Trapped surfaces in vacuum arising dynamically from mild incoming radiation

An, Xinliang; Luk, Jonathan

doi:10.4310/atmp.2017.v21.n1.a1

Cited by 34 publications

(104 citation statements)

References 18 publications

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“…Moreover, from the above remarks, we conclude that the same smallness holds for all permutations of the operatorsX [1,3] L acting on along 1 0 . To obtain the desired smallness of…”

Section: Smallness Assumptionsmentioning

confidence: 82%

“…Our strategy of propagating the smallness of some quantities while simultaneously allowing derivatives transversal to the characteristics to be large has roots in the similar approach taken by Christodoulou [16] in his celebrated proof of the formation of trapped surfaces in solutions to the Einstein-vacuum equations and in the related works [3,33,37,38,[47][48][49]. Similar strategies have been used [51,61,62,65] to prove global existence results for semilinear wave equations verifying the null condition in regimes that allow for large transversal derivatives.…”

Section: Overview Of the Analysismentioning

confidence: 92%

“…Eikonal functions have also been used as central ingredients in proofs of low-regularity well-posedness for quasilinear wave equations; see, for example, [35,36,58,63]. 3 The dynamic vectorfield frame at two distinct points in P t u , where 0 < u < 1…”

Section: Overview Of the Analysismentioning

confidence: 99%

“…To bound the term (T ) P (3) [ ], we also need to use the estimates (10.11) and (10.13), which allow us to bound the first two terms in braces on RHS (3.14c). Note that since no absolute value is taken on LHS (11.38), we are free to replace the factor (X μ)/μ from RHS (3.14c) with the factor [X μ] + /μ (which we bounded in (10.13)).…”

Section: Lemma 1111 (Pointwise Bounds For the Error Terms Generated mentioning

confidence: 99%

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Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

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“…Moreover, from the above remarks, we conclude that the same smallness holds for all permutations of the operatorsX [1,3] L acting on along 1 0 . To obtain the desired smallness of…”

Section: Smallness Assumptionsmentioning

confidence: 82%

Section: Overview Of the Analysismentioning

confidence: 92%

Section: Overview Of the Analysismentioning

confidence: 99%

Section: Lemma 1111 (Pointwise Bounds For the Error Terms Generated mentioning

confidence: 99%

See 2 more Smart Citations

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

et al. 2016

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“…However, it was subsequently proven [48] that in vacuum spacetimes a trapped surface can dynamically form from regular initial data that does not have a trapped surface (this theorem was generalized in [49,50]). A sequence of marginally outer trapped surfaces, in which the area approaches zero and hence forms an apparent horizon, were then constructed within a region up to the "center" of gravitational collapse in the full 4D vacuum GR field equations [51]. The existence of marginally outer trapped surfaces are important for proving the positive mass theorem and the Penrose inequality [52] (see below).…”

Section: Open Problems In General Relativitymentioning

confidence: 99%

Mathematical general relativity

Coley

2019

Gen Relativ Gravit

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We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein-Yang-Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics.

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Examples of Naked Singularity Formation in Higher-Dimensional Einstein-Vacuum Spacetimes

Zhang

2017

Ann. Henri Poincaré

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Abstract. The vacuum Einstein equations in 5+1 dimensions are shown to admit solutions describing naked singularity formation in gravitational collapse from nonsingular asymptotically locally flat initial data that contain no trapped surface. We present a class of specific examples with topology R 3+1 × S 2 . Thanks to the Kaluza-Klein dimensional reduction, these examples are constructed by lifting continuously selfsimilar solutions of the 4-dimensional Einstein-scalar field system with a negative exponential potential. The latter solutions are obtained by solving a 3-dimensional autonomous system of first-order ordinary differential equations with a combined analytic and numerical approach. Their existence provides a new test-bed for weak cosmic censorship in higher-dimensional gravity. In addition, we point out that a similar attempt of lifting Christodoulou's naked singularity solutions of massless scalar fields fails to capture formation of naked singularities in 4+1 dimensions, due to a diverging Kretschmann scalar in the initial data.Mathematics Subject Classification (2010). Primary 83C75; Secondary 83E15, 35Q75.

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Trapped surfaces in vacuum arising dynamically from mild incoming radiation

Cited by 34 publications

References 18 publications

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

Mathematical general relativity

Examples of Naked Singularity Formation in Higher-Dimensional Einstein-Vacuum Spacetimes

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