Advances in Theoretical and Mathematical Physics

Causal properties of AdS-isometry groups I: causal actions and limit sets

Thierry Barbot

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We study the causality relation in the 3-dimensional anti-de Sitter space AdS and its conformal boundary $\mathrm{Ein}_2$. To any closed achronal subset $\Lambda$ in $\mathrm{Ein}_2$ we associate the invisible domain $E(\Lambda)$ from $\Lambda$ in AdS. We show that if $\Gamma$ is a torsion-free discrete group of isometries of AdS preserving $\Lambda$ and is non-elementary (for example, not abelian) then the action of $\Gamma$ on $E(\Lambda)$ is free, properly discontinuous and strongly causal. If $\Lambda$ is a topological circle then the quotient space $M_\Lambda(\Gamma) = \Gamma\setminus E(\Lambda)$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $S$ such that the induced metric on $S$ is complete. In a forthcoming paper we study the case where $\Gamma$ is elementary and use the results of the present paper to define a large family of AdS-spacetimes including all the previously known examples of BTZ multi-black holes.

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Adv. Theor. Math. Phys., Volume 12, Number 1 (2008), 1-66.

First available in Project Euclid: 10 January 2008

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Barbot, Thierry. Causal properties of AdS-isometry groups I: causal actions and limit sets. Adv. Theor. Math. Phys. 12 (2008), no. 1, 1--66.

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See also

  • See: Thierry Barbot. Causal properties of AdS-isometry groups II: BTZ multi-black-holes. Adv. Theor. Math. Phys., vol. 12, no. 6 (2008), 1209-1257.