Black hole formation in the grazing collision of high-energy particles

TL;DR

The paper tackles horizon formation in grazing collisions of high-energy particles within $D$-dimensional gravity, using the Eardley-Giddings construction to locate apparent horizons by solving interior Poisson equations on a $(D-2)$-dimensional plane and determining the maximal impact parameter $b_{\\text{max}}$ and the cross-section factor $F(D)$ in $\\sigma_{\\text{b.h.production}}=F(D)\\pi r_h^2(2\\mu)$. It finds that an apparent horizon forms for $b \\\lesssim 1.5\\, r_h(\\mu)$ and that $b_{\\text{max}}$ grows with dimension $D$, with horizon shapes becoming more prolate at higher $D$; the horizon mass fraction $M_{\\text{A.H.}}/(2\\mu)$ decreases with $D$, implying more energy escapes into extra dimensions. The study shows $H_D^{\\text{A.H.}}(b_{\\text{max}})\\sim 1.4$–$1.6$ and $\\mathcal{H}_D^{\\text{A.H.}}(b)$ near unity, supporting the $(D-3)$-volume conjecture as a better horizon-formation criterion than the hoop conjecture in higher dimensions, and highlights the potential need for a rigorous definition of the $(D-3)$-volume. Overall, the work informs collider phenomenology and the role of extra dimensions in gravitational collapse by providing dimension-dependent horizon criteria and energy-trapping behavior.

Abstract

We numerically investigate the formation of D-dimensional black holes in high-energy particle collision with the impact parameter and evaluate the total cross section of the black hole production. We find that the formation of an apparent horizon occurs when the distance between the colliding particles is less than 1.5 times the effective gravitational radius of each particles. Our numerical result indicates that although both the one-dimensional hoop and the (D-3)-dimensional volume corresponding to the typical scale of the system give a fairly good condition for the horizon formation in the higher-dimensional gravity, the (D-3)-dimensional volume provide a better condition to judge the existence of the horizon.

Figures (6)

• Figure 1: The shape of the apparent horizon $\mathcal{C}$ on $(x_1,x_2)$-plane in the collision plane $u=v=0$ for $D=4,...,7$. Incoming particles are located on the horizontal line $x_2=0$. Values of $b/r_0$ are $0$($\bullet$), $0.4$($\circ$), $0.7$($\blacklozenge$), $1.0$($\lozenge$, shown only in $D=6,7$), and $b_{\text{max}}/r_0$($\star$). As the distance $b$ between two particles increases, the radius of $\mathcal{C}$ decreases.
• Figure 2: The relation between the impact parameter $b$ and the minimum radius $r_{\text{min}}$ of $\mathcal{C}$ for $D=4,...,11$. The value of $b_{\text{max}}/r_0$ grows as $D$ increases.
• Figure 3: The value of $b_{\text{max}}$ (crosses) as a function of the spacetime dimension $D$. The dotted line $r_h(2\mu)$ which comes from the hoop conjecture. Although the ratio $b_{\text{max}}/r_h(2\mu)$ takes the value around unity, the hoop conjecture does not explain the increase of $b_{\text{max}}/r_h(2\mu)$ with $D$. The solid line $b_{\text{max}}/r_h(2\mu)=1.5r_h(\mu)/r_h(2\mu)\sim 2^{-1/(D-3)}$ gives a good fit of $b_{\text{max}}$.
• Figure 4: The relation between the horizon mass $M_{\text{A.H.}}/2\mu$ and the impact parameter $b$ for $D=4,..., 11$. $M_{\text{A.H.}}/2\mu$ becomes small as $D$ increases. The black hole in the higher dimensional spacetime can trap small amount of the energy.
• Figure 5: The value of $H_D^{\text{A.H.}}$ as a function of $b/r_h(2\mu)$ for $D=4,..., 11$. The circles show the values at $b=b_{\text{max}}$. $H_D^{\text{A.H.}}(b_{\text{max}})$ becomes large as $D$ increases.