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

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 $\lambda$ scatters off a strong shock of energy $\nu >>\lambda$. Calculations of gravitational radiation at first, second,... order in ($\frac{\lambda }{\nu }$) translate, when boosted back to the center-of-mass frame, into calculations of the coefficients $a_0\left(\frac{\hat{\tau }}{\mu }\right),a_2\left(\frac{\hat{\tau }}{\mu }\right)$,... in the convergent series expansion $c_0(\hat{\tau },\hat{\theta })=\Sigma {n=0\mathrm{}}^{\infty }{a}_{2n}\left(\frac{\hat{\tau }}{\mu }\right){sin}^{2n}\hat{\theta }$ expected for the news function $c_0$, where $\hat{\tau }$ is a retarded time coordinate, $\hat{\theta }$ is the angle from the symmetry axis, and $\mu$ is the energy of each incoming black hole in the center-of-mass frame. In paper I, $a_0\left(\frac{\hat{\tau }}{\mu }\right)$ was computed and $a_2\left(\frac{\hat{\tau }}{\mu }\right)$ was obtained as an integral expression too complicated to be tractable numerically. In the present paper a simpler expression for $a_2\left(\frac{\hat{\tau }}{\mu }\right)$ 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, $a_2\left(\frac{\hat{\tau }}{\mu }\right)$ can be extracted after removing, by a gauge transformation, the $\frac{\left(\ln r\right)}{r}$ 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.