Abstract
Killing horizons which can be such for two or more linearly independent Killing vectors are studied. We provide a rigorous definition and then show that the set of Killing vectors sharing a Killing horizon is a Lie algebra of dimension at most the dimension of the spacetime. We prove that one cannot attach different surface gravities to such multiple Killing horizons, as they have an essentially unique nonzero surface gravity (or none). always contains an Abelian (sub)algebra—whose elements all have vanishing surface gravity—of dimension equal to or one less than dim . There arise only two inequivalent possibilities, depending on whether or not the nonzero surface gravity exists. We show the connection with near-horizon geometries, and also present a linear system of PDEs, the master equation, for the proportionality function on the horizon between two Killing vectors of a multiple Killing horizon, with its integrability conditions. We provide explicit examples of all possible types of multiple Killing horizons, as well as a full classification of them in maximally symmetric spacetimes.
Highlights
The notion of Killing horizon captures the idea that a Killing vector ξ in a spacetime (M, g) may change causal character precisely on a null hypersurface
We show the connection with Near Horizon geometries, and present a linear system of PDEs, the master equation, for the proportionality function on the horizon between two Killing vectors of a multiple Killing horizon, with its integrability conditions
Killing horizons play a fundamental role in general relativity, in particular in the context of black holes in equilibrium: By Hawking’s rigidity theorem the event horizon of a stationary, asymptotically flat black hole spacetime, is a Killing horizon
Summary
The notion of Killing horizon captures the idea that a Killing vector ξ in a spacetime (M, g) may change causal character precisely on a null hypersurface. In the first case we call the MKH fully degenerate, in the latter one non-fully degenerate or just non-degenerate This result states, in particular, that to any MKH one can ascribe a single non-zero surface gravity (or temperature) and this is associated to a single Killing generator (up to scale, naturally). Another general property obtained is that, letting n + 1 denote the spacetime dimension, the maximal dimension m of the Lie algebra AH is n in the fully degenerate case while it is n + 1 in the non-degenerate case. Small Latin indices i, j, . . . will enumerate the different Killing vectors of multiple Killing horizons and will take values in {1, . . . , m}, where m ≤ n + 1
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