Second-Order Gravitational Self-Force

Abstract

Using a rigorous method of matched asymptotic expansions, I derive the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass (neglecting effects of internal structure). The motion is found to be geodesic in a certain locally defined regular geometry satisfying Einstein's equation at second order. I outline a method of numerically obtaining both the metric of that regular geometry and the complete second-order metric perturbation produced by the body.PACS numbers: 04.25.Nx, 04.30.Db Introduction. The governing equation of general relativity, the Einstein field equation (EFE), describes how bodies influence spacetime curvature and move within the resultant curved geometry. Yet, since the seminal work of Einstein, Infeld, and Hoffman in 1938 [1], study of this nonlinear problem of motion has largely focused on the post-Newtonian limit of slow motion and weak fields. In the strong-field regime, bodies have instead typically been approximated as test bodies moving in a spacetime that is unaffected by them. Only within the last fifteen years [2,3] has there arisen an analytical description of the gravitational backreaction: the body's perturbative effect on spacetime geometry and that perturbation's effect on the body's motion. In the case of a small mass m, this backreaction is called the gravitational self-force, and it is now well understood at linear order in m [4][5][6][7]. Beyond its foundational role, the self-force is also potentially of great astrophysical importance, as it describes the evolution of extreme-mass-ratio inspirals (EMRIs), in which a stellar black hole or neutron star spirals into a supermassive black hole. Such systems are predicted to be key sources for the planned gravitational wave detector LISA [8], and they will afford both a unique probe of strong-field dynamics and a map of the spacetime near black holes. The self-force also provides an essential point of comparison with other treatments of the problem of motion: it complements post-Newtonian theory [9,10] and fully nonlinear numerical simulations [10,11], both of which are ill-suited to extreme mass ratios; and it fixes mass-dependent parameters in Effective One Body (EOB) theory [12][13][14].

Tidal invariants for compact binaries on quasicircular orbits

“…It seems plausible that a similar interpretation may be possible at higher orders [e.g., Oðμ 2 Þ]. Formulations of the second-order problem by Pound [70], Gralla [71] and Detweiler [72] have laid a foundation. Recent progress in overcoming certain practical and technical barriers [73,74] suggests that secondorder results are imminent.…”

Is motion under the conservative self-force in black hole spacetimes an integrable Hamiltonian system?

“…Scheme 2 is said to arise from 3 via a reduction of order procedure: because the self-forced worldline and the osculating geodesic agree to zeroth order in the mass ratio, the former can be replaced with the latter in calculating the self-force to first order. The relative accuracies and computational merits of 2 and variants of 3 are subtle issues, entangled with formulating and implementing a consistent second-order analog of the MiSaTaQuWa result [11][12][13][14], which will not be addressed here.…”

Comparison between self-force and post-Newtonian dynamics: Beyond circular orbits

“…B. The 3PN prediction (4.50c) could be compared with future calculations of the second-order GSF [73][74][75][76][77][78]. Finally, we may express the result (4.50b) for the 3PN expansion of the GSF contribution to the generalized redshift by means of the usual parametrization of bound timelike geodesic orbits in Schwarzschild in terms of the semilatus rectum p and eccentricity e (see Sec.…”