Kerr-de Sitter Universe

Abstract

It is now widely accepted that the universe as we understand it is accelerating in expansion and fits the de Sitter model rather well. As such, a realistic assumption of black holes must place them on a de Sitter background and not Minkowski as is typically done in General Relativity. The most astrophysically relevant black hole is the uncharged, rotating Kerr solution, a member of the more general Kerr-Newman metrics. A generalization of the rotating Kerr black hole to a solution of the Einstein's equation with a cosmological constant $\Lambda$ was discovered by Carter \cite{DWDW}. It is typically referred to as the Kerr-de Sitter spacetime. Here, we discuss the horizon structure of this spacetime and its dependence on $\Lambda$. We recall that in a $\La>0$ universe, the term `extremal black hole' refers to a black hole with angular momentum $J > M^2 $. We obtain explicit numerical results for the black hole's maximal spin value and get a distribution of admissible Kerr holes in the ($\Lambda$, spin) parameter space. We look at the conformal structure of the extended spacetime and the embedding of the 3-geometry of the spatial hypersurfaces. In analogy with Reissner-Nordstr\"{o}m -de Sitter spacetime, in particular by considering the Kerr-de Sitter causal structure as a distortion of the Reissner-Nordstr\"{o}m-de Sitter one, we show that spatial sections of the extended spacetime are 3-spheres containing 2-dimensional topologically spherical sections of the horizons of Kerr holes at the poles. Depending on how a $t=$ constant 3-space is defined these holes may be seen as black or white holes (four possible combinations).