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Summary and discussion

We have investigated the temporal evolution of the apparent horizon for high energy particle collisions. The apparent horizon which encloses the two particles appears at $ T=T_c$. Its radius increases in time and reaches $ \rho _0$ at $ T=0$. We calculated $ H_D$ and found that $ H_D$ decreases as $ D$ increases. This means that if we increase the space-time dimension, the size of the hoop which enclose the system should be much smaller than $ 2\pi r_{\text{h}}$. Therefore, the formation of the apparent horizon becomes more difficult for larger $ D$. On the other hand, $ H_4=H_D\cdot r_{\text{h}}/2G_4M$ gives a large value $ \sim 10^{32}$ regardless of the decrease in $ H_D$. This is because the horizon radius $ r_{\text{h}}$ becomes far larger than $ 2G_4 M$. As the horizon radius corresponds to the length scale which enclose the system, this leads to the conclusion that a black hole is easily formed in the TeV scale scenario.

Finally we discuss the validity of the hoop conjecture. Obviously, $ H_4$ does not give the picture of the hoop conjecture because its value at the horizon formation is far larger than unity. The ratio $ H_D$ 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 $ C$ and the mass $ M$ to evaluate $ H_D$ and $ H_4$. 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 $ C$ 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 $ H_4$ with Hawking's quasi-local mass $ M_{\text{H}}(\text{S})$ [8] becomes a better parameter to judge the horizon formation for the system with motions. We must calculate $ H^{(\text{H})}_4(\text{S})=C(\text{S})/4\pi G_4 M_{\text{H}}(\text{S})$ 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 $ H^{(\text{H})}_D\lesssim 1$ becomes a condition for the horizon formation. The value $ H_D$ would decrease as $ D$ increases even if we use the quasi-local mass because $ H_D$ should reflect the decrease in $ \vert T_c\vert/\rho_0$.

Although we can regard $ C/2\pi r_{\text{h}}(M)\lesssim 1$ as the condition for the horizon formation in $ D$-dimensional gravity, it does not give a unique condition. The topology of apparent horizon is not restricted to be $ S^{D-2}$ surface in a multi-dimensional space-time. Emparan and Reall derived the solution of rotating black ring in $ D=5$ [9]. For apparent horizon which does not have $ S^{D-2}$ topology, the criterion for its formation may take another form. Our criterion $ C/2\pi r_h(M)\lesssim 1$ is applicable only to the horizon with $ S^{D-2}$ topology.

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


next up previous
Next: Bibliography Up: High-energy head-on collisions of Previous: Hoop conjecture
Yasusada Nambu
2002-08-23