Covariant diagonalization of the perfect fluid stress-energy tensor

TL;DR

The paper addresses diagonalizing the perfect-fluid stress-energy tensor in the presence of vorticity by constructing covariant tetrads that render $T_{munu}$ diagonal at every spacetime point. It develops a velocity curl extremal field $xi_{munu}$ with a local complexion $alpha$ and uses it to build a covariant tetrad basis that diagonalizes $T_{munu}$ while preserving geometric clarity. The key contributions include a complete procedure for covariant diagonalization, the construction of an Euler-hypersurface orthogonal congruence for Cauchy evolution, and the application to the Carter-Lichnerowicz equation (including the barotropic/isentrope case). This framework provides a powerful tool for analyzing relativistic hydrodynamics in astrophysical settings and offers a transparent link between inertia, vorticity, and gravity within a covariant tetrad formalism.

Abstract

We introduce new tetrads that manifestly and covariantly diagonalize the stress-energy tensor for a perfect fluid with vorticity at every spacetime point. This new tetrad can be applied and introduce simplification in the analysis of astrophysical relativistic problems where vorticity is present through the Carter-Lichnerowicz equation. This new tetrad also enables the construction in a simple fashion of Euler and Coordinate observers relevant to the Cauchy evolution of many hydrodynamical relativistic problems in the case with no vorticity and the presence of a symmetry where the tetrads are completely analogous to the case with vorticity. We also discuss the origin of inertia in this special case from the standpoint of our new local tetrads.