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

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 γ. 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 ν off which a weak shock with energy λ≪ν scatters. This yields a singular perturbation problem, in which the Einstein field equations are solved by expanding in powers of λ/ν around flat space-time. When viewed back in the center-of-mass frame, this gives a good description of the regions of the space-time in which gravitational radiation propagates at small angles θ^ 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 ${\mathit{c}}_0$(τ^,θ^) describing the gravitational radiation is expected to have a convergent series expansion ${\mathit{c}}_0$(τ^,θ^) =${\mathit{tsum}}_{\mathit{n}=0\mathrm{}}^{\mathrm{\infty }}$${\mathit{a}}_{2\mathit{n}}$(τ^)${\sin }^{2\mathit{n}}$θ^, where τ^ is a retarded time coordinate. First-order perturbation theory gives an expression for ${\mathit{a}}_0$(τ^) in agreement with that found previously by studying the finite-γ collisions. Second-order perturbation theory gives ${\mathit{a}}_2$(τ^) 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 ${\mathit{c}}_0$(τ^,θ^), then the final mass can be determined from knowledge only of ${\mathit{a}}_0$(τ^) and ${\mathit{a}}_2$(τ^). This leads to an interesting test of the cosmic censorship hypothesis. The numerical calculation of ${\mathit{a}}_2$(τ^) 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.