Self-force in Non-vacuum Spacetimes: Theory and Applications




Zimmerman, Peter

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University of Guelph


In the past, the formulation of the gravitational self-force problem has been restricted to background spacetimes which are devoid of additional matter fields. The solutions describing these geometries obey the vacuum Einstein field equations and the motion of test-particles is geodesic. While the vacuum formulation may be adequate for characterizing extreme mass ratio inspirals around black holes, which form the most promising candidates for the proposed space-based gravitational wave detector (e)LISA, many phenomena require contending with extra fields. For example, one may wish to consider the motion of a small charged body through the magnetosphere of a larger object as it inspirals, which would require modeling the combined effects of the electromagnetic and gravitational perturbations created by the small body. The non-vacuum gravitational self-force will also play an important role in describing the motion of small bodies in alternative theories of gravity, where the gravitational field is mediated by both the spacetime metric and additional fields. Moreover, the non-vacuum self-force is needed to test whether a Reissner-Nordro ̈m black hole can be driven to an overcharged state by bombarding it with a charged particle. In this thesis, we provide a foundational framework for tackling these problems through a sequence of formal derivations of the first-order self-force in non-vacuum spacetimes containing additional matter fields with integer little group representations. Specifically, we present two derivations: one based on regular solutions to the linearized field equations, and the other following from effective field theory principles. Both derivations utilize a novel “meta-index” notation to collect the fields into a single“super-field” which streamlines the analysis. The formalism is then applied to the scalarvac and electrovac spacetimes as concrete scenarios. We then derive the first-order self-force in scalar-tensor theory of gravity and mention the prospect of constraining the theory using the result. We also find that both the mass and effective charge of the particle in scalar-tensor theory evolve due to the self-field, indicating that scalarization phenomena are exhibited in the extreme mass ratio inspiral problem.



self-force, non-vacuum spacetime, relativistic motion, classical effective field theory