Black topologies production in extra dimensions

We investigate the configuration resulting after a collision of gravitational sources in a higher dimensional space with extra dimensions. Evidence is found that as the energy increases, there is a phase transition in the topology of the black object that is being formed: from the black hole to the black string topology. An intuitive mechanism for the way the transition takes place is proposed. The transition occurs at a finite value of the energy where an upper and a lower bound is found. Furthermore, at low energies the compact dimension behaves as an extended one while at high energies the extra dimension seems to decouple. Finally, the implications about the Gregory-Laflamme instability, the implications to the accelerators as well as holographic implications are discussed.

Figure 1

Plot of the boundary $C$ of the trapped surface in the two extreme limits as found from analytical considerations. Horizontal axis: $\theta$, vertical axis: $u=r/R$ (the same applies for the plots of the figures that follow). Top plot refers to low energies and yields to a $D^4$ topology (BH). As $x$ increases in the range $x={10}^2$, ${10}^3$, ${10}^3$, ${10}^5$ the circles become smaller. Bottom plot refers to high energies (BS) with topology $D^3\times S^1$. As $x$ decreases in the range $x={10}^{-2}$, ${10}^{-2.6}$, ${10}^{-3.2}$, ${10}^{-4}$, the lines move upwards.

Figure 2

Trapped boundary for the two distinct boundary conditions (8); dashed and solid curves correspond to the first and second boundary condition respectively. Top panel: ${\varphi }_h^{x\gg 1}=C^{x\gg 1}=-2^{7/6}R{x}^{-1/3}$ and $x=1$, 8, ${10}^2$,${10}^3$. The pairs of curves shrink as $x$ increases and soon they merge (third pair of curves for $x=8$; low energy limit that results to a BH). Bottom panel: ${\varphi }_h^{x\ll 1}=C^{x\ll 1}=-2^{3/2}R{x}^{-1/2}$ and $x=0.1$, 0.035, 0.02. The pairs of curves move upwards as $x$ decreases and soon they merge (top pair of curves with $x=0.02$; high energy limit that results to a BS).

Figure 3

As the energy increases, it is expected that the “nearby” BHs will grow and approach each other.

Figure 4

The curves result from Eq. (11). The value of $x$ decreases from the top to the bottom plot. In the top plot, the trapped horizon which is not convex, covers the whole compact dimension because it extends up to values of $\theta =\pm \pi$. Increasing the energy further (two bottom plots), seems to yield to a phase transition: the resulting BH begins to wind the compact dimension changing the topology from $D^4$ to $D^3\times S^1$. This is a nonuniform BS topology. As the energy increases further ($x$ decreases), the curves move upwards ending up to straight lines independent on $\theta$ and matching those of Fig. 1 (uniform BS configuration). It is pointed out that the shaded areas consisting of the two (half) ellipses left and right of $r=0$ are not a part of the trapped surface. There presence is to shield the image charges (see Sec. 5).

Figure 5

Trapped boundary resulting from first condition of (6) with ${\varphi }_h^{x\ll 1}=C^{x\ll 1}=-2^{3/2}R{x}^{-1/2}$. As energy increases ($x$ decreases), the curves move upwards ending up to straight lines and matching those of Fig. 1 (uniform BS configuration). During the $x$ evolution, the trapped area remains convex (in each subinterval: $[(2n-1)\pi ,(2n+1)\pi ]$, $n\in \mathbb{Z}$) unlike the plots of Fig. 4.