Improved analysis of black hole formation in high-energy particle collisions

We investigate formation of an apparent horizon (AH) in high-energy particle collisions in four- and higher-dimensional general relativity, motivated by TeV-scale gravity scenarios. The goal is to estimate the prefactor in the geometric cross section formula for the black hole production. We numerically construct AHs on the future light cone of the collision plane. Since this slice lies to the future of the slice used previously by Eardley and Giddings [Phys. Rev. D 66, 044011 (2002)] and by one of us and Nambu [Phys. Rev. D 67, 024009 (2003)], we are able to improve the prefactor estimates. The black hole production cross section increases by 40%–70% in the higher-dimensional cases, indicating larger black hole production rates in future-planned accelerators than previously estimated. We also determine the mass and the angular momentum of the final black hole state, as allowed by the area theorem.

Figure 1

Schematic picture of the spacetime of colliding high-energy particles.

Figure 2

Schematic picture of the spacetime of colliding high-energy particles with $(D-3)$ dimensions suppressed. The schematic shape of AH on the new slice ($u>0,v=0$ and $v>0,u=0$) is shown by solid lines, while the AH on the old slice ($u<0,v=0$ and $v<0,u=0$) is shown by dashed lines. Dotted lines indicate coordinate singularities.

Figure 3

Schematic picture of the boundaries $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$. We should solve for $h(r,\varphi )$ in the region surrounded by the two boundaries.

Figure 4

The shape of $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$ for $b/r_0=0.4,0.6,0.8,0.841$ in the $D=4$ case. Two dots in each figure indicate the location of incoming particles. The old-slice AHs in 17 for $b/r_0=0.4,0.6,0.8$ are shown by gray lines. We could not find the AH for $b/r_0=0.842$.

Figure 5

The shapes of $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$ for $b=0.4,0.9,1.1,1.145$ in the $D=5$ case. Two dots in each figure indicate the location of the incoming particles. The old-slice AHs of 18 for $b/r_0=0.4,0.9$ are shown by gray lines. We could not find the AH for $b/r_0=1.146$.

Figure 6

The shape of $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$ for $b/r_0=0.4,1.0,1.3,1.333$ in the $D=6$ case. Two dots in each figure indicate the location of the incoming particles. The old-slice AHs of 18 for $b=0.4,1.0$ are shown by gray lines. We could not find the AH for $b/r_0=1.334$.

Figure 7

Three-dimensional plots of the AH shape function at $b=b_{\max }$ for $D=4,5,$ and $6$. The AH becomes taller for larger $D$ as a consequence of the growing power exponent in the boundary condition (18).

Figure 8

The relation between $b/r_0$ and $r_{\min }/r_0$, where $r_{\min }=g_{\mathrm{in}}(\pi /2)$. The maximal impact parameter occurs at $d{r}_{\min }/db=-\infty$.

Figure 9

Summary of the $b_{\max }/r_h(2\mu )$ values (○) and the previous ${\widehat{b}}_{\max }/r_h(2\mu )$ values (×) for $D=4,\dots ,11$.

Figure 10

The rigorous lower bound $M_{lb}$ of the final irreducible mass (black lines) and the indicator of the trapped energy $M_{\mathrm{AH}}$ (light gray lines) for $D=4,\dots ,11$. The previous value of the AH mass ${\widehat{M}}_{\mathrm{AH}}$ of 17, 18 is also shown by dark gray lines.

Figure 11

The regions (i), (ii), and (iii) in the $(\xi ,\zeta )$-plane for $b=0.4,0.6,0.8$ in the $D=4$ case.

Figure 12

The regions (i), (ii), and (iii) in the $(\xi ,\zeta )$-plane for $b=0.5,1.0,1.145$ in the $D=5$ case.

Figure 13

The regions (ii) and (iii) in the $(\xi ,\zeta )$-plane for $b=0.5,1.0,1.3$ in the $D=6$ case.

Figure 14

The regions (ii) and (iii) in the $(\xi ,\zeta )$-plane for $b=0.5,1.0,1.5$ in the $D=9$ case.