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Prescriptions for numerical self-force calculations have traditionally beendesigned for frequency-domain or (1+1) time-domain codes which employ a modedecomposition to facilitate in carrying out a delicate regularization scheme.This has prevented self-force analyses from benefiting from the powerful suiteof tools developed and used by numerical relativists for simulations of theevolution of comparable-mass black hole binaries. In this work, we revisit apreviously-introduced (3+1) method for self-force calculations, and demonstrateits viability by applying it to the test case of a scalar charge moving in acircular orbit around a Schwarzschild black hole. Two (3+1) codes originallydeveloped for numerical relativity applications were independently employed,and in each we were able to compute the two independent components of theself-force and the energy flux correctly to within $< 1%$. We also demonstrateconsistency between $t$-component of the self-force and the scalar energy flux.Our results constitute the first successful calculation of a self-force in a(3+1) framework, and thus open opportunities for the numerical relativitycommunity in self-force analyses and the perturbative modeling ofextreme-mass-ratio inspirals.

Self-force with (3+1) codes: a primer for numerical relativists