Sufficient Conditions for Trapped-Surface Formation

The null convergence equation suggests that enough incoming shear can make an outgoing light-front converge, but converting that idea into a closed trapped surface is a nonlinear existence problem. A formation theorem must control the spacetime up to the candidate surface while proving that both expansions are negative there. Its assumptions are sufficient, deliberately quantitative, and far stronger than a necessary characterization of collapse.

Required background. Hypersurface normals and tangential projectors supplies null normals, induced metrics, and second fundamental forms.

Helpful background. Black-Hole Geometry and Mechanics develops trapped surfaces and singularity theorems; Black-Hole Perturbation Theory supplies shear and radiative data; Formulations of General Relativity explains characteristic initial data; and Computational Relativity connects analytic conditions with horizon searches.

A characteristic formation theorem

Work in a smooth, four-dimensional vacuum spacetime satisfying $\operatorname{Ric}(g)=0$. Prescribe characteristic initial data on two intersecting null hypersurfaces: an incoming hypersurface $\underline H_0$ that agrees with a truncated backward Minkowski cone, and an outgoing hypersurface $H_1$ carrying shear $\hat\chi_0$ over a short affine interval $0\leq\underline u\leq\delta$. Their intersections $S_{u,\underline u}$ are topological two-spheres. The null coordinates, normalization, differentiability, and angular derivatives are parts of the hypotheses.

Measure all lengths in units of the initial round-sphere radius, so its value is $1$. A representative theorem of An and Luk introduces dimensionless amplitudes $a,b$ satisfying $b_0\leq b\leq a$ and $\delta a^{1/2}b\lt 1$. It assumes a controlled sum of angular derivatives,

$$\sum_{i\leq7} \left\lVert\nabla^i\hat\chi_0\right\rVert_ {L^\infty_{\underline u}L^2(S_{1,\underline u})} \leq a^{1/2},$$

and a lower bound on incoming shear energy along every generator,

$$\inf_{\vartheta\in S_{1,0}} \int_0^\delta \lvert\hat\chi_0(\underline u',\vartheta)\rvert^2 \,d\underline u' \geq 4b\delta a^{1/2}.$$

Theorem. For a universal sufficiently large $b_0$, the vacuum development remains regular in the stated double-null region and the sphere at $u=b\delta a^{1/2}$, $\underline u=\delta$ is trapped. This is Theorem 1.3 of An and Luk 2017, pp. 4–5, PDF. The exact constants and coordinate normalization belong to the statement; the theorem is not the slogan “large radiation makes a black hole.”

The hypotheses allow data large in a scaling-critical norm while retaining enough weighted control to reach the vertex-like region where focusing becomes strong. A related anisotropic theorem can replace a uniform angular lower bound by radiation concentrated near one generator, at the price of deforming the candidate surface; see Klainerman, Luk, and Rodnianski 2014, Theorem 3, pp. 3–5, PDF.

Proof architecture

First, impose a double-null gauge and decompose the vacuum Einstein equations into null structure equations for the expansions, shears, torsion, and connection coefficients, together with null Bianchi equations for curvature. The short interval creates a hierarchy: some components are large in $\delta$, but their integrated effect and weighted combinations remain controlled.

Second, establish an existence region reaching $u\sim\delta a^{1/2}b$. Weighted energy estimates for curvature and transport estimates for connection coefficients prevent a caustic or loss of regularity before the target sphere. Renormalized curvature variables remove products that would otherwise exceed the available scale-invariant bounds. An and Luk outline this bootstrap and renormalization strategy in An and Luk 2017, §1.3, pp. 10–14, PDF; the estimates occupy the later sections of the paper.

Third, integrate the outgoing-expansion equation. Schematically,

$$\nabla_4\theta_{(\ell)} =-\frac12\theta_{(\ell)}^2 -\lvert\hat\chi\rvert^2 -2\omega\,\theta_{(\ell)}$$

in vacuum. The shear-energy lower bound forces $\theta_{(\ell)}\lt 0$ on the target sphere, while the controlled incoming geometry keeps $\theta_{(k)}\lt 0$. The upper bounds matter as much as the lower bound: they justify transporting the initial radiation estimate to the target without losing the solution.

Finally, the candidate is a smooth closed two-sphere and the inequalities hold pointwise over it. That last uniformity is precisely what a statement about focusing along one selected geodesic would not provide. In the anisotropic construction, a deformation argument supplies a differently shaped closed surface instead.

Consequence and failed converse

Once a compact trapped surface exists, Penrose’s theorem gives future null geodesic incompleteness if the spacetime has a noncompact Cauchy surface and satisfies the null convergence and causality hypotheses. The original implication and its global assumptions are stated in Penrose 1965, pp. 57–59. It does not locate an event horizon, prove curvature blow-up, or classify the incomplete geodesic.

The converse fails in several ways. Data that violate the displayed uniform lower bound may still form a trapped surface through anisotropic concentration, matter focusing, or a different scale hierarchy. A black-hole region can also be defined globally without a chosen foliation ever presenting the particular round sphere used by this theorem. Conversely, satisfying a heuristic integrated-energy estimate without the regularity and upper bounds does not invoke the theorem: the evolution might become uncontrolled before the proposed surface.

Raychaudhuri’s equation alone is therefore not a formation proof. It follows a congruence after one has supplied it and assumes the geometry exists along the integration interval. The hard steps are controlling the coupled Einstein equations, closing the surface, and obtaining both signs everywhere.

Check your understanding

Suppose the shear-energy lower bound holds on $90\%$ of the angular generators but vanishes on the remaining cap. Can one apply the uniform An–Luk theorem stated above? Does failure to apply imply that no trapped surface forms?

Solution

No. The infimum over angles sees the cap and the uniform lower-bound hypothesis fails. But failure of a sufficient hypothesis is not a nonformation theorem. The data might satisfy an anisotropic formation criterion after deforming the candidate surface, or another nonlinear mechanism might produce a trapped surface. One must check the hypotheses of an appropriate theorem or solve the evolution.

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.
• Klainerman, Sergiu, Jonathan Luk, and Igor Rodnianski. “A Fully Anisotropic Mechanism for Formation of Trapped Surfaces in Vacuum.” Inventiones Mathematicae 198 (2014): 1–26. doi:10.1007/s00222-014-0490-0.
• Penrose, Roger. “Gravitational Collapse and Space-Time Singularities.” Physical Review Letters 14 (1965): 57–59. doi:10.1103/PhysRevLett.14.57.