2.6 I+-regularity

The classification theory of stationary black holes requires that the spacetime under consideration satisfies various global regularity conditions. These are captured by the following definition:

Definition 2.1 Let (M, g ) be a spacetime containing an asymptotically-flat end, or a KK-asymptotically-flat end 𝒮ext, and let k be a stationary Killing vector field on M. We will say that (M, g,k) is I+-regular if k is complete, if the domain of outer communications ⟨⟨Mext ⟩⟩ is globally hyperbolic, and if ⟨⟨Mext ⟩⟩ contains a spacelike, connected, acausal hypersurface 𝒮 ⊃ 𝒮ext, the closure --- 𝒮 of which is a topological manifold with boundary, consisting of the union of a compact set and of a finite number of asymptotic ends, such that the boundary --- --- ∂𝒮 := 𝒮 ∖ 𝒮 is a topological manifold satisfying

--- ∂𝒮 ⊂ ℰ + := ∂⟨⟨Mext ⟩⟩ ∩ I+(Mext ), (2.23 )
with --- ∂𝒮 meeting every generator of + ℰ precisely once. (See Figure 2View Image.)
View Image

Figure 2: The hypersurface 𝒮 from the definition of I+-regularity.

The “+ I” of the name is due to the + I appearing in (2.23View Equation).

Some comments about the definition are in order. First, one requires completeness of the orbits of the stationary Killing vector to have an action of ℝ on M by isometries. Next, global hyperbolicity of the domain of outer communications is used to guarantee its simple connectedness, to make sure that the area theorem holds, and to avoid causality violations as well as certain kinds of naked singularities in ⟨⟨Mext⟩⟩. Further, the existence of a well-behaved spacelike hypersurface is a prerequisite to any elliptic PDEs analysis, as is extensively needed for the problem at hand. The existence of compact cross-sections of the future event horizon prevents singularities on the future part of the boundary of the domain of outer communications, and eventually guarantees the smoothness of that boundary. The requirement Eq. (2.23View Equation) might appear somewhat unnatural, as there are perfectly well-behaved hypersurfaces in, e.g., the Schwarzschild spacetime, which do not satisfy this condition, but there arise various technical difficulties without this condition. Needless to say, all those conditions are satisfied by the Kerr–Newman and the Majumdar–Papapetrou (MP) solutions.

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