Skip to main content
SpringerLink
Log in

Reference-Ellipsoid and Normal Gravity Field in Post-Newtonian Geodesy

  • Chapter
  • First Online:
Book cover Relativistic Geodesy

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 196))

  • 1098 Accesses

Abstract

Modern geodesy is undergoing a crucial transformation from the Newtonian paradigm to the Einstein theory of general relativity. This is motivated by advances in developing quantum geodetic sensors including gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of geoid and multipolar structure of Earth’s gravitational field. At the same time, Very Long Baseline Interferometry, Satellite Laser Ranging and Global Navigation Satellite System have achieved an unprecedented level of accuracy in measuring spatial coordinates of reference points of the International Terrestrial Reference Frame and the world height system. The main geodetic reference standard to which gravimetric measurements of Earth’s gravitational field are referred, is called normal gravity field which is represented in the Newtonian gravity by the field of a uniformly rotating, homogeneous Maclaurin ellipsoid having mass and quadrupole momentum equal to the total mass and (tide-free) quadrupole moment of the gravitational field of Earth. The present chapter extends the concept of the normal gravity field from the Newtonian theory to the realm of general relativity. We focus on the calculation of the post-Newtonian approximation of the normal field that would be sufficiently precise for near-future practical applications. We show that in general relativity the level surface of homogeneous and uniformly rotating fluid is no longer described by the Maclaurin ellipsoid in the most general case but represents an axisymmetric spheroid of the fourth order (PN spheroid) with respect to the geodetic Cartesian coordinates. At the same time, admitting post-Newtonian inhomogeneity of mass density in the form of concentric elliptical shells allows us to preserve the level surface of the fluid as an exact ellipsoid of rotation. We parametrize the mass density distribution and the level equipotential surface with two parameters which are intrinsically connected to the existence of the residual gauge freedom, and derive the post-Newtonian normal gravity field of the rotating spheroid both inside and outside of the rotating fluid body. The normal gravity field is given, similarly to the Newtonian gravity, in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce a more conventional post-Newtonian multipolar expansion of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy as it does not match the normal gravity field of the Maclaurin ellipsoid already in the Newtonian limit. We derive the post-Newtonian generalization of the Somigliana formula for the normal gravity field measured on the surface of the rotating PN spheroid and employed in practical work for measuring the Earth gravitational field anomalies. Finally, we discuss the possible choice of the gauge-dependent parameters of the normal gravity field model for practical applications and compare it with the existing EGM2008 model of gravitational field.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The minus sign in definition of the proper time appears because \(ds^2<0\) due to the choice of the metric signature shown in (6a)–(6c).

  2. 2.

    We remind that the associated Legendre functions of the imaginary argument, \(z=x+iy\), are defined for all z except at a cut line along the real axis, \(-1\le x\le 1\). The associated Legendre functions of a real argument are defined only on the cut line, \(-1\le x\le 1\) [69, Section 12.10].

  3. 3.

    Definition of the associated Legendre polynomials adopted in the present chapter follows [72, Sec. 8.81]. It differs by a factor \((-1)^m\) from the definition of the associated Legendre polynomials adopted in the book [10].

  4. 4.

    Post-newtonian definitions of mass, center of mass, and other multipole moments can be found, for example, in [5].

  5. 5.

    Notice that \(\mathcal{D}^z\not =0\) but we don’t need this component for calculating \(V^+\).

  6. 6.

    Our method is partially overlapping with a similar development given in [10, Section 2.9].

  7. 7.

    It is equivalent to a constant \(L_G=W_0/c^2=6.969290134\times 10^{-10}\) that determines the difference between TT and TCG time scales (see [20, 27] or [5, Appendix C.2, Resolution B1.9.]).

  8. 8.

    For more details about the gauge transformations of the post-Newtonian spheroid the reader is referred to [3, Section 4].

  9. 9.

    One should notice that in classic geodesy the Somigliana formula is usually expressed in terms of the geographic latitude \(\Phi \) on ellipsoid that is related to the ellipsoidal angle \(\theta \) by \(\theta =\beta -\pi /2\), and, \(a\tan \beta =b\tan \Phi \), [10, Eq. 2-77].

  10. 10.

    For example, in case of a rigidly rotating homogeneous perfect fluid the relation, \(\epsilon =\epsilon (\rho ,\omega )\), is simply given by the Maclaurin formula (86).

References

  1. S.M. Kopejkin, Relativistic manifestations of gravitational fields in gravimetry and geodesy. Manuscr. Geod. 16, 301–312 (1991)

    ADS  Google Scholar 

  2. S.M. Kopeikin, E.M. Mazurova, A.P. Karpik, Towards an exact relativistic theory of Earth’s geoid undulation. Phys. Lett. A 379, 1555–1562 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. S.M. Kopeikin, Reference ellipsoid and geoid in chronometric geodesy. Front. Astron. Space Sci. 3(5), 5 (2016)

    ADS  Google Scholar 

  4. S. Kopeikin, W. Han, E. Mazurova, Post-Newtonian reference ellipsoid for relativistic geodesy. Phys. Rev. D 93(4), 044069 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  5. S. Kopeikin, M. Efroimsky, G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley, Berlin, 2011), xxxii+860 pp

    Book  MATH  Google Scholar 

  6. P. Vaníček, E.J. Krakiwsky, Geodesy, the Concepts, 2nd edn. (Amsterdam, North Holland, 1986), xv+697 pp

    Google Scholar 

  7. B. Hofmann-Wellenhof, H. Moritz, Physical Geodesy (Springer, Berlin, 2006)

    Google Scholar 

  8. W. Torge, J. Müller, Geodesy, 4th edn. (De Gruyter, Berlin, 2012), 433 pp

    Google Scholar 

  9. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1975), xiii+402 pp

    Chapter  Google Scholar 

  10. W.A. Heiskanen, H. Moritz, Physical Geodesy (W. H. Freeman, San Francisco, 1967), 364 pp

    Google Scholar 

  11. G. Petit, B. Luzum, IERS conventions. IERS Tech. Note 36, 179 pp. (2010)

    Google Scholar 

  12. K. Sośnica, D. Thaller, A. Jäggi, R. Dach, G. Beutler, Sensitivity of Lageos orbits to global gravity field models. Artif. Satell. 47, 47–65 (2012)

    Article  ADS  Google Scholar 

  13. P.L. Bender, R.S. Nerem, J.M. Wahr, Possible future use of laser gravity gradiometers. Space Sci. Rev. 108, 385–392 (2003)

    Article  ADS  Google Scholar 

  14. N.K. Pavlis, S.A. Holmes, S.C. Kenyon, J.K. Factor, The development and evaluation of the Earth gravitational model 2008 (EGM2008). J. Geophys. Res. Solid Earth 117, B04406 (2012)

    Article  ADS  Google Scholar 

  15. N.K. Pavlis, S.A. Holmes, S.C. Kenyon, J.K. Factor, Correction to “the development and evaluation of the Earth gravitational model 2008 (EGM2008)”. J. Geophys. Res. Solid Earth 118, 2633–2633 (2013)

    Article  ADS  Google Scholar 

  16. Fu Lee-Lueng, On the decadal trend of global mean sea level and its implication on ocean heat content change. Front. Mar. Sci. 3, 37 (2016)

    Google Scholar 

  17. H.-P. Plag, M. Pearlman (eds.), Global Geodetic Observing System (Springer, Dordrecht, 2009), 322 pp

    Google Scholar 

  18. L.-L. Fu, B.J. Haines, The challenges in long-term altimetry calibration for addressing the problem of global sea level change. Adv. Space Res. 51, 1284–1300 (2013)

    Article  ADS  Google Scholar 

  19. Z. Altamimi, P. Rebischung, L. Métivier, X. Collilieux, ITRF2014: a new release of the international Terrestrial Reference Frame modeling nonlinear station motions. J. Geophys. Res. Solid Earth 121, 6109–6131 (2016)

    Article  ADS  Google Scholar 

  20. J. Müller, D. Dirkx, S.M. Kopeikin, G. Lion, I. Panet, G. Petit, P.N.A.M. Visser, High performance clocks and gravity field determination. Space Sci. Rev. 214(5), 1–31 (2018)

    ADS  Google Scholar 

  21. R. Bondarescu, M. Bondarescu, G. Hetényi, L. Boschi, P. Jetzer, J. Balakrishna, Geophysical applicability of atomic clocks: direct continental geoid mapping. Geophys. J. Int. 191, 78–82 (2012)

    Article  ADS  Google Scholar 

  22. E. Mai, J. Müller, General remarks on the potential use of atomic clocks in relativistic geodesy. ZFV - Zeitschrift fur Geodasie, Geoinformation und Landmanagement 138(4), 257–266 (2013)

    Google Scholar 

  23. E. Mai, Time, atomic clocks, and relativistic geodesy. Report No 124, Deutsche Geodátische Kommission der Bayerischen Akademie der Wissenschaften (DGK) (2014), 128 pp., http://dgk.badw.de/fileadmin/docs/a-124.pdf

  24. E. Hackmann, C. Lämmerzahl, Generalized gravitomagnetic clock effect. Phys. Rev. D 90(4), 044059 (2014)

    Article  ADS  Google Scholar 

  25. J.M. Cohen, B. Mashhoon, Standard clocks, interferometry, and gravitomagnetism. Phys. Lett. A 181, 353–358 (1993)

    Article  ADS  Google Scholar 

  26. V.F. Fateev, S.M. Kopeikin, S.L. Pasynok, Effect of irregularities in the earth’s rotation on relativistic shifts in frequency and time of earthbound atomic clocks. Meas. Tech. 58, 647–654 (2015)

    Article  Google Scholar 

  27. M. Soffel, S.A. Klioner, G. Petit, P. Wolf, S.M. Kopeikin, P. Bretagnon, V.A. Brumberg, N. Capitaine, T. Damour, T. Fukushima, B. Guinot, T.-Y. Huang, L. Lindegren, C. Ma, K. Nordtvedt, J.C. Ries, P.K. Seidelmann, D. Vokrouhlický, C.M. Will, C. Xu, The IAU 2000 resolutions for astrometry, celestial mechanics, and metrology in the relativistic framework: explanatory supplement. Astron. J. (USA) 126, 2687–2706 (2003)

    Article  ADS  Google Scholar 

  28. M. Soffel, S. Kopeikin, W.-B. Han, Advanced relativistic VLBI model for geodesy. J. Geod. 91(7), 783–801 (2017)

    Article  ADS  Google Scholar 

  29. V.A. Brumberg, S.M. Kopeikin, Relativistic equations of motion of the earth’s satellite in the geocentric frame of reference. Kinematika i Fizika Nebesnykh Tel 5, 3–8 (1989)

    ADS  MathSciNet  Google Scholar 

  30. V.A. Brumberg, S.M. Kopejkin, Relativistic reference systems and motion of test bodies in the vicinity of the Earth. Nuovo Cim. B Ser. 103, 63–98 (1989)

    Article  ADS  Google Scholar 

  31. T. Damour, M. Soffel, C. Xu, General-relativistic celestial mechanics. IV. Theory of satellite motion. Phys. Rev. D 49, 618–635 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  32. A. San Miguel, Numerical integration of relativistic equations of motion for Earth satellites. Celest. Mech. Dyn. Astron. 103, 17–30 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. U. Kostić, M. Horvat, A. Gomboc, Relativistic positioning system in perturbed spacetime. Class. Quantum Gravity 32(21), 215004 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  34. K.-M. Roh, B.-K. Choi, The effects of the IERS conventions (2010) on high precision orbit propagation. J. Astron. Space Sci. 31, 41–50 (2014)

    Article  ADS  Google Scholar 

  35. K.-M. Roh, S.M. Kopeikin, J.-H. Cho, Numerical simulation of the post-Newtonian equations of motion for the near Earth satellite with an application to the LARES satellite. Adv. Space Res. 58, 2255–2268 (2016)

    Article  ADS  Google Scholar 

  36. I. Ciufolini, E.C. Pavlis, A confirmation of the general relativistic prediction of the Lense-Thirring effect. Nature 431, 958–960 (2004)

    Article  ADS  Google Scholar 

  37. I. Ciufolini, Dragging of inertial frames. Nature 449, 41–47 (2007)

    Article  ADS  Google Scholar 

  38. I. Ciufolini, E.C. Pavlis, A. Paolozzi, J. Ries, R. Koenig, R. Matzner, G. Sindoni, K.H. Neumayer, Phenomenology of the lense-thirring effect in the solar system: measurement of frame-dragging with laser ranged satellites. New Astron. 17, 341–346 (2012)

    Article  ADS  Google Scholar 

  39. V.G. Gurzadyan, I. Ciufolini, A. Paolozzi, A.L. Kashin, H.G. Khachatryan, S. Mirzoyan, G. Sindoni, Satellites testing general relativity: residuals versus perturbations. Int. J. Mod. Phys. D 26, 1741020 (2017)

    Article  ADS  Google Scholar 

  40. K.S. Thorne, R.D. Blandford, Black holes and the origin of radio sources, in Extragalactic Radio Sources. IAU Symposium, vol. 97, ed. by D.S. Heeschen, C.M. Wade (1982), pp. 255–262

    Google Scholar 

  41. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, 1969), ix+252 pp

    Google Scholar 

  42. A.N. Petrov, S.M. Kopeikin, R.R. Lompay, B. Tekin, Metric Theories of Gravity: Perturbations and Conservation Laws (De Gruyter, Berlin, 2017), xxiv+597 pp

    Google Scholar 

  43. L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev. Relativ. 17, 2 (2014)

    Article  ADS  MATH  Google Scholar 

  44. M. Soffel, F. Frutos, On the usefulness of relativistic space-times for the description of the Earth’s gravitational field. J. Geod. 90(12), 1345–1357 (2016)

    Article  ADS  Google Scholar 

  45. D. Philipp, V. Perlick, D. Puetzfeld, E. Hackmann, C. Lämmerzahl, Definition of the relativistic geoid in terms of isochronometric surfaces. Phys. Rev. D 95(10), 104037 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  46. M. Oltean, R.J. Epp, P.L. McGrath, R.B. Mann, Geoids in general relativity: geoid quasilocal frames. Class. Quantum Gravity 33(10), 105001 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. K.D. Krori, P. Borgohain, Uniform-density cold neutron stars in general relativity. J. Phys. Math. Gen. 8, 512–520 (1975)

    Article  ADS  Google Scholar 

  48. J. Ponce de León, Fluid spheres of uniform density in general relativity. J. Math. Phys. 27, 271–276 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. L. Lindblom, Static uniform-density stars must be spherical in general relativity. J. Math. Phys. 29, 436–439 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. J.N. Islam, Rotating Fields in General Relativity (Cambridge University Press, Cambridge, 1985), 127 pp

    Google Scholar 

  51. E. Gourgoulhon, An introduction to the theory of rotating relativistic stars. Lectures Given at the Compstar 2010 School (Caen, 8–16 Feb 2010) (2010)

    Google Scholar 

  52. J.L. Friedman, N. Stergioulas, Rotating Relativistic Stars (Cambridge University Press, Cambridge, 2013), 438 pp

    Google Scholar 

  53. S. Chandrasekhar, The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. I. The Maclaurin spheroids and the virial theorem. Astrophys. J. 142, 1513–1518 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  54. S. Chandrasekhar, The post-Newtonian effects of general relativity on the equilibrium of uniformly rotating bodies. II. The deformed figures of the Maclaurin spheroids. Astrophys. J. 147, 334–352 (1967)

    Article  ADS  MathSciNet  Google Scholar 

  55. N.P. Bondarenko, K.A. Pyragas, On the equilibrium figures of an ideal rotating liquid in the post-Newtonian approximation of general relativity. II: Maclaurin’s P-ellipsoid. Astrophys. Space Sci. 27, 453–466 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. D. Petroff, Post-Newtonian Maclaurin spheroids to arbitrary order. Phys. Rev. D 68(10), 104029 (2003)

    Article  ADS  Google Scholar 

  57. G.L. Clark, The gravitational field of a rotating nearly spherical body. Philos. Mag. 39(297), 747–778 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  58. P. Teyssandier, Rotating stratified ellipsoids of revolution and their effects on the dragging of inertial frames. Phys. Rev. D 18, 1037–1046 (1978)

    Article  ADS  Google Scholar 

  59. H. Cheng, G.-X. Song, C. Huang, The internal and external metrics of a rotating ellipsoid under post-Newtonianian approximation. Chin. Astron. Astrophys. 31, 192–204 (2007)

    Article  ADS  Google Scholar 

  60. C.M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, UK, 1993), xi+396 pp

    Google Scholar 

  61. S. Kopeikin, I. Vlasov, Parametrized post-Newtonian theory of reference frames, multipolar expansions and equations of motion in the N-body problem. Phys. Rep. 400, 209–318 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  62. M.H. Soffel, Relativity in Astrometry, Celestial Mechanics and Geodesy (Springer, Berlin, 1989), xiv+208 pp

    Chapter  Google Scholar 

  63. V.A. Brumberg, Essential Relativistic Celestial Mechanics (Adam Hilger, Bristol, 1991), x+263 pp

    Google Scholar 

  64. A.D. Rendall, Convergent and divergent perturbation series and the post-Minkowskian approximation scheme. Class. Quantum Gravity 7(5), 803–812 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  65. J. Müller, M. Soffel, S.A. Klioner, Geodesy and relativity. J. Geod. 82, 133–145 (2008)

    Article  ADS  MATH  Google Scholar 

  66. V.A. Fock, The Theory of Space, Time and Gravitation, 2nd edn. (Macmillan, New York, 1964); Translated from the Russian by N. Kemmer, xii+448 pp

    Google Scholar 

  67. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972)

    Google Scholar 

  68. A.P. Lightman, W.H. Press, R.H. Price, S.A. Teukolsky, Problem Book in Relativity and Gravitation (Princeton University Press, Princeton, 1975), xiv+603 pp

    Google Scholar 

  69. G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists, 4th edn. (Academic Press, San Diego, 1995), xviii+1029 pp

    Chapter  Google Scholar 

  70. E.W. Hobson, The Theory of Spherical and Elliptical Harmonics (Cambridge University Press, Cambridge, 1931), vi+500 pp

    Google Scholar 

  71. V. Pohánka, Gravitational field of the homogeneous rotational ellipsoidal body: a simple derivation and applications. Contrib. Geophys. Geod. 41, 117–157 (2011)

    Article  ADS  Google Scholar 

  72. I.S. Gradshteyn, I.M. Ryzhik, in Table of Integrals, Series and Products, 4th edn., ed. by Y.V. Geronimus, M.Y. Tseytlin (Academic Press, New York, 1965); First appeared in 1942 as MT15 in the Mathematical tables series of the National Bureau of Standards

    Google Scholar 

  73. I. Ciufolini, J.A. Wheeler, Gravitation and Inertia (Princeton University Press, Princeton, 1995), 512 pp

    Google Scholar 

  74. S.M. Kopeikin, Gravitomagnetism and the speed of gravity. Int. J. Mod. Phys. D 15, 305–320 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. S.M. Kopeikin, The gravitomagnetic influence on Earth-orbiting spacecrafts and on the lunar orbit, in General Relativity and John Archibald Wheeler, vol. 367, Astrophysics and Space Science Library, ed. by I. Ciufolini, R.A.A. Matzner (Springer, Berlin, 2010)

    Chapter  Google Scholar 

  76. B.H. Hager, M.A. Richards, Long-wavelength variations in Earth’s geoid - physical models and dynamical implications. Philos. Trans. R. Soc. Lond. Ser. A 328, 309–327 (1989)

    Article  ADS  Google Scholar 

  77. J.-L. Tassoul, Theory of Rotating Stars (Princeton University Press, Princeton, 1979), xvi+508 pp

    Google Scholar 

  78. P. Pizzetti, Principii della teoria meccanica della figura dei pianeti (E. Spoerri, Pisa, 1913), xiii+251 pp

    Google Scholar 

  79. R.O. Hansen, Multipole moments of stationary space-times. J. Math. Phys. 15, 46–52 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. K.S. Thorne, Multipole expansions of gravitational radiation. Rev. Mod. Phys. 52, 299–339 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  81. L. Blanchet, T. Damour, Radiative gravitational fields in general relativity. I - general structure of the field outside the source. Philos. Trans. R. Soc. Lond. Ser. A 320, 379–430 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. H. Quevedo, Multipole moments in general relativity - static and stationary vacuum solutions. Fortschritte der Physik 38, 733–840 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  83. T. Damour, B.R. Iyer, Multipole analysis for electromagnetism and linearized gravity with irreducible Cartesian tensors. Phys. Rev. D 43, 3259–3272 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  84. L. Blanchet, On the multipole expansion of the gravitational field. Class. Quantum Gravity 15, 1971–1999 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  85. C. Jekeli, The exact transformation between ellipsoidal and spherical harmonic expansions. Manuscr. Geod. 13, 106–113 (1988)

    Google Scholar 

  86. C. Snow, Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd edn. (US Government Printing Office, Washington, 1952), xi+427 pp

    Google Scholar 

  87. M. Soffel, R. Langhans, Space-Time Reference Systems (Springer, Berlin, 2013), xiv+314 pp

    Book  MATH  Google Scholar 

  88. S. Chandrasekhar, J.C. Miller, On slowly rotating homogeneous masses in general relativity. Mon. Not. Roy. Astron. Soc. 167, 63–80 (1974)

    Article  ADS  Google Scholar 

  89. J.M. Bardeen, A reexamination of the post-Newtonian Maclaurin spheroids. Astrophys. J. 167, 425 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  90. R. Meinel, M. Ansorg, A. Kleinwächter, G. Neugebauer, D. Petroff, Relativistic Figures of Equilibrium (Cambridge University Press, Cambridge, 2008), p. ix+218 pp

    Book  MATH  Google Scholar 

  91. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein’s Field Equations, 2nd edn. (Cambridge University Press, Cambridge, 2003), xxix+701 pp

    Google Scholar 

  92. W.C. Hernandez, Material sources for the Kerr metric. Phys. Rev. 159, 1070–1072 (1967)

    Article  ADS  Google Scholar 

  93. J.L. Hernandez-Pastora, L. Herrera, Interior solution for the Kerr metric. Phys. Rev. D 95(2), 024003 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  94. C. Jiang, W. Lin, Harmonic metric for Kerr black hole and its post-Newtonian approximation. Gen. Relativ. Gravit. 46, 1671 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  95. W. Lin, C. Jiang, Exact and unique metric for Kerr-Newman black hole in harmonic coordinates. Phys. Rev. D 89(8), 087502 (2014)

    Article  ADS  Google Scholar 

  96. H. Essén, The physics of rotational flattening and the point core model. Int. J. Geosci. 5, 555–570 (2014)

    Article  Google Scholar 

  97. A. Krasinski, Ellipsoidal space-times, sources for the Kerr metric. Ann. Phys. 112, 22–40 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  98. T. Wolf, G. Neugebauer, About the non-existence of perfect fluid bodies with the Kerr metric outside. Class. Quantum Gravity 9, L37–L42 (1992)

    Article  ADS  MATH  Google Scholar 

  99. C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity (Springer, Singapore, 2017), xv+340 pp

    Chapter  MATH  Google Scholar 

Download references

Acknowledgements

I thank Physikzentrum Bad Honnef for hospitality and Wilhelm and Else Heraeus Stiftung for providing generous travel support to deliver a talk at 609 WE-Heraeus-Seminar “Relativistic Geodesy: Foundations and Applications” (13.03. - 19.03.2016). This work contributes to the research project “Spacetime Metrology, Clocks and Relativistic Geodesy” [http://www.issibern.ch/teams/spacetimemetrology/] sponsored by the International Space Science Institute (ISSI) in Bern, Switzerland.

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, University of Missouri, 322 Physics Bldg, Columbia, MO, 65211, USA

    Sergei Kopeikin

Authors
  1. Sergei Kopeikin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergei Kopeikin .

Editor information

Editors and Affiliations

  1. ZARM, University of Bremen, Bremen, Germany

    Dirk Puetzfeld

  2. ZARM, University of Bremen, Bremen, Germany

    Claus Lämmerzahl

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kopeikin, S. (2019). Reference-Ellipsoid and Normal Gravity Field in Post-Newtonian Geodesy. In: Puetzfeld, D., Lämmerzahl, C. (eds) Relativistic Geodesy. Fundamental Theories of Physics, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-030-11500-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation