52 The Quantum Structure of Space and Tame
in the future of certain types of Cauchy data.
Thus unlike classical Yang-Mills theory and scalar fields theories with renor-
malisable potentials, Leibniz-Laplace Determinism breaks down for General
Relativity. It can at best be an effective theory.
A-2 < A-1
The area of a closed trapped 2-surface decreases in both the inward and the
outward directions if pushed to the future along its two lightlike normals.
The singularity theorem give very little information about the nature of the sin-
gularities, in effect they deny the existence of timelike or null geodesically complete
spacetimes. The reason for the incompleteness is not predicted.
It is widely believed that incompleteness is due to divergences of curvature invari-
ants 2, or the components of the curvature in certain privileged, example parallelly
propagated frames.
The singularity theorems also fail to predict the scale at which singularities arise.
Indeed classical general relativity has no in built scale,
g/lv + x2g,v (1)
with X constant is a symmetry of the theory.
Many people believe that the resolution of the problem of singularities will come
from modifications of the Einstein equations due to Quantum Gravity at the Planck
scale, but this is by no means obvious. The necessary modification could, in prin-
ciple, have nothing to do with quantum mechanics.. How-
ever examples tend to show that some singularities still remain, e.g. those in singular
pp-waves which are in effect solutions of almost all theories of gravity. In addition
many, but not all admit ghosts.
Well motivated modifications of Einstein’s theory include supergravity theories
and the low energy limits of String theories.
It might for example entail the introduction of higher curvature termslor possibly Born-Infeld theory
2but these may all vanish, e.g. for singular pp waves
3For example Born-Infeld Gravity [26]