Horizon Topology, Branching, and Jumps

Horizon “topology change” can mean three inequivalent things: a change in spatial cross-sections of the global event horizon, the birth or merger of marginally trapped tubes, or a discontinuous change in which MOTS is outermost on successive slices. Theorems constrain each only under stated hypotheses. A binary merger can therefore exhibit branch creation and apparent-horizon area jumps while strictly stable outer cross-sections remain two-spheres under the topology theorem’s energy and regularity hypotheses.

Helpful background. The horizon taxonomy is developed in Black-Hole Geometry and Mechanics; late-time deformation belongs to Black-Hole Perturbation Theory; foliations and energy conditions come from Formulations of General Relativity; numerical branch tracking is treated in Computational Relativity; and the binary regime is organized in Relativistic Two-Body Dynamics.

Topology theorem and hypotheses

Use the cited theorem’s normalization $G_{ab}=T^{\rm eff}_{ab}$, where $T^{\rm eff}_{ab}=8\pi G T_{ab}$ absorbs the usual Einstein coupling. Let $(\mathcal M^{n+1},g)$, $n\geq3$, satisfy the dominant energy condition, let $V^n$ be spacelike, and let $S^{n-1}\subset V$ be smooth, closed, two-sided, and marginally outer trapped. Galloway and Schoen first assume that $S$ is an outer apparent horizon: no homologous outer-trapped surface lies outside it. Their proof also applies when $S$ is stably outermost, meaning that the principal stability eigenvalue—denoted $\lambda_1$ in their indexing and $\lambda_0$ elsewhere in this chapter—obeys $\lambda_1\geq0$. Then $S$ is of positive Yamabe type unless its induced metric is Ricci flat and both its outgoing null second fundamental form $\chi^{(K)}_{AB}$ and the flux $T^{\rm eff}_{ab}U^aK^b$ vanish identically. Strict stability, $\lambda_1>0$, removes this exception Galloway and Schoen 2006, Theorem 2.1 and concluding remark 1, pp. 2–5.

In four spacetime dimensions, a compact orientable two-surface of positive Yamabe type has genus zero and is topologically $S^2$. The exceptional Ricci-flat induced metric is flat and permits a torus when the stability, null-curvature, and flux terms all saturate. Thus stable outermostness alone does not exclude that equality case; strict stability does. This is not a theorem about every marginal surface: inner unstable branches need not inherit the conclusion, higher dimensions allow other positive-Yamabe topologies, and a cosmological term must be included in the effective right-hand side before its energy sign is tested. The merger application here is vacuum with $\Lambda=0$.

Proof architecture

The argument begins with the first-variation operator for $\theta_{(\ell)}$. Stability supplies a positive principal eigenfunction. Dividing by that eigenfunction and integrating the stability inequality over $S$ converts the differential operator into an inequality involving the intrinsic scalar curvature, $T_{ab}U^aK^b$, $|\chi^{(K)}|^2$, the principal eigenvalue, and nonnegative gradient terms. The dominant energy condition fixes the flux sign. Strict positivity follows unless the principal eigenvalue and every term in the Ricci-flat exception vanish.

In two dimensions the Gauss–Bonnet theorem then gives

$$\int_S {}^{(2)}R\,dA=4\pi\chi_{\rm E}(S)=8\pi(1-g),$$

where $g$ is the genus. Strict positivity forces $g=0$. The decisive steps are therefore stability, the energy-sign inequality, and Gauss–Bonnet—not a visual assumption that a black hole “ought to be round.” Remove stable outermostness or the sign conditions and the conclusion does not follow.

Branches and apparent jumps

The theorem permits several MOTSs on one slice and can constrain more than one stable branch; it does not force every inner unstable branch to be spherical. At common-horizon formation an outer and inner common branch can be born together. Later, inner and individual branches may approach, self-intersect in a projection, or annihilate in a solution-space diagram. These are changes in the set of solutions of an elliptic equation, not necessarily changes in the topology of a single smooth world tube.

An apparent-horizon area can jump when the identity of the outermost MOTS switches between branches. If $S_1$ is outermost on one slice and a newly available enclosing $S_2$ becomes outermost on the next, the reported area changes discontinuously even though each branch is smooth in its own parameter and both surfaces are $S^2$. The detailed head-on merger studied by Pook-Kolb and collaborators resolves precisely such multiple branches and continuing inner surfaces Pook-Kolb et al. 2021, §§ III–VI.

Event horizons require another description. Their null generators can enter through a crease or caustic set; away from that set the horizon is a smooth null hypersurface. Spatial sections chosen to intersect the endpoint set differently can display a coalescence of spheres or a transient torus even when standard late sections are spheres Siino 1998, Theorems III.5–III.6, pp. 10–11. Such a slice-dependent section does not contradict topological censorship or the stable-MOTS theorem because their objects and hypotheses differ.

Scope and failure boundaries

The safe conclusion is conditional: a compact orientable stably outermost MOTS in four-dimensional Einstein gravity with the stated regularity and dominant-energy assumptions is spherical unless the induced metric is flat and the null second fundamental form and energy flux vanish; strict stability excludes that exceptional case. Branch multiplicity and outermost-surface switching remain possible. The theorem does not prove uniqueness of a MOTS, smoothness through a bifurcation, or slicing independence of an apparent-horizon history. It also does not constrain AdS horizons by the same sign argument or establish the topology of every inner marginal surface.

Numerical diagrams add further limitations. A branch can be missed if the finder is seeded only near the outer solution; coordinate embeddings can mimic self-intersection; and finite resolution can blur a zero stability eigenvalue. Accepting a topology or jump claim requires converged intrinsic geometry, tracked eigenvalues, multiple initial guesses, and an explicit foliation.

Transfer check

Suppose a code finds a toroidal marginal surface inside a spherical outermost MOTS during a merger satisfying the dominant energy condition. Does the stable-MOTS topology theorem show that the torus is spurious?

Solution

No. For a stably outermost MOTS, the positive-Yamabe conclusion has a Ricci-flat, zero-null-curvature, zero-flux exception; strict stability removes that exception. An inner toroidal solution may instead be unstable and therefore outside the statement. The proper checks are to compute its intrinsic topology and principal stability eigenvalue, test convergence and slice dependence, and verify that it is not being claimed as the strictly stable outermost surface. If it were strictly stable and outermost with all hypotheses satisfied, then the toroidal result would signal a failed assumption or a numerical error.

References

• Galloway, Gregory J., and Richard Schoen. “A Generalization of Hawking’s Black Hole Topology Theorem to Higher Dimensions.” Communications in Mathematical Physics 266 (2006): 571–576. doi:10.1007/s00220-006-0019-z.
• Pook-Kolb, Daniel, Ofek Birnholtz, José Luis Jaramillo, Badri Krishnan, and Erik Schnetter. “Horizons in a Binary Black Hole Merger I: Geometry and Area Increase.” Physical Review D 103 (2021): 026005. doi:10.1103/PhysRevD.103.026005.
• Siino, Masaru. “Topology of Event Horizon.” Physical Review D 58 (1998): 104016. doi:10.1103/PhysRevD.58.104016.