Quasilocal linear momentum in black-hole binaries

We propose a quasilocal formula for the linear momentum of black-hole horizons inspired by the formalism of quasilocal horizons. We test this formula using two complementary configurations: (i) by calculating the large orbital linear momentum of the two black holes in an unequal-mass, zero-spin, quasicircular binary and (ii) by calculating the very small recoil momentum imparted to the remnant of the head-on collision of an equal-mass, anti-aligned-spin binary. We obtain results consistent with the horizon trajectory in the orbiting case, and consistent with the net radiated linear momentum for the much smaller head-on recoil velocity.

Figure 1

The $x$ and $y$ components and magnitude of the individual horizon momentum computed for the unequal-mass black-hole binary with mass ratio $q=3/8$ until merger at $t\sim 164M$. Here $P$ denotes the quasilocal momentum and $\overrightarrow{V}=d\overrightarrow{y}/dt$.

Figure 2

The remnant recoil velocity for the head-on test as determined by the isolated horizon formula. The panel on the left shows $P_y$ versus resolution for $\eta =1$. Note the rapid convergence of the asymptotic plateau. The panel on the right shows $P_y$ versus $\eta$ for $h=M/32$. The expected value of the recoil is $P_y/M=(20.4\pm 0.5)\mathrm{km}{\mathrm{s}}^{-1}$.

Figure 3

The recoil velocity of the remnant (bottom) and error (top) for the orbiting binary versus resolution for $\eta =2$.