On a covariant 2+2 formulation of the initial value problem in general relativity

Abstract

The initial value problems in general relativity are considered from a geometrical standpoint. First of all a covariant 3+1 formalism is developed for the investigation of non space-like initial value problems. This involves an analysis of the problem of locally imbedding a family of null hypersurfaces in space-time. More precisely, an intrinsic affine connection is constructed on each hypersurface, a rigging is then introduced, and the resulting Gauss-Codazzi equations are derived. The fundamental differences between foliations of space-time into spacelike and null hypersurfaces are demonstrated, and the difficulties of applying this work to non space-like initial value problems is discussed. The major part of this thesis is concerned with the development of a covariant 2+2 formalism in which space-time is foliated by space-like 2-surfaces. This foliation is then rigged by a suitably chosen pair of directions in the orthogonal time-like 2-surface elements. The resulting 2+2 break-up of the Einstein vacuum field equations is then used to investigate space-like, characteristic and mixed initial value problems. In each case the so-called conformal 2-structure (essentially the conformal metric of the foliation into space-like 2-surfaces) is identified as explicitly embodying the true gravitational degrees of freedom. By means of the formalism, the geometrical significance of both the physically meaningful initial data and the various possible gauge choices is made clear. Finally, a Lagrangian formulation is included which supports the role of the conformal 2-structure as the dynamical variables of the pure gravitational field.

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