Gravitational radiation in black-hole collisions at the speed of light. I. Perturbation treatment of the axisymmetric collision

Abstract

In this and the two following papers II and III we study the axisymmetric collision of two black holes at the speed of light, with a view to understanding the more realistic collision of two black holes with a large but finite incoming Lorentz factor y. The curved radiative region of the space-time, produced after the two incoming impulsive plane-fronted shock waves have collided, is treated using perturbation theory, following earlier work by Curtis and Chapman. The collision is viewed in a frame to which a large Lorentz boost has been applied, giving a strong shock with energy v off which a weak shock with energy A. && v scatters. This yields a singular perturbation problem, in which the Einstein field equations are solved by expanding in powers of A, /v around flat space-time. When viewed back in the center-ofmass frame, this gives a good description of the regions of the space-time in which gravitational radiation propagates at small angles L9 but a large distance from the symmetry axis, near each shock as it continues to propagate, having been 'distorted and deflected in the initial collision. The news function co(v, 8) describing the gravitational radiation is expected to have a convergent series expansion co(r, 9) = g"" Oa, "(r)sin "8, where r" is a retarded time coordinate. First-order perturbation theory gives an expression for ao(~) in agreement with that found previously by studying the finite-y collisions.Second-order perturbation theory gives a&( f) as a complicated integral expression. A new mass-loss formula is derived, which shows that if the end result of the collision is a single Schwarzschild black hole at rest, plus gravitational radiation which is (in a certain precise sense) accurately described by the above series for co(~, 8), then the final mass can be determined from knowledge only of ao(~) and a, (~). This leads to an interesting test of the cosmic censorship hypothesis. The numerical calculation of a, (~) is made practicable by analytical simplifications described in the following paper II, where the perturbative field equations are reduced to a system in only two independent variables. Results are presented in the concluding paper III, which discusses the implications for the energy emitted and the nature of the radiative space-time.PACS number(s): 04.30.+x, 97.60.Lf

Causality violation, gravitational shockwaves and UV completion

“…The shockwave geometry can be analysed in terms of the four regions I, II, III and IV shown in figure 4. The geometry in regions I, II and III is actually flat whereas in region IV the collision curves the spacetime in a way which is difficult to analyse [33]. However, if we take the shockwaves to be particle shockwaves (having the same energy µ for simplicity) and the impact parameter such that Gµ/L 1, then we expect that the curvature in region IV will be small.…”

High-energy head-on collisions of particles and the hoop conjecture

“…Combining two shock waves, we can set up the high energy collision. This system was originally developed by D'Eath and Payne [4]. The black hole formation with an impact parameter for D = 4 was investigated by Eardley and Giddings [3], and they showed that the apparent horizon which encloses two particles exists at the instance of collision for sufficiently small impact parameter.…”

Transplanckian bremsstrahlung and black hole production

“…Apart from [11], where gravitational bremsstrahlung of soft photons was studied in the context of string theory, the existing estimates of gravitational radiation either refer to phase (ii), or are based on the assumption of an already existing BH (e.g. radiation from particles falling into the BH [12]), on results of linearized theory relevant only to the case of non-gravitational scattering [8], on weakly relativistic numerical simulations [13] or again on collisions of waves in 4D [14]. For a related discussion see also [15].…”