Transitions, Jumps, and Multiple Horizons

Dynamical spacetimes can contain several marginally outer trapped surfaces on one slice and several marginal tubes through one region. A newly found outermost surface can appear far outside the previously tracked one, producing a jump in the apparent-horizon record. This need not be a discontinuity of spacetime geometry: branches can be connected through a smooth tube that folds relative to the chosen time slicing.

Required background. Hypersurface normals and tangential projectors supplies surface deformations and tube signature.

Helpful background. Black-Hole Geometry and Mechanics develops apparent and dynamical horizons; Black-Hole Perturbation Theory supplies stability spectra; Formulations of General Relativity explains slicing dependence; and Computational Relativity develops multi-branch horizon searches.

Pair creation and stability

Let $\Theta(t,h)=0$ be the nonlinear marginal-surface equation for a graph $h$ on slice $\Sigma_t$. Its derivative in the surface direction is the MOTS stability operator $L_S$. When the principal eigenvalue is positive, $L_S$ is invertible and the implicit-function theorem gives a locally unique smooth branch $h(t)$.

At a generic fold, the principal eigenvalue passes through zero. A schematic reduction to the critical mode has

$$0=\alpha(t-t_0)+\beta a^2+\cdots,$$

where $a$ is branch amplitude. On one side there is no nearby solution; at $t_0$ one degenerate solution; and on the other side two branches

$$a_\pm\simeq\pm \sqrt{-\frac{\alpha}{\beta}(t-t_0)}.$$

One branch is typically outer and stable, the other inner and unstable. Because $da/dt$ diverges at the fold, parameterizing the smooth tube by slicing time is ill-conditioned even when another tube parameter is regular.

Andersson, Mars, and Simon prove smooth continuation under strict stability and explicitly explain how selecting only globally outermost leaves produces jumps in Andersson, Mars, and Simon 2005, pp. 1–4, PDF. Their theorem does not claim global uniqueness of every marginal tube.

A merger branch structure

In a binary merger, two individual MOTSs can coexist with a newly created common pair. The outer common branch expands and relaxes toward the remnant horizon. The inner common branch contracts toward the individual surfaces, can change signature, and may develop self-intersections. During the interval of coexistence, “the horizon” is ambiguous unless the branch and purpose are stated.

The detailed head-on calculation of Pook-Kolb and collaborators finds a common MOTS that bifurcates into inner and outer branches before the individual horizons touch. The branch sequence, stability behavior, signature, and area are documented in Pook-Kolb et al. 2021, §§I–III, pp. 1–8, PDF, with the area-law qualifications in §§IV–VI, pp. 8–15, PDF.

This example separates three events: appearance of a common MOTS pair, contact of individual MOTSs, and late relaxation of the outer common horizon. None is automatically the event-horizon merger time. Their coordinate times change with slicing.

Area along the correct branch

The area law applies to a smooth future outer spacelike dynamical horizon with its orientation fixed. It does not say that area must increase when one concatenates unrelated branches in increasing coordinate time. An inner tube may be timelike or fail $\theta_{(k)}\lt 0$ on part of a leaf, so the usual positive-flux conclusion is unavailable there.

Nor should one add horizon areas without identifying the physical quantity. Before a common horizon appears, $A_1+A_2$ is a useful sum of two quasi-local areas. Afterward, $A_{\rm outer}$ belongs to another surface enclosing both. Interaction energy prevents $\sqrt{A_1}+\sqrt{A_2}$ or the sum of irreducible masses from being a conserved total.

To follow a branch, store more than centroid and area: retain the surface embedding, expansion residuals, ingoing expansion, stability eigenvalue, intrinsic multipoles, signature, and overlap with the preceding eigenfunction. A horizon finder initialized only from the previous outermost surface can miss an outer branch born outside it. Multi-start searches and continuation in a modified expansion help expose the full family.

What a jump means physically

The outermost-MOTS jump changes the quasi-local boundary used for excision and budgets. Matter between the old and new surfaces becomes enclosed at once in that bookkeeping, even though it did not cross a single smooth tube at that coordinate instant. Applying a local flux law across the jump without the connecting geometry creates a false residual.

The appropriate response is to identify a smooth branch sequence or treat the two surfaces as distinct boundaries and state the discontinuity. Event-horizon generators offer a different, global merger picture. That boundary is null where differentiable but need not be smooth at the crease set where new generators join; it does not erase the quasi-local branch structure.

Check your understanding

At time $t_0$ a common outer and inner MOTS pair appears with principal eigenvalue zero. A code reports the outer area jumping from an individual horizon area $A_1$ to a common area $A_c$. May one infer an instantaneous flux $(A_c-A_1)/(t-t_0)$ through one horizon?

Solution

No. The reported outermost surface has switched branches, and slicing time is singular as a branch parameter at the fold. The two areas refer to different surfaces. A flux law requires a specified smooth tube segment and evolution vector; one must reconstruct the connecting branches or record the change as a boundary reclassification rather than an instantaneous crossing flux.

References

• Andersson, Lars, Marc Mars, and Walter Simon. “Local Existence of Dynamical and Trapping Horizons.” Physical Review Letters 95 (2005): 111102. doi:10.1103/PhysRevLett.95.111102.
• 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.