Exponential stretch-rotation formulation of Einstein’s equations

We study a tensorial exponential transformation of a three-dimensional metric of space-like hypersurfaces embedded in a four-dimensional space-time, γij = e ikmmekejknn , where φk are logarithms of the eigenvalues of γij , θk are rotation angles, and ǫijk is a fully antisymmetric symbol. Evolution part of Einstein’s equations, formulated in terms of φk and θk, describes time evolution of the metric at every point of a hyper-surface as a continuous stretch and rotation of a local coordinate system in a tangential space. The exponential stretch-rotation (ESR) transformation generalizes particular exponential transformations used previously in cases of spatial symmetry. The ESR 3+1 formulation of Einstein’s equations may have certain advantages for long-term stable integration of these equations. Code 6404, Naval Research Laboratory, Washington, DC 20375 Theoretical Astrophysics Center Juliane Maries Vej 30, 2100 Copenhagen, Denmark University Observatory, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Astro Space Center of P.N. Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117810, Russia NORDITA, Blegdamsvej 17, 2100, Copenhagen, Denmark