Abstract
We give an overview of the derivation of multipolar equations of motion of extended test bodies for a wide set of gravitational theories beyond the standard general relativistic framework. The classes of theories covered range from simple generalizations of General Relativity, e.g., encompassing additional scalar fields, to theories with additional geometrical structures which are needed for the description of microstructured matter. Our unified framework even allows to handle theories with nonminimal coupling to matter, and thereby for a systematic test of a very broad range of gravitational theories.
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Notes
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Notice a different conventional sign, as compared to our previous work [26].
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Acknowledgments
We would like to thank A. Trautman (University of Warsaw), W.G. Dixon (University of Cambridge), J. Madore (University of Paris South), and W. Tulczyjew (INFN Napoli) for sharing their insights into gravitational multipole formalisms and discussing their pioneering works with us. Furthermore, we would like to thank F.W. Hehl (University of Cologne) for fruitful discussion on gauge gravity models, in particular on Metric-Affine Gravity (MAG). This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant LA-905/8-1/2 and SFB 1128/1 (D.P.).
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Appendices
Appendix
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 Tables 3, 4 and 5.
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. As is well-known, the number of such permutations is equal to \(k!\). The sign factor depends on whether a permutation is even (\(|\pi | = 0\)) or odd (\(|\pi | = 1\)). The number of independent components of the totally symmetric tensor \(T_{(a_1\ldots a_k)}\) of rank \(k\) in \(n\) dimensions is equal to the binomial coefficient \({\left( {\begin{array}{c}n-1+k\\ k\end{array}}\right) } = (n-1+k)!/[k!(n-1)!]\), whereas the number of independent components of the totally antisymmetric tensor \(T_{[a_1\ldots a_k]}\) of rank \(k\) in \(n\) dimensions is equal to the binomial coefficient \({\left( {\begin{array}{c}n\\ k\end{array}}\right) } = n!/[k!(n-k)!]\). For example, for a second rank tensor \(T_{ab}\) the symmetrization yields a tensor \(T_{(ab)} = {\frac{1}{2}}(T_{ab} + T_{ba})\) with 10 independent components, and the antisymmetrization yields another tensor \(T_{[ab]} = {\frac{1}{2}}(T_{ab} - T_{ba})\) with 6 independent components.
In the derivation of the equations of motion we made use of the bitensor formalism, see, e.g., [7, 8, 80] for introductions and references. In particular, the world-function is defined as an integral \(\sigma (x,y) := \frac{1}{2} \epsilon \left( \int \limits _x^y d\tau \right) ^2\) over the geodesic curve connecting the spacetime points \(x\) and \(y\), where \(\epsilon = \pm 1\) for timelike/spacelike curves. Note that our curvature conventions differ from those in [7, 80]. Indices attached to the world-function always denote covariant derivatives, at the given point, i.e. \(\sigma _y:= \widetilde{\nabla }_y \sigma \), hence we do not make explicit use of the semicolon in case of the world-function. The parallel propagator by \(g^y{}_x(x,y)\) allows for the parallel transportation of objects along the unique geodesic that links the points \(x\) and \(y\). For example, given a vector \(V^x\) at \(x\), the corresponding vector at \(y\) is obtained by means of the parallel transport along the geodesic curve as \(V^y = g^y{}_x(x,y)V^x\). For more details see, e.g., [7, 8] or Sect. 5 in [80]. A compact summary of useful formulas in the context of the bitensor formalism can also be found in the appendices A and B of [26].
We start by stating, without proof, the following useful rule for a bitensor \(B\) with arbitrary indices at different points (here just denoted by dots):
Here a coincidence limit of a bitensor \(B_{\ldots }(x,y)\) is a tensor
determined at \(y\). Furthermore, we collect the following useful identities:
B Covariant Expansions
Here we briefly summarize the covariant expansions of the second derivative of the world-function, and the derivative of the parallel propagator:
The coefficients \(\alpha , \beta , \gamma \) in these expansions are polynomials constructed from the Riemann curvature tensor and its covariant derivatives. The first coefficients read as follows:
In addition, we also need the covariant expansion of a vector:
C Explicit Form
Here we make contact with our notation in [51] to facilitate a direct comparison to the results in there.
We introduce the auxiliary variables
Their derivatives
can be straightforwardly evaluated by using the expansions from the previous appendix.
In terms of (258) and (259) the integrated conservation laws (127) and (128) take the form:
This form allows for a direct comparison to (29) and (30) in [51]. Explicitly, in terms of (258) and (259) the integrated moments from (141)–(143) are given by:
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Obukhov, Y.N., Puetzfeld, D. (2015). Multipolar Test Body Equations of Motion in Generalized Gravity Theories. In: Puetzfeld, D., Lämmerzahl, C., Schutz, B. (eds) Equations of Motion in Relativistic Gravity. Fundamental Theories of Physics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-18335-0_2
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