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Encyclopedia of Geodesy pp 1–6Cite as

Operational Significance of the Deviation Equation in Relativistic Geodesy

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Part of the book series: Encyclopedia of Earth Sciences Series ((EESS))

Definitions

Deviation equation . Second-order differential equation for the 4-vector which measures the distance between reference points on neighboring world lines in spacetime manifolds.

Relativistic geodesy . Science representing the Earth (or any planet), including the measurement of its gravitational field, in a four-dimensional curved spacetime using differential-geometric methods in the framework of Einstein’s theory of gravitation (general relativity).

Introduction

How does one measure the gravitational field in Einstein’s theory? What is the foundation of relativistic gradiometry? The deviation equation gives answers to these fundamental questions.

In Einstein’s theory of gravitation, i.e., general relativity, the gravitational field manifests itself in the form of the Riemannian curvature tensor Rabcd (Synge, 1960). This 4th-rank tensor can be defined as a measure of the noncommutativity of the parallel transport process of the underlying spacetime manifold (Synge and...

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References

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  • Synge, J. L., 1926. The first and second variations of the length integral in Riemannian space. Proceedings of the London Mathematical Society, 25, 247.

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Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant PU 461/1-1 (D.P.). The work of Y.N.O. was partially supported by PIER (“Partnership for Innovation, Education and Research” between DESY and Universität Hamburg) and by the Russian Foundation for Basic Research (Grant No. 16-02-00844-A).

Author information

Authors and Affiliations

  1. ZARM, University of Bremen, Bremen, Germany

    Dirk Puetzfeld

  2. Theoretical Physics Laboratory, Nuclear Safety Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri N. Obukhov

Authors
  1. Dirk Puetzfeld
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  2. Yuri N. Obukhov
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Corresponding author

Correspondence to Dirk Puetzfeld .

Editor information

Editors and Affiliations

  1. Geodetic Institute, University of Stuttgart, Stuttgart, Germany

    Erik Grafarend

Explicit Solutions

Explicit Solutions

A.1 General Spacetime

$$ 01:{R}_{1010}=\frac{3}{4}{{}^{\left(1,1\right)}A}_1{c}_{10}^{-2}, $$
(14)
$$ 02:{R}_{2010}=\frac{3}{4}{{}^{\left(1,1\right)}A}_2{c}_{10}^{-2}, $$
(15)
$$ 03:{R}_{3010}=\frac{3}{4}{{}^{\left(1,1\right)}A}_3{c}_{10}^{-2}, $$
(16)
$$ 04:{R}_{2020}=\frac{3}{4}{{}^{\left(1,2\right)}A}_2{c}_{10}^{-2}, $$
(17)
$$ 05:{R}_{3020}=\frac{3}{4}{{}^{\left(1,2\right)}A}_3{c}_{10}^{-2}, $$
(18)
$$ 06:{R}_{3030}=\frac{3}{4}{{}^{\left(1,3\right)}A}_3{c}_{10}^{-2}, $$
(19)
$$ 07:{R}_{2110}=\frac{3}{4}{{}^{\left(2,1\right)}A}_2{c}_{21}^{-1}{c}_{20}^{-1}-{R}_{2010}{c}_{21}^{-1}{c}_{20}, $$
(20)
$$ 08:{R}_{3110}=\frac{3}{4}{{}^{\left(2,1\right)}A}_3{c}_{21}^{-1}{c}_{20}^{-1}-{R}_{3010}{c}_{21}^{-1}{c}_{20}, $$
(21)
$$ 09:{R}_{0212}=\frac{3}{4}{{}^{\left(3,1\right)}A}_0{c}_{32}^{-2}+{R}_{2010}{c}_{32}^{-1}{c}_{30}, $$
(22)
$$ {\displaystyle \begin{array}{c}10:{R}_{1212}=\frac{3}{4}{{}^{\left(2,2\right)}A}_2{c}_{21}^{-2}-{R}_{2020}{c}_{20}^2{c}_{21}^{-2}\\ {}\qquad -2{R}_{0212}{c}_{21}^{-1}{c}_{20},\end{array}} $$
(23)
$$ 11:{R}_{3220}=\frac{3}{4}{{}^{\left(3,2\right)}A}_3{c}_{32}^{-1}{c}_{30}^{-1}-{R}_{3020}{c}_{32}^{-1}{c}_{30}, $$
(24)
$$ 12:{R}_{0313}=\frac{3}{4}{{}^{\left(4,1\right)}A}_0{c}_{43}^{-2}+{R}_{3010}{c}_{43}^{-1}{c}_{40}, $$
(25)
$$ {\displaystyle \begin{array}{c}13:{R}_{1313}=\frac{3}{4}{{}^{\left(2,3\right)}A}_3{c}_{21}^{-2}-{R}_{3030}{c}_{20}^2{c}_{21}^{-2}\\ {}\qquad -2{R}_{0313}{c}_{21}^{-1}{c}_{20},\end{array}} $$
(26)
$$ 14:{R}_{0323}=\frac{3}{4}{{}^{\left(4,2\right)}A}_0{c}_{43}^{-2}+{R}_{3020}{c}_{43}^{-1}{c}_{40}, $$
(27)
$$ {\displaystyle \begin{array}{c}15:{R}_{2323}=\frac{3}{4}{{}^{\left(4,2\right)}A}_2{c}_{43}^{-2}-{R}_{2020}{c}_{43}^{-2}{c}_{40}^2\\ {}\qquad +2{R}_{3220}{c}_{43}^{-1}{c}_{40},\end{array}} $$
(28)
$$ {\displaystyle \begin{array}{lll}16:{R}_{3132}=\frac{3}{8}{{}^{\left(5,3\right)}A}_3{c}_{52}^{-1}{c}_{51}^{-1}-\frac{1}{2}{R}_{3030}{c}_{52}^{-1}{c}_{51}^{-1}{c}_{50}^2\\ {}& -{R}_{0313}{c}_{52}^{-1}{c}_{50}-{R}_{0323}{c}_{51}^{-1}{c}_{50}\\ {}& \quad -\frac{1}{2}{R}_{1313}{c}_{52}^{-1}{c}_{51}-\frac{1}{2}{R}_{2323}{c}_{52}{c}_{51}^{-1},\end{array}} $$
(29)
$$ {\displaystyle \begin{array}{lll}17:{R}_{1213}=\frac{3}{8}{{}^{\left(6,1\right)}A}_1{c}_{63}^{-1}{c}_{62}^{-1}-\frac{1}{2}{R}_{1010}{c}_{63}^{-1}{c}_{62}^{-1}{c}_{60}^2\\ {}& +{R}_{2110}{c}_{63}^{-1}{c}_{60}+{R}_{3110}{c}_{62}^{-1}{c}_{60}\\ {}& \quad -\frac{1}{2}{R}_{1212}{c}_{63}^{-1}{c}_{62}-\frac{1}{2}{R}_{1313}{c}_{63}{c}_{62}^{-1},\end{array}} $$
(30)
$$ 18:{R}_{0231}=\frac{1}{4}{{}^{\left(4,1\right)}A}_2{c}_{40}^{-1}{c}_{43}^{-1}-\frac{1}{4}{{}^{\left(2,2\right)}A}_3{c}_{20}^{-1}{c}_{21}^{-1}+\frac{1}{3}\left({R}_{3020}{c}_{20}{c}_{21}^{-1}+{R}_{3121}{c}_{21}{c}_{20}^{-1}-{R}_{2010}{c}_{40}{c}_{43}^{-1}-{R}_{2313}{c}_{43}{c}_{40}^{-1}\right), $$
(31)
$$ 19:{R}_{0312}=\frac{1}{4}{{}^{\left(4,1\right)}A}_2{c}_{40}^{-1}{c}_{42}^{-1}+\frac{1}{2}{{}^{\left(2,2\right)}A}_3{c}_{20}^{-1}{c}_{21}^{-1}-\frac{1}{3}\left(2{R}_{3020}{c}_{20}{c}_{21}^{-1}+2{R}_{3121}{c}_{21}{c}_{20}^{-1}+{R}_{2010}{c}_{40}{c}_{43}^{-1}+{R}_{2313}{c}_{43}{c}_{40}^{-1}\right), $$
(32)
$$ 20:{R}_{3212}=\frac{3}{4}{{}^{\left(4,1\right)}A}_3{c}_{20}^{-1}{c}_{21}^{-1}{c}_{50}{c}_{52}^{-1}-\frac{3}{4}{{}^{\left(5,2\right)}A}_3{c}_{51}^{-1}{c}_{52}^{-1}+{R}_{3121}{c}_{52}^{-1}\left({c}_{51}-{c}_{50}{c}_{21}{c}_{20}^{-1}\right)+{R}_{3220}{c}_{50}{c}_{51}^{-1}+{R}_{3020}{c}_{50}{c}_{52}^{-1}\left({c}_{50}{c}_{51}^{-1}-{c}_{20}{c}_{21}^{-1}\right). $$
(33)

A.2 Vacuum Spacetime

$$ 01:{C}_{1010}=\frac{3}{4}{{}^{\left(1,1\right)}A}_1{c}_{10}^{-2}, $$
(34)
$$ 02:{C}_{2010}=\frac{3}{4}{{}^{\left(1,1\right)}A}_2{c}_{10}^{-2}, $$
(35)
$$ 03:{C}_{3010}=\frac{3}{4}{{}^{\left(1,1\right)}A}_3{c}_{10}^{-2}, $$
(36)
$$ 04:{C}_{2020}=\frac{3}{4}{{}^{\left(1,2\right)}A}_2{c}_{10}^{-2}, $$
(37)
$$ 05:{C}_{3020}=\frac{3}{4}{{}^{\left(1,2\right)}A}_3{c}_{10}^{-2}, $$
(38)
$$ 06:{C}_{2110}=\frac{3}{4}{{}^{\left(2,1\right)}A}_2{c}_{21}^{-1}{c}_{20}^{-1}-{C}_{2010}{c}_{21}^{-1}{c}_{20}, $$
(39)
$$ 07:{C}_{3110}=\frac{3}{4}{{}^{\left(2,1\right)}A}_3{c}_{21}^{-1}{c}_{20}^{-1}-{C}_{3010}{c}_{21}^{-1}{c}_{20}, $$
(40)
$$ 08:{C}_{0212}=\frac{3}{4}{{}^{\left(3,1\right)}A}_0{c}_{32}^{-2}+{C}_{2010}{c}_{32}^{-1}{c}_{30}, $$
(41)
$$ 09:{C}_{0231}=\frac{1}{4}{{}^{\left(4,1\right)}A}_2{c}_{40}^{-1}{c}_{43}^{-1}-\frac{1}{4}{{}^{\left(2,2\right)}A}_3{c}_{20}^{-1}{c}_{21}^{-1}\qquad+\frac{1}{3}{C}_{3020}\left({c}_{20}{c}_{21}^{-1}+{c}_{21}{c}_{20}^{-1}\right)\qquad -\frac{1}{3}{C}_{2010}\left({c}_{40}{c}_{43}^{-1}+{c}_{43}{c}_{40}^{-1}\right), $$
(42)
$$ 10:{C}_{0312}=\frac{1}{4}{{}^{\left(4,1\right)}A}_2{c}_{40}^{-1}{c}_{42}^{-1}+\frac{1}{2}{{}^{\left(2,2\right)}A}_3{c}_{20}^{-1}{c}_{21}^{-1}-\frac{2}{3}{C}_{3020}\left({c}_{20}{c}_{21}^{-1}+{c}_{21}{c}_{20}^{-1}\right)+\frac{1}{3}{C}_{2010}\left({c}_{40}{c}_{43}^{-1}+{c}_{43}{c}_{40}^{-1}\right). $$
(43)

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Puetzfeld, D., Obukhov, Y.N. (2019). Operational Significance of the Deviation Equation in Relativistic Geodesy. In: Grafarend, E. (eds) Encyclopedia of Geodesy. Encyclopedia of Earth Sciences Series. Springer, Cham. https://doi.org/10.1007/978-3-319-02370-0_164-1

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  • DOI: https://doi.org/10.1007/978-3-319-02370-0_164-1

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  • Print ISBN: 978-3-319-02370-0

  • Online ISBN: 978-3-319-02370-0

  • eBook Packages: Springer Reference Earth and Environm. ScienceReference Module Physical and Materials ScienceReference Module Earth and Environmental Sciences

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