Gravitational wave luminosity and net momentum flux in head-on mergers of black holes: Radiative patterns and mode mixing

Abstract

We show that gravitational wave radiative patterns from a point test particle falling radially into a Schwarzschild black hole, as derived by Davis, Ruffini, Press and Price [M. Davis et al., Phys. Rev. Lett. 27, 1466 (1971).], are present in the nonlinear regime of head-on mergers of black holes. We use the Bondi-Sachs characteristic formulation and express the gravitational wave luminosity and the net momentum flux in terms of the news functions. We then evaluate the ($\ensuremath{-}2$)-spin-weighted $\ensuremath{\ell}$-multipole decomposition of these quantities via exact expressions valid in the nonlinear regime and defined at future null infinity. Our treatment is made in the realm of Robinson-Trautman dynamics, with characteristic initial data corresponding to the head-on merger of two black holes. We consider mass ratios in the range $0.01\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1$. We obtain the exponential decay with $\ensuremath{\ell}$ of the total energy contributed by each multipole $\ensuremath{\ell}$, with an accurate linear correlation in the log-linear plot of the points up to $\ensuremath{\alpha}\ensuremath{\simeq}0.7$. Above this mass ratio the contribution of the odd modes to the energy decreases faster than that of the even modes, leading to the breaking of the linear correlation; for $\ensuremath{\alpha}=1$ the energy in all odd modes is zero. The dominant contribution to the total radiated energy comes from the quadrupole mode $\ensuremath{\ell}=2$ corresponding, for instance, to about $\ensuremath{\simeq}84%$ for small mass ratios up to $\ensuremath{\simeq}99.8%$ for the limit case $\ensuremath{\alpha}=1$. The total rescaled radiated energy ${E}_{W}^{\text{total}}/{m}_{0}{\ensuremath{\alpha}}^{2}$ decreases linearly with decreasing $\ensuremath{\alpha}$, yielding for the point particle limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ the value $\ensuremath{\simeq}0.0484$, about 5 times larger than the result of Davis et al. [1]. The mode decomposition of the net momentum flux and of the associated gravitational wave impulses results in an adjacent-even-odd mode-mixing pattern. We obtain that the impulses contributed by each $(\ensuremath{\ell},\ensuremath{\ell}+1)$ mixed mode also accurately satisfy the exponential decay with $\ensuremath{\ell}$, for the whole mass ratio domain considered, $0.01\ensuremath{\le}\ensuremath{\alpha}l1$. The (2, 3) mode contributions to the total impulses are dominant. The mode-mixing effect can also be seen in the decomposition of the net kick velocity imparted to the system by the gravitational wave emission. The mixed mode impulses reach a maximum at $\ensuremath{\alpha}\ensuremath{\simeq}0.7$; for $\ensuremath{\alpha}g0.7$ the impulses decrease and are zero in the equal mass case, due to the decrease to zero of the odd modes of the news functions.