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EXPONENTIAL STRETCH–ROTATION FORMULATION OF EINSTEIN'S EQUATIONS

ALEXEI KHOKHLOV

Department of Astronomy and Astrophysics, The University of Chicago, 5640 South Ellis Ave., Chicago, IL 60637, USA

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and

IGOR NOVIKOV

Niels Bohr Institute and NORDITA, Blegdamsvej 17, DK-2100, Copenhagen, Denmark

Astro Space Center of P.N. Lebedev Physical Institute, Profsouznaya 84/32, Moscow, 117810, Russia

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https://doi.org/10.1142/S021827180400458XCited by:3 (Source: Crossref)

Abstract

We study an exponential stretch–rotation (ESR) transformation, γ_ij=e^∊_ikmθ_me^ϕ_ke^-∊_jknθ_n, of a three-dimensional metric γ_ij of space-like hypersurfaces embedded in a four-dimensional space–time, where ϕ_k are logarithms of the eigenvalues of γ_ij, θ_k 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 γ_ij and its differentials to ϕ_k, θ_k 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.

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