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Exponential stretch-rotation (ESR) formulation of general relativity

A. M. Khokhlov·I. D. Novikov
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Abstract

We study a tensorial exponential transformation of a three-dimensional metric of space-like hypersurfaces embedded in a four-dimensional space-time, $γ_{ij} = e^{ε_{ikm}θ_m} e^{ϕ_k} e^{-ε_{jkn}θ_n}$, where $ϕ_k$ are logarithms of the eigenvalues of $γ_{ij}$, $θ_k$ are rotation angles, and $ε_{ijk}$ is a fully anti-symmetric symbol. Evolution part of Einstein's equations, formulated in terms of $ϕ_k$ and $θ_k$, 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.

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A. M. Khokhlov, I. D. Novikov. 2003-03-18. Exponential stretch-rotation (ESR) formulation of general relativity. https://doi.org/10.1142/s021827180400458x

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