Quantum Gravity : Mathematical Models and Experimental Bounds

Quantum Gravity — A Short Overview 3
To probe, for example, the Planck length with contemporary accelerators one
would have to built a machine of the size of our Milky Way. Direct observations
are thus expected to come mainly from the astrophysical side — the early universe
and black holes. In addition there may of course occur low-energy effects such as
a small violation of the equivalence principle [5, 3]. The search for such effects is
often called ‘quantum gravity phenomenology’. The irrelevance of quantum gravity
in usual astrophysical investigations can be traced back to the huge discrepancy
between the Planck mass and the proton mass: It is the small constant
α g = Gm2 pr
c
=
(
mpr
m P
) 2
≈ 5.91 × 10 −39 , (5)
where m pr denotes the proton mass, that enters astrophysical quantities of interest
such as stellar masses and stellar lifetimes.
On the road towards quantum gravity it is important to study all levels of
interaction between quantum systems and the gravitational field. The first level,
where experiments exist, concerns the level of quantum mechanical systems interacting
with Newtonian gravity [7]. The next level is quantum field theory on a
given (usually curved) background spacetime. Here one has a specific prediction:
Black holes radiate with a temperature proportional to (‘Hawking radiation’),
T BH =
κ
2πk B c , (6)
where κ is the surface gravity. For a Schwarzschild black hole this temperature
reads
c 3
( )
T BH =
8πk B GM ≈ 6.17 × M⊙
10−8 K . (7)
M
The black hole shrinks due to Hawking radiation and possesses a finite lifetime.
The final phase, where γ-radiation is being emitted, could be observable. The temperature
(7) is unobservably small for black holes that result from stellar collapse.
One would need primordial black holes produced in the early universe because they
could possess a sufficiently low mass, cf. the review by Carr in [3]. For example,
black holes with an initial mass of 5 × 10 14 g would evaporate at the present stage
of the universe. In spite of several attempts, no experimental hint for black-hole
evaporation has been found. Primordial black holes can result from density fluctuations
produced during an inflationary epoch. However, they can only be produced
in sufficient numbers if the scale invariance of the power spectrum is broken at
some scale, cf. [8] and references therein.
Since black holes radiate thermally, they also possess an entropy, the ‘Bekenstein–Hawking
entropy’, which is given by the expression
S BH = k Bc 3 A
4G
= k B
A
4l 2 P
, (8)