Locally Inertial Approximations of Balance Laws Arising in (1+1)-Dimensional General Relativity

Abstract

An elementary model of (1+1)-dimensional general relativity, known as “${\cal R}={\cal T}$” and mainly developed by Mann and coworkers in the early 1990s, is set up in various contexts. Its formulation, mostly in isothermal coordinates, is derived and a relativistic Euler system of self-gravitating gas coupled to a Liouville equation for the metric's conformal factor is deduced. First, external field approximations are carried out: both a Klein--Gordon equation is studied along with its corresponding density, and a Dirac one inside a hydrostatic gravitational field induced by a static, piecewise constant mass repartition. Finally, the coupled Euler--Liouville system is simulated, by means of a locally inertial Godunov scheme: the gravitational collapse of a static random initial distribution of density is displayed. Well-balanced discretizations rely on the treatment of source terms at each interface of the computational grid, hence the metric remains flat in every computational cell.