Radiation from a $D$-dimensional collision of shock waves: proof of first order formula and angular factorisation at all orders

TL;DR

This paper analytically confirms the first order inelasticity formula for D-dimensional shock wave collisions and demonstrates angular factorization at all perturbation orders, revealing a hidden symmetry and clarifying the relation between perturbation order and angular series truncation.

Contribution

It proves the exactness of the first order inelasticity formula and establishes angular factorization at all orders in perturbation theory for D-dimensional shock wave collisions.

Findings

First order inelasticity formula is exact.

Angular dependence factorizes at all perturbation orders.

Truncation of angular series relates to metric perturbation order.

Abstract

In two previous papers we have computed the inelasticity $\epsilon$ in a head-on collision of two $D$-dimensional Aichelburg-Sexl shock waves, using perturbation theory to calculate the geometry in the future light-cone of the collision. The first order result, obtained as an accurate numerical fit, yielded the remarkably simple formula $\epsilon_{\rm 1st\, order} = 1/2 - 1/D$. Here we show, analytically, that this result is exact in first order perturbation theory. Moreover, we clarify the relation between perturbation theory and an angular series of the inelasticity's angular power around the symmetry axis of the collision $(\theta = 0,\pi)$. To establish these results, firstly, we show that at null infinity the angular dependence factorises order by order in perturbation theory, as a result of a hidden symmetry. Secondly, we show that a consistent truncation of the angular series in powers of $\sin^2 \theta$ at some order $O(n)$ requires knowledge of the metric perturbations up to $O(n+1)$. In particular, this justifies the isotropy assumption used in first order perturbation theory. We then compute, analytically, all terms that contribute to the inelasticity and depend linearly on the initial conditions (surface terms), including second order contributions.