Exponential stretch-rotation (ESR) formulation of general relativity

We study a tensorial exponential transformation of a three-dimensional metric of space-like hypersurfaces embedded in a four-dimensional space-time, $\gamma_{ij} = e^{\epsilon_{ikm}\theta_m} e^{\phi_k} e^{-\epsilon_{jkn}\theta_n}$,where$\phi_k$ are logarithms of the eigenvalues of ${\gamma }_{ij}$, ${\theta }_k$ are rotation angles, and ${ϵ}_{ijk}$ is a fully anti-symmetric symbol. Evolution part of Einstein’s equations, formulated in terms of ${\varphi }_k$ and ${\theta }_k$, 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.