No abstract

The authors show that, by employing the Bianchi identities, the field equations and symmetry conditions, one can reduce the upper bound on the order of the covariant derivative of the Riemann tensor needed to provide a Karlhede classification of a type D vacuum spacetime.

A covariant 2 + 2 formalism is developed in which space-time is decomposed into a family of spacelike two-surfaces and their orthogonal timelike two-surface elements. The resulting 2 + 2 breakup of the Einstein vacuum field equations is then used to investigate covariant formulations of spacelike, characteristic, and mixed initial-value problems. In each case the so-called conformal two-structure (essentially the conformal metric of a family of spacelike two-surfaces) is identified as the freely specifiable initial data. The formalism makes clear the geometrical significance of both the initial data and the various choices of gauge variables. A Lagrangian formulation is included which supports the role of the conformal two-structure as dynamical variables of the pure gravitational field.

This is the first in a series of papers concerning a project to set up a computer database of exact solutions in general relativity which can be accessed and updated by the user community. In this paper, we briefly discuss the Cartan-Karlhede invariant classification of geometries and the significance of the standard form of a spinor. We then present algorithms for putting the Weyl spinor, Ricci spinor and general spinors into standard form.

In this paper, we suggest that what we shall call the conformal 2-structure may, in an appropriate coordinate system, serve to embody the two gravitational degrees of freedom of the Einstein (vacuum) field equations. The conformal 2-structure essentially gives information concerning the manner in which a family of 2-surfaces is embedded in a 3-surface. We show that, formally at least, this prescription works for the exact plane and cylindrical gravitational wave solutions, for the double-null and null-timelike characteristic initial value problems, and for the usual Cauchy spacelike initial value problem. We conclude with a preliminary consideration of a two-plus-two breakup of the field equations aimed at unifying these and other initial value problems; and a discussion of some aspirations and remaining problems of this approach. value problem as analyzed by Bondi et at. , 6 Sachs, 1 and 2447