Gravitational radiation in black-hole collisions at the speed of light. II. Reduction to two independent variables and calculation of the second-order news function

P. D. D'Eath and P. N. Payne
Phys. Rev. D 46, 675 – Published 15 July 1992
PDFExport Citation

Abstract

This paper describes analytical simplifications which make feasible the numerical calculation of the second-order news function, which gives partial information about the angular distribution of gravitational radiation emitted in the axisymmetric collision of two black holes at the speed of light. In the preceding paper, paper I, the curved radiative region of the space-time, produced after the collision of the two incoming plane-fronted shock waves, was treated using perturbation theory by making a large Lorentz boost to a frame in which a weak shock of energy λ scatters off a strong shock of energy ν>>λ. Calculations of gravitational radiation at first, second,... order in (λν) translate, when boosted back to the center-of-mass frame, into calculations of the coefficients a0(τ^μ),a2(τ^μ),... in the convergent series expansion c0(τ^,θ^)=Σn=0a2n(τ^μ)sin2nθ^ expected for the news function c0, where τ^ is a retarded time coordinate, θ^ is the angle from the symmetry axis, and μ is the energy of each incoming black hole in the center-of-mass frame. In paper I, a0(τ^μ) was computed and a2(τ^μ) was obtained as an integral expression too complicated to be tractable numerically. In the present paper a simpler expression for a2(τ^μ) is derived, using the property that the perturbative field equations may all be reduced to equations in only two independent variables, because of a conformal symmetry at each order in perturbation theory. The Green's function for the perturbative field equations is found by reduction from the retarded flat-space Green's function in four dimensions, leading to expressions in terms of two variables for the second-order radiative metric components. From these, a2(τ^μ) can be extracted after removing, by a gauge transformation, the (lnr)r terms present in the second-order metric in the harmonic gauge used here (r being a radial coordinate). Numerical results are presented in the following paper, paper III, which discusses the implications for the energy emitted and the nature of the radiative space-time.

  • Received 4 February 1992

DOI:https://doi.org/10.1103/PhysRevD.46.675

©1992 American Physical Society

Authors & Affiliations

P. D. D'Eath

  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England

P. N. Payne

  • 59, The Avenue, Nedlands 6009, Western Australia

References (Subscription Required)

Click to Expand
Issue

Vol. 46, Iss. 2 — 15 July 1992

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review D

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×