On the Applicability of the Geodesic Deviation Equation in General Relativity
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.
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.
- 4.D. Philipp, V. Perlick, C. Lämmerzahl, K. Deshpande, On geodesic deviation in Schwarzschild spacetime, in IEEE Metrology for Aerospace (2015), p. 198Google Scholar
- 6.W.G. Dixon, The new mechanics of Myron Mathisson and its subsequent development, in Equations of Motion in Relativistic Gravity, ed. by D. Puetzfeld et al. Fundamental theories of Physics, vol. 179 (Springer, Berlin, 2015), p. 1Google Scholar
- 14.C. Misner, K. Thorne, J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973)Google Scholar
- 19.E. Hackmann, C. Lämmerzahl, Geodesic equation in Schwarzschild-(anti-)de Sitter space-times: analytical solutions and applications. Phys. Rev. D 78 (2008)Google Scholar
- 23.W. Zimdahl, Shirokov effect and gravitationally induced supercurrents. Exp. Tech. Phys. 33, 403 (1985)Google Scholar