Skip to main content
SpringerLink
Log in

Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge–Wheeler and Zerilli equations obtained recently by Anderson et al. (Ann. Henri Poincaré 21:61–813, 2020) are sufficient to obtain the conformal scattering.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Andersson, L., Blue, P., Wang, J.: Morawetz estimate for linearized gravity in Schwarzschild. Ann. Henri Poincaré 21, 761–813 (2020)

    Article  MathSciNet  Google Scholar 

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

  3. Angelopoulos, Y., Aretakis, S., Gajic, D.: A non-degenerate scattering theory for the wave equation on extremal Reissner–Nordström. Commun. Math. Phys. 380, 323–408 (2020)

    Article  Google Scholar 

  4. Baez, J.C., Segal, I.E., Zhou, Z.F.: The global Goursat problem and scattering for nonlinear wave equations. J. Funct. Anal. 93(2), 239–269 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1983)

    MATH  Google Scholar 

  7. Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: A scattering theory for the wave equation on Kerr black hole exteriors. Ann. Sci. Ec. Norm. Super. 51(2), 371–486 (2018)

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. 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), 1–118 (2019)

  10. Giorgi, E.: Boundedness and decay for the Teukolsky system of spin 2 on Reissner–Nordström spacetime: the case of small charge. arXiv:1811.03526

  11. Giorgi, E.: The linear stability of Reissner–Nordström spacetime: the full subextremal range. arXiv:1910.05630

  12. Friedlander, F.G.: On the radiation field of pulse solutions of the wave equation I. Proc. Roy. Soc. Ser. A 269, 53–65 (1962)

    MathSciNet  MATH  Google Scholar 

  13. Friedlander, F.G.: On the radiation field of pulse solutions of the wave equation II. Proc. Roy. Soc. Ser. A 279, 386–394 (1964)

    MathSciNet  MATH  Google Scholar 

  14. Friedlander, F.G.: On the radiation field of pulse solutions of the wave equation III. Proc. Roy. Soc. Ser. A 299, 264–278 (1967)

    MathSciNet  MATH  Google Scholar 

  15. Friedlander, F.G.: Radiation fields and hyperbolic scattering theory. Math. Proc. Camb. Phil. Soc. 88, 483–515 (1980)

    Article  MathSciNet  Google Scholar 

  16. Friedlander, F.G.: Notes on the wave equation on asymptotically Euclidean manifolds. J. Funct. Anal. 184, 1–18 (2001)

    Article  MathSciNet  Google Scholar 

  17. Hörmander, L.: A remark on the characteristic Cauchy problem. J. Funct. Anal. 93, 270–277 (1990)

    Article  MathSciNet  Google Scholar 

  18. Joudioux, J.: Conformal scattering for a nonlinear wave equation. J. Hyperbolic Differ. Equ. 9(1), 1–65 (2012)

    Article  MathSciNet  Google Scholar 

  19. Joudioux, J.: Hörmander’s method for the characteristic Cauchy problem and conformal scattering for a non linear wave equation. Lett. Math. Phys. 110, 1391–1423 (2020). https://doi.org/10.1007/s11005-020-01266-0

    Article  MathSciNet  MATH  Google Scholar 

  20. Kehle, C., Shlapentokh-Rothman, Y.: A scattering theory for linear waves on the interior of Reissner–Nordström black holes. Ann. Henri Poincaré 20, 1583–1650 (2019)

    Article  MathSciNet  Google Scholar 

  21. Mason, L.J., Nicolas, J.-P.: Conformal scattering and the Goursat problem. J. Hyperbolic Differ. Equ. 1(2), 197–233 (2004)

    Article  MathSciNet  Google Scholar 

  22. Mokdad, M.: Conformal scattering of Maxwell fields on Reissner–Nordström-de Sitter black hole spacetimes. Ann. Inst. Fourier 69(5), 2291–2329 (2019)

    Article  MathSciNet  Google Scholar 

  23. Mason, L.J., Nicolas, J.-P.: Regularity an space-like and null infinity. J. Inst. Math. Jussieu 8(1), 179–208 (2009)

    Article  MathSciNet  Google Scholar 

  24. Nicolas, J.-P.: Non linear Klein–Gordon equation on Schwarzschild-like metrics. J. Math. Pures Appl. 74, 35–58 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Nicolas, J.-P.: On Lars Hörmander’s remark on the characteristic Cauchy problem. Ann. Inst. Fourier 56(3), 517–543 (2006)

  26. Nicolas, J.-P.: Conformal scattering on the Schwarzschild metric. Ann. Inst. Fourier 66(3), 1175–1216 (2016)

    Article  MathSciNet  Google Scholar 

  27. Nicolas, J.-P., Pham, T.X.: Peeling on Kerr spacetime: linear and non linear scalar fields. Ann. Henri Poincaré 20(10), 3419–3470 (2019)

    Article  MathSciNet  Google Scholar 

  28. Penrose, R.: Conformal approach to infinity. In: De Witt, B.S., De Witt, C.M. (eds.) Relativity, groups and topology, Les Houches 1963. Gordon and Breach, New-York (1964)

    Google Scholar 

  29. Penrose, R., Rindler, W.: Spinors and Space-Time, Vol. I & II, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984–1986)

  30. Pham, T.X.: Peeling and conformal scattering on the spacetimes of the general relativity, Phd’s thesis, Brest university (France) (4/2017). https://tel.archives-ouvertes.fr/tel-01630023/document

  31. Pham, T.X.: Peeling of Dirac field on Kerr spacetime. J. Math. Phys. 61, 032501 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  33. Taujanskas, G.: Conformal scattering of the Maxwell-scalar field system on de Sitter space. J. Hyperbolic Differ. Equ. 16(04), 743–791 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of informatics, faculty of mathematics, Thuyloi University, 175 Tay Son, Dong Da, Ha Noi, Vietnam

    Truong Xuan Pham

Authors
  1. Truong Xuan Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Truong Xuan Pham.

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

Pham, T.X. Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime. Ann Glob Anal Geom 60, 589–608 (2021). https://doi.org/10.1007/s10455-021-09789-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-021-09789-y

Keywords

Mathematics Subject Classification

Search

Navigation