(1) |

where and are the line element and volume of the unit -sphere and is the -dimensional gravitational constant. Aichelburg-Sexl solution [6] is found by taking the limit of large boost and small mass with the fixed total energy . The resulting metric represents a massless particle moving in the direction with the speed of light:

where and . depends only on the transverse radius and takes the form

for | (3) | |

for | (4) |

A delta function appeared in (2) shows that two coordinate systems are discontinuously connected on . The continuous coordinate system can be introduced by

(5) | ||

where is the Heaviside step function and is the -dimensional flat-space derivative. We can superpose the two solutions to obtain the exact geometry outside the future light cone of the collision of the shocks:

(6) |

where

(7) |

Here is the point in flat -space that is transverse to the direction of particle motion.

The apparent horizon is defined as a closed spacelike -surface on which the outer null geodesic congruence have zero convergence. It was shown that the apparent horizon exists in the union of the two shock waves, and [3]. This apparent horizon consists of two flat discs with radii

(8) |

and gives a characteristic scale for each dimension .

2002-08-23