Feynman Path Integral Formulation

162 5 Semiclassical Gravity
One can then compute the total number of outgoing particles created in the frequency
range dω around ω. It is given by
∫
dω · dω ′ |β ωω ′| 2 , (5.98)
which is infinite since β ωω ′ goes like 1/ √ ω ′ for large ω ′ . The infinity in the total
number of created particles indicates a steady rate of particle emission from the
black hole, continuing for an infinite amount of time.
A precise relationship between the magnitudes of α ωω ′ and β ωω ′ can be computed
by the following argument based on the analytic properties of the amplitude.
The p ω solutions just constructed is zero on past null infinity I − for large values
of v. Its Fourier transform will therefore be analytic in the upper half ω ′ plane, but
the complex amplitude α ωω ′ contains a factor (−iω ′ ) iωκ with a logarithmic branch
point at ω ′ = 0. To obtain β ωω ′ from α ωω ′ one needs to analytically continue α ωω ′
around the singularity, which implies for large ω ′ the relationship
|α ωω ′ | = e πωκ |β ωω ′ | . (5.99)
This last equality then implies, for a given mode of frequency ω and for a fixed
ω ′ ≡ ω 0 , the following frequency distribution of outgoing particles
|β ωω0 | 2
|α ωω0 | 2 −|β ωω0 | 2 = 1
e πωκ − 1 , (5.100)
where the denominator |α ωω0 | 2 −|β ωω0 | 2 describes the fraction of particles that
enters the collapsing body. It leads therefore to an emission probability
N (ω) =
1
e πωκ − 1 . (5.101)
But this is precisely the relationship one would expect between absorption and emission
cross sections for a body with temperature T = 1/2πκ = 1/8πMG.
Other types of fields can be considered, such as the electromagnetic field or the
linearized gravitational field, and one finds also in these cases a thermal radiation
spectrum. If the particles are fermions, such as neutrinos, then the spectrum can be
shown to be of the Fermi-Dirac type (e πωκ + 1) −1 . If the fermions have mass m,
then ω will contain a rest mass contribution. The thermal emission will then be very
small unless T = 1/8πGM is greater than m. Finally there is the very interesting
and difficult general issue of the extent to which these results are universal, that
is independent of the detailed nature, history and composition of the black hole
(Fredenhagen and Haag, 1990).