On the Applicability of the Geodesic Deviation Equation in General Relativity

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.

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

$$\begin{aligned} T_{(a_1\ldots a_k)}:= & {} {\frac{1}{k!}}\sum _{I=1}^{k!}T_{\pi _I\!\{a_1\ldots a_k\}},\end{aligned}$$

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$$\begin{aligned} T_{[a_1\ldots a_k]}:= & {} {\frac{1}{k!}}\sum _{I=1}^{k!}(-1)^{|\pi _I|}T_{\pi _I\!\{a_1\ldots a_k\}}, \end{aligned}$$

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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:

$$\begin{aligned}&2 A^{c_1 \ldots c_k}{}_{d_1 \ldots d_l ; [ba] } \equiv 2 \nabla _{[a} \nabla _{b]} A^{c_1 \ldots c_k}{}_{d_1 \ldots d_l} \nonumber \\&= \sum ^{k}_{i=1} R_{abe}{}^{c_i} A^{c_1 \ldots e \ldots c_k}{}_{d_1 \ldots d_l} - \sum ^{l}_{j=1} R_{abd_j}{}^{e} A^{c_1 \ldots c_k}{}_{d_1 \ldots e \ldots d_l}. \end{aligned}$$

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