The Quantum Structure of Space and Time (293 pages)

60 The Quantum Structure of Space and Time

The situation improves if one considers the Kaluza-Klein monopoles of Gross
Perry [3] and Sorkin [4].

ds2 = dt2 + V-l(dz5 i w&2)2 + Vdx2, (13)

grad V = curlw (14)

Periodicity of x5 imposes a quantisation condition on the Klauza-Klein charges
and on the magnetic monopole moment.

In addition, the singularities of the four-dimensional metric receive a Higher

Dimensional Resolution: they are mere coordinate artifacts in five dimensions.

In this way, they evade the Pauli-Einstein theorem.
Gibbons, Horowitz and Townsend [a] have shown that higher dimensional reso-
lutions are quite common. However the problem of singularities and the ultimate
outcome of gravitational collapse and Hawking evaporation cannot be solved in this
this way. Moreover, the solution is unstable in the sense that

is an exact, time-dependent solution.

This is a particular example of the general tendency of higher dimensions to

undergo gravitational collapse [17], which is fatal to the dimensional reduction pro-

gramme unless some means can be found to stabilize the various 'moduli 'fields.

These examples also underscore the need for a theory of initial conditions

in order to understand cosmology and the initial singularity or big bang. As em-

phasised by Penrose among others, elementary thermodynamic arguments indicate

that the Universe began in a very special state and even proponents of eternal in-

flation have had to concede, following Borde, Guth and Vilenkin [28] , that eternity

is past incomplete. In other words if inflation is past eternal then spacetime is

geodesically incomplete. Penrose's Weyl Curvature Hypothesis postulates a

connection between gravitational entropy and Weyl curvature' and hence demands

of the universe that the initial singularity has vanishing, or possibly finite, Weyl

curvature, as in F-L-R-W. models.In general such singularities are called isotropic

and Tod and co-workers [22] [23] [24] have proven existence and uniqueness results

for the associated Cauchy problem. However there is as yet no derivation of this

condition from something deeper and it remains a purely classical viewpoint.

Hartle and Hawking's No Boundary Proposal achieves a similar purpose

but at the expense of leaving the realm of Lorentzian metrics. In principle this is

8despite the example of de-Sitter spacetime