Applications of Arnowitt-Deser-Misner quantization of some metrics with at-least-two-parameter isometry groups

Abstract

The physical situations of spacetimes in general relativity with cylindrical symmetry, with plane symmetry, and the spacetime in the interaction regions of two colliding plane gravitational waves are mathematically related by the existence of an isometry group with at least two parameters in each case. We extend the work of Kucha\ifmmode \check{r}\else \v{r}\fi{}, Berger, and Gowdy to include the Arnowitt-Deser-Misner (ADM) quantization of cylindrical spacetimes with a quantized scalar field source, plane spacetimes with a quantized scalar field source, and the interaction region of two colliding gravitational plane waves with noncompact invariant hypersurfaces. The inhomogeneous nonvacuum metrics are interesting because they provide the first example of the quantization of arbitrarily strong interacting gravitational and quantum matter fields in the usual framework of a quantum field theory. Tentative conclusions are drawn concerning the "Landau conjecture" of a "gravitational renormalization" of the usual special-relativistic quantum field-theoretic divergences, and conclusions concerning the effect of quantum gravity on the classically predicted general-relativistic singularities are also presented. Finally, the extremely simple model of plane symmetry with scalar field source is closely related to the spherically symmetric situation, and represents an analogous physical mechanism to the particle production by Hawking's "exploding black holes." A program involving ADM quantization that can extend the "exploding black hole" calculations to include back reaction and quantum-gravity effects is presented.