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.

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$\gg${}m1. More precisely, we construct the function huuR,L(x)$\equiv${}h$\mu${}$\nu${}R,Lu$\mu${}u$\nu${} (related to Detweiler's gauge-invariant ``redshift'' variable), where h$\mu${}$\nu${}R,L($\propto${}m1) is the regularized metric perturbation in the Lorenz gauge, u$\mu${} is the four-velocity of m1 in the background Schwarzschild metric of m2, and x$\equiv${}[Gc-3(m1+m2)$\Omega${}]2/3 is an invariant coordinate constructed from the orbital frequency $\Omega${}. In particular, we explore the behavior of huuR,L just outside the ``light ring'' at x=13 (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 $\nu${}$\equiv${}m1m2/(m1+m2)2, of the main radial potential A(u,$\nu${})=1-2u+$\nu${}a(u)+O($\nu${}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<13 (thereby extending previous results limited to u$\le${}15). We find that a(u) diverges like a(u)$\approx${}0.25(1-3u)-1/2 at the light-ring limit, u$\rightarrow${}(13)-, 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<13 (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=16, as well as the associated O($\nu${}) 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($\nu${}) piece of a second EOB radial potential D\ifmmode\bar\else\textasciimacron\fi{}(u)=1+$\nu${}d\ifmmode\bar\else\textasciimacron\fi{}(u)+O($\nu${}2). Combining these results with our present global analytic representation of a(u), we numerically compute d\ifmmode\bar\else\textasciimacron\fi{}(u) on the interval 0<u$\le${}16.

We calculate the gravitational self-force acting on a pointlike particle of mass $\mu$, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; we then compute the (finite) backreaction force from each of the multipole modes of the perturbation; finally, we apply the “mode-sum” method to obtain the total, physical self-force. The temporal component of the self-force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The radial component of the self-force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the $O(\mu )$ correction to the specific energy and angular momentum parameters (in the Lorenz gauge), as well as the $O(\mu )$ shift in the orbital frequency (which is gauge invariant).

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(<