### 5.5 Black hole collisions

Interesting numerical evidence for critical phenomena in the grazing collision of two black holes has
been found by Pretorius [179]. He evolved initial data for two equal mass nonrotating black
holes, with a reflection symmetry through the orbital plane, parameterised by their relative
boost at constant impact parameter. The threshold in initial data space is between data which
merge immmediately and those which do not (although they will merge later for initial data
which are bound). The critical solution is a circular orbit that loses 1 – 1.5% of the total energy
per orbit through radiation. On both sides of the threshold, the number of orbits scales as
for . The simulations are currently limited by numerical accuracy to and
, but Pretorius conjectures that the total energy loss and hence the number of
orbits is limited only by the irreducible mass of the initial data, a much larger number. In
particular, he speculates that for highly boosted initial data such that the total energy of the initial
data is dominated by kinetic energy, almost all the energy can be converted into gravitational
radiation.
The standard dynamical systems picture of critical collapse, with the critical solution an attractor in the
threshold hypersurface, appears to be consistent with these observation. Pretorius compares his data with
the unstable circular geodesics in the spacetime of the hypothetical rotating black hole that would
result if merger occurred promptly. These give orbital periods, and their linear perturbations
give a critical exponent, in rough agreement with the numerical values for the full black hole
collision.

Pretorius does not comment on the nature of the critical solution, but because of the mass loss through
gravitational radiation it cannot be strictly stationary. The mass loss would be compatible with
self-similarity, with a helical homothetic vector field, but exact self-similarity would be compatible with the
presence of black holes only in the infinite boost limit. Therefore the critical solution is likely to be more
complicated, and can perhaps be written as an expansion with an exactly stationary or homothetic
spacetime as the leading term.

Pretorius also speculates that these phenomena generalise to generic initial data with unequal masses
and black hole spins which are not aligned. This seems uncertain, given the claim by Levin [146] that the
threshold of immediate merger is fractal if the spins are not aligned, and that the system is therefore
chaotic. However, Levin’s analysis is based on a 2nd order post-Newtonian approximation to general
relativity, in which there is no radiation reaction, while the rapid energy loss observed here may suppress
chaos. Nevertheless, the phase space is much bigger when the orbit is not confined to an orbital
plane, and so the critical solution observed here may not be an attractor in the full critical
surface.

Sperhake et al. [191] push the numerics towards higher energies emitted and conjecture that
the merger of nonspinning black holes can in principle yield a black hole arbitrarily close to
extremal Kerr. See also [187]. Update