Norichika Sago

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

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

Gravitational self-force correction to the innermost stable circular orbit of a Schwarzschild black hole

1+1

Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring

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.

Two approaches for the gravitational self force in black hole spacetime: Comparison of numerical results

The innermost stable circular orbit (ISCO) of a test particle around a Schwarzschild black hole of mass M has (areal) radius r_{isco}=6MG/c;{2}. If the particle is endowed with mass micro(<<M), it experiences a gravitational self-force whose conservative piece alters the location of the ISCO. Here we calculate the resulting shifts Deltar_{isco} and DeltaOmega_{isco} in the ISCOs radius and frequency, at leading order in the mass ratio micro/M. We obtain, in the Lorenz gauge, Deltar_{isco}=-3.269(+/-0.003)microG/c;{2} and DeltaOmega_{isco}/Omega_{isco}=0.4870(+/-0.0006)micro/M. We discuss the implications of our result within the context of the extreme-mass-ratio binary inspiral problem.

Evolution of inspiral orbits around a Schwarzschild black hole

We compute the conservative piece of the gravitational self-force (GSF) acting on a particle of mass m1 as it moves along an (unstable) circular geodesic orbit between the innermost stable orbit and the light ring of a Schwarzschild black hole of mass m2?m1. More precisely, we construct the function huuR,L(x)?h??R,Lu?u? (related to Detweiler’s gauge-invariant “redshift” variable), where h??R,L(?m1) is the regularized metric perturbation in the Lorenz gauge, u? is the four-velocity of m1 in the background Schwarzschild metric of m2, and x?[Gc-3(m1+m2)?]2/3 is an invariant coordinate constructed from the orbital frequency ?. In particular, we explore the behavior of huuR,L just outside the “light ring” at x=1/3 (i.e., r=3Gm2/c2), where the circular orbit becomes null. Using the recently discovered link between huuR,L and the piece a(u), linear in the symmetric mass ratio ??m1m2/(m1+m2)2, of the main radial potential A(u,?)=1-2u+?a(u)+O(?2) of the effective-one-body (EOB) formalism, we compute from our GSF data the EOB function a(u) over the entire domain 0<u<1/3 (thereby extending previous results limited to u?1/5). We find that a(u) diverges like a(u)?0.25(1-3u)-1/2 at the light-ring limit, u?(1/3)-, explain the physical origin of this divergent behavior, and discuss its consequences for the EOB formalism. We construct accurate global analytic fits for a(u), valid on the entire domain 0<u<1/3 (and possibly beyond), and give accurate numerical estimates of the values of a(u) and its first three derivatives at the innermost stable circular orbit u=1/6, as well as the associated O(?) shift in the frequency of that orbit. In previous work we used GSF data on slightly eccentric orbits to compute a certain linear combination of a(u) and its first two derivatives, involving also the O(?) piece of a second EOB radial potential D? (u)=1+?d? (u)+O(?2). Combining these results with our present global analytic representation of a(u), we numerically compute d? (u) on the interval 0<u?1/6

Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism

Recently, two independent calculations have been presented of finite-mass (“self-force”) effects on the orbit of a point mass around a Schwarzschild black hole. While both computations are based on the standard mode-sum method, they differ in several technical aspects, which makes comparison between their results difficult—but also interesting. Barack and Sago [Phys. Rev. D 75, 064021 (2007)] invoke the notion of a self-accelerated motion in a background spacetime, and perform a direct calculation of the local self-force in the Lorenz gauge (using numerical evolution of the perturbation equations in the time domain); Detweiler [Phys. Rev. D 77, 124026 (2008)] describes the motion in terms a geodesic orbit of a (smooth) perturbed spacetime, and calculates the metric perturbation in the Regge-Wheeler gauge (using frequency-domain numerical analysis). Here we establish a formal correspondence between the two analyses, and demonstrate the consistency of their numerical results. Specifically, we compare the value of the conservative O(?) shift in ut (where ? is the particle’s mass and ut is the Schwarzschild t component of the particle’s four-velocity), suitably mapped between the two orbital descriptions and adjusted for gauge. We find that the two analyses yield the same value for this shift within mere fractional differences of ?10-5–10-7 (depending on the orbital radius)—comparable with the estimated numerical error.

Periastron advance in black-hole binaries.

We present results from calculations of the orbital evolution in eccentric binaries of nonrotating black holes with extreme mass-ratios. Our inspiral model is based on the method of osculating geodesics, and is the first to incorporate the full gravitational self-force (GSF) effect, including conservative corrections. The GSF information is encapsulated in an analytic interpolation formula based on numerical GSF data for over a thousand sample geodesic orbits. We assess the importance of including conservative GSF corrections in waveform models for gravitational-wave searches.

Adiabatic evolution of orbital parameters in kerr spacetime

Using a recently presented numerical code for calculating the Lorenz-gauge gravitational self-force (GSF), we compute the O(m) conservative correction to the precession rate of the small-eccentricity orbits of a particle of mass m moving around a Schwarzschild black hole of mass M?m. Specifically, we study the gauge-invariant function ?(x), where ? is defined as the O(m) part of the dimensionless ratio (??r/???)2 between the squares of the radial and azimuthal frequencies of the orbit, and where x=[Gc-3(M+m)???]2/3 is a gauge-invariant measure of the dimensionless gravitational potential (mass over radius) associated with the mean circular orbit. Our GSF computation of the function ?(x) in the interval 0

Adiabatic Radiation Reaction to Orbits in Kerr Spacetime

The general relativistic (Mercury-type) periastron advance is calculated here for the first time with exquisite precision in full general relativity. We use accurate numerical relativity simulations of spinless black-hole binaries with mass ratios 1/8≤m(1)/m(2)≤1 and compare with the predictions of several analytic approximation schemes. We find the effective-one-body model to be remarkably accurate and, surprisingly, so also the predictions of self-force theory [replacing m(1)/m(2)→m(1)m(2)/(m(1)+m(2))(2)]. Our results can inform a universal analytic model of the two-body dynamics, crucial for ongoing and future gravitational-wave searches.

Gravitational wave memory of gamma-ray burst jets

We investigate the adiabatic orbital evolution of a point particle in Kerr spacetime due to the emission of gravitational waves. In the case that the timescale of the orbital evolution is sufficiently smaller than the characteristic timescale of orbits, the evolution of orbits is characterized by the rates of change of three constants of motion, the energy E, the azimuthal angular momentum L, and the Carter constant Q. We can evaluate the rates of change of E and L from the fluxes of the energy and the angular momentum at infinity and on the event horizon, employing the balance argument. However, for the Carter constant, we cannot use the balance argument because we do not know the conserved current associated with it. Recently, Mino proposed a new method of evaluating the average rate of change rate of the Carter constant by using the radiative field. In a previous paper, we developed a simplified scheme for determining the evolution of the Carter constant based on Mino’s proposal. In this paper we describe our scheme in more detail and derive explicit analytic formulae for the rates of change of the energy, the angular momentum and the Carter constant.