Gravitational self-force on a particle in eccentric orbit around a Schwarzschild black hole

Abstract

We present a numerical code for calculating the local gravitational self-force acting on a pointlike particle in a generic (bound) geodesic orbit around a Schwarzschild black hole. The calculation is carried out in the Lorenz gauge: For a given geodesic orbit, we decompose the Lorenz-gauge metric perturbation equations (sourced by the delta-function particle) into tensorial harmonics, and solve for each harmonic using numerical evolution in the time domain (in 1+1 dimensions). The physical self-force along the orbit is then obtained via mode-sum regularization. The total self-force contains a dissipative piece as well as a conservative piece, and we describe a simple method for disentangling these two pieces in a time-domain framework. The dissipative component is responsible for the loss of orbital energy and angular momentum through gravitational radiation; as a test of our code we demonstrate that the work done by the dissipative component of the computed force is precisely balanced by the asymptotic fluxes of energy and angular momentum, which we extract independently from the wave-zone numerical solutions. The conservative piece of the self-force does not affect the time-averaged rate of energy and angular-momentum loss, but it influences the evolution of the orbital phases; this piece is calculated here for the first time in eccentric strong-field orbits. As a first concrete application of our code we recently reported the value of the shift in the location and frequency of the innermost stable circular orbit due to the conservative self-force [Phys. Rev. Lett. 102, 191101 (2009)]. Here we provide full details of this analysis, and discuss future applications.

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

“…How h R αβ may be computed in practice, on a Schwarzcshild background, is discussed, for example, in Ref. [23].…”

“…Soon after, Barack and Sago considered two more such "conservative" invariant quantities, namely, the frequency of the innermost stable circular orbit (ISCO) and the rate of periastron advance [23,24]. These results led to a plethora of comparisons between PN, GSF and numerical relativity [25][26][27][28][29][30] and the subsequent refinement of EOB theory [31][32][33][34].…”

Sources of Gravitational Waves: Theory and Observations

“…To obtain physically meaningful results, one needs to combine the MiSaTaQuWa equation of motion with the metric perturbations h µν to obtain gauge invariant quantities that can be related to physical observables. Although considerable progress has been made in the last several years to develop methods to calculate the metric perturbation and GSF at first order [238,239], the majority of the work has focused on computing the GSF on a particle that moves on a fixed worldline of the background spacetime -for example for a static particle [240], radial [241], circular [242,243] and eccentric [244,245] geodesics in Schwarzschild. Methods to compute the GSF on a particle orbiting a Kerr BH have been proposed (e.g., see [246]) and actual implementations are underway.…”

Periastron Advance in Black-Hole Binaries

“…Modelling the dynamics of these systems requires going beyond the geodesic approximation, by taking into account the back-reaction effect due to the interaction of the small object with its own gravitational perturbation. This "gravitational self-force" (GSF) effect has recently been computed for generic (bound) geodesic orbits around a Schwarzschild black hole [21][22][23]. In particular, the O(q) correction to the test-mass result K Schw has been derived [24].…”