EXPONENTIAL STRETCH–ROTATION FORMULATION OF EINSTEIN'S EQUATIONS

Abstract

We study an exponential stretch–rotation (ESR) transformation, γij=e∊ikmθmeϕke-∊jknθn, of a three-dimensional metric γij of space-like hypersurfaces embedded in a four-dimensional space–time, where ϕk are logarithms of the eigenvalues of γij, θk are rotation angles, and ∊ijk is a fully anti-symmetric symbol. This tensorial exponential transformation generalizes particular exponential transformations used previously in cases of spatial symmetry. General formulae are derived that relate γij and its differentials to ϕk, θk and their differentials in a compact form. The evolution part of Einstein's equations formulated in terms of ESR variables describes time evolution of the metric at every point of a hyper-surface as a continuous stretch and rotation of a triad associated with the main axes of the metric tensor in a tangential space. An ESR 3+1 formulation of Einstein's equations may have certain advantages for long-term stable integration of these equations.