Covariant 2+2 formulation of the initial-value problem in general relativity

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.