Feynman Path Integral Formulation

168 5 Semiclassical Gravity
Fig. 5.2 Pictorial representation
of the wrinkled surface of
a quantum-mechanical black
hole. The geometry of the
horizon is rendered rough by
strong short distance quantum
fluctuations in the metric.
Since these are described by a
massless particle, one would
expect the microscopic geometry
to be self-similar, perhaps
best described by a gravitational
Hausdorff dimension,
in analogy with the quantummechanical
Feynman path of
a non-relativistic particle.
horizon then changes to a scale-invariant spectrum for length scales smaller than
some critical value.
This would imply that, in a quantum theory of gravity, the classical notion of
a sharp black hole horizon at r = 2MG would be superseded at shorter distances
by a geometrically more complex object, an entity whose shape is far from smooth
on small scales. The horizon would appear to be essentially classical up to some
distance scale, below which its self-similar (or fractal) nature would start to emerge
(in the Newtonian approximation this scale is of the order of (MG 2 ) 1/3 ). Thus its
surface area would no longer be given by the Euclidean result; instead it would
depend, just like the overall size of a random walk or a Wiener path, on the scale at
which it is measured.
Since these short distance fluctuations are essentially described by a massless
particle, one would expect equally the microscopic geometry to be self-similar or
fractal, and therefore to be best described by a suitable Hausdorff dimension, in
analogy with the quantum-mechanical Feynman path of a non-relativistic particle.
Furthermore, one would expect that the difference in surface area on microscopic
scales, comparable to the ultraviolet cutoff, and on the larger classical scale would be
described by an (ultraviolet divergent) renormalization factor, A R = Z A (Λ) · A 0 (Λ),
in analogy to charge and wavefunction renormalization in QED.