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Hoop conjecture

Now we examine the difference of the horizon formation for various spacetime dimension using the hoop conjecture. The hoop conjecture gives the criterion of black hole formation in the 4-dimensional general relativity [5]. It states that an apparent horizon forms when and only when the mass $ M$ of the system gets compacted into a region of which circumference $ C$ satisfies

$\displaystyle H_4\equiv C/4\pi G_4 M\lesssim 1.$ (21)

As $ 4\pi G_4
M$ is the circumference of the 4-dimensional Schwarzschild horizon, we can expect that the criterion of black hole formation in the $ D$-dimensional Einstein gravity is given by

$\displaystyle H_D\equiv C/2\pi r_{\text{h}}(M)\lesssim 1,$ (22)

where $ r_{\text{h}}(M)$ is the Schwarzschild radius of $ D$-dimensional spacetime. This criterion was implicitly used to estimate the total cross section for black hole production via non-head-on collisions [2].

To calculate the ratio $ H_D$ and $ H_4$, we must specify the definition of the mass of the system. In this paper, we use total energy $ E=2\mu$ as the mass of the system. The circumference $ C$ is defined as minimum length which encloses two particles. We take the loop as shown in FIG. 4 and calculate $ C$ by taking the limit $ c\rightarrow 0$. $ C$ reduces to $ 4\vert T\vert$ which is the twice the distance of two particles. The value of $ H_D$ at $ T=T_c$ is shown in TABLE I. As $ D$ increases, the value of $ H_D$ decreases and the mass $ M$ must be compacted into the region with smaller circumference $ C$ than $ 2\pi r_{\text{h}}$ to produce a black hole. This reflects the decrease in $ \vert T_c\vert/\rho_0$ with increase in $ D$.

Figure 3: The apparent horizon for $ D=5$ at $ T/\rho _0=-0.227,-0.2,0 $. The dark line is the horizon at $ T=T_c=-0.227\rho _0$, and light line is the horizon at $ T=0$. The unit of the axis is $ \rho _0$.
\includegraphics[width=0.5\linewidth]{fig3.eps}

Figure 4: The closed loop to calculate the circumference. We calculate $ C$ by taking $ c\rightarrow 0$.
\includegraphics[width=0.5\linewidth]{fig4.eps}

The value of $ H_4$ at $ T=T_c$ is also shown in TABLE I. This result can be written as

$\displaystyle H_4=F(D)\frac{(G_DE)^{1/(D-3)}}{G_4E},$ (23)

where $ F(D)=0.03\sim 0.2$. The $ D$-dimensional gravitational constant is related to the Planck energy as

$\displaystyle M_p^{D-2}=\frac{(2\pi)^{D-4}}{4\pi G_D}.$ (24)

Using this formula, Eq.(23) becomes

  $\displaystyle H_4=F(D)\left(\frac{M_4}{M_p}\right)^2 \left(\frac{8\pi^2M_p}{E}\right)^{\frac{D-4}{D-3}}.$ (25)

If the Planck energy is TeV scale, $ M_4/M_p$ is $ \sim 10^{16}$ and $ H_4$ becomes $ \sim 10^{32}$. Thus the mass does not need to be compacted into a small region of which circumference is $ C\lesssim 4\pi G_4M$ to produce a black hole.


Table 1: The value of $ H_4$ and $ H_D$ at $ T=T_c$ for $ D=4\sim 11$.
$ D$ $ H_D$ $ H_4$
$ 4$ $ 0.1773 $ $ 0.1773 $
$ 5$ $ 0.1567$ $ 0.0722 \left[(G_5E)^{1/2}/(G_4E)\right]$
$ 6$ $ 0.1348$ $ 0.0527 \left[(G_6E)^{1/3}/(G_4E)\right]$
$ 7$ $ 0.1176$ $ 0.0444 \left[(G_7E)^{1/4}/(G_4E)\right]$
$ 8$ $ 0.1042$ $ 0.0396 \left[(G_8E)^{1/5}/(G_4E)\right]$
$ 9$ $ 0.0936$ $ 0.0364 \left[(G_9E)^{1/6}/(G_4E)\right]$
$ 10$ $ 0.0849$ $ 0.0340 \left[(G_{10}E)^{1/7}/(G_4E)\right]$
$ 11$ $ 0.0777$ $ 0.0321 \left[(G_{11}E)^{1/8}/(G_4E)\right]$


next up previous
Next: Summary and discussion Up: High-energy head-on collisions of Previous: Time slicing and apparent
Yasusada Nambu
2002-08-23