2.4 Domains of outer communications, event horizons

For t ∈ ℝ let ϕt[k ] : M → M denote the one-parameter group of diffeomorphisms generated by k; we will write ϕt for ϕt[k] whenever ambiguities are unlikely to occur.

Recall that I− (Ω), respectively J− (Ω ), is the set covered by past-directed timelike, respectively causal, curves originating from Ω, while ˙I− denotes the boundary of I−, etc. The sets I+, etc., are defined as − I, etc., after changing time-orientation. See [143Jump To The Next Citation Point, 16, 256, 236, 266, 66] and references therein for details of causality theory.

Consider an asymptotically-flat, or KK-asymptotically-flat, spacetime with a Killing vector k, which is timelike on the asymptotic end 𝒮ext. The exterior region Mext and the domain of outer communications ⟨⟨M ⟩⟩ ext, for which we will also use the abbreviation d.o.c., are then defined as (see Figure 1View Image)

View Image

Figure 1: 𝒮ext, Mext, together with the future and the past of Mext. One has Mext ⊂ I±(Mext ), even though this is not immediately apparent from the figure. The domain of outer communications is the intersection I+(Mext ) ∩ I− (Mext ); compare with Figure 2View Image.
+ − ⟨⟨Mext ⟩⟩ = I (∪◟tϕt(◝𝒮◜ext)◞) ∩ I (∪tϕt(𝒮ext)). (2.11 ) =:Mext
The black-hole region ℬ and the black-hole event horizon + ℋ are defined as
ℬ = M ∖ I− (M ), ℋ + = ∂ℬ. ext

The white-hole region 𝒲 and the white-hole event horizon − ℋ are defined as above after changing time orientation:

𝒲 = M ∖ I+(Mext ), ℋ − = ∂𝒲 , ℋ = ℋ + ∪ ℋ − .
It follows that the boundaries of ⟨⟨Mext⟩⟩ are included in the event horizons. We set
± ± + − ℰ = ∂⟨⟨Mext ⟩⟩ ∩ I (Mext ), ℰ = ℰ ∪ ℰ . (2.12 )
There is considerable freedom in choosing the asymptotic region 𝒮ext. However, it is not too difficult to show that I±(M ) ext, and hence ⟨⟨M ⟩⟩ ext, ℋ ± and ℰ±, are independent of the choice of 𝒮 ext whenever the associated Mext’s overlap.

By standard causality theory, an event horizon is the union of Lipschitz null hypersurfaces. It turns out that event horizons in stationary spacetimes satisfying energy conditions are as smooth as the metric allows [76Jump To The Next Citation Point, 69Jump To The Next Citation Point]; thus, smooth if the metric is smooth, analytic if the metric is.


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