Distorted five-dimensional vacuum black hole

In this paper we study how the distortion generated by a static and neutral distribution of external matter affects a five-dimensional Schwarzschild-Tangherlini black hole. A solution representing a particular class of such distorted black holes admits an ${\mathbb{R}}^1\times U(1)\times U(1)$ isometry group. We show that there exists a certain duality transformation between the black hole horizon surface and the stretched singularity surface. The space-time near the distorted black hole singularity has the same topology and Kasner exponents as those of a five-dimensional Schwarzschild-Tangherlini black hole. We calculate the maximal proper time of free fall of a test particle from the distorted black hole horizon to its singularity and find that, depending on the distortion, it can be less, equal to, or greater than that of a Schwarzschild-Tangherlini black hole of the same horizon area. This implies that due to the distortion, the singularity of a Schwarzschild-Tangherlini black hole can come close to its horizon. A relation between the Kretschmann scalar calculated on the horizon of a five-dimensional static, asymmetric, distorted black hole and the trace of the square of the Ricci tensor of the horizon surface is derived.

Figure 1

The Hopf coordinates $(\lambda ,\chi ,\varphi )$. The fixed points of the Killing vectors ${\xi }_{(\varphi )}^{\alpha }$ and ${\xi }_{(\chi )}^{\alpha }$ belong to the axes defined by $\lambda =0$ and $\lambda =\pi /2$, respectively. The coordinate origin $O$ is a fixed point of the isometry group $U_{\chi }(1)\times U_{\varphi }(1)$. Planes 1, 2, and 3, embedded into four-dimensional space, are orthogonal to each other.

Figure 2

Conformal diagram for the $(\psi ,\theta )$ plane of orbits corresponding to the black hole interior. Arrows illustrate propagation of future-directed null rays. Points $A$ and $B$ connected by one of such rays are symmetric with respect to the central point $C(\pi /2,\pi /2)$.

Figure 3

Intrinsic curvature invariants of the horizon surface. (a) Dimensionless Ricci scalar. (b) The trace of the square of the Ricci tensor. Dipole-monopole distortion: $a_1=-1/5$, $b_1=0$ (line 1), $a_1=1/5$, $b_1=0$ (line 2). Quadrupole-quadrupole distortion: $a_2=b_2=-1/7$ (line 3), $a_2=b_2=1/7$ (line 4). The horizontal dashed lines represent the dimensionless Ricci scalar and the trace of the square of the Ricci tensor of a Schwarzschild-Tangherlini black hole.

Figure 4

Rotational curves of the horizon surface. (a) Section $(\theta ,\chi )$. (b) Section $(\theta ,\varphi )$. Dipole-monopole distortion: $a_1=-1/5$, $b_1=0$ (line 1), $a_1=1/5$, $b_1=0$ (line 2). Quadrupole-quadrupole distortion: $a_2=b_2=-1/7$ (line 3), $a_2=b_2=1/7$ (line 4). Regions of the sections embedded into pseudo-Euclidean space are illustrated by dashed lines. Dotted arcs of unit radius represent the horizon surface of an undistorted black hole.

Figure 5

The maximal proper time $\tau$ in units of $R_o$. (a) The maximal proper time $\tau {|}_{\theta =0}$ for the dipole-monopole distortion. (b) The maximal proper time $\tau {|}_{\theta =\pi }$ for the dipole-monopole distortion. (c) The maximal proper time $\tau {|}_{\theta =0}=\tau {|}_{\theta =\pi }$ for the quadrupole-quadrupole distortion.