Deformed hyperbolic black holes

Black holes with planar or hyperbolic horizons are known to exist in anti-de Sitter (AdS) space, alongside the usual ones with spherical horizons. In this paper, we consider a one-parameter generalization of these black holes that is contained in the AdS C-metric. In terms of the domain-structure analysis recently developed for such solutions, these black holes have a domain in the shape of a triangle. It is shown that the horizons of these black holes are deformed hyperbolic spaces, with the new parameter controlling the amount of deformation. The space-times are static and completely regular outside the horizons. We argue that these black holes are hyperbolic analogues of the “slowly accelerating” spherical black holes known to exist in AdS space.

Figure 1

The curves of $F(x)$ and $G(x)$ for (a) $\mu <0$, and (b) $\mu >0$.

Figure 2

The parameter range as a plot of $\nu$ versus $\mu$. The four shaded regions A–D are those which describe a Lorentzian space-time in the coordinate range $-1<x<y<0$.

Figure 3

The domain structure for Region A. The $x$-axis is in the horizontal direction, while the $y$-axis is in the vertical direction. Shaded areas correspond to coordinate ranges with Lorentzian signature, with the darker shade representing the coordinate range of particular interest $-1<x<y<0$. The diagonal line corresponds to $x=y$, while the edges of the plot correspond to $x,y=\pm \infty$.

Figure 4

The domain structure for the various parts of Region B: (a) Region B1, (b) Region B2, (c) Region B3.

Figure 5

The domain structure for the various parts of Region C: (a) Region C1, (b) Region C2, (c) Region C3.

Figure 6

The domain structure for the various parts of Region D: (a) Region D1, (b) Region D2, (c) Region D3, (d) Region D4.

Figure 7

Examples of the horizon geometries of the deformed spherical black holes, as embeddings in three-dimensional Euclidean space. The units of the axes are defined only up to an overall scale. The full geometry can be visualized as the surface of revolution of these curves around the horizontal axis. The curves plotted are for $\mu =4$ and for $\nu$ values as indicated in the legend.

Figure 8

Examples of the horizon geometries of the deformed hyperbolic black holes, as plots of $\sqrt{g_{\varphi \varphi }^{\mathrm{H}}}$ versus $x$. The full geometry can be visualized as the surface of revolution of these curves around the $x$-axis. The solid, dashed and dotted curves have $(\mu ,\nu )=(0.5,0.9)$, (1, 1.8) and (4, 3.9), respectively. The last case corresponds to a black globule. Since it has a positive scalar curvature, the globular part can actually be embedded in three-dimensional Euclidean space (cf. Fig. 11).

Figure 9

The interpretation of the various parts of the $\nu =1+\mu$ boundary of the $\mu –\nu$ plot of Fig. 2, when the limit $\nu \to 1+\mu$ is taken as described in the text.

Figure 10

The domain structures for the case $\mu =0$ with (a) $-1<\nu <0$; and (b) $0<\nu <1$.

Figure 11

Examples of horizon geometries for $\mu =4$ and for values of $\nu$ at and around the critical value $2\sqrt{\mu }$, as embeddings in three-dimensional Euclidean space. As usual, the full geometry can be visualized as the surface of revolution of these curves around the horizontal axis.

Figure 12

The closed curve $K=0$ in the $\mu –\nu$ plot of Fig. 2. The shaded regions outside this curve with $K>0$ are parameters which have a correspondence to those in the traditional form (43). The labeling of the shaded regions follows that of Fig. 2.