Newtonian hydrodynamic equations with relativistic pressure and velocity

TL;DR

This paper introduces a new approximation method that incorporates relativistic pressure and velocity into Newtonian hydrodynamics, bridging the gap between Newtonian and relativistic regimes in nonlinear scenarios.

Contribution

It derives a consistent set of equations from Einstein's gravity that include relativistic effects in Newtonian hydrodynamics, valid in fully nonlinear situations.

Findings

Equations recover the proper special relativity limit without gravity.

The approximation complements post-Newtonian methods.

Applicable to fully nonlinear astrophysical phenomena.

Abstract

We present a new approximation to include fully general relativistic pressure and velocity in Newtonian hydrodynamics. The energy conservation, momentum conservation and two Poisson's equations are consistently derived from Einstein's gravity in the zero-shear gauge assuming weak gravity and action-at-a-distance limit. The equations show proper special relativity limit in the absence of gravity. Our approximation is complementary to the post-Newtonian approximation and the equations are valid in fully nonlinear situations.