Trapped Surfaces and Horizon Formation

Trapped surfaces connect local light-front convergence to the onset of strong gravitational collapse. This chapter builds that connection in layers: the two null expansions, stability of a marginal surface, rigorous sufficient conditions for formation, the assembly of marginal leaves into tubes, the distinction among horizon notions, and the numerical methods that locate them in evolving data.

Helpful background. Black-Hole Geometry and Mechanics distinguishes the horizon notions used here, and Hypersurface Normals and Tangential Projectors supplies the nested induced geometry and normal decompositions.

Enter this chapter

The basic object is a smooth closed spacelike two-surface in a four-dimensional spacetime. Future and outward orientations must be declared. The global metric convention is $(-,+,+,+)$, and extrinsic-curvature signs are stated where they enter a $3+1$ formula. These conventions matter for formulas; the signs and zero sets of expansions remain invariant under positive reciprocal rescalings of the null normals.

Read the chapter in this order:

1. Null Expansions and Trapped Surfaces constructs $\theta_{(\ell)}$ and $\theta_{(k)}$ and distinguishes trapped, outer trapped, and marginal surfaces.
2. Stability of Marginally Outer Trapped Surfaces uses the principal eigenvalue of an elliptic operator to prove local continuation.
3. Sufficient Conditions for Trapped-Surface Formation explains how controlled incoming radiation dynamically produces a closed trapped sphere.
4. Emergence of Marginal Tubes in Collapse relates flux, stability, and the tube’s causal signature.
5. Event Horizons, Apparent Horizons, and Trapping Boundaries matches each horizon definition to the question it can answer.
6. Locating Horizons in Dynamical Data turns the expansion equation into tested surface-finding workflows.

From expansion to formation

For future null normals $\ell\cdot k=-1$ and induced two-metric $q_{ab}$,

$$\theta_{(\ell)}=q^{ab}\nabla_a\ell_b, \qquad \theta_{(k)}=q^{ab}\nabla_a k_b.$$

A trapped surface has both expansions negative; a MOTS has $\theta_{(\ell)}=0$. These are surface properties, not labels attached to spacetime points. A positive boost changes the expansion magnitudes but preserves their signs.

Raychaudhuri’s equation shows why shear and positive null energy focus outgoing rays. It does not by itself prove formation: a theorem must construct and control the spacetime to a smooth closed surface on which both signs hold everywhere. The characteristic results presented here make that distinction explicit. They impose quantitative regularity, upper bounds, scale conditions, and a shear-energy lower bound. An and Luk’s theorem and proof architecture are stated in An and Luk 2017, Theorem 1.3, pp. 4–5, and §9, pp. 93–98, PDF.

Stability, tubes, and branches

Linearizing $\theta_{(\ell)}$ under an outward surface deformation gives a generally non-self-adjoint elliptic operator $L_S$. Its real principal eigenvalue controls outer stability. Strict positivity makes $L_S$ invertible, so an implicit-function theorem continues the MOTS through nearby slices. The same proof does not cover a degenerate leaf with zero principal eigenvalue; branches may meet or turn there.

If marginal leaves form a tube and $\mathcal V^a=\ell^a-Ck^a$ is tangent to it, preservation of $\theta_{(\ell)}=0$ determines $C$. Under the null energy and outer conditions, matter flux or outgoing shear gives $C\gt 0$ and a spacelike tube; vanishing flux gives the null limit. In spherical Vaidya collapse this reduces to the direct check that $r=2Gm(v)$ is spacelike for $m'(v)\gt 0$.

The outermost surface on successive slices can jump between branches. That observation reconciles discontinuous apparent-horizon histories with smooth marginal tubes in spacetime. Andersson, Mars, and Simon give the stability theorem, causal conclusion, and branch picture in Andersson, Mars, and Simon 2005, pp. 1–4, PDF.

Local and global horizons

An event horizon is the global boundary $\partial J^-(\mathscr I^+)$ and depends on the entire future development. An apparent horizon is defined relative to a spatial slice. A future outer trapping horizon adds a transverse-variation sign; a dynamical horizon adds spacelike signature. Ashtekar and Krishnan state these definitions, their distinct inputs, and their equilibrium relations in Ashtekar and Krishnan 2004, §§ 2.1–2.3, pp. 9–20. In standard stationary cross-sections the event, apparent, and null trapping horizons agree; the null horizon is a limiting state, not a spacelike dynamical horizon. During accretion these boundaries separate.

Consequently, “horizon formation time” is incomplete without naming the horizon and time slicing. A MOTS can be found during an evolution, while an event horizon is normally located retrospectively. A null marginal tube is not automatically an event horizon, and failure of one horizon finder is not proof that the corresponding geometric surface is absent.

Standards for trustworthy location

An apparent-horizon computation must expose its $3+1$ sign conventions, topology and star-shapedness assumptions, interpolation, initial guess, angular resolution, nonlinear tolerance, residual, branch, and invariant outputs. It must refine both the spacetime and surface grids. Event-horizon reconstruction additionally varies seed time and shape, generator density, time sampling, and caustic treatment. Thornburg’s review separates these algorithms and their limitations in Thornburg 2007, §§2–3, pp. 11–59, PDF.

The minimum validation ladder is Schwarzschild, Kerr or another stationary distorted-coordinate test, a spherical dynamical solution, at least three resolutions, and an independent implementation or alternative method. Areas, spins, multipoles, expansion residuals, and stability data are scientific outputs; coordinate radius alone is not.

Once a smooth future marginal tube and its branch have been identified, Dynamical Horizons and Flux Laws develops the area, angular-momentum, energy, and multipole balances carried by that geometry.

Review the chapter

A horizon finder reports a new outer MOTS pair. The outer leaf has principal eigenvalue $\lambda_1\gt 0$, nonzero outgoing shear, $\theta_{(k)}\lt 0$, and satisfies the null energy condition. What local conclusions follow, and which global conclusion does not?

Solution

Strict stability gives a smooth local marginal tube through the outer leaf in the prescribed foliation. With the null energy condition it is achronal, and nonzero outgoing shear makes it spacelike near that leaf. Its area therefore grows in the outward evolution direction. None of this proves that the tube is the event horizon or locates $\partial J^-(\mathscr I^+)$; that requires the complete future causal development.

References

• An, Xinliang, and Jonathan Luk. “Trapped Surfaces in Vacuum Arising Dynamically from Mild Incoming Radiation.” Advances in Theoretical and Mathematical Physics 21 (2017): 1–120. doi:10.4310/ATMP.2017.v21.n1.a1.
• 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.
• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. doi:10.12942/lrr-2004-10.
• Thornburg, Jonathan. “Event and Apparent Horizon Finders for 3+1 Numerical Relativity.” Living Reviews in Relativity 10 (2007): 3. doi:10.12942/lrr-2007-3.