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Morawetz Estimate for Linearized Gravity in Schwarzschild

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Abstract

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge–Wheeler–Zerilli–Moncrief system. Applying the technique introduced in Andersson and Blue (Ann Math 182(2):787–853, 2015), we prove an integrated local energy decay estimate for both the Regge–Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the \(r^p\) hierarchy estimates (Dafermos and Rodnianski, in: Exner (ed) XVIth international congress on mathematical physics, World Scientic, London, pp 421–433, 2009; Schlue in Anal PDE 6:515–600, 2013) to prove that both the Regge–Wheeler and Zerilli variables decay as \(t^{-\frac{3}{2}}\) in fixed regions of r.

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Notes

  1. Observe that equation (3.23) in [3] is missing a minus sign in front of \(x^2(x+d)^2\partial _x^2{\tilde{v}}\), but the rest of the argument there is correct.

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C (1964)

  2. Andersson, L., Bäckdahl, T., Blue, P., Ma, S.: Stability for linearized gravity on the Kerr spacetime (2019). arXiv:1903.03859

  3. Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr space-time. Ann. Math. 182(2), 787–853 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bardeen, J.M., Press, W.H.: Radiation fields in the Schwarzschild background. J. Math. Phys. 14, 7–13 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  5. Blue, P., Soffer, A.: Semilinear wave equations on the Schwarzschild manifold I: local decay estimates. Adv. Differ. Equ. 8, 595–614 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Blue, P., Soffer, A.: The wave equation on the Schwarzschild metric II. Local decay for the spin-2 Regge–Wheeler equation. J. Math. Phys. 46(012502), 9 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Blue, P., Sterbenz, J.: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Commun. Math. Phys. 268(2), 481–504 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  8. Chandrasekhar, S.: On the equations governing the perturbations of the Schwarzschild black hole. Proc. R. Soc. Lond. A Math. 343, 289–298 (1975)

    ADS  MathSciNet  Google Scholar 

  9. Chandrasekhar, S.: The mathematical theory of black holes, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, Reprint of the 1992 edition (1998)

  10. Clarkson, C.A., Barret, R.K.: Covariant perturbations of Schwarzschild black holes. Class. Quantum Gravity 20, 3855–3884 (2003)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Dafermos, M., Rodnianski, I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. Invent. Math. 185(3), 467–559 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  13. Dafermos, M., Rodnianski, I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162, 381–457 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  14. Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, XVIth International Congress on Mathematical Physics (P. Exner, ed.), World Scientific, London, pp. 421–433 (2009)

  15. Dafermos, M., Rodnianski, I.: The red-shift effect and radiation decay on black hole spacetimes. Commun. Pure Appl. Math. 62, 859–919 (2009)

    Article  MathSciNet  Google Scholar 

  16. Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. In Evolution Equations, Clay Mathematics Proceedings, vol. 17, Amer. Math. Soc., Providence, RI, pp. 97–205 (2013)

  17. Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case \(|a|<M\). Ann. Math. 183(3), 787–913 (2016)

    Article  MathSciNet  Google Scholar 

  18. Donninger, R., Schlag, W., Soffer, A.: On pointwise decay of linear waves on a Schwarzschild black hole background. Commun. Math. Phys. 309, 51–86 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  19. Finster, F., Kamran, N., Smoller, F., Yau, S.T.: Decay of solutions of the wave equation in the Kerr geometry. Commun. Math. Phys. 264, 465–503 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  20. Hung, P., Keller, J.: Linear stability of Schwarzschild spacetime subject to axial perturbations (2016). arXiv:1610.08547

  21. Hung, P., Keller, J., Wang, M.: Linear stability of Schwarzschild spacetime: the Cauchy Problem of Metric Coefficients (2017). arXiv:1702.02843

  22. Jezierski, J.: Energy and angular momentum of the weak gravitational waves on the Schwarzschild background-Quasilocal gauge-invariant formulation. General Relativ. Gravitat. 31, 1855–1890 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  23. Luk, J.: Improved decay for solutions to the linear wave equation on a Schwarzschild black hole. Ann. Henri Poincaré 11, 805–880 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  24. Ma, S.: Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole I: Maxwell field (2017). arXiv:1705.06621

  25. Ma, S.: Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity (2017). arXiv:1708.07385

  26. Metcalfe, J., Tataru, D., Tohaneanu, M.: Price’s law on nonstationary space-times. Adv. Math. 230, 995–1028 (2012)

    Article  MathSciNet  Google Scholar 

  27. Moncrief, V.: Gravitational perturbations of spherical symmetric systems. I. The exterior problem. Ann. Phys. 88, 323–342 (1975)

    Article  ADS  Google Scholar 

  28. Moncrief, V.: Spacetime symmetries and linearization stability of the Einstein equations. J. Math. Phys. 16, 493–498 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  29. Moschidis, G.: The \(r^p\)-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications. Ann. PDE 2, 6 (2016)

    Article  MathSciNet  Google Scholar 

  30. Pasqualotto, F.: The spin \(\pm \)1 Teukolsky equations and the Maxwell system on Schwarzschild. arXiv:1612.07244v2

  31. Regge, T., Wheeler, John A: Stability of a Schwarzschild singularity. Phys. Rev. 108, 1063–1069 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  32. Sarbach, O., Tiglio, M.: Gauge-invariant perturbations of Schwarzschild black holes in horizon-penetrating coordinates. Phys. Rev. D 64(15), 084016 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  33. Schlue, V.: Decay of linear waves on higher dimensional Schwarzschild black holes. Anal. PDE 6, 515–600 (2013)

    Article  MathSciNet  Google Scholar 

  34. Tataru, D., Tohaneanu, M.: A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. IMRN 2011(2), 248–292 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Tataru, D.: Local decay of waves on asymptotically flat stationary space-times. Am. J. Math. 135, 361–401 (2013)

    Article  MathSciNet  Google Scholar 

  36. Teukolsky, S.A.: Perturbations of a rotating black hole. I. Fundamental equations for gravitational, electromagnetic, and Neutrino-field perturbations. Astrophys. J. 185, 635–648 (1973)

    Article  ADS  Google Scholar 

  37. Vishveshwara, C.V.: Stability of the Schwarzschild metric. Phys. Rev. D 1, 2870–2879 (1970)

    Article  ADS  Google Scholar 

  38. Zerilli, F.J.: Effective potential for even-parity Regge–Wheeler gravitational perturbation equations. Phys. Rev. Lett. 24(13), 737 (1970)

    Article  ADS  Google Scholar 

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Acknowledgements

We are grateful to Steffen Aksteiner, Siyuan Ma, and Vincent Moncrief for many helpful discussions and suggestions. J.W. was supported by a Humboldt Foundation postdoctoral fellowship at the Albert Einstein Institute during the period 2014–2016, when part of this work was done. She is also supported by the Fundamental Research Funds for the Central Universities (Grant No. 20720170002) and NSFC (Grant No. 11701482).

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Authors and Affiliations

  1. Albert Einstein Institute, Am Mühlenberg 1, 14476, Potsdam, Germany

    Lars Andersson

  2. The School of Mathematics and the Maxwell Institute, University of Edinburgh, Edinburgh, UK

    Pieter Blue

  3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Jinhua Wang

Authors
  1. Lars Andersson
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  2. Pieter Blue
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  3. Jinhua Wang
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Correspondence to Jinhua Wang.

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Communicated by Mihalis Dafermos.

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Andersson, L., Blue, P. & Wang, J. Morawetz Estimate for Linearized Gravity in Schwarzschild. Ann. Henri Poincaré 21, 761–813 (2020). https://doi.org/10.1007/s00023-020-00886-5

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