To simplify the analysis, we follows the method adopted by Eardley and Giddings [3]. First, the tension of the brane which is expected to be the Planck scale can be negligible if the center of mass energy is substantially larger than the Planck scale. Second, the geometry of the extra dimensions plays no essential role if the geometrical scales of the extra dimensions are large compared to the horizon radius for the center of mass energy. Thus we consider the head-on collisions in -dimensional Einstein gravity. The metric with a high-energy point particle is obtained by infinitely boosting the Schwarzschild black hole metric with the fixed total energy . The resulting system becomes a massless point particle accompanied by a plane-fronted gravitational shock wave which is the Lorentz-contracted longitudinal gravitational field of the particle. Combining two shock waves, we can set up the high energy collision. This system was originally developed by D'Eath and Payne [4]. The black hole formation with an impact parameter for was investigated by Eardley and Giddings [3], and they showed that the apparent horizon which encloses two particles exists at the instance of collision for sufficiently small impact parameter.

We examine the head-on collisions using the different slicing of the spacetime: we expect that the apparent horizon forms before the collision of two particles. We construct the solutions of the apparent horizons analytically and discuss how the dimension affects the formation of the horizon from the viewpoint of the hoop conjecture [5].

2002-08-23