Geoids in general relativity: geoid quasilocal frames

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.