Collisions of unequal mass black holes and the point particle limit

Numerical relativity has seen incredible progress in the last years, and is being applied with success to a variety of physical phenomena, from gravitational wave research and relativistic astrophysics to cosmology and high-energy physics. Here we probe the limits of current numerical setups, by studying collisions of unequal mass, nonrotating black holes of mass ratios up to $1:100$ and making contact with a classical calculation in general relativity: the infall of a pointlike particle into a massive black hole. Our results agree well with the predictions coming from linearized calculations of the infall of pointlike particles into nonrotating black holes. In particular, in the limit that one hole is much smaller than the other, and the infall starts from an infinite initial separation, we recover the point-particle limit. Thus, numerical relativity is able to bridge the gap between fully nonlinear dynamics and linearized approximations, which may have important applications. Finally, we also comment on the “spurious” radiation content in the initial data and the linearized predictions.

Figure 1

Convergence analysis for the $l=2$ multipole of the gravitational wave signal for simulation $q=1/4$, $D=16.53M$ (upper) and simulation $q=1/100$, $D=9.58$ (lower panel). In both cases we show the higher resolution differences (solid black) together with the lower resolution result rescaled for second (dashed red lines) and fourth-order convergence (dotted blue lines).

Figure 2

Rescaled wave forms for mass ratios $q=1/4$ (top), $q=1/10$ (center) and $q=1/100$ (bottom panels) for $l=2$ (upper) and $l=3$ (lower half of each panel), for two different initial separations. Also shown is the wave form in the PP limit (black solid lines).