Initial-value problem of general relativity. I. General formulation and physical interpretation

The initial-value equations of Einstein's theory of general relativity are formulated as a system of four coupled quasilinear elliptic equations. These equations result from a covariant orthogonal decomposition of symmetric tensors and a generalized technique of conformal deformation of initial data. Mathematical properties and global integrability conditions of the equations are discussed. Physical interpretation of the independent and dependent data is given for both spatially closed and asymptotically flat initial-data sets. In the latter case, the four dependent functions constitute long-range scalar and vector potentials which determine the total mass and total linear and angular momenta of an isolated system. The definitions of linear and angular momenta suggest a unique extension to asymptotically flat three-spaces of the group of translations and rotations of flat three-space. In turn, the "almost symmetries" thus defined lead to Gaussian theorems expressing the equality of certain surface and volume integrals for total linear and angular momenta. An interpretation of the scalar and vector potentials for closed three-spaces is also given. In the Appendix we treat the special case of conformally flat initial data.