Black Hole Formation from the Collision of Plane-Fronted Gravitational Waves

This paper introduces a new effort to study the collision of plane-fronted gravitational waves in four dimensional, asymptotically flat spacetime, using numerical solutions of the Einstein equations. The pure vacuum problem requires singular, Aichelburg-Sexl type sources to achieve finite energy solutions, which are problematic to treat both mathematically and numerically. Instead then, we use null (massless) particles to source non-trivial geometry within the initial wave fronts. The main purposes of this paper are to (a) motivate the problem, (b) introduce methods for numerically solving the Einstein equations coupled to distributions of collisionless massless or massive particles, and (c) present a first result on the formation of black holes in the head-on collision of axisymmetric distributions of null particles. Regarding the last-named, initial conditions are chosen so that a black hole forms promptly, with essentially no matter escaping the collision. This can be interpreted as approaching the ultra-relativistic collision problem from within an infinite boost limit, but where the matter distribution is spread out, and thus non-singular. We find results that are consistent with earlier perturbative calculations of the collision of Aichelburg-Sexl singularities, as well as numerical studies of the high-speed collision of boson stars, black holes, and fluid stars: a black hole is formed containing most of the energy of the spacetime, with the remaining $15\pm1\%$ of the initial energy radiated away as gravitational waves. The methods developed here could be relevant for other problems in strong field gravity and cosmology that involve particle distributions of matter.