Isolated black holes without ${\mathbb{Z}}_2$ isometry

A mechanism to construct asymptotically flat, isolated, stationary black hole (BH) spacetimes with no ${\mathbb{Z}}_2$ ($\mathrm{No}\mathbb{Z}$) isometry is described. In particular, the horizon geometry of such $\mathrm{No}\mathbb{Z}$ BHs does not have the usual north-south (reflection) symmetry. We discuss two explicit families of models wherein $\mathrm{No}\mathbb{Z}$ BHs arise. In one of these families, we exhibit the intrinsic horizon geometry of an illustrative example by isometrically embedding it in Euclidean 3-space, resulting in an “egg-like” shaped horizon. This asymmetry leaves an imprint in the $\mathrm{No}\mathbb{Z}$ BH phenomenology, for instance in its lensing of light; but it needs not be manifest in the BH shadow, which in some cases can be analytically shown to retain a ${\mathbb{Z}}_2$ symmetry. Light absorption and scattering due to an isotropic source surrounding a $\mathrm{No}\mathbb{Z}$ BH endows it with a non-zero momentum, producing an asymmetry triggered BH rocket effect.

Figure 1

Scalar field amplitude obtained from (2) with $\alpha =1$, using the electromagnetic source (3) of a KN background. The contour lines are level sets and are clearly not ${\mathbb{Z}}_2$ invariant (i.e., as $\theta \to \pi -\theta$).

Figure 2

Euclidean embeddings of the intrinsic horizon geometry for rotating dyonic BHs in Kaluza Klein theory. (Top panel) A 3D plot for the solution with $(J;Q,P)=(0.035;1.87,0.01)$. (Bottom panel) 2D plots (constant azimuthal coordinate) for a sequence of solutions with ($J;Q,P$) varying between ($0.54;0.23,1.01$) (largest) and ($0.065;0.02,1.8$) (smallest).

Figure 3

(Top panel) Deformation $\epsilon$ $vs$ the dimensionless parameter $\xi /M^2$ for several values of $\alpha$. (Bottom panel) Scalar field level sets for the solution highlighted as a dot in the upper panel.

Figure 4

Lensing and shadow due to a $\mathrm{No}\mathbb{Z}$ dyonic BH with $(J,Q,P)\simeq (0.22,0.15,1.5)$. The green (light blue) color light is emitted by the north (south) far away celestial sphere. The image is for an observer on the $\theta =\pi /2$ plane. The inset shows the shadow contour and its ${\mathbb{Z}}_2$ reflection. Both curves coincide. Thus, despite the lack of ${\mathbb{Z}}_2$ of the lensing, the shadow is ${\mathbb{Z}}_2$ symmetric.

Figure 5

Illustration of the thrust imparted on a $\mathrm{No}\mathbb{Z}$ BH due to an isotropic light source (celestial sphere).