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Combined gravitational and electromagnetic self-force on charged particles in electrovac spacetimes

We consider the self-force on a charged particle moving in a curved spacetime with a background electromagnetic field, extending previous studies to situations in which gravitational and electromagnetic perturbations are comparable. The formal expression $f^{ret}_α$ for the self-force on a particle, written in terms of the retarded perturbed fields, is divergent, and a renormalization is needed to find the particle's acceleration at linear order in its mass $m$ and charge $e$. We assume that, as in previous work in a Lorenz gauge, the renormalization for accelerated motion comprises an angle average and mass renormalization. Using the short distance expansion of the perturbed electromagnetic and gravitational fields, we show that the renormalization is equivalent to that obtained from a mode sum regularization in which one subtracts from the expression for the self-force in terms of the retarded fields a singular part field comprising only the leading and subleading terms in the mode sum. The most striking part of our result, arising from a remarkable cancellation, is that the renormalization involves no mixing of electromagnetic and gravitational fields. In particular, the renormalized mass is obtained by subtracting (1) the purely electromagnetic contribution from a point charge moving along an accelerated trajectory and (2) the purely gravitational contribution from a point mass moving along the same trajectory. In a mode-sum regularization, the same cancellation implies that the required regularization parameters are sums of their purely electromagnetic and gravitational values.