Multipolar Test Body Equations of Motion in Generalized Gravity Theories

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.

Appendices

Appendix

A Conventions and Symbols

Table 3 Directory of symbols

Table 4 Directory of symbols

Table 5 Directory of 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

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

(235)

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

(236)

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):

$$\begin{aligned} \left[ B_{\ldots } \right] _{;y} = \left[ B_{\ldots ; y} \right] + \left[ B_{\ldots ; x} \right] . \end{aligned}$$

(237)

Here a coincidence limit of a bitensor $B_{\ldots }(x,y)$ is a tensor

$$\begin{aligned} \left[ B_{\ldots } \right] = \lim \limits _{x\rightarrow y}\,B_{\ldots }(x,y), \end{aligned}$$

(238)

determined at $y$. Furthermore, we collect the following useful identities:

$$\begin{aligned}&\sigma _{y_0 y_1 x_0 y_2 x_1} = \sigma _{y_0 y_1 y_2 x_0 x_1} = \sigma _{x_0 x_1 y_0 y_1 y_2 }, \end{aligned}$$

(239)

$$\begin{aligned}&g^{x_1 x_2} \sigma _{x_1} \sigma _{x_2} = 2 \sigma = g^{y_1 y_2} \sigma _{y_1} \sigma _{y_2}, \end{aligned}$$

(240)

$$\begin{aligned}&\left[ \sigma \right] =0, \quad \left[ \sigma _x \right] = \left[ \sigma _y \right] = 0, \end{aligned}$$

(241)

$$\begin{aligned}&\left[ \sigma _{x_1 x_2} \right] = \left[ \sigma _{y_1 y_2} \right] = g_{y_1 y_2}, \quad \left[ \sigma _{x_1 y_2} \right] = \left[ \sigma _{y_1 x_2} \right] = - g_{y_1 y_2}, \end{aligned}$$

(242)

$$\begin{aligned}&\left[ \sigma _{x_1 x_2 x_3} \right] = \left[ \sigma _{x_1 x_2 y_3} \right] = \left[ \sigma _{x_1 y_2 y_3} \right] = \left[ \sigma _{y_1 y_2 y_3} \right] = 0,\end{aligned}$$

(243)

$$\begin{aligned}&\left[ g^{x_0}{}_{y_1} \right] = \delta ^{y_0}{}_{y_1}, \quad \left[ g^{x_0}{}_{y_1 ; x_2} \right] = \left[ g^{x_0}{}_{y_1 ; y_2} \right] = 0, \end{aligned}$$

(244)

$$\begin{aligned}&\left[ g^{x_0}{}_{y_1 ; x_2 x_3} \right] = \frac{1}{2} \widetilde{R}{}^{y_0}{}_{y_1 y_2 y_3}. \end{aligned}$$

(245)

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:

$$\begin{aligned} \sigma ^{y_0}{}_{x_1}= & {} g^{y'}{}_{x_1}\biggl ( -\,\delta ^{y_0}{}_{y'} +\,\sum \limits _{k=2}^\infty \,{\frac{1}{k!}}\,\alpha ^{y_0}{}_{y'y_2\!\ldots \!y_{k+1}}\sigma ^{y_2}\cdots \sigma ^{y_{k+1}}\biggr )\!,\end{aligned}$$

(246)

$$\begin{aligned} \sigma ^{y_0}{}_{y_1}= & {} \delta ^{y_0}{}_{y_1} -\,\sum \limits _{k=2}^\infty \,{\frac{1}{k!}}\,\beta ^{y_0}{}_{y_1y_2\ldots y_{k+1}} \sigma ^{y_2}\!\cdots \!\sigma ^{y_{k+1}}, \end{aligned}$$

(247)

$$\begin{aligned} g^{y_0}{}_{x_1 ; x_2}= & {} g^{y'}{\!}_{x_1} g^{y''}{\!}_{x_2}\biggl ({\frac{1}{2}} \widetilde{R}{}^{y_0}{}_{y'y''y_3}\sigma ^{y_3}\!+\!\sum \limits _{k=2}^\infty \,{\frac{1}{k!}}\,\gamma ^{y_0}{}_{y'y''y_3\ldots y_{k+2}}\sigma ^{y_3}\!\cdots \!\sigma ^{y_{k+2}}\!\biggr )\!,\nonumber \\ \end{aligned}$$

(248)

$$\begin{aligned} g^{y_0}{}_{x_1 ; y_2}= & {} g^{y'}{\!}_{x_1} \biggl ({\frac{1}{2}} \widetilde{R}{}^{y_0}{}_{y'y_2y_3}\sigma ^{y_3}\!+\!\sum \limits _{k=2}^\infty \,{\frac{1}{k!}}\,\gamma ^{y_0}{}_{y'y_2y_3\ldots y_{k+2}}\sigma ^{y_3}\!\cdots \!\sigma ^{y_{k+2}}\!\biggr ).\end{aligned}$$

(249)

$$\begin{aligned} G^{Y_0}{}_{X_1 ; x_2}= & {} G^{Y'}{\!}_{X_1} g^{y''}{\!}_{x_2} \sum \limits _{k=1}^\infty \,{\frac{1}{k!}}\,\gamma ^{Y_0}{}_{Y'y''y_3\ldots y_{k+2}}\sigma ^{y_3}\!\cdots \!\sigma ^{y_{k+2}}, \end{aligned}$$

(250)

$$\begin{aligned} G^{Y_0}{}_{X_1 ; y_2}= & {} G^{Y'}{\!}_{X_1} \sum \limits _{k=1}^\infty \,{\frac{1}{k!}}\,\gamma ^{Y_0}{}_{Y'y_2y_3\ldots y_{k+2}}\sigma ^{y_3}\!\cdots \!\sigma ^{y_{k+2}}. \end{aligned}$$

(251)

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:

$$\begin{aligned} \alpha ^{y_0}{}_{y_1y_2y_3}= & {} - \frac{1}{3} \widetilde{R}{}^{y_0}{}_{(y_2y_3)y_1},\end{aligned}$$

(252)

$$\begin{aligned} \beta ^{y_0}{}_{y_1y_2y_3}= & {} \frac{2}{3}\widetilde{R}{}^{y_0}{}_{(y_2y_3)y_1},\end{aligned}$$

(253)

$$\begin{aligned} \alpha ^{y_0}{}_{y_1y_2y_3y_4}= & {} \frac{1}{2} \widetilde{\nabla }_{(y_2}\widetilde{R}{}^{y_0}{}_{y_3y_4)y_1},\end{aligned}$$

(254)

$$\begin{aligned} \beta ^{y_0}{}_{y_1y_2y_3y_4}= & {} - \frac{1}{2} \widetilde{\nabla }_{(y_2} \widetilde{R}{}^{y_0}{}_{y_3y_4)y_1},\end{aligned}$$

(255)

$$\begin{aligned} \gamma ^{y_0}{}_{y_1y_2y_3y_4}= & {} \frac{1}{3} \widetilde{\nabla }_{(y_3} \widetilde{R}{}^{y_0}{}_{|y_1|y_4)y_2}. \end{aligned}$$

(256)

In addition, we also need the covariant expansion of a vector:

$$\begin{aligned} A_x = g^{y_0}{}_x\,\sum \limits _{k=0}^\infty \,{\frac{(-1)^k}{k!}} \, A_{y_0;y_1\ldots y_k}\,\sigma ^{y_1}\cdots \sigma ^{y_k}. \end{aligned}$$

(257)

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

$$\begin{aligned} \Phi ^{y_1\ldots y_ny_0}{}_{x_0}:= & {} \sigma ^{y_1} \cdots \sigma ^{y_n} g^{y_0}{}_{x_0},\end{aligned}$$

(258)

$$\begin{aligned} \Psi ^{y_1\ldots y_ny_0y'}{}_{x_0x'}:= & {} \sigma ^{y_1} \cdots \sigma ^{y_n} g^{y_0}{}_{x_0}g^{y'}{}_{x'}. \end{aligned}$$

(259)

Their derivatives

$$\begin{aligned} \Psi ^{y_1\ldots y_ny_0y'}{}_{x_0x';z}= & {} \sum ^{n}_{a=1}\sigma ^{y_1}\cdots \sigma ^{y_a}{}_z\cdots \sigma ^{y_n}g^{y_0}{}_{x_0}g^{y'}{}_{x'} \nonumber \\&+ \sigma ^{y_1} \cdots \sigma ^{y_n}\left( g^{y_0}{}_{x_0;z}g^{y'}{}_{x'} + g^{y_0}{}_{x_0}g^{y'}{}_{x';z}\right) ,\end{aligned}$$

(260)

$$\begin{aligned} \Phi ^{y_1\ldots y_ny_0}{}_{x_0;z}= & {} \sum ^{n}_{a=1}\sigma ^{y_1}\cdots \sigma ^{y_a}{}_z\cdots \sigma ^{y_n}g^{y_0}{}_{x_0}\nonumber \\&+ \sigma ^{y_1} \cdots \sigma ^{y_n}\,g^{y_0}{}_{x_0;z}, \end{aligned}$$

(261)

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:

$$\begin{aligned}&{\frac{D}{\textit{ds}}} \int \Psi ^{y_1\ldots y_ny_0y'}{}_{x_0x'}{\mathfrak S}^{x_0 x' x_2} d\Sigma _{x_2} = \nonumber \\&\int \Psi ^{y_1\ldots y_ny_0y'}{}_{x_0x'}\Big ( - U_{x''''}{}^{x'}{}_{x''}{}^{x_0}{}_{x'''}{\mathfrak S}^{x'' x''' x''''} + {\mathfrak T}^{x' x_0} - {\mathfrak t}^{x' x_0}\Big ) w^{x_2} d\Sigma _{x_2} \nonumber \\&+\,\int \Psi ^{y_1\ldots y_ny_0y'}{}_{x_0x';x''}{\mathfrak S}^{x_0 x' x''}w^{x_2} d\Sigma _{x_2}\nonumber \\&+ \int v^{y_{n+1}}\Psi ^{y_1\ldots y_ny_0y'}{}_{x_0x';y_{n+1}}{\mathfrak S}^{x_0 x' x_2}d\Sigma _{x_2}, \end{aligned}$$

(262)

$$\begin{aligned}&{\frac{D}{\textit{ds}}} \int \Phi ^{y_1\ldots y_ny_0}{}_{x_0}{\mathfrak T}^{x_0 x_2} d\Sigma _{x_2} = \int \Phi ^{y_1\ldots y_ny_0}{}_{x_0} \Big (-V_{x''}{}^{x_0}{}_{x'}{\mathfrak T}^{x' x''} \nonumber \\&- R^{x_0}{}_{x''' x' x''}{\mathfrak S}^{x' x'' x'''} - \frac{1}{2} Q^{x_0}{}_{x'' x'}{\mathfrak t}^{x' x''} -\,A^{x_0}{\mathfrak L}_\mathrm{mat}\Big ) w^{x_2} d\Sigma _{x_2} \nonumber \\&+ \int \Phi ^{y_1\ldots y_ny_0}{}_{x_0;x'}{\mathfrak T}^{x_0 x'}w^{x_2} d\Sigma _{x_2} + \int v^{y_{n+1}}\Phi ^{y_1\ldots y_ny_0}{}_{x_0;y_{n+1}}{\mathfrak T}^{x_0 x_2} d\Sigma _{x_2}. \nonumber \\ \end{aligned}$$

(263)

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:

$$\begin{aligned} p^{y_1\ldots y_n y_0}:= & {} (-1)^n\!\!\int \limits _{\Sigma (\tau )}\!\!\Phi ^{y_1\ldots y_n y_0}{}_{x_0}{\mathfrak T}^{x_0 x_1}d\Sigma _{x_1},\end{aligned}$$

(264)

$$\begin{aligned} k^{y_2\ldots y_{n+1} y_0 y_1}:= & {} (-1)^n\!\!\int \limits _{\Sigma (\tau )}\!\!\Psi ^{{y_2}\ldots {y_{n+1} y_0 y_1}}{}_{x_0 x_1}{\mathfrak T}^{x_0 x_1}w^{x_2}d\Sigma _{x_2},\end{aligned}$$

(265)

$$\begin{aligned} h^{y_2\ldots y_{n+1}y_0 y_1}:= & {} (-1)^n\!\!\int \limits _{\Sigma (\tau )}\!\!\Psi ^{y_2 \ldots y_{n+1}y_0 y_1}{}_{x_0 x_1 }{\mathfrak S}^{x_0 x_1 x_2}d\Sigma _{x_2},\end{aligned}$$

(266)

$$\begin{aligned} q^{y_3\ldots y_{n+2}y_0 y_1 y_2}:= & {} (-1)^n\!\!\int \limits _{\Sigma (\tau )}\!\!\Psi ^{y_3 \ldots y_{n+2} y_0 y_1}{}_{x_0 x_1} g^{y_2}{}_{x_2}{\mathfrak S}^{x_0 x_1 x_2 }w^{x_3}d\Sigma _{x_3},\end{aligned}$$

(267)

$$\begin{aligned} \mu ^{y_2\ldots y_{n+1} y_0 y_1}:= & {} (-1)^n\!\!\int \limits _{\Sigma (\tau )}\!\!\Psi ^{y_2 \ldots y_{n+1} y_0 y_1}{}_{x_0 x_1}{\mathfrak t}^{x_0 x_1}w^{x_2}d\Sigma _{x_2},\end{aligned}$$

(268)

$$\begin{aligned} \xi ^{y_1\ldots y_{n}}:= & {} (-1)^n\!\!\int \limits _{\Sigma (\tau )}\!\!\sigma ^{y_1}\cdots \sigma ^{y_{n}}{\mathfrak L}_\mathrm{mat}w^{x_2}d\Sigma _{x_2}. \end{aligned}$$

(269)