Potentials for general-relativistic geodesy

TL;DR

The paper develops a fully relativistic framework for geodesy by showing that stationary spacetimes are described by two scalar potentials, $\phi$ (gravitoelectric) and $\varpi$ (gravitomagnetic), plus a spatial metric, rather than a single Newtonian potential. It introduces the Ernst formulation, where $\mathcal{E}=e^{2\phi}+i\varpi$ encapsulates the vacuum solution, and demonstrates how these potentials define isochronometric surfaces and a relativistic geoid suitable for clock- and interferometer-based measurements. The authors provide explicit expressions for $\phi$ and $\varpi$ in several spacetimes (Schwarzschild, Kerr, NUT, Kerr-NUT, rotating $q$-metric) to illustrate how rotation, multipoles, and exotic parameters influence the geodetic surfaces and their coordinate role. They discuss operational realizations (redshift, Sagnac, Schiff effect, Gravity Probe B, atom interferometry) and emphasize the practical potential and current limitations of observing gravitomagnetic effects on Earth, as well as future prospects for space-based GR geodesy and multipole modeling.

Abstract

Geodesy in a Newtonian framework is based on the Newtonian gravitational potential. The general-relativistic gravitational field, however, is not fully determined by a single potential. The vacuum field around a stationary source can be decomposed into two scalar potentials and a tensorial spatial metric, which together serve as the basis for general-relativistic geodesy. One of the scalar potentials is a generalization of the Newtonian potential while the second one describes the influence of the rotation of the source on the gravitational field for which no non-relativistic counterpart exists. In this paper the operational realizations of these two potentials, and also of the spatial metric, are discussed. For some analytically given spacetimes the two potentials are exemplified and their relevance for practical geodesy on Earth is outlined.

Figures (3)

• Figure 1: The gravitoelectric (blue dotted lines) and gravitomagnetic (red lines) potentials for a Schwarzschild spacetimes. The crosshatched area is the region inside the black-hole horizon and the gray shaded area is the region where the potentials are not defined because the Killing vector field is spacelike.
• Figure 2: The gravitoelectric (blue dotted lines) and gravitomagnetic (red lines) potentials for Kerr spacetimes. The crosshatched area is the region inside the outer horizon and the gray shaded area is the region where the potentials are not defined because the Killing vector field is spacelike.
• Figure 3: The gravitoelectric (blue dotted lines) and gravitomagnetic (red lines) potentials for the (Kerr-)NUT spacetime and for the $q$-metric. The crosshatched area is the region inside a black-hole horizon, the black area is the part bounded by a naked singularity, and the gray shaded area is the region where the potentials are not defined because the Killing vector field is spacelike.