Multipolar Test Body Equations of Motion in Generalized Gravity Theories
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
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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.
1 Introduction In this work we present a general unified multipolar framework, which enables us to derive equations of motion of extended test bodies for a wide range of gravitational theories. The framework presented here can be applied to theories which significantly go beyond General Relativity (GR), and range from the most straightforward extensions of GR, like scalar-tensor theories, to theories with additional geometrical structures and nonminimal coupling. The multipolar method which we employ here can be thought of as the direct generalization of the ideas pioneered by Mathisson [1], Papapetrou [2], and Dixon [3–6] to the case of generalized gravity theories. As sketched in Fig. 1, the main aim of such methods is to find a simplified local description of the motion of extended test bodies in terms of a suitable set of multipolar moments, which catches the essential properties of the body at the chosen order of approximation. Y.N. Obukhov (B) Theoretical Physics Laboratory, Nuclear Safety Institute, Russian Academy of Sciences, B.Tulskaya 52, Moscow 115191, Russia e-mail: [email protected] D. Puetzfeld ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany e-mail: [email protected] URL: http://puetzfeld.org © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_2
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Fig. 1 General idea behind multipolar approximation schemes: The world-tube of a body is replaced by a representative world-line L, whereas the original energy-momentum tensor T ab is substituted by a set of multipole moments m ab··· along this world-line. Such a multipolar description simplifies the equations of motion. This is achieved by consideration of only a finite set of moments. Different flavors of multipolar approximation schemes exist in the literature, in this work we define the moments à la Dixon in [3]
In this work, we use a covariant multipolar description, based on Synge’s worldfunction formalism [7, 8]. The multipolar moments to be introduced, can be viewed as a direct generalization of the moments first introduced by Dixon in [3]. Central to the derivation of the equations of motion, by means of a multipolar method, is the knowledge of the corresponding conservation laws of the underlying gravity theory. In General Rela
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