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A Scale-Critical Trapped Surface Formation Criterion: A New Proof Via Signature for Decay Rates

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Abstract

We provide a self-contained proof of a trapped surface formation theorem, which simplifies the previous results by Christodoulou and by An–Luk. Our argument is based on a systematic approach for the scale-critical estimates in An–Luk and it connects Christodoulou’s short-pulse method and Klainerman–Rodnianski’s signature counting argument to the peeling properties previously studied in the small-data regime such as Klainerman–Nicolo. In particular this allows us to avoid elliptic estimates and geometric renormalizations, and gives us new technical improvements and simplifications.

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Notes

  1. The detailed construction of double null foliation will be explained in Sect. 2.1.

  2. A 2-surface is called a trapped surface if its area element is infinitesimally decreasing along both families of null geodesics emanating from the surface

  3. Metric of Minkowskian spacetime in spherical coordinates: \(g_{M}=-dt^2+dr^2+r^2(d\theta ^2+\sin \theta ^2 d\phi ^2)\).

  4. Metric of Minkowskian spacetime in stereographic coordinates: \(g_{M}=-dt^2+dr^2+\frac{4r^4}{(r^2+\theta _1^2+\theta _2^2)^2}(d\theta _1^2+d\theta _2^2)\). In Sect. 8, we will do a scaling argument in stereographic coordinates.

  5. Metric of Schwarzschild spacetime: \(g_S=-(1-\frac{2M}{r})dt^2+({1-\frac{2M}{r}})^{-1}dr^2+r^2(d\theta ^2+\sin \theta ^2 d\phi ^2)\). Here M is a constant. In \(g_S\) all metric components are independent of t, Schwarzschild spacetime is static, i.e. not changing with t.

  6. The classic approach to deriving energy estimates with Bel–Robinson tensors as in [6] and [13] is avoid, since for higher order energy estimates there are many more terms (including borderline terms) from deformation tensors would appear.

  7. Letting \(a=\delta ^{-1}\), in a finite region they recover Christodoulou’s main result of [6].

  8. It comes from spacetime conformal compactification. See [11].

  9. In this article, we use l.o.t. to mean lower order terms.

  10. More details will be provided in Section 2.5.

  11. For a general double null foliation, we have the gauge freedom of choosing how to extend \(\Omega \) along \(H_{u_{\infty }}\) and \({\underline{H}}_0\). In this paper, we extend \(\Omega \equiv 1\) on both \(H_{u_{\infty }}\) and \({\underline{H}}_0\).

  12. On \({\underline{H}}_0\), we have \(\Omega =1\) and \({{\mathcal {L}}} /\,_{{\underline{L}}} \theta ^A=\frac{\partial }{\partial u}\theta ^A\).

  13. That’s because (1.1) is a geometric PDE system and it respects some natural scalings.

  14. See full details in Chapter 16 of [6] or [17] or Sect. 10 of [8] for a beautiful exposition.

  15. See Remark 14.

  16. Recall \({\underline{L}}=\Omega e_3\).

  17. From conformal compactification.

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Acknowledgements

XA has been supported by the startup grant from National University of Singapore under Project Number R-146-000-269-133.

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  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Xinliang An

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An, X. A Scale-Critical Trapped Surface Formation Criterion: A New Proof Via Signature for Decay Rates. Ann. PDE 8, 3 (2022). https://doi.org/10.1007/s40818-021-00114-1

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