Finally we discuss the validity of the hoop conjecture. Obviously, does not give the picture of the hoop conjecture because its value at the horizon formation is far larger than unity. The ratio also does not give the picture of the hoop conjecture because its value at the horizon formation is much smaller than unity. However, we used the rough estimated values of the circumference and the mass to evaluate and . The energy of shock wave with a high-energy particle is distributed in the transverse direction of the motion, and our estimation of the circumference is too small because the region surrounded by this circumference does not enclose much of the gravitational energy. In our previous paper [7], we stated that with Hawking's quasi-local mass [8] becomes a better parameter to judge the horizon formation for the system with motions. We must calculate for all surfaces S and then take the minimum value of them. Even if the Hawking mass in multi-dimensional space-time has not been calculated in this paper, we expect that becomes a condition for the horizon formation. The value would decrease as increases even if we use the quasi-local mass because should reflect the decrease in .

Although we can regard as the condition for the horizon formation in -dimensional gravity, it does not give a unique condition. The topology of apparent horizon is not restricted to be surface in a multi-dimensional space-time. Emparan and Reall derived the solution of rotating black ring in [9]. For apparent horizon which does not have topology, the criterion for its formation may take another form. Our criterion is applicable only to the horizon with topology.

The authors would like to thank Akira Tomimatsu and Masaru Shibata for helpful discussions.

2002-08-23