Global embedding of the Kerr black hole event horizon into hyperbolic 3-space

An explicit global and unique isometric embedding into hyperbolic 3-space, $H^3$, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into $H^3$ of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the $U(1)$ isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into $H^3$, for arbitrary values of the angular momentum. For this example, considering a quotient of $H^3$ by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding that cannot be made global.

Figure 1

$f(u)$ (left) and Gauss curvature (right) for the Kerr-Newman solution, for fixed mass $M=1$ and various values of the angular momentum $J$. The embedding in ${\mathbb{E}}^3$ fails when the $f(u)$ becomes negative; this function is bounded by $-1$ for any $J$. Note that there are regions where the curvature is positive but the embedding fails.

Figure 2

Profile of the embedding in ${\mathbb{E}}^3$ (left, first presented in 25) and $H^3$ (right, using $k=1$) of the Kerr horizon, for fixed mass $M=1$ and various values of the angular momentum $J$. For the hyperbolic embedding only the shape is relevant, not the overall scaling, since the latter depends on an arbitrary integration constant.

Figure 3

Embedding the round 2-sphere ($J=0$) into $H^3$ for $k=1$ (left) and $k=3$ (right) and various values of $\alpha$. For $k=1$ ($k=3$) the embedding cannot (can, taking an appropriate $\alpha$) be fitted completely inside the fundamental domain (11)—shaded region. We display the $x$, $z$ plane—as in Fig. 4, 7, 8—but note that the embedded surface has a $U(1)$ isometry in the $x-y$ plane.

Figure 4

Embedding the round 2-sphere ($J=0$) into $H^3$ for $\alpha =8$ and various values of $k$. The embedding can, taking the value of $k$ in an appropriate range, be fitted completely inside the fundamental domain (11)—shaded region.

Figure 5

Embedding the Kerr horizon for fixed mass $M=1$ and various values of the angular momentum $J$ ($J=0$, 0.86, 1) in ${\mathbb{R}}^3$. In the extremal case the embedding covers only a region of the horizon around the equator [8].

Figure 6

Embedding the Kerr horizon for fixed mass $M=1$ and various values of the angular momentum $J$ ($J=0$, 0.86, 1) in $H^3$, with $k=1$. The embedding covers the whole of the horizon, even in the extremal case. Note, however, that the ${\mathbb{Z}}_2$ symmetry around the equator is lost, when the angular momentum is turned on. This is to be expected, since it is not a symmetry of the embedding space. But surprisingly the symmetry remains in the $J=0$ case.

Figure 7

Embedding the Kerr-Newman event horizon for $M=1$ and $J=0.95$ into $H^3$ for various values of $k$ and of an additional integration constant $\alpha$, fixed at $\alpha ={\alpha }_{\mathrm{touch}}$. No choice can fit the embedding completely inside the fundamental domain (11) [shaded region].

Figure 8

Embedding the Kerr-Newman event horizon for $M=1$, $J=0.8725$ (left) and $J=0.8735$ (right) into $H^3$ for various values of $k$, including for each case $k_{\max }$ and $\alpha ={\alpha }_{\mathrm{touch}}$. In the former case the embedding can still be fitted inside the fundamental domain (11) (shaded region), but not in the latter case, which is explicit in the detail (middle).

Figure 9

Function $f(u)$ for $\zeta =100$ (left) and $\zeta =10$ (right), fixed mass $M=1$ and various values of the angular momentum $J$. The left plot corresponds to a large distance and therefore resembles Fig. 1 (i.e. the case of an isolated black hole), except at the North pole ($u=1$), where the strut meets the horizon.

Figure 10

Function $f(u)$ for $\zeta =5$ (left) and $\zeta =2.2$ (right), fixed mass $M=1$ and various values of the angular momentum $J$. As the distance decreases, the function becomes less sensitive to the angular momentum, and approaches that of a nonrotating black hole, albeit deformed near the North pole.

Figure 11

Profile of the embedding in ${\mathbb{E}}^3$ (left, first presented in 25) and $H^3$ (right, using $k=1$) of the double-Kerr horizon, for fixed mass and angular momentum $M=1=J$ and various values of the physical distance in between the two black holes. At the north pole there is a “strut,” and the embedding always fails. The grey (black) lines correspond to the counter-rotating (corotating) case.

Figure 12

Embedding the lower black hole in double-Kerr horizon for fixed mass and angular momentum $M=1=J$ and various values of the physical distance $d$ ($d=25$, 10, 3) in ${\mathbb{E}}^3$. As the distance decreases: i) the South pole gets covered by the embedding (this is a consequence of the smaller angular velocity of the black hole); ii) there is a growing patch around the North pole that is not covered by the embedding, manifesting the deformation induced by the strut, which becomes stronger as the interaction between the two black holes becomes stronger.

Figure 13

Embedding the double-Kerr horizon for fixed mass and angular momentum $M=1=J$ and various values of the physical distance $d$ ($d=25$, 10, 3) in $H^3$, with $k=1$. Unlike the Euclidean embedding, the South pole is always covered by the hyperbolic embedding. As in the Euclidean embedding, there is a growing patch around the North pole that is not covered by the hyperbolic embedding, manifesting the deformation induced by the strut. The patch does not appear smaller than in the Euclidean embedding.