Quantum-gravity fluctuations and the black-hole temperature
Eur. Phys. J. C
Quantum-gravity fluctuations and the black-hole temperature
Shahar Hod 0 1
0 The Hadassah Institute , 91010 Jerusalem , Israel
1 The Ruppin Academic Center , 40250 Emeq Hefer , Israel
Bekenstein has put forward the idea that, in a quantum theory of gravity, a black hole should have a discrete energy spectrum with concomitant discrete line emission. The quantized black-hole radiation spectrum is expected to be very different from Hawking's semi-classical prediction of a thermal black-hole radiation spectrum. One naturally wonders: Is it possible to reconcile the discrete quantum spectrum suggested by Bekenstein with the continuous semi-classical spectrum suggested by Hawking? In order to address this fundamental question, in this essay we shall consider the zero-point quantum-gravity fluctuations of the black-hole spacetime. In a quantum theory of gravity, these spacetime fluctuations are closely related to the characteristic gravitational resonances of the corresponding black-hole spacetime. Assuming that the energy of the black-hole radiation stems from these zero-point quantum-gravity fluctuations of the black-hole spacetime, we derive the effective temperature of the quantized black-hole radiation spectrum. Remarkably, it is shown that this characteristic temperature of the discrete (quantized) black-hole radiation agrees with the well-known Hawking temperature of the continuous (semiclassical) black-hole spectrum. One of the most remarkable theoretical predictions of modern physics is Hawking's celebrated result that black holes are not completely black [1]. According to Hawking's semi-classical analysis, a black hole is quantum mechanically unstable-it emits continuous thermal radiation whose characteristic temperature is given by
-
Here M is the mass of the Schwarzschild black hole. (We use
gravitational units in which G = c = 1.)
It should be stressed, however, that Hawkings derivation
of the continuous black-hole radiation spectrum is restricted
to the semi-classical regime: the matter fields are treated
quantum mechanically but the spacetime (and, in particular,
A
SBH = 4h .
the black hole itself) are treated classically. One therefore
expects to find important new features in the character of
the black-hole radiation spectrum once quantum properties
of the black hole itself are properly taken into account.1 It
is therefore appropriate to regard the Hawking temperature
(1) as the semi-classical (SC) temperature of the continuous
black-hole radiation:
The quantization of black holes was first proposed in the
seminal work of Bekenstein [2, 3]. The original quantization
procedure was based on the physical observation that the
surface area of a black hole behaves as a classical adiabatic
invariant [2, 3]. In the spirit of the Ehrenfest principle [4],
any classical adiabatic invariant corresponds to a quantum
entity with a discrete spectrum, Bekenstein suggested that
the horizon area A of a quantum black hole should have a
discrete spectrum of the form
n = 1, 2, 3, . . . .
Here is an unknown fudge factor which was introduced
in [2, 3].
In order to determine the value of the coefficient ,
Mukhanov and Bekenstein [5, 6] have suggested, in the spirit
of the BoltzmannEinstein formula in statistical physics
[4], to relate gn exp[SBH(n)] to the number of
blackhole micro-states that correspond to a particular external
black-hole macro-state. Here SBH is the black-hole entropy,
which is related to its surface area A by the thermodynamic
geometric relation [1, 2]
1 This state of affairs is reminiscent of atomic spectroscopy: according
to the classical laws of electrodynamics an atom should have a
continuous emission spectrum, whereas quantum mechanics dictates a discrete
line emission from the atom.
The emission of a gravitational quantum from the black hole
results in a change M = hR [see Eq. (7)] in the
blackhole mass. Using the first-law of black-hole
thermodynamics, A = 32 M M ,2 one finds the fundamental change
A = 4 ln 3 h in the Schwarzschild black-hole surface area.
Taking cognizance of Eqs. (3) and (8), one finally obtains the
quantized area spectrum:
An = 4h ln 3 n; n = 1, 2, 3, . . . .
It is worth emphasizing again that the black-hole area
spectrum (9) is consistent both with the area-entropy
thermodynamic relation (4) for black holes, with the Boltzmann
Einstein formula (5) in statistical physics, and with the Bohr
correspondence principle (7) [7].
One therefore concludes that, in a quantum theory of
gravity, a Schwarzschild black hole has a discrete energy (mass)
spectrum of the form:3
Mn =
2 Here we have used the relation A = 16 M2 for the Schwarzschild
3 See footnote 2.
The statistical degeneracy [see Eqs. (3) and (4)]
gn exp[SBH(n)] = exp
of the nth black-hole area level has to be an integer for every
integer n. This physical requirement dictates the relation [5,
6]
for the fudge factor , where the unknown constant k should
be an integer.
Determining th (...truncated)