Horizons in 2+1 dimensional collapse of particles

Event horizons are generated by null geodesics. They are therefore natural candidates for applying a time development equation of the Raychaudhuri type. But in the present context we focus on the non-smooth regions of horizons and other features that we describe by geometrical construction, a kind of virtual use of the Raychaudhuri equation. We hope that our material is nevertheless of interest to readers of this volume. The presence of an event horizon is the defining characteristic of a black hole. When a black hole is formed by gravitational collapse an event horizon starts at some stage of the time development, spreads out increasing its area until it encompasses all the dynamical features, and eventually becomes stationary. The final black hole is then one of the small family of “hairless” types. The most active and interesting period in the life of a black hole is the time near the formation of the horizon. Because the horizon is a global property of the spacetime and cannot be characterized locally, this interesting period is typically studied only in numerically generated spacetime regions that extend over a long time. In two spaceand one time-dimensions (2+1 D) the situation is much simplified for several reasons. There is a simple kind of matter that can collapse gravitationally and form black holes, namely point particles; the geometry of spacetimes with such collapsing matter is known exactly, at least in principle; and there are no gravitational waves emitted in the collapse that may allow the final black hole state to be reached only asymptotically in time. Thus, in the collapse of 2+1 D particles, the formative stage of the horizon lasts only a finite time and is over as soon as the horizon has passed all the particles. These simplifying features in a 2+1 D spacetime of course make it somewhat unrealistic as a model for 3+1 D collapse, but one would still expect that many of the lessons one learns in 2+1 dimensions survive, in some form, in 3+1 dimensions. In this paper we focus on the horizon formation when two particles in 2+1 D collapse head-on. In order to form a true black hole the particles must have sufficient mass-energy and there must be a negative cosmological constant. We first recall the equations of motion for test particles, and we then discuss a model process for the case of a vanishing cosmological constant. The changes in the horizon’s behavior during the active period for negative cosmological constant are shown to be relatively minor. We also mention the collapse of a particle into a black hole and the case of more than two collapsing particles.