Relativistic geoid

Gravity potential and relativistic effects

authored by: Dennis Philipp, Eva Hackmann, Claus Lämmerzahl, Jürgen Müller
Abstract: The Earth's geoid is one of the most essential and fundamental concepts to provide a gravity field-related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions (including the geoid) within Einstein's theory of general relativity is inevitable. Building on the theoretical construction of isochronometric surfaces and the so-called redshift potential for clock comparison, we define a relativistic gravity potential as a generalization of (post-)Newtonian notions. This potential exists in any stationary configuration with rigidly corotating observers, and it is the same as realized by local plumb lines. In a second step, we employ the gravity potential to define the relativistic geoid in direct analogy to the Newtonian understanding. In the respective limit, the framework allows to recover well-known (post-) Newtonian results. For a better illustration and proper interpretation of the general relativistic gravity potential and geoid, some particular examples are considered. Explicit results are derived for exact vacuum solutions to Einstein's field equation as well as a parametrized post-Newtonian model. Comparing the Earth's Newtonian geoid to its relativistic generalization is a very subtle problem, but of high interest. An isometric embedding into Euclidean three-dimensional space is an appropriate solution and allows a genuinely intrinsic comparison. With this method, the leading-order differences are determined, which are at the mm level.
Organisation(s): Institute of Geodesy
QuantumFrontiers
External Organisation(s): University of Bremen
Fraunhofer Institute for Digital Medicine (MEVIS)
Carl von Ossietzky University of Oldenburg
Type: Article
Journal: Physical Review D
Volume: 101
No. of pages: 16
ISSN: 2470-0010
Publication date: 17.03.2020
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Physics and Astronomy (miscellaneous)
Electronic version(s): https://arxiv.org/abs/1912.10159 (Access: Open)
https://doi.org/10.1103/PhysRevD.101.064032 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{75799be981524b56ba50049b59a33c09,
title = "Relativistic geoid: Gravity potential and relativistic effects",
abstract = "The Earth's geoid is one of the most essential and fundamental concepts to provide a gravity field-related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions (including the geoid) within Einstein's theory of general relativity is inevitable. Building on the theoretical construction of isochronometric surfaces and the so-called redshift potential for clock comparison, we define a relativistic gravity potential as a generalization of (post-)Newtonian notions. This potential exists in any stationary configuration with rigidly corotating observers, and it is the same as realized by local plumb lines. In a second step, we employ the gravity potential to define the relativistic geoid in direct analogy to the Newtonian understanding. In the respective limit, the framework allows to recover well-known (post-) Newtonian results. For a better illustration and proper interpretation of the general relativistic gravity potential and geoid, some particular examples are considered. Explicit results are derived for exact vacuum solutions to Einstein's field equation as well as a parametrized post-Newtonian model. Comparing the Earth's Newtonian geoid to its relativistic generalization is a very subtle problem, but of high interest. An isometric embedding into Euclidean three-dimensional space is an appropriate solution and allows a genuinely intrinsic comparison. With this method, the leading-order differences are determined, which are at the mm level.",
author = "Dennis Philipp and Eva Hackmann and Claus L{\"a}mmerzahl and J{\"u}rgen M{\"u}ller",
note = "Funding information: We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) under Germanys Excellence Strategy EXC-2123/1 (Project-ID: 390837967) and through the Research Training Group 1620 “Models of Gravity.” This work was also supported by the German Space Agency DLR with funds provided by the Federal Ministry of Economics and Technology (BMWi) under Grant No. DLR 50WM1547 and the DLR institute for Satellite Geodesy and Inertial Sensing. The authors would like to thank Volker Perlick, Dirk Puetzfeld, Heiner Denker, Domenico Giulini, and Sergei Kopeikin for helpful discussions.",
year = "2020",
month = mar,
day = "17",
doi = "10.1103/PhysRevD.101.064032",
language = "English",
volume = "101",
journal = "Physical Review D",
issn = "2470-0010",
publisher = "American Institute of Physics",
number = "6",
}