Hoop conjecture and the horizon formation cross section in Kaluza-Klein spacetimes

We analyze momentarily static initial data sets of the gravitational field produced by two-point sources in five-dimensional Kaluza-Klein spacetimes. These initial data sets are characterized by the mass, the separation of sources and the size of an extra dimension. Using these initial data sets, we discuss the condition for black hole formation, and propose a new conjecture which is a hybrid of the four-dimensional hoop conjecture and the five-dimensional hyperhoop conjecture. By using the new conjecture, we estimate the cross section of black hole formation due to collisions of particles in Kaluza-Klein spacetimes. We show that the mass dependence of the cross section gives us information about the size and the number of the compactified extra dimensions.

Figure 1

Apparent horizons for $m^2/l=0.0025$ and $a=0.02m^{1/2}$, $0.07m^{1/2}$, and $0.08m^{1/2}$ from left to right. Solid lines represent cover-all horizons, and dashed lines represent each independent horizons of two-point sources. In the right panel, where $a=0.08m^{1/2}$, no cover-all horizon exists. The horizontal and vertical axes represent $x=r\sin \theta$ and $z=r\cos \theta$, respectively.

Figure 2

Areas of the two-dimensional geodetic surfaces $\mathcal{A}$ on apparent horizons are depicted as functions of $a/m^{1/2}$ in the cases of $m/l^2=0.0025$, 1, 4, and 100. $V_2={\mathcal{A}}_{\psi =0}$, the maximum value of $\mathcal{A}$ is not a monotonic increasing function of $a/m^{1/2}$ in the cases $m/l^2=1$, 4, 100. The critical values of ${\mathcal{A}}_{\psi =0}$ become much larger than the critical size $32G_{5}M/3$ in these cases.

Figure 3

Length of closed one-dimensional geodesic curves $\mathcal{C}$ on apparent horizons are depicted as functions of $a/m^{1/2}$ in the cases of $m/l^2=0.0025$, 1, 4, and 100. The critical values of $V_1={\mathcal{C}}_{\psi =0}^{\varphi =0}$ become much larger than the critical size $2G_{5}M/l^2$ in the $m/l^2=0.0025$ case.

Figure 4

Values of $W$ are depicted as functions of $al/m$ for the cases of $m/l^2=0.0025$, 1, 4, 100. $W$ is a monotonic increasing function of $al/m$ and $W(a_{\mathrm{cr}})\sim 1$ in all cases.

Figure 5

The cross section ${\sigma }_{\mathrm{p}}$ of five-dimensional Kaluza-Klein black hole formation which is estimated by the hybrid hoop conjecture is depicted as a function of $G_{5}M/l^2$.

Figure 6

The cross section of black hole formation ${\sigma }_{\mathrm{p}}$, which is estimated by the hybrid hoop conjecture, is depicted as a function of the center of mass energy $M$. The total dimension is 10 and six of them are compactified in each scale. Two directions, which contribute to resolving the hierarchy problem, are compactified in ${10}^{-2}\mathrm{cm}$. Another two of the compactified directions have the size ${10}^{-15}\mathrm{cm}$, and the remaining two dimensions have the size ${10}^{-17}\mathrm{cm}$. The power exponent of the cross section changes around ${\sigma }_{\mathrm{p}}=20{\pi }^2/3\times ({10}^{-15}\mathrm{cm}{)}^2$ and ${\sigma }_{\mathrm{p}}=21{\pi }^2/3\times ({10}^{-17}\mathrm{cm}{)}^2$ which are given by Eq. (57). The numbers of effective dimensions $D_{\mathrm{eff}}$ are given by 6, 8, 10 in the regions I, II, and III, respectively. The power exponent of ${\sigma }_{\mathrm{p}}$ in each region is given by $2/(D_{\mathrm{eff}}-3)$ on the mass scale.

Figure 7

$\mathcal{A}$: Areas of the two-dimensional geodetic surfaces on apparent horizons.