Exponential stretch-rotation (ESR) formulation of general relativity

TL;DR

This paper introduces an exponential stretch-rotation (ESR) formulation of general relativity, representing the metric as a combination of local stretch and rotation, potentially improving stability in numerical simulations.

Contribution

The paper develops a novel ESR 3+1 formulation of Einstein's equations using tensorial exponential transformations, extending previous symmetry-based methods.

Findings

Potential for more stable long-term numerical integration.

Generalization of exponential transformations for arbitrary spatial metrics.

Framework applicable to various spacetime symmetries.

Abstract

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 $\epsilon_{ijk}$ is a fully anti-symmetric symbol. Evolution part of Einstein's equations, formulated in terms of $\phi_k$ 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.