Approaches to Quantum Gravity

Questions and answers• Q - L. Crane - to T. Thiemann:In order to apply the canonical approach to General Relativity, it is necessaryto choose a spacelike foliation of the spacetime. Is it important that a generalspacetime does not admit such a foliation? For example, spacetimes with blackholes in them do not admit such foliations, or at least not ones with physical timefunctions and constant topology. Does this manifest itself indirectly in some ofthe problems of the LQG approach?– A - T. Thiemann:By a well known theorem due to Geroch, every globally hyperbolic spacetimeadmits a foliation by spacelike hypersurfaces. Global hyperbolicity is aphysical requirement that is motivated by being able to have a well posedinitial value formulation of General Relativity. Hence, classically there isabsolutely no loss in making this assumption. In particular, spacetimes withblack holes are certainly globally hyperbolic, in fact the black hole theoremsdue to Penrose and Hawking have global hyperbolicity in their assumptions(for Schwarzschild use Kruskal coordinates to see it explicitly).LQG starts from this classical framework and so one may think that it cannotdeal with topology change. However, very beautifully this is not the case: vectorsin the LQG Hilbert space are superpositions of spin network states. Thesedescribe polymerlike excitations of the gravitational field on finite graphs.Consider the volume operator of LQG associated with some spatial region.If that region has empty intersection with the given graph then the volumevanishes. Physically this means that the given state assigns no volume tothat region, i.e. that there is a hole in that hypersurface. Hence we see thattopology change is all over the place in LQG. The reason why this happensis that in order to mathematically define the classical Einstein equations wemust assume that the metric is everywhere non-degenerate. However, thatrequirement can be totally relaxed in the quantum formulation. Notice that332