Brill-Lindquist-Riemann sums and their limits

TL;DR

This paper investigates the convergence of discretized astrophysical models, called Brill-Lindquist-Riemann sums, towards a continuous dust universe, analyzing geometric limits and horizon properties in a relativistic context.

Contribution

It establishes existence and uniqueness of horizons in Brill-Lindquist metrics and explores geometric limits, including scalar curvature jumps, in the convergence process.

Findings

Existence and uniqueness of horizons near point sources.

Analysis of Gromov-Hausdorff and intrinsic flat limits of the geometries.

Examples of scalar curvature discontinuities in the limits.

Abstract

This article commences a study of convergence of discretized point-object configurations, which we call Brill-Lindquist-Riemann sums, towards a charged dust continuum from the perspective of relativistic initial data. We are motivated by the interpretation of Brill-Lindquist-Riemann sums as collections of relatively isolated astrophysical bodies such as stars and galaxies in the universe, and the interpretation of the dust continuum as the universe itself. Our work begins by establishing the existence and the uniqueness of horizons/minimal surfaces of Brill-Lindquist metrics in the vicinity of the point-sources ("stars"). We then study the geometries of the regions exterior to said minimal surfaces, and discuss their Gromov-Hausdorff and intrinsic flat limit. An interesting and purely geometric byproduct of our work are examples in which the scalar curvature jumps upon taking Gromov-Hausdorff and /or intrinsic flat limits.