Radiation from a D-dimensional collision of shock waves: first order perturbation theory

We study the spacetime obtained by superimposing two equal Aichelburg-Sexl shock waves in D dimensions traveling, head-on, in opposite directions. Considering the collision in a boosted frame, one shock becomes stronger than the other, and a perturbative framework to compute the metric in the future of the collision is setup. The geometry is given, in first order perturbation theory, as an integral solution, in terms of initial data on the null surface where the strong shock has support. We then extract the radiation emitted in the collision by using a D-dimensional generalisation of the Landau-Lifschitz pseudo-tensor and compute the percentage of the initial centre of mass energy, epsilon, emitted as gravitational waves. In D=4 we find epsilon=25.0%, in agreement with the result of D'Eath and Payne. As D increases, this percentage decreases monotonically, reaching 18.3% in D=10. Our result is always within the bound obtained from apparent horizons by Penrose, in D=4, yielding 29.3%, and Eardley and Giddings, in D> 4, which increases monotonically with dimension, reaching 41.2% in D=10. We also present the wave forms and provide a physical interpretation for the observed peaks, in terms of the null generators of the shocks.