Special relativistic hydrodynamics with gravitation

TL;DR

This paper derives the equations for special relativistic hydrodynamics coupled with weak gravity from Einstein's general relativity, providing a new framework for understanding such systems.

Contribution

It introduces the first consistent derivation of relativistic hydrodynamics with Poisson-type gravity, including anisotropic stress, from Einstein's equations.

Findings

Derived hydrodynamic equations in maximal slicing.

Presented equations in the first post-Newtonian approximation.

Included anisotropic stress in the formulation.

Abstract

The special relativistic hydrodynamics with weak gravity is hitherto unknown in the literature. Whether such an asymmetric combination is possible was unclear. Here, the hydrodynamic equations with Poisson-type gravity considering fully relativistic velocity and pressure under the weak gravity and the action-at-a-distance limit are consistently derived from Einstein's general relativity. Analysis is made in the maximal slicing where the Poisson's equation becomes much simpler than our previous study in the zero-shear gauge. Also presented is the hydrodynamic equations in the first post-Newtonian approximation, now under the {\it general} hypersurface condition. Our formulation includes the anisotropic stress.