Non-relativistic limit analysis of the Chandrasekhar-Thorne relativistic Euler equations with physical vacuum

TL;DR

This paper rigorously analyzes the non-relativistic limit of cylindrically symmetric relativistic Euler equations with vacuum states, showing convergence to classical Euler solutions at a rate proportional to 1/c^2.

Contribution

It provides the first rigorous proof of the non-relativistic limit for these equations with vacuum states, extending formal analyses to a solid mathematical foundation.

Findings

Uniform a priori estimates for solutions as c increases

Convergence of relativistic solutions to classical Euler solutions at rate 1/c^2

Validates formal non-relativistic limit analysis for vacuum states

Abstract

Our results provide a first step to make rigorous the formal analysis in terms of $\frac{1}{c^2}$ proposed by Chandrasekhar \cite{Chandra65b}, \cite{Chandra65a}, motivated by the methods of Einstein, Infeld and Hoffmann, see Thorne \cite{Thorne1}. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations, when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light $c$ and suitably small time $T,$ we obtain uniform, with respect to $c,$ \lq\lq a priori\rq\rq estimates for the local smooth solutions. Moreover, the smooth solutions of the cylindrically symmetric relativistic Euler equations converge to the solutions of the classical compressible Euler equation, at the rate of order $\frac{1}{c^2}$.