The MPD Equations in Analytic Perturbative Form
A longstanding approach to the dynamics of extended objects in curved space-time is given by the Mathisson-Papapetrou-Dixon (MPD) equations of motion for the so-called “pole-dipole approximation.” This paper describes an analytic perturbation approach to
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Abstract A longstanding approach to the dynamics of extended objects in curved space-time is given by the Mathisson-Papapetrou-Dixon (MPD) equations of motion for the so-called “pole-dipole approximation.” This paper describes an analytic perturbation approach to the MPD equations via a power series expansion with respect to the particle’s spin magnitude, in which the particle’s kinematic and dynamical degrees of freedom are expressible in a completely general way to formally infinite order in the expansion parameter, and without any reference to pre-existent spacetime symmetries in the background. An important consequence to emerge from the formalism is that the particle’s squared mass and spin magnitudes can shift based on a classical analogue of “radiative corrections” due to spin-curvature coupling, whose implications are investigated. It is explicitly shown how to solve for the linear momentum and spin angular momentum for the spinning particle, up to second order in the expansion. As well, this paper outlines two distinct approaches to address the study of many-body dynamics of spinning particles in curved space-time. The first example is the spin modification of the Raychaudhuri equation for worldline congruences, while the second example is the computation of neighbouring worldlines with respect to an arbitrarily chosen reference worldline.
1 Introduction It is an undeniable fact that, for all practical purposes, macroscopic astrophysical objects in the Universe possess classical spin angular momentum during their formation and long-term evolution. In addition, it has been known for a long time that relativistic astrophysical objects, such as black holes and neutron stars with spin, also theoretically exist within the theory of General Relativity, with strong observational evidence to effectively confirm their presence within the Universe. Further to this, it becomes very relevant to consider, for example, the physical consequences involving the spin interaction of relativistic systems, such as the orbital dynamics of rapidly D. Singh (B) Department of Physics, University of Regina, Regina, SK S4S 0A2, Canada e-mail: [email protected] © 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_5
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spinning neutron stars around supermassive black holes expected to exist in the centres of most galaxies. Therefore, it is vital to properly understand the dynamics of extended bodies in curved space-time that incorporates classical spin. The first attempt to introduce classical spin within the framework of General Relativity was presented by Mathisson [1], who demonstrated the existence of equations of motion involving an interaction due to the direct coupling of the Riemann curvature tensor with the moving particle’s spin. Besides the so-called “pole-dipole approximation” for the case of a spinning point dipole as the leading order spindependent term alon
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