Adapted gauge to a quasilocal measure of the black holes recoil

Abstract

We explore different gauge choices in the moving puncture formulation in order to improve the accuracy of a linear momentum measure evaluated on the horizon of the remnant black hole produced by the merger of a binary. In particular, motivated by constant values studies, we design a gauge via a variable shift parameter $m\eta(\vec{r}(t))$ such that it takes a low asymptotic (and at the orbiting punctures) value, while about the standard value of 2 at the final hole horizon. This choice then follows the remnant black hole as it moves due to its net recoil velocity. We find that this choice keeps the accuracy of the binary evolution and, once the asymptotic value of the parameter $m\eta$ is chosen about or below 1.0, it produces more accurate results for the recoil velocity than the corresponding evaluation of the radiated linear momentum at infinity, for typical numerical resolutions. We also find that the choice of the $\partial_t$-gauge (at our working resolutions) is more accurate in this regard of computing recoil velocities than the $\partial_0$-gauge. Detailed studies of an unequal mass $q=m_1/m_2=1/3$ nonspinning binary are provided and then verified for other mass ratios $(q=1/2,1/5)$ and spinning $(q=1)$ binary black hole mergers.