Duke Mathematical Journal

High-frequency backreaction for the Einstein equations under polarized U(1)-symmetry

Cécile Huneau and Jonathan Luk

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Abstract

Known examples in plane symmetry or Gowdy symmetry show that, given a 1-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stress-energy-momentum tensor. We consider this phenomenon under polarized U(1)-symmetry—a much weaker symmetry than most of the known examples—such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized U(1)-symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large and appears to be the first construction of such examples with more than two families.

Article information

Source
Duke Math. J., Volume 167, Number 18 (2018), 3315-3402.

Dates
Received: 30 June 2017
Revised: 15 May 2018
First available in Project Euclid: 16 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1542337536

Digital Object Identifier
doi:10.1215/00127094-2018-0035

Mathematical Reviews number (MathSciNet)
MR3881199

Zentralblatt MATH identifier
07009768

Subjects
Primary: 35Q75: PDEs in connection with relativity and gravitational theory

Keywords
general relativity

Citation

Huneau, Cécile; Luk, Jonathan. High-frequency backreaction for the Einstein equations under polarized $\mathbb{U}(1)$ -symmetry. Duke Math. J. 167 (2018), no. 18, 3315--3402. doi:10.1215/00127094-2018-0035. https://projecteuclid.org/euclid.dmj/1542337536


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References

  • [1] L. Andersson, N. Gudapati and J. Szeftel, Global regularity for the $2+1$ dimensional equivariant Einstein-wave map system, Ann. PDE 3 (2017), no. 13.
  • [2] D. R. Brill and J. B. Hartle, Method of the self-consistent field in general relativity and its application to the gravitational geon, Phys. Rev. B 135 (1964), 271–278.
  • [3] G. A. Burnett, The high-frequency limit in general relativity, J. Math. Phys. 30 (1989), 90–96.
  • [4] Y. Choquet-Bruhat, Construction de solutions radiatives approchées des équations d’Einstein, Comm. Math. Phys. 12 (1969), 16–35.
  • [5] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford Math. Monogr., Oxford Univ. Press, New York, 2009.
  • [6] M. Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, Comm. Pure Appl. Math. 58 (2005), 445–504.
  • [7] M. Dafermos and J. Luk, The interior of dynamical vacuum black holes, I: The $C^{0}$-stability of the Kerr Cauchy horizon, preprint, arXiv:1710.01722v1 [gr-qc].
  • [8] S. R. Green and R. M. Wald, New framework for analyzing the effects of small scale inhomogeneities in cosmology, Phys. Rev. D 83 (2011), no. 084020.
  • [9] S. R. Green and R. M. Wald, Examples of backreaction of small scale inhomogeneities in cosmology, Phys. Rev. D 87 (2013), no. 124037.
  • [10] P. A. Hogan and T. Futamase, Some high-frequency spherical gravity waves, J. Math. Phys. 34 (1993), 154–169.
  • [11] C. Huneau, Constraint equations for $3+1$ vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case, Ann. Henri Poincaré 17 (2016), 271–299.
  • [12] C. Huneau, Stability of Minkowski space–time with a translation space-like Killing field, Ann. PDE 4 (2018), no. 12.
  • [13] C. Huneau and J. Luk, Einstein equations under polarized $\mathbb{U}(1)$ symmetry in an elliptic gauge, Comm. Math. Phys. 361 (2018), 873–949.
  • [14] D. Ida, No black-hole theorem in three-dimensional gravity, Phys. Rev. Lett. 85 (2000), 3758–3760.
  • [15] R. A. Isaacson, Gravitational radiation in the limit of high frequency, I: The linear approximation and geometrical optics, Phys. Rev. 166 (1968), 1263–1271.
  • [16] R. A. Isaacson, Gravitational radiation in the limit of high frequency, II: Nonlinear terms and the effective stress tensor, Phys. Rev. 166 (1968), 1272–1279.
  • [17] S. Klainerman, I. Rodnianski and J. Szeftel, The bounded $L^{2}$ curvature conjecture, Invent. Math. 202 (2015), 91–216.
  • [18] J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, EMS Monogr. Math. 5, Eur. Math. Soc. (EMS), Zürich, 2012.
  • [19] J. Lott, Backreaction in the future behavior of an expanding vacuum spacetime, Classical Quantum Gravity 35 (2018), no. 035010.
  • [20] J. Lott, Collapsing in Einstein flow, Ann. Henri Poincaré 19 (2018), 2245–2296.
  • [21] J. Luk and S.-J. Oh, Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data, I: The interior of the black hole region, preprint, arXiv:1702.05715v1 [gr-qc].
  • [22] J. Luk and S.-J. Oh, Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data, II: The exterior of the black hole region, preprint, arXiv:1702.05716v1 [gr-qc].
  • [23] J. Luk and I. Rodnianski, Local propagation of impulsive gravitational waves, Comm. Pure Appl. Math. 68 (2015), 511–624.
  • [24] J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math. 5 (2017), 435–570.
  • [25] J. Luk and I. Rodnianski, High frequency limits in general relativity, in preparation.
  • [26] M. A. H. MacCallum and A. H. Taub, The averaged Lagrangian and high-frequency gravitational waves, Comm. Math. Phys. 30 (1973), 153–169.
  • [27] R. C. McOwen, The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math. 32 (1979), 783–795.
  • [28] G. Moschidis, A proof of the instability of AdS for the Einstein–null dust system with an inner mirror, preprint, arXiv:1704.08681v1 [gr-qc].
  • [29] E. Poisson and W. Israel, Internal structure of black holes, Phys. Rev. D 41 (1990), 1796–1809.
  • [30] J. Sterbenz and D. Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys. 298 (2010), 231–264.
  • [31] S. J. Szybka, K. Głód, M. J. Wyrebowski, and A. Konieczny, Inhomogeneity effect in Wainwright-Marshman space-times, Phys. Rev. D 89 (2014), no. 044033.
  • [32] S. J. Szybka and M. J. Wyrebowski, Backreaction for Einstein-Rosen waves coupled to a massless scalar field, Phys. Rev. D 94 (2016), no. 024059.
  • [33] T. Tao, Global regularity of wave maps, II: Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443–544.
  • [34] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math. 106, Amer. Math. Soc., Providence, 2006.