MG14 - Talk detail

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I describe the procedure of Hamiltonian reduction of general relativity of 4 dimensional spacetimes under no symmetry assumptions using the (2+2) formalism. The privileged spacetime coordinates are such that the physical time is the {\it area} element of the spatial cross section of out-going null hypersurfaces, and the physical radial coordinate is defined by {\it equipotential} surfaces on a given spacelike hypersurface of constant physical time. In the privileged coordinates, the Einstein's constraints are completely solved to determine the non-zero physical Hamiltonian and momentum densities in terms of the physical degrees of freedom of gravitational field, which are the conformal two metric and its conjugate momentum. The physical degrees of freedom are subject to a topological constraint that dictates the spatial topology of a compact two dimensional cross section of a null hypersurface be either a two sphere or a torus. The physical Hamiltonian is local, and has an explicit dependence on the physical time. I present Hamilton's equations of motion which follow from the non-zero physical Hamiltonian, and find that they are identical to Einstein's equations written in the privileged coordinates. This proves that the Hamiltonian reduction proposed in this paper is a self-consistent procedure. As an application of the Hamiltonian reduction, I present three exact solutions, i.e. the Minkowski spacetime, the plane symmetric solution of Taub, and the general Kasner solution by solving Hamilton's equations of motion governed by the physical Hamiltonian. This work may be regarded as a generalization of the ADM Hamiltonian reduction of midi-superspace to 4 dimensional spacetimes with no isometries.

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