Skip to main content
Log in

The Linear Stability of Reissner–Nordström Spacetime for Small Charge

  • Manuscript
  • Published:
Annals of PDE Aims and scope Submit manuscript

Abstract

In this paper, we prove the linear stability to gravitational and electromagnetic perturbations of the Reissner–Nordström family of charged black holes with small charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-Nordström solution arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To obtain decay of the solution, one must add a residual pure gauge solution which is proved to be itself controlled from initial data. Our results rely on decay statements for the Teukolsky system of spin \(\pm \,2\) and spin \(\pm \,1\) satisfied by gauge-invariant null-decomposed curvature components, obtained in earlier works. These decays are then exploited to obtain polynomial decay for all the remaining components of curvature, electromagnetic tensor and Ricci coefficients. In particular, the obtained decay is optimal in the sense that it is the one which is expected to hold in the non-linear stability problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

  2. Bardeen, J.M., Press, W.H.: Radiation fields in the Schwarzschild background. J. Math. Phys. 14, 7–19 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  3. Blue, P.: Decay of the Maxwell field on the Schwarzschild manifold. J. Hyperbolic Differ. Equ. 5(4), 807–856 (2008)

    Article  MathSciNet  Google Scholar 

  4. Bondi, H., van der Burg, M.G.J., Metzner, A.W.K.: Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems. Proc. R. Soc. Ser. A 269, 21–52 (1962)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Chandrasekhar, S.: The mathematical theory of black holes. Oxford University Press, Oxford (1983)

    MATH  Google Scholar 

  6. Chandrasekhar, S.: On the Equations Governing the Perturbations of the Reissner-Nordström Black Hole. Proc. R. Soc. Lond. A 365, 453–65 (1979)

    Article  ADS  Google Scholar 

  7. Chandrasekhar, S., Xanthopoulos, B.C.: On the metric perturbations of the Reissner-Nordström black hole. Proc. R. Soc. Lond. A 367, 1–14 (1979)

    Article  ADS  Google Scholar 

  8. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Princeton Math. Series 41. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  9. 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 

  10. Dafermos, M., Holzegel, G., Rodnianski, I.: Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case \(|a|\ll M\). Ann. PDE 5, 2 (2019)

    Article  MathSciNet  Google Scholar 

  11. 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 

  12. Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: XVIth international congress on mathematical physics, 421–433 (2009)

  13. Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. In: Evolution equations, Clay Mathematics Proceedings, Vol. 17, pp. 97–205. Amer. Math. Soc. (2013)

  14. Fernández Tío, J.M., Dotti, G.: Black hole nonmodal linear stability under odd perturbations: the Reissner–Nordström case. Phys. Rev. D 95(12), 124041 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. Giorgi, E.: Boundedness and decay for the Teukolsky system of spin \(\pm 2\) on Reissner-Nordström spacetime: the case \(|Q| \ll M\). arXiv preprint arXiv:1811.03526 (2018)

  16. Giorgi, E.: Boundedness and decay for the Teukolsky-type equation of spin \(\pm 1\) on Reissner-Nordström spacetime: the \(\ell =1\) spherical mode. Class. Quantum Grav. 36, 205001 (2019)

    Article  ADS  Google Scholar 

  17. Griffiths, J.B., Podolsky, J.: Exact space–times in Einstein’s general relativity. Cambridge monographs on mathematical physics. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  18. Häfner, D., Hintz, P., Vasy, A.: Linear stability of slowly rotating Kerr black holes. arXiv preprint arXiv:1906.00860 (2019)

  19. Hintz, P., Vasy, A.: The global non-linear stability of the Kerr-de Sitter family of black holes. Acta Math. 220, 1–206 (2018)

    Article  MathSciNet  Google Scholar 

  20. Hintz, P.: Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes. Ann. PDE 4(1), 11 (2018)

    Article  MathSciNet  Google Scholar 

  21. Hung, P.-K., Keller, J., Wang, M.-T.: Linear stability of Schwarzschild spacetime: the cauchy problem of metric coefficients. arXiv preprint arXiv:1702.02843 (2017)

  22. Hung, P.-K.: The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part. arXiv preprint arXiv:1803.03881 (2018)

  23. Johnson, T.: On the linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge. arXiv preprint arXiv:1803.04012 (2018)

  24. Johnson, T.: The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge. arXiv preprint arXiv:1810.01337 (2018)

  25. Klainerman, S., Szeftel, J.: Global non-linear stability of Schwarzschild spacetime under polarized perturbations. arXiv preprint arXiv:1711.07597 (2017)

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

  27. Moncrief, V.: Odd-parity stability of a Reissner-Nordström black hole. Phys. Rev. D 9, 2707 (1974)

    Article  ADS  Google Scholar 

  28. Moncrief, V.: Stability of Reissner–Nordström black holes. Phys. Rev. D 10, 1057 (1974)

    Article  ADS  Google Scholar 

  29. Moncrief, V.: Gauge-invariant perturbations of Reissner–Nordström black holes. Phys. Rev. D 12, 1526 (1974)

    Article  ADS  Google Scholar 

  30. Pasqualotto, F.: The spin \(\pm 1\) Teukolsky equations and the Maxwell system on Schwarzschild. Ann. Henri Poincaré 20, 1263–323 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  32. Wald, R.M.: Construction of solutions of gravitational, electromagnetic, or other perturbation equations from solutions of decoupled equations. Phys. Rev. Lett. 41, 203 (1978)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank Sergiu Klainerman and Mu-Tao Wang for their guidance and support. The author is also grateful to Jérémie Szeftel, Pei-Ken Hung and Federico Pasqualotto for helpful discussions. The author is grateful to the anonymous referee for several helpful suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Elena Giorgi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Giorgi, E. The Linear Stability of Reissner–Nordström Spacetime for Small Charge. Ann. PDE 6, 8 (2020). https://doi.org/10.1007/s40818-020-00082-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40818-020-00082-y

Keywords

Navigation