Global structure of Kerr–de Sitter spacetimes

The global structure of the family of Kerr--de Sitter spacetimes is reexamined. Taking advantage of the natural length scale set by the cosmological constant $\mathrm{\ensuremath{\Lambda}}>0$, conditions on the parameters $(\mathrm{\ensuremath{\Lambda}},M,{a}^{2})$ have been found, so that a Kerr--de Sitter spacetime either describes a black hole with well-separated horizons, or describes degenerate configurations where two or more horizons coincide. As long as the rotation parameter ${a}^{2}$ is subject to the constraint ${a}^{2}\mathrm{\ensuremath{\Lambda}}\ensuremath{\ll}1$, and the mass parameter $M$ is subject to ${a}^{2}[1+O({a}^{2}\mathrm{\ensuremath{\Lambda}}{)}^{2})]<{M}^{2}<\frac{1}{9\mathrm{\ensuremath{\Lambda}}}[1+2{a}^{2}\mathrm{\ensuremath{\Lambda}}+O({a}^{2}\mathrm{\ensuremath{\Lambda}}{)}^{2})]$, then a Kerr--de Sitter spacetime with parameters in these ranges describes a black hole possessing an inner horizon separated from an outer horizon and the hole is embedded within a pair of cosmological horizons. Still for ${a}^{2}\mathrm{\ensuremath{\Lambda}}\ensuremath{\ll}1$, but assuming that either ${M}^{2}>\frac{1}{9\mathrm{\ensuremath{\Lambda}}}[1+2{a}^{2}\mathrm{\ensuremath{\Lambda}}+O({a}^{2}\mathrm{\ensuremath{\Lambda}}{)}^{2})]$ or ${M}^{2}<{a}^{2}[1+O({a}^{2}\mathrm{\ensuremath{\Lambda}}{)}^{2})]$, the Kerr--de Sitter spacetime describes a ringlike singularity enclosed by two cosmological horizons. A Kerr--de Sitter spacetime may also describe configurations where the inner, the outer and one of the cosmological horizons coincide. However, we found that this coalescence occurs provided ${M}^{2}\mathrm{\ensuremath{\Lambda}}\ensuremath{\sim}1$ and due to the observed smallness of $\mathrm{\ensuremath{\Lambda}}$, these configurations are probably irrelevant in astrophysical settings. Extreme black holes, i.e. black holes where the inner horizon coincides with the outer black hole horizon, are also admitted. We have found that in the limit ${M}^{2}\mathrm{\ensuremath{\Lambda}}\ensuremath{\ll}1$ and ${a}^{2}\mathrm{\ensuremath{\Lambda}}\ensuremath{\ll}1$, extreme black holes occur, provided ${a}^{2}={M}^{2}(1+O(\mathrm{\ensuremath{\Lambda}}{M}^{2}))$. Finally a coalescence between the outer and the cosmological horizon, although in principle possible, is likely to be unimportant at the astrophysical level, since this requires ${M}^{2}\mathrm{\ensuremath{\Lambda}}\ensuremath{\sim}1$. Our analysis shows that as far as the structure of the horizons is concerned, the family of Kerr--de Sitter spacetimes exhibits similar structure to the Reissner--Nordstrom--de Sitter family of spacetimes.

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