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 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 ${\epsilon }_{1st\mathrm{order}}=1/2-1/D$. 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.

arXiv comment21 pages

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Cited by in corpus (5)

• High Energy Collisions of Black Holes Numerically Revisited
• The high-energy collision of black holes in higher dimensions
• Numerical Relativity and High Energy Physics: Recent Developments
• Gravitational radiation in massless-particle collisions
• Radiation from a D-dimensional collision of shock waves: Two dimensional reduction and Carter-Penrose diagram