Domains of Outer Communications and Interior Boundaries

The domain of outer communications is the causally accessible exterior attached to a chosen asymptotic end: events there can both receive influence from the distant past and communicate with the distant future. This is the region classified by stationary uniqueness theorems. Its spacetime boundary is associated with event horizons, whereas an axis becomes a boundary only after symmetry reduction. When nonzero charge or spin supplies an inner Cauchy horizon, it lies behind the exterior event horizon. Keeping those roles distinct prevents coordinate diagrams from silently changing the theorem’s domain.

Helpful background. Exact Solutions of Einstein’s Equations supplies maximal extensions and named examples; Causality and Global Structure supplies chronological sets and conformal infinity; The Einstein Equation fixes the matter model used by classification; Killing Fields and Isometry Groups supplies stationary reduction; and Black-Hole Geometry and Mechanics orients the event-horizon definition.

The exterior defined from both time directions

Choose one asymptotically flat end with past and future null infinities $\mathscr I^-$ and $\mathscr I^+$. Its domain of outer communications is the connected exterior component

$$\mathcal D =I^+(\mathscr I^-)\cap I^-(\mathscr I^+).$$

The use of chronological sets makes $\mathcal D$ open. A point belongs to it only if some future-directed timelike curve can reach it from the asymptotic past and some future-directed timelike curve can continue from it to the asymptotic future. In a stationary formulation one can equivalently thicken a spatial asymptotic end by the stationary flow and take the intersection of its chronological future and past. Chruściel, Costa, and Heusler give this precise construction in Chruściel, Costa, and Heusler 2012, §2.4, eqs. (2.11)–(2.12).

The chosen end matters. The eternal Schwarzschild extension has two asymptotically flat exteriors; each is the domain of outer communications relative to its own infinity. For one selected end, $\mathcal D$ is the corresponding region $r\gt2GM$, not the union of that exterior with the black-hole interior or the other exterior. In a collapse spacetime there need not be a past white-hole horizon, yet the accessible exterior is still selected by both causal directions.

For positive-mass subextremal Reissner–Nordström, Kerr, and Kerr–Newman black holes, $\mathcal D$ is likewise the connected region outside $r_+$. Kerr’s ergoregion partly lies inside $\mathcal D$ when the spin is nonzero: inability to remain static is not inability to escape. The definition is causal, not a sign condition on the norm of the stationary Killing field.

Event-horizon boundaries and conformal infinity

The future event horizon relative to the chosen end is

$$\mathcal H^+=\partial J^-(\mathscr I^+),$$

and the past horizon is the time-reversed boundary $\mathcal H^-=\partial J^+(\mathscr I^-)$. Appropriate portions of these horizons form the boundary of $\mathcal D$ in the physical spacetime. They are not included in the open set $\mathcal D$, although theorem statements often work with a closure whose boundary contains smooth horizon cross-sections.

Null infinity has a different status. It is a boundary in the conformal completion, not a set of physical spacetime points. Asymptotic flatness controls the behavior of fields as one approaches it and normalizes the stationary time translation used to define mass and surface gravity. Thus a typical exterior problem has one limiting boundary at infinity and another at the horizon, but only the latter is a hypersurface in the physical extension.

An axial fixed-point set is not a boundary of the four-dimensional spacetime either. Near a regular rotation axis, the apparent degeneration of the azimuthal coordinate is the same polar-coordinate phenomenon as the origin of a plane. After quotienting a stationary axisymmetric spacetime by its time and rotation symmetries, however, the axis projects to an edge of the two-dimensional orbit space. Horizon segments and asymptotic segments also become orbit-space boundaries. Uniqueness arguments impose different regularity data on these edges, so confusing the reduced boundary with a physical wall changes the geometry.

Cauchy horizons are not exterior boundaries

On the charged subextremal Reissner–Nordström branch or the rotating subextremal Kerr branch, the inner root $r_-$ is a Cauchy horizon in the idealized maximal solution. It bounds a domain of dependence for suitable initial data and marks a loss of unique prediction if one permits further extensions. But it lies inside the future event horizon $r_+$ and therefore is not a boundary component of the domain of outer communications associated with the original asymptotic end. At the joint Schwarzschild endpoint, the formal value $r_-=0$ is the central curvature singularity, not a Cauchy horizon.

This distinction can be read directly from causal accessibility. Points just outside $r_+$ can communicate with $\mathscr I^+$; points in the black-hole interior cannot, whether or not they have crossed an available $r_-$. Whenever an inner Cauchy horizon exists, the outer event horizon is encountered first along an infalling worldline. Calling both roots “boundaries of the exterior” erases that order.

The smooth inner horizon of an exact stationary metric also does not supply a stability theorem. Perturbations are blueshifted there, and the resulting interior may differ strongly from the analytic extension. Visser separates the outer, inner, and chronology structures of Kerr in Visser 2009, §§9–12. The domain used by stationary classification deliberately avoids depending on that unresolved deep-interior continuation.

Hypotheses that make the exterior theorem-ready

The definition of $\mathcal D$ alone assumes neither stationarity nor field equations. A uniqueness theorem adds substantial hypotheses. One precise axisymmetric electrovacuum result assumes a four-dimensional, stationary, axisymmetric, asymptotically flat Einstein–Maxwell spacetime with a complete stationary Killing field and an $I^+$-regular exterior. Here $I^+$-regularity includes global hyperbolicity of $\mathcal D$ and a connected acausal spacelike hypersurface whose closure meets each relevant future- horizon generator once. The full definition and boundary condition appear in Chruściel, Costa, and Heusler 2012, §2.6, Definition 2.1.

Under those conditions, the axisymmetric uniqueness theorem classifies the domain of outer communications by a Weinstein electrovacuum solution; if the event horizon is connected, the exterior is the Kerr–Newman domain. This is a theorem about the exterior geometry. It does not assert that the analytic region beyond an inner horizon is physically realized.

The proof architecture explains why $\mathcal D$ is natural. Global regularity and topology control the orbit space; quotienting by stationarity and axial symmetry reduces the Einstein–Maxwell equations to an elliptic harmonic-map problem. Horizon, axis, and infinity supply distinct boundary data in that reduced problem. A divergence identity compares two candidate solutions, and regularity makes the boundary terms vanish, forcing equality. The connected-horizon condition then selects Kerr–Newman. The theorem and this argument are summarized in Chruściel, Costa, and Heusler 2012, §3.2.7, Theorem 3.2.

Every hypothesis matters. With different matter models, stationary black holes can carry additional fields; in higher dimensions, black rings show that spherical Kerr-like classification fails; with disconnected degenerate components, Majumdar–Papapetrou solutions provide an electrovacuum limitation to a one-horizon slogan. These are counterexamples to overbroad statements, not failures of the precisely stated theorem.

Locate each boundary in subextremal Kerr

For a selected asymptotic end of rotating subextremal Kerr, $0\lt\lvert a\rvert\lt GM$, classify the following: $r\gt r_+$, $r=r_+$, the rotation axis in the exterior, conformal infinity, and $r=r_-$. Which belong to the open domain of outer communications, which bound it in spacetime, and which become boundaries only in a completion or symmetry quotient?

Solution

The connected region $r\gt r_+$ is the open domain of outer communications; it includes the exterior part of the ergoregion. The future component of $r=r_+$ is the spacetime event-horizon boundary of that domain, and the past component plays the time-reversed role in the eternal extension.

The regular rotation axis at $\sin\theta=0$ consists of physical points inside the exterior, not a spacetime boundary. It becomes an edge after quotienting by axial symmetry. Null and spatial infinity belong to the conformal completion and encode asymptotic boundary conditions, but they are not points of the physical manifold. The inner surface $r=r_-$ is a Cauchy horizon behind $r_+$; it is not a boundary of the selected exterior domain. This classification is exactly what permits the reduced proof problem to use axis, horizon, and infinity data without importing the inner extension.

Killing Horizons and Bifurcate Geometry next develops the local geometry of the null boundary and its preferred stationary generator.

References

• Chruściel, Piotr T., João Lopes Costa, and Markus Heusler. “Stationary Black Holes: Uniqueness and Beyond.” Living Reviews in Relativity 15 (2012): 7. DOI.
• Visser, Matt. “The Kerr Spacetime: A Brief Introduction.” In The Kerr Spacetime, edited by David L. Wiltshire, Matt Visser, and Susan M. Scott. Cambridge: Cambridge University Press, 2009. arXiv:0706.0622v3.