Formation of Trapped Surfaces from Regular Data

The singularity theorem assumes that a trapped surface already exists. A formation theorem begins earlier: its characteristic initial hypersurfaces are regular and untrapped, yet the vacuum Einstein evolution concentrates enough incoming gravitational radiation to make both future null expansions negative on a later sphere. Christodoulou’s short-pulse construction and its refinements solve this genuinely nonlinear problem without inserting matter or a trapped boundary by hand.

Required background. Analysis for Mathematical Relativity supplies bootstrap and weighted-energy estimates; timelike congruences and spatial projection supplies null expansion, shear, and Raychaudhuri equations; and the Einstein equation fixes the vacuum system.

Helpful background. Causality and Global Structure clarifies trapped sets and incompleteness, while PDEs and Green Functions for Gravitation provides the energy-estimate language.

Characteristic data with no initial trapped sphere

Use a double-null foliation with outgoing hypersurfaces $H_u$, incoming hypersurfaces $\underline H_{\underline u}$, and spheres $S_{u,\underline u}=H_u\cap\underline H_{\underline u}$. Fix null vectors by

$$g(e_3,e_4)=-2.$$

The null second fundamental forms are $\chi$ and $\underline\chi$; their traces $\operatorname{tr}\chi$ and $\operatorname{tr}\underline\chi$ are the two future expansions. A sphere is future trapped when both are negative.

The finite characteristic problem prescribes flat data on one incoming cone $\underline H_0$ and a short burst of shear $\widehat\chi_0$ on an outgoing cone $H_0$, supported in $0\le\underline u\le\delta$. The pulse has small duration but large derivative scale. To display the scaling without hiding dimensions, choose a fixed reference length $L$, write the physical pulse duration as $\delta=\varepsilon L$, and use orthonormal shear components. A schematic profile is

$$\widehat\chi_0(\underline u,\vartheta) =\frac{\varepsilon^{-1/2}}{L} f\!\left(\frac{\underline u}{\varepsilon L},\vartheta\right), \qquad 0\le\underline u\le\varepsilon L.$$

Its pointwise size grows as $\delta\to0$, while the flux along each generator,

$$\mathcal F(\vartheta) =\int_0^\delta \lvert\widehat\chi_0(\underline u,\vartheta)\rvert^2 \,d\underline u,$$

equals $L^{-1}\int_0^1|f(s,\vartheta)|^2ds$, independent of $\varepsilon$. Shear has dimensions length$^{-1}$ and this flux has dimensions length$^{-1}$. This is the central scaling: the data are smooth for every fixed $\delta\gt0$ and contain a controlled, finite amount of incoming radiation. The initial spheres can all have positive outgoing expansion, so the conclusion is dynamical formation rather than propagation of a pre-existing trapped set.

The short-pulse conclusion

For a concrete quantitative statement, use An and Luk’s normalization in this paragraph: all lengths are divided by the initial radius, the initial outgoing cone is $H_1$, and $u$ decreases toward the future along the flat incoming cone $\underline H_0$. Thus $r=u$ on $\underline H_0$, with the intersection sphere at $u=1$. Prescribe smooth vacuum characteristic data, Minkowskian on $\underline H_0$, and let $\delta$ now denote the dimensionless pulse duration on $H_1$. Assume

$$\sum_{i=0}^{7} \|\nabla^i\widehat\chi_0\|_{L^\infty_{\underline u}L^2(S_{1,\underline u})} \le a^{1/2}.$$

Here angular covariant derivatives and norms use the sphere metric in the source’s double-null gauge. There is a universal sufficiently large $b_0$ such that, if $b_0\le b\le a$ and $\delta a^{1/2}b\lt1$, the solution is regular on $b\delta a^{1/2}\le u\le1$, $0\le\underline u\le\delta$. If also

$$\inf_\vartheta\int_0^\delta |\widehat\chi_0(\underline u,\vartheta)|^2d\underline u \ge4b\delta a^{1/2},$$

then $S_{b\delta a^{1/2},\delta}$ is trapped:

$$\operatorname{tr}\chi\lt0, \qquad \operatorname{tr}\underline\chi\lt0.$$

This is An and Luk 2017, Theorem 1.3, pp. 4–5 (PDF). Derivative upper bounds keep the solution regular; the generator-wise flux lower bound forces the sign change. A flux lower bound without the existence estimates is not a formation theorem.

To select examples with no trapped sphere on the initial cones, choose the profile and $a$ first, then take $\delta a$ sufficiently small. The upper angular-derivative bound controls the pointwise shear, so its integral on the initial radius-one cone is small and the outgoing expansion stays positive there. Trapping occurs deeper inside, at radius comparable to $b\delta a^{1/2}$. Initial absence of trapping must be checked separately; it is not implied by a lower flux bound alone.

Klainerman and Rodnianski give a finite-cone proof under an enlarged admissible data class and with fewer controlled curvature derivatives Klainerman–Rodnianski 2012, Theorems 2.6–2.7 and Remark 2, pp. 13–14. An and Luk later relax the incoming-radiation requirement to data large only in a scaling-critical norm and extend the controlled existence region An–Luk 2017, Theorem 1.3 and Corollary 1.4, pp. 4–5. These are open conditions within their stated weighted data spaces; they are not claims that every large wave forms a black hole.

Why the outgoing expansion changes sign

Return to the original $H_0$ notation and dimensional radii. Define the nonaffinity coefficient by $\nabla_{e_4}e_4=-2\omega e_4$. The vacuum optical equation is

$$\nabla_4\operatorname{tr}\chi +\frac12(\operatorname{tr}\chi)^2 =-\lvert\widehat\chi\rvert^2 -2\omega\operatorname{tr}\chi.$$

The shear term has a favorable sign for focusing. If one freezes lower-order errors, shear transport approximately preserves $r\widehat\chi$ along the incoming direction. With initial radius $r_0$ and final radius $r=r(u,0)$, Raychaudhuri integration then suggests

$$\operatorname{tr}\chi(u,\delta,\vartheta) \approx \frac{2}{r(u,0)} -\frac{r_0^2}{r(u,0)^2} \int_0^\delta \lvert\widehat\chi_0\rvert^2\,d\underline u.$$

The factor $r_0^2$ is essential: each term has dimensions length$^{-1}$. Equivalently, the length $\mathcal E(\vartheta)=r_0^2\mathcal F(\vartheta)$ must exceed approximately $2r$ with a margin larger than the controlled errors. Setting $r_0=1$ after nondimensionalizing gives the frequently used normalized formula. The quantitative theorem above supplies an actual margin and an existence region; this heuristic alone supplies neither.

Moving inward decreases $r$ and amplifies the second term. A sufficiently large flux therefore overturns the initially positive flat-space expansion. The incoming expansion starts negative and remains negative under the accompanying estimates. This heuristic identifies the mechanism, but it is not the proof: all Ricci coefficients and curvature components are coupled, and the pulse has large derivatives.

The proof closes a hierarchy adapted to the pulse scaling:

1. Existence through the pulse. Weighted energies control curvature in a double-null slab despite components of size $\delta^{-1/2}$ or worse.
2. Transport and elliptic estimates. Null structure equations propagate Ricci coefficients; Hodge systems on $S_{u,\underline u}$ recover angular derivatives without losing the pulse weights.
3. Renormalization. Dangerous curvature–connection products are grouped so that Bianchi identities expose cancellations and avoid a derivative loss.
4. Focusing estimate. The actual shear flux is compared with its prescribed initial value; error terms are shown smaller than the positive margin in the lower bound.
5. Sign verification. Both expansions are evaluated on the same final sphere. Negative average expansion is not enough; the estimate is pointwise in angle for a trapped sphere.

Uniform, anisotropic, and mild-radiation criteria

Christodoulou’s original criterion imposes a lower flux bound along every generator Christodoulou 2009, Theorem 17.1, p. 588 of the 2008 preprint (PDF). That uniform condition is sufficient, not necessary. A localized beam may focus only part of a sphere; Klainerman, Luk, and Rodnianski combine localized short-pulse estimates with a deformation of the candidate surface to obtain a fully anisotropic criterion Klainerman–Luk–Rodnianski 2013, main theorems. The deformed surface reaches farther into the focused region, and an elliptic inequality controls how its shape redistributes the expansion.

These refinements prevent two common overstatements. First, angularly uniform radiation is not required. Second, “arbitrarily small data form trapped surfaces” is meaningless without a norm: the data can be small in one scale-dependent quantity and large in the critical flux that drives focusing. The principal limitation is equally important: these criteria describe carefully prepared characteristic data and do not classify generic large spacelike Cauchy data.

Formation is also not the same as a final-state theorem. Combining the trapped sphere with Penrose focusing yields an incomplete null geodesic under the global hypotheses, but does not prove that the exterior settles to Kerr or that the entire future boundary is classified.

Check the pulse scaling

Work in units $L=1$ for this exercise. Let $\widehat\chi_0(\underline u)=\delta^{-1/2}f(\underline u/\delta)$ on $0\le\underline u\le\delta$. Show that its $L^2$ flux is independent of $\delta$, while its pointwise amplitude diverges. Why is ordinary pointwise small-data theory not the relevant estimate?

Solution

With $s=\underline u/\delta$,

$$\int_0^\delta \lvert\widehat\chi_0\rvert^2\,d\underline u =\int_0^1\lvert f(s)\rvert^2\,ds,$$

which is independent of $\delta$. But $\lVert\widehat\chi_0\rVert_{L^\infty} =\delta^{-1/2}\lVert f\rVert_{L^\infty}$ diverges. The proof must therefore use weighted, component-sensitive null estimates: treating every coefficient as a uniformly small pointwise perturbation would discard precisely the finite focusing flux.

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. arXiv.
• Christodoulou, Demetrios. The Formation of Black Holes in General Relativity. EMS Monographs in Mathematics. Zürich: European Mathematical Society, 2009. DOI. Open PDF, 2008 preprint.
• Klainerman, Sergiu, and Igor Rodnianski. “On the Formation of Trapped Surfaces.” Acta Mathematica 208 (2012): 211–333. arXiv.
• Klainerman, Sergiu, Jonathan Luk, and Igor Rodnianski. “A Fully Anisotropic Mechanism for Formation of Trapped Surfaces in Vacuum.” Inventiones Mathematicae 198 (2014): 1–26. arXiv.