Gravitational Self-Force Correction to the Binding Energy of Compact Binary Systems

Using the first law of binary black-hole mechanics, we compute the binding energy $E$ and total angular momentum $J$ of two nonspinning compact objects moving on circular orbits with frequency $\Omega$, at leading order beyond the test-particle approximation. By minimizing $E(\Omega )$ we recover the exact frequency shift of the Schwarzschild innermost stable circular orbit induced by the conservative piece of the gravitational self-force. Comparing our results for the coordinate-invariant relation $E(J)$ to those recently obtained from numerical simulations of comparable-mass nonspinning black-hole binaries, we find a remarkably good agreement, even in the strong-field regime. Our findings confirm that the domain of validity of perturbative calculations may extend well beyond the extreme mass-ratio limit.

Figure 1

In the upper panel, the (specific) binding energy $\widehat{E}=E/\mu$ of an equal-mass black-hole binary is shown as a function of the (dimensionless) total angular momentum $\widehat{J}=J/(\mu m)$, as computed in numerical-relativity (“NR”), in PN theory (3PN), in the EOB model [“EOB(3PN)”], in the test-particle approximation (Schw), and including the conservative gravitational self-force (“$\mathrm{GSF}q$” and “$\mathrm{GSF}\nu$”). The 3PN, EOB(3PN), and test-particle curves show cusps at their respective ISCO/MECO; the lower branches correspond to stable circular orbits, while the upper branches correspond to unstable circular orbits. We might find similar branches for the GSF curves when more data closer to the light-ring become available. The “$\mathrm{GSF}q$” and “$\mathrm{GSF}\nu$” curves are only shown in the region where numerical data for the self-force are available (i.e., $r\ge 5m$, corresponding to $\widehat{J}\simeq 3.126$ for the “$\mathrm{GSF}\nu$” model and to $\widehat{J}\simeq 1.896$ for the “$\mathrm{GSF}q$” model). The differences between the various models and the NR result, $\Delta \widehat{E}\equiv \widehat{E}-{\widehat{E}}_{\mathrm{NR}}$, are shown in the lower panel (down to the ISCO/MECO, when that is present), with the exception of the “$\mathrm{GSF}q$” model, which quickly grows beyond the plot range. The shaded area represents the error affecting the NR results.