Abstract
Within the theory of General Relativity, we study the solution and range of applicability of the standard geodesic deviation equation in highly symmetric spacetimes. In the Schwarzschild spacetime, the solution is used to model satellite orbit constellations and their deviations around a spherically symmetric Earth model. We investigate the spatial shape and orbital elements of perturbations of circular reference curves. In particular, we reconsider the deviation equation in Newtonian gravity and then determine relativistic effects within the theory of General Relativity by comparison. The deviation of nearby satellite orbits, as constructed from exact solutions of the underlying geodesic equation, is compared to the solution of the geodesic deviation equation to assess the accuracy of the latter. Furthermore, we comment on the so-called Shirokov effect in the Schwarzschild spacetime and limitations of the first order deviation approach.
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Notes
- 1.
Planets do not possess any net charge, therefore we do not consider charged solutions like, for example, the Reissner–Nordstrøm spacetime.
- 2.
- 3.
Note the misprint in Eq. (4.14) in [13], where actually the inverse value of the correct result is shown and we assume this to be simply a typo.
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Acknowledgements
The present work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant PU 461/1-1 (D.P.), the Sonderforschungsbereich (SFB) 1128 Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q), and the Research Training Group 1620 Models of Gravity. We also acknowledge support by the German Space Agency DLR with funds provided by the Federal Ministry of Economics and Technology (BMWi) under grant number DLR 50WM1547.
The authors would like to thank V. Perlick, J.W. van Holten, and Y.N. Obukhov for valuable discussions.
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A Conventions and Symbols
A Conventions and Symbols
In the following we summarize our conventions, and collect some frequently used formulas. A directory of symbols used throughout the text can be found in Table 3. The signature of the spacetime metric is assumed to be \((+,-,-,-)\). Latin indices \(i,j,k,\ldots \) are spacetime indices and take values \(0 \ldots 3\). For an arbitrary k-tensor \(T_{a_1 \ldots a_k}\), the symmetrization and antisymmetrization are defined by
where the sum is taken over all possible permutations (symbolically denoted by \(\pi _I\!\{a_1\ldots a_k\}\)) of its k indices.
The covariant derivative defined by the Riemannian connection is conventionally denoted by the nabla or by the semicolon: \(\nabla _a =\) “\( {}_{;a}\)”. Our conventions for the Riemann curvature are as follows:
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Philipp, D., Puetzfeld, D., Lämmerzahl, C. (2019). On the Applicability of the Geodesic Deviation Equation in General Relativity. 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_13
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