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

TL;DR

The paper develops a time-domain, Lorenz-gauge gravitational self-force code for a point particle on eccentric bound geodesics around a Schwarzschild black hole, solving the linearized Einstein equations in 1+1D with a tensor-harmonic decomposition and mode-sum regularization. It separates the SF into dissipative and conservative pieces, validates the dissipative piece by balancing local work with asymptotic energy and angular-momentum fluxes, and computes the first full strong-field conservative SF for eccentric orbits, including the ISCO shift. The approach combines a high-order finite-difference scheme, careful treatment of monopole/dipole modes via extended homogeneous solutions, and two independent ISCO-shift calculations (extrapolation in the $p$–$e$ plane and $e$-expansion of the field equations), yielding robust results and a path toward SF-informed gravitational-wave templates. The work significantly advances the capability to model extreme-mass-ratio inspirals in the strong-field regime and provides a benchmark for future Kerr calculations and EOB/post-Newtonian comparisons, with practical implications for waveform modeling and tests of general relativity in the strong-field regime.

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.\ {\bf 102}, 191101 (2009)]. Here we provide full details of this analysis, and discuss future applications.

Figures (10)

• Figure 1: Parameter space for bound geodesics in Schwarzschild spacetime. The (dimensionless) "semi-latus rectum" $p$ and "eccentricity" $e$ are defined in Eq. (\ref{['eq:def-pe']}). Bound geodesics have $e\geq 0$ and $p>6+2e$. Points along the separatrix $p=6+2e$ represent marginally unstable orbits. Stable circular orbits lie along the axis $e=0$ for $p\geq 6$. The point $(p,e)=(6,0)$ is the ISCO.
• Figure 2: Numerical domain: a staggered 1+1-dimensional mesh in null coordinates $v,u$. $r_*$ is the standard Schwarzschild 'tortoise' radial coordinate. The dotted line represents the trajectory of a typical eccentric orbit. In actual implementation the mesh is, of course, much finer than it is depicted here.
• Figure 3: Grid points involved in constructing our finite-difference scheme. The point C is the center of the cell over which we integrate the field equations, as described in the text. The dimensions of each grid cell are $\Delta v\times\Delta u=h\times h$.
• Figure 4: Same as in Fig. \ref{['fig:grid']}, but now point C is located near the particle's worldline, represented by the dashed line. The finite-difference scheme for this case is described in the text under "near-orbit cell".
• Figure 5: Illustration of the four cases considered in formulating the finite-difference equation for grid cells traversed by the particle's worldline (here represented by dashed lines). The center of the cell is at $(u,v)=(u_c,v_c)$, and in each case we indicate the $u,v$ coordinates of the two points where the particle enters/leaves the cell.