum electrodynamics. The particles have
only a brief existence before they are forced by energy conservation to recombine and vanish. On
the right the photon may escape since the black hole can provide energy to the system