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

## Abstract

In two previous papers [1, 2] we have computed the inelasticity *ϵ* 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 *ϵ*_{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 (*θ* = 0*, π*). 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}*θ* at some order \( \mathcal{O}(n) \) requires knowledge of the metric perturbations up to \( \mathcal{O}\left(n+1\right) \). 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.

### Keywords

Large Extra Dimensions Black Holes### References

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