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Characteristic Gluing to the Kerr Family and Application to Spacelike Gluing

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Abstract

This is the third paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We provide full details of our characteristic gluing (including the 10 charges) of strongly asymptotically flat data to the data of a suitably chosen Kerr spacetime. The choice of the Kerr spacetime crucially relies on relating the 10 charges to the ADM energy, linear momentum, angular momentum and the center-of-mass. As a corollary, we obtain an alternative proof of the Corvino-Schoen spacelike gluing construction for strongly asymptotically flat spacelike initial data.

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References

  1. Angelopoulos, Y., Aretakis, S., Gajic, D.: Price’s law and precise late-time asymptotics for subextremal Reissner–Nordström black holes. arXiv:2102.11888, 65 pp

  2. Angelopoulos, Y., Aretakis, S., Gajic, D.: Late-time tails and mode coupling of linear waves on Kerr spacetimes. arXiv:2102.11884, 100 pp

  3. Angelopoulos, Y., Aretakis, S., Gajic, D.: Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds. Adv. Math. 375, 107363, 139 pp (2020)

  4. Angelopoulos, Y., Aretakis, S., Gajic, D.: Horizon hair of extremal black holes and measurements at null infinity. Phys. Rev. Lett. 121(13), 131102 (2018)

    Article  ADS  Google Scholar 

  5. Angelopoulos, Y., Aretakis, S., Gajic, D.: Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes. Adv. Math. 323, 529–621 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aretakis, S.: Horizon instability of extremal black holes. Adv. Theor. Math. Phys. 19, 507–530 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aretakis, S.: Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II. Ann. Henri Poincaré 12, 1491–1538 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Aretakis, S.: The characteristic gluing problem and conservation laws for the wave equation on null hypersurfaces. Ann. PDE 3 (2017). arXiv:1310.1365

  9. Aretakis, S.: On a foliation-covariant elliptic operator on null hypersurface. Int. Math. Res. Not. 15, 6433–6469 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Aretakis, S.: Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I. Commun. Math. Phys. 307, 17–63 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Aretakis, S., Czimek, S., Rodnianski, I.: The characteristic gluing problem for the Einstein equations and applications. arXiv:2107.02441, 31 pp

  12. Aretakis, S., Czimek, S., Rodnianski, I.: The characteristic gluing problem for the Einstein vacuum equations. Linear and non-linear analysis. arXiv:2107.02449, 102 pp

  13. Arnowitt, R., Deser, R., Misner, C.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122, 997–1006 (1961)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Ashtekar, A., Hansen, R.: A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity. J. Math. Phys. 19(7), 1542–1566 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  15. Bruhat, Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles nonlinéaires. Acta Math. 88, 141–225 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Burko, L., Khanna, G., Sabharwal, S.: Scalar and gravitational hair for extreme Kerr black holes. Phys. Rev. D 103, 021502 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  17. Christodoulou, D.: Mathematical Problems of General Relativity I. European Mathematical Society, Zurich Lectures in Advanced Mathematics, 157 pp (2008)

  18. Christodoulou, D.: The formation of black holes in general relativity. European Mathematical Society, Zürich, x+589 pp (2009)

  19. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, 41. Princeton University Press, Princeton, x+514 pp (1993)

  20. Chruściel, P.: A remark on the positive-energy theorem. Class. Quantum Gravity 3(6), L115–L121 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Chruściel, P., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. Fr. (N.S.) 94, vi+103 (2003)

  22. Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Corvino, J., Schoen, R.: On the asymptotics for the vacuum Einstein constraint equations. J. Differ. Geom. 73(2), 185–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Czimek, S.: An extension procedure for the constraint equations. Ann. PDE 4(1), Art. 2, 122 pp (2018)

  25. Czimek, S., Graf, O.: The canonical foliation on null hypersurfaces in low regularity. Ann. PDE 69 pp. arXiv:1909.07345

  26. Dafermos, M., Holzegel, G., Rodnianski, I.: The linear stability of the Schwarzschild solution to gravitational perturbations. Acta Math. 222(1), 1–214 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dain, S., Friedrich, H.: Asymptotically flat initial data with prescribed regularity at infinity. Commun. Math. Phys. 222, 569–609 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Fischer, A., Marsden, J.: The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. Commun. Math. Phys. 28, 138 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hawking, S., Ellis, G.: The Large Scale Structure of Space-Time (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (1973). https://doi.org/10.1017/CBO9780511524646

  30. Herzlich, M.: Computing asymptotic invariants with the Ricci tensor on asymptotically flat and asymptotically hyperbolic manifolds. Ann. Henri Poincaré 17, 3605–3617 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Huang, L.: On the center of mass in general relativity. Stud. Adv. Math., 51, pt. 1. vol. 2, pp. 575–591. AMS (2012)

  32. Huang, L.: On the center of mass of isolated systems with general asymptotics. Class. Quantum Gravity 26, 015012, 25 pp (2009)

  33. Hughes, T., Kato, T., Marsden, J.: Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal. 63, 273–394 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Klainerman, S., Nicolò, F.: The evolution problem of general relativity. Prog. Math. Phys. 25, xiv + 385 pp. Birkhäuser Boston, Inc. (2003)

  35. Komar, A.: Covariant conservation laws in general relativity. Phys. Rev. 113(3), 934–936 (1959)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Landau, L., Lifshitz, E.: The Classical Theory of Fields. Pergamon Press, Oxford (1951)

    MATH  Google Scholar 

  37. Luk, J.: On the local existence for the characteristic initial value problem in general relativity. Int. Math. Res. Not. IMRN 20, 4625–4678 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Luk, J., Rodnianski, I.: Local propagation of impulsive gravitational waves. Commun. Pure Appl. Math. 68(4), 511–624 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ma, S., Zhang, L.: Sharp decay estimates for massless Dirac fields on a Schwarzschild background. arXiv:2008.11429 (2020)

  40. Miao, P., Tam, L.: Evaluation of the ADM mass and center of mass via the Ricci tensor. Proc. A.M.S. 144(2), 753–761 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute of Mathematical Sciences Lecture Notes, New York (1974)

    MATH  Google Scholar 

  42. Schoen, R., Yau, S.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Schoen, R., Yau, S.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Wald, R.: General Relativity. University of Chicago Book Press (1984). https://doi.org/10.7208/chicago/9780226870373.001.0001

  45. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80(3), 381–402 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

S.A. acknowledges support through the NSERC Grant 502581 and the Ontario Early Researcher Award. S.C. acknowledges support through the NSF Grant DMS-1439786 of the Institute for Computational and Experimental Research in Mathematics (ICERM). I.R. acknowledges support through NSF Grants DMS-2005464, DMS-1709270 and a Simons Investigator Award.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St George Street, Toronto, ON, Canada

    Stefanos Aretakis

  2. Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109, Leipzig, Germany

    Stefan Czimek

  3. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544, USA

    Igor Rodnianski

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  1. Stefanos Aretakis
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  2. Stefan Czimek
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Correspondence to Stefan Czimek.

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Aretakis, S., Czimek, S. & Rodnianski, I. Characteristic Gluing to the Kerr Family and Application to Spacelike Gluing. Commun. Math. Phys. 403, 275–327 (2023). https://doi.org/10.1007/s00220-023-04800-y

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