Black Hole Evaporation

The discovery of a temperature seen by an accelerated fiducial observer adds a new dimension to the equivalence principle. We can expect that identical thermal effects will occur near the horizon of a very massive black hole. However, in the case of a black hole a new phenomenon can take place evaporation. Unlike the Rindler case, the thermal atmosphere is not absolutely confined by the centrifugal potential. The particles of the thermal atmosphere will gradually leak through the barrier and carry off energy in the form of thermal radiation. A good qualitative understanding of the process can be obtained from the Rindler quantum field theory by observing two facts:

1) The Rindler time ω is related to the Schwarzschild time t by the equation

 (1.1)

Thus a field quantum with Rindler frequency ν_R is seen by the distant Schwarzschild observer to have a red shifted frequency ν

 (1.2)

The implication of this fact is that the temperature of the thermal atmosphere is reckoned to be red shifted also. Thus the temperature as seen by the distant observer is

 (1.3)

a form first calculated by Stephen Hawking.

2) The centrifugal barrier which is described in the Rindler theory by the potential k² exp(2u) is modified at distances r ≈ 3MG. In particular the maximum value that V takes on for angular momentum zero is

 (1.4)

Any s-wave quanta with frequencies of order (V_max)1/2 = 3√3/32MG or greater will easily escape the barrier. Since the average energies of massless particles in thermal equilibrium at temperature T is of course of order T , equation 1.3 indicates that some of the s-wave particles will easily escape to infinity. Unless the black hole is kept in equilibrium by incoming radiation it will lose energy to its surroundings.

Particles of angular momenta higher than s-waves cannot easily escape because the potential barrier is higher than the thermal scale. The black hole is like a slightly leaky cavity containing thermal radiation. Most quanta in the thermal atmosphere have high angular momenta and reflect off the walls of the cavity. A small fraction of the particles carry very low angular momenta. For these particles, the walls are semi-transparent and the cavity slowly radiates its energy. This is the process first discovered by Hawking and is referred to as Hawking radiation.

The above description of Hawking radiation does not depend in any essential way on the free field approximation. Indeed it only makes sense if there are interactions of sufficient strength to keep the system in equilibrium during the course of the evaporation. In fact, most discussions of Hawking radiation rely in an essential way on the free field approximation, and ultimately lead to absurd results. At the end of the next lecture, we will discuss one such absurdity.