Towards an exact relativistic theory of Earth's geoid undulation

Kopeikin, Sergei M.; Mazurova, Elena; Karpik, Alexander P.

doi:10.1016/j.physleta.2015.02.046

Cited by 21 publications

(38 citation statements)

References 35 publications

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“…Exact concept of the relativistic geoid in general relativity that is not limited to the first post-Newtonian approximation has been discussed in Kopeikin et al (2015) and Oltean et al (2015). We consider the Earth as made up of the background matter with small perturbations which reflect the actual mass distribution, stresses, and velocity flow of Earth's matter with respect to the rigidly rotating frame of reference.…”

Section: Relativistic Geoidmentioning

confidence: 99%

“…The surface p = 0 is called the level surface. Relativistic Euler equation derived for the rigidly rotating fluid body tells us that the level surface coincides with the equipotential surface of the postNewtonian gravitational potentialW which is expressed in terms of the centrifugal and gravitational potentials by the following expression (Kopejkin, 1991;Kopeikin et al, 2011Kopeikin et al, , 2015)…”

Section: Post-newtonian Equation Of the Level Surfacementioning

confidence: 99%

“…Because the potentialW is to be constant on the level surface (Kopeikin et al, 2015), the numerical coefficientsK 1 andK 2 must vanish. The first condition,K 1 = 0, yields a relation between the angular velocity of rotation, ω, and the eccentricity, e, of the rotating fluid ellipsoid:…”

Section: Post-newtonian Equation Of the Level Surfacementioning

confidence: 99%

“…Combining the data of the chronometric leveling with those of the conventional geodetic techniques will allow us to determine the normal heights of reference points with an unprecedented accuracy (Bondarescu et al, 2012). An adequate physical interpretation of this type of measurements requires more precise theoretical definition of geoid (Kopeikin et al, 2015;Oltean et al, 2015) and is inconceivable without an accompanying development of the corresponding mathematical algorithms accounting for the major relativistic effects in the description of the equipotential level surface and its, so-called, normal) gravity field (Hofmann-Wellenhof and Moritz, 2006;Torge and Müller, 2012).…”

Section: Introductionmentioning

confidence: 99%

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Reference Ellipsoid and Geoid in Chronometric Geodesy

Kopeikin

2016

Front. Astron. Space Sci.

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Chronometric geodesy applies general relativity to study the problem of the shape of celestial bodies including the earth, and their gravitational field. The present paper discusses the relativistic problem of construction of a background geometric manifold that is used for describing a reference ellipsoid, geoid, the normal gravity field of the earth and for calculating geoid's undulation (height). We choose the perfect fluid with an ellipsoidal mass distribution uniformly rotating around a fixed axis as a source of matter generating the geometry of the background manifold through the Einstein equations. We formulate the post-Newtonian hydrodynamic equations of the rotating fluid to find out the set of algebraic equations defining the equipotential surface of the gravity field. In order to solve these equations we explicitly perform all integrals characterizing the interior gravitational potentials in terms of elementary functions depending on the parameters defining the shape of the body and the mass distribution. We employ the coordinate freedom of the equations to choose these parameters to make the shape of the rotating fluid configuration to be an ellipsoid of rotation. We derive expressions of the post-Newtonian mass and angular momentum of the rotating fluid as functions of the rotational velocity and the parameters of the ellipsoid including its bare density, eccentricity and semi-major axes. We formulate the post-Newtonian Pizzetti and Clairaut theorems that are used in geodesy to connect the parameters of the reference ellipsoid to the polar and equatorial values of force of gravity. We expand the post-Newtonian geodetic equations characterizing the reference ellipsoid into the Taylor series with respect to the eccentricity of the ellipsoid, and discuss the small-eccentricity approximation. Finally, we introduce the concept of relativistic geoid and its undulation with respect to the reference ellipsoid, and discuss how to calculate it in chronometric geodesy by making use of the anomalous gravity potential.

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Section: Relativistic Geoidmentioning

confidence: 99%

Section: Post-newtonian Equation Of the Level Surfacementioning

confidence: 99%

Section: Post-newtonian Equation Of the Level Surfacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reference Ellipsoid and Geoid in Chronometric Geodesy

Kopeikin

2016

Front. Astron. Space Sci.

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“…Recently, extensive work has been done aiming at developing an exact relativistic theory of Earth's geoid undulation [26], as well as developing a theory of the reference level surface in the context of post-Newtonian gravity [27,28]. This goes beyond the problem of the realization of a reference isochronometric surface and tackles the tough work of extending all concepts of classical physical geodesy (see, e.g., [23]) in the framework of general relativity.…”

Section: The Chronometric Geoidmentioning

confidence: 99%