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
n ≃ − γ ln|p − p∗| (48 )
for γ ≃ 0.31– 0.38. The simulations are currently limited by numerical accuracy to n < 5 and ΔE ∕E < 0.08, 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

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