Nonlinear gravitational self-force: Second-order equation of motion

Abstract

When a small, uncharged, compact object is immersed in an external background spacetime, at zeroth order in its mass it moves as a test particle in the background. At linear order, its own gravitational field alters the geometry around it, and it moves instead as a test particle in a certain effective metric satisfying the linearized vacuum Einstein equation. In the letter [Phys. Rev. Lett. 109, 051101 (2012)], using a method of matched asymptotic expansions, I showed that the same statement holds true at second order: if the object's leading-order spin and quadrupole moment vanish, then through second order in its mass it moves on a geodesic of a certain smooth, locally causal vacuum metric defined in its local neighbourhood. Here I present the complete details of the derivation of that result. In addition, I extend the result, which had previously been derived in gauges smoothly related to Lorenz, to a class of highly regular gauges that should be optimal for numerical self-force computations.

Self-force effects on the marginally bound zoom-whirl orbit in Schwarzschild spacetime

“…Namely, the only gauge perturbation that shifts the CoM location is the one generated by ξ − α(3) ; and, with ξ − α (3) normalized as in Eq. (67), it does so by an amount of −1 in the x direction.…”

“…(67), it does so by an amount of −1 in the x direction. It is now a good time to return to the question of the physical interpretation of ξ − α (3) . That is made clear by examining the form of this generator at r → ∞, in Lorenzian coordinates:…”

“…A bound-orbit configuration in black-hole perturbation theory does not admit an obvious (local) notion of conserved energy, as it lacks a local time-translation symmetry (except in the geodesic limit). 3 An unbound orbit, on the other hand, has a vanishing interaction potential at t → −∞ (and also at t → +∞, if the orbit scatters back to infinity), and therefore a readily identifiable (Bondi-type) invariant mass and binding energy. This direct handle on the energetics of the scattering process is invaluable in establishing a common language between self-force and other approaches (e.g., PN or EOB), which must be based on 2 Perhaps a sole exception is the early work in [28], which considered a radial infall trajectory into a Schwarzschild black hole as a first test case, concentrating on method development.…”

“…This direct handle on the energetics of the scattering process is invaluable in establishing a common language between self-force and other approaches (e.g., PN or EOB), which must be based on 2 Perhaps a sole exception is the early work in [28], which considered a radial infall trajectory into a Schwarzschild black hole as a first test case, concentrating on method development. 3 See, however, our discussion below of the first law of binary blackhole mechanics, where a time-averaged notion of such energy is introduced, neglecting dissipation. a catalogue of physically unambiguous, gauge-invariant calculable quantities.…”

Second-Order Self-Force Calculation of Gravitational Binding Energy in Compact Binaries

“…Working at second order comes with numerous challenges. At the foundational level, the point-particle approximation for the small object fails, and only in 2012 were viable formulations of second-order SF theory derived [27][28][29][30], after some years of preparatory work [31][32][33][34][35]. At the level of concrete implementation, new effects (and new obstacles) arise on both large temporal and spatial scales [36] and on small scales near the small object [37].…”