Geometrostatics: the geometry of static space-times

TL;DR

This paper introduces a geometric framework called geometrostatics for analyzing static isolated spacetimes in general relativity, providing new formulas, interpretations, and convergence to Newtonian physics.

Contribution

It develops localized formulas for mass and center of mass, interprets level sets of the lapse function, and establishes uniqueness and force notions in geometrostatic systems.

Findings

Formulas for ADM-mass and center of mass that converge to Newtonian limits

A new physical interpretation of the lapse function's level sets

Uniqueness results for geometrostatic configurations

Abstract

We present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center of mass of geometrostatic systems. We then explain the pseudo-Newtonian character of these formulas and show that they converge to Newtonian mass and center of mass in the Newtonian limit, respectively, using Ehlers' frame theory. Moreover, we present a novel physical interpretation of the level sets of the canonical lapse function and apply it to prove uniqueness results. Finally, we suggest a notion of force on test particles in geometrostatic space-times.