Gravitational self-force on generic bound geodesics in Kerr spacetime

Abstract

In this work we present the first calculation of the gravitational self-force on generic bound geodesics in Kerr spacetime to first order in the mass-ratio. That is, the local correction to equations of motion for a compact object orbiting a larger rotating black hole due to its own impact on the gravitational field. This includes both dissipative and conservative effects. Our method builds on and extends earlier methods for calculating the gravitational self-force on equatorial orbits. In particular we reconstruct the local metric perturbation in the outgoing radiation gauge from the Weyl scalar ψ4, which in turn is obtained by solving the Teukolsky equation using semi-analytical frequency domain methods. The gravitational self-force is subsequently obtained using (spherical) l-mode regularization.We test our implementation by comparing the large l-behaviour against the analytically known regularization parameters. In addition we validate our results be comparing the long-term average changes to the energy, angular momentum, and Carter constant to changes to these constants of motion inferred from the gravitational wave flux to infinity and down the horizon.

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

“…At the other end of the orbit, for r p → ∞, the azimuthal angle ϕ p approaches a constant limiting value: combining (4) and (5) we find dϕ p /dr p ∼ (M/r 3 p ) 1/2 , and hence ϕ p → ϕ ∞ + O(M/r p ) 1/2 . In that sense, the ZEZO is asymptotically "radial" at infinity.…”

“…In Sec. VIII we shall go through this calculation and show how these EOB predictions are in full agreement 5 We shall discuss in detail below the relation between the notions of energy and angular momentum in the first law (and in EOB dynamics), and the usual Arnowitt-Deser-Misner (ADM), or Bondi, ones. The first law of binaries was originally formulated in a PN context.…”

“…The ZEZO is defined by the requirements that (i) r p → 0 for r p → ∞, and (ii) the trajectory asymptotes to a circular orbit in either the infinite future or the infinite past, i.e.ṙ p = 0 =r p for some r p = R. The first requirement determines E using Eq. (5), and subsequently the second requirement determines L and R using (5) and (6). One obtains…”

“…The equation of motion for this worldline is now known through O(η 2 ) in the local effective acceleration [1-3] 1 , and a program for computing the self-force and its effects in astrophysically relevant binaries has been ongoing for over two decades. Recent achievements include numerical calculations of the firstorder self-force [O(η) self-acceleration] for generic bound orbits in Kerr geometry [5], and a first direct calculation of a second-order effect of the self-force [O(η 2 ) selfacceleration] [6]. Ref.…”

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

“…These observations have been enabled by (and have validated) our theoretical models of such systems, without which the observed waves would be unintelligible. Great strides continue to be made in these models throughout the binary parameter space: in the venerable post-Newtonian (PN) theory [4,5], which applies for binaries with large orbital separations; in numerical relativity (NR) [6], which applies for small separations and comparable masses; and in post-geodesic, gravitational self-force (SF) theory [7,8], which applies for disparate masses.…”