Geoids in General Relativity: Geoid Quasilocal Frames

TL;DR

This paper introduces a relativistic concept of geoid as a specific quasilocal frame in general relativity, compares it with Newtonian geoid models, and calculates relativistic corrections for measurable quantities.

Contribution

It formulates a relativistic geoid using quasilocal frames and analyzes its properties and corrections in simple spacetimes, extending geoid concepts beyond Newtonian gravity.

Findings

Relativistic geoid defined as a quasilocal frame boundary.

Computed general-relativistic corrections to geometric quantities.

Compared relativistic and Newtonian geoid models in simple spacetime scenarios.

Abstract

We develop, in the context of general relativity, the notion of a geoid -- a surface of constant "gravitational potential". In particular, we show how this idea naturally emerges as a specific choice of a previously proposed, more general and operationally useful construction called a quasilocal frame -- that is, a choice of a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume. We study the geometric properties of these geoid quasilocal frames, and construct solutions for them in some simple spacetimes. We then compare these results -- focusing on the computationally tractable scenario of a non-rotating body with a quadrupole perturbation -- against their counterparts in Newtonian gravity (the setting for current applications of the geoid), and we compute general-relativistic corrections to some measurable geometric quantities.