Limit of Kerr–de Sitter spacetime with infinite angular-momentum parameter $a$

We consider the limit $a\to \infty$ of the Kerr–de Sitter spacetime. The spacetime is a Petrov type-D solution of the vacuum Einstein field equations with a positive cosmological constant $\mathrm{\Lambda }$, vanishing Mars-Simon tensor and conformally flat $ℐ$. It possesses an Abelian 2-dimensional group of symmetries whose orbits are spacelike or timelike in different regions, and it includes, as a particular case, de Sitter spacetime. The global structure of the solution is analyzed in detail, with particular attention to its Killing horizons: they are foliated by noncompact marginally trapped surfaces of finite area, and one of them “touches” the curvature singularity, which resembles a null 2-dimensional surface. Outside the region between these horizons there exist trapped surfaces that again are noncompact. The solution contains, apart from $\mathrm{\Lambda }$, a unique free parameter which can be related to the angular momentum of the nonsingular horizon in a precise way. A maximal extension of the (axis of the) spacetime is explicitly built. We also analyze the structure of $ℐ$, whose topology is ${\mathbb{R}}^3$.

Figure 1

Schematic conformal diagram of the axis $x=0$. As usual, null lines are drawn at 45° and the future direction is upwards. In this diagram we assume that the parameter $m$ is positive, and that ${\partial }_r$ is a null vector field pointing to the future too. The two Killing horizons intersect the axis at the null geodesics labeled as $r=r_1$ and $r=0$. Notice that $r_1<0$. Past null infinity, where $r\to -\infty$, is denoted by ${ℐ}^{-}$, while future null infinity ($r\to \infty$) is ${ℐ}^{+}$. We know from the considerations on [3] that both $ℐ$’s have the topology of ${\mathbb{S}}^3\backslash \{p\}={\mathbb{R}}^3$. Null geodesics traveling along the vector field ${\partial }_r$ extend all the way from ${ℐ}^{-}$ to ${ℐ}^{+}$ and are complete. The second family of null geodesics—parallel, and including, the lines marking the horizons—are incomplete to the future if they have $r<0$, while they are incomplete to the past if they have $r>r_1$. Therefore, this metric is extendible across the two borders marked by $v=\pm \infty$.

Figure 2

A conformal diagram of the axis $x=0$ similar to the previous one and with the same conventions, but now assuming that ${\partial }_r$ is past pointing (keeping $m>0$). Again the intersection of the Killing horizons with the axis are the null geodesics labeled as $r=r_1$ and $r=0$, with $r_1<0$. Past null infinity ${ℐ}^{-}$ is now defined by $r\to \infty$ while ${ℐ}^{+}$ is given by $r\to -\infty$, and as before both $ℐ$’s have the topology of ${\mathbb{S}}^3\backslash \{p\}={\mathbb{R}}^3$. Null geodesics traveling along the vector field $-{\partial }_r$ extend all the way from ${ℐ}^{-}$ to ${ℐ}^{+}$ and are complete. The second family of null geodesics are incomplete to the future if they have $r>r_1$, while they are incomplete to the past if they have $r<0$ so that this Lorentzian manifold is extendible across the two borders marked by $v=\pm \infty$. It should be noted that this spacetime can be understood as a time reversal of that shown in Fig. 1. Moreover, one can smoothly join this diagram with that of Fig. 1 on both sides to produce an extension of the axis of the solution, as explained in Fig. 3.

Figure 3

A conformal diagram of a maximal extension of the axis $x=0$ keeping the same conventions as before, but now with ${\partial }_r$ changing causal orientation depending on the patch—and always keeping $m>0$. This extension is obtained by simply gluing a diagram of the type shown in Fig. 2 to the left and right borders of the original diagram given in Fig. 1. Still, this extended axis can be further extended to the left and to the right, by now joining two copies, one at each side, of the diagram in Fig. 1, and this process can be repeated indefinitely. The dots are not part of the $ℐ$’s.