Definition of the relativistic geoid in terms of isochronometric surfaces

Abstract

We present a definition of the geoid that is based on the formalism of general relativity without approximations; i.e. it allows for arbitrarily strong gravitational fields. For this reason, it applies not only to the Earth and other planets but also to compact objects such as neutron stars. We define the geoid as a level surface of a time-independent redshift potential. Such a redshift potential exists in any stationary spacetime. Therefore, our geoid is well defined for any rigidly rotating object with constant angular velocity and a fixed rotation axis that is not subject to external forces. Our definition is operational because the level surfaces of a redshift potential can be realized with the help of standard clocks, which may be connected by optical fibers. Therefore, these surfaces are also called "isochronometric surfaces." We deliberately base our definition of a relativistic geoid on the use of clocks since we believe that clock geodesy offers the best methods for probing gravitational fields with highest precision in the future. However, we also point out that our definition of the geoid is mathematically equivalent to a definition in terms of an acceleration potential, i.e. that our geoid may also be viewed as a level surface orthogonal to plumb lines. Moreover, we demonstrate that our definition reduces to the known Newtonian and post-Newtonian notions in the appropriate limits. As an illustration, we determine the isochronometric surfaces for rotating observers in axisymmetric static and axisymmetric stationary solutions to Einstein's vacuum field equation, with the Schwarzschild metric, the Erez-Rosen metric, the q-metric and the Kerr metric as particular examples.

Schwarzschild radial perturbations in Eddington–Finkelstein and Painlevé–Gullstrand coordinates

“…1. The series representation of the function Ξ ωsl (x) is absolutely convergent on the interval |x| < 1.…”

“…1 and give the solutions of the perturbation equation for a massive scalar field on Schwarzschild background spacetime in EF coordinates. The propagation of such a massive scalar field is described by the Klein-Gordon equation with a mass term:…”

“…While the regular solution can be used to investigate scattering of radiation or to calculate quasi-normal modes, the second solution describes Hawking radiation and is, therefore, connected to black hole evaporation as shown in Ref. 1. We have further shown that using an analytic continuation procedure the bounded domain of convergence of the local solution (i.e., the unit circle around the horizon in the complex plane) can be expanded to the entire open interval between the origin and spatial infinity.…”

“…The series representation of the function Ξ ωsl (x) is absolutely convergent on the interval |x| < 1. 1,14 (v) Full solution of the radial equation.…”

Relativistic geoid: Gravity potential and relativistic effects

“…The essential steps to do so have already been outlined in Refs. [12,15,16], in which the relativistic geoid is defined in terms of isochronometric surfaces, the level sets of a so-called stationary redshift potential for Killing congruences. We will, to a large extent, use the results in these references and incorporate them into the definition of a relativistic gravity potential in the next section.…”