Radiation from a $D$-dimensional collision of gravitational shock waves

Abstract

Classically, if two highly boosted particles collide head-on, a black hole is expected to form whose mass may be inferred from the gravitational radiation emitted during the collision. If this occurs at trans-Planckian energies, it should be well described by general relativity. Furthermore, if there exist hidden extra dimensions, the fundamental Planck mass may well be of the order of the TeV and thus achievable with current or future particle accelerators. By modeling the colliding particles as Aichelburg-Sexl shock waves on a flat, $D$-dimensional background, we devise a perturbative framework to compute the space-time metric in the future of the collision. Then, a generalisation of Bondi's formalism is employed to extract the gravitational radiation and compute the inelasticity of the collision: the percentage of the initial centre-of-mass energy that is radiated away. Using the axial symmetry of the problem, we show that this information is encoded in a single function of the transverse metric components - the news function. We then unveil a hidden conformal symmetry which exists at each order in perturbation theory and thus makes the problem effectively two-dimensional. Moreover, it allows for the factorisation of the angular dependence of the news function, i.e. the radiation pattern at null infinity, and clarifies the correspondence between the perturbative series of the metric and an angular expansion off the collision axis. The first-order estimate, or isotropic term, is computed analytically and yields a remarkable simple formula for the inelasticity for any $D$. Higher-order terms, however, require the use of a computer for numerical integration. We study the integration domain and compute, numerically, the Green's functions and the sources, thus paving the way for the computation of the inelasticity in a future work.