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2008

A Rigorous Derivation of Gravitational Self-force

Gralla, Samuel E., Wald, Robert M.

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There is general agreement that the MiSaTaQuWa equations should describe the motion of a "small body" in general relativity, taking into account the leading order self-force effects.

However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features.
For example, all previous derivations have invoked, without proper justification, the step of "Lorenz gauge relaxation", wherein the linearized Einstein equation is written down in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed--thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations.
In this paper, we analyze the issue of "particle motion" in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics, $g_{ab} (\lambda)$, corresponding to having a body (or black hole) that is "scaled down" to zero size and mass in an appropriate manner.

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