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Communications in Mathematical Physics

, Volume 358, Issue 2, pp 437–520 | Cite as

A Proof of Friedman’s Ergosphere Instability for Scalar Waves

  • Georgios Moschidis
Article

Abstract

Let \({(\mathcal{M}^{3+1},g)}\) be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion \({\mathscr{E}}\) and no future event horizon \({\mathcal{H}^{+}}\). In Friedman (Commun Math Phys 63(3):243–255, 1978), Friedman observed that, on such spacetimes, there exist solutions \({\varphi}\) to the wave equation \({\square_{g}\varphi=0}\) such that their local energy does not decay to 0 as time increases. In addition, Friedman provided a heuristic argument that the energy of such solutions actually grows to \({+\infty}\). In this paper, we provide a rigorous proof of Friedman’s instability. Our setting is, in fact, more general. We consider smooth spacetimes \({(\mathcal{M}^{d+1},g)}\), for any \({d\ge2}\), not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary \({\partial\mathscr{E}}\) of \({\mathscr{E}}\) on a small neighborhood of a point \({p\in\partial\mathscr{E}}\). This condition always holds if \({(\mathcal{M},g)}\) is analytic in that neighborhood of p, but it can also be inferred in the case when \({(\mathcal{M},g)}\) possesses a second Killing field \({\Phi}\) such that the span of \({\Phi}\) and the stationary Killing field T is timelike on \({\partial\mathscr{E}}\). We also allow the spacetimes \({(\mathcal{M},g)}\) under consideration to possess a (possibly empty) future event horizon \({\mathcal{H}^{+}}\), such that, however, \({\mathcal{H}^{+}\cap\,\,\mathscr{E}=\emptyset}\) (excluding, thus, the Kerr exterior family). As an application of our theorem, we infer an instability result for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first investigated numerically by Oliveira et al. in (Phys Rev D 89(12):124008, 2014). Furthermore, as a side benefit of our proof, we provide a derivation, based entirely on the vector field method, of a Carleman-type estimate on the exterior of the ergoregion for a general class of stationary and asymptotically flat spacetimes. Applications of this estimate include a Morawetz-type bound for solutions \({\varphi}\) of \({\square_{g}\varphi=0}\) with frequency support bounded away from \({{\omega}=0}\) and \({{\omega}=\pm\infty}\).

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References

  1. 1.
    Alinhac S., Baouendi M.: A non uniqueness result for operators of principal type. Math. Z. 220(1), 561–568 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andersson L., Blue P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. Ann. Math. 182(3), 787–853 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Blue P., Soffer A.: Semilinear wave equations on the Schwarzschild manifold I: local decay estimates. Adv. Differ. Equ. 8(3), 595–614 (2003)zbMATHMathSciNetGoogle Scholar
  4. 4.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Butterworth E., Ipser J.: On the structure and stability of rapidly rotating fluid bodies in general relativity. I-The numerical method for computing structure and its application to uniformly rotating homogeneous bodies. Astrophys. J. 204, 200–223 (1976)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cardoso V., Pani P., Cadoni M., Cavaglia M.: Ergoregion instability of ultracompact astrophysical objects. Phys. Rev. D 77(12), 124044 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    Chandrasekhar S.: Solutions of two problems in the theory of gravitational radiation. Phys. Rev. Lett. 24(11), 611–615 (1970)ADSCrossRefGoogle Scholar
  8. 8.
    Christodoulou D., Klainerman S.: The Global Nonlinear Stability of the Minkowski Space, volume 1 of Princeton Mathematical Series. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  9. 9.
    Comins, N., Schutz, B.: On the ergoregion instability. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 364, pp. 211–226. The Royal Society (1978)Google Scholar
  10. 10.
    Dafermos, M., Holzegel, G., Rodnianski, I.: The linear stability of the Schwarzschild solution to gravitational perturbations (2016). arXiv preprint arXiv:1601.06467
  11. 11.
    Dafermos M., Rodnianski I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162(2), 381–457 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dafermos, M., Rodnianski, I.: A note on energy currents and decay for the wave equation on a Schwarzschild background (2007). arXiv preprint arXiv:0710.0171
  13. 13.
    Dafermos M., Rodnianski I.: The red-shift effect and radiation decay on black hole spacetimes. Commun. Pure Appl. Math. 62(7), 859–919 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a|<<M or axisymmetry (2010). arXiv preprint arXiv:1010.5132
  15. 15.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. In: Evolution Equations, Clay Mathematics Proceedings, vol. 17, pp. 97–205 (2013)Google Scholar
  17. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Datchev K.: Quantitative limiting absorption principle in the semiclassical limit. Geom. Funct. Anal. 24(3), 740–747 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Eskin G.: Superradiance initiated inside the ergoregion. Rev. Math. Phys. 28(10), 1650025 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Friedman J.L.: Ergosphere instability. Commun. Math. Phys. 63(3), 243–255 (1978)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Friedman J.L., Lockitch K.: Gravitational-wave driven instability of rotating relativistic stars. Prog. Theor. Phys. Suppl. 136, 121–134 (1999)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Friedman J.L., Schutz B.: Secular instability of rotating Newtonian stars. Astrophys. J. 222, 281–296 (1978)ADSCrossRefGoogle Scholar
  23. 23.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, vol. 224. Springer, Berlin (2001)zbMATHGoogle Scholar
  24. 24.
    Holmgren E.: Über Systeme von linearen partiellen Differentialgleichungen. Öfversigt af Kongl, Vetenskaps-Academien Förhandlinger 58, 91–103 (1901)Google Scholar
  25. 25.
    Kay B.S., Wald R.M.: Linear stability of Schwarzschild under perturbations which are non-vanishing on the bifurcation 2-sphere. Class. Quantum Gravity 4(4), 893 (1987)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Kokkotas K., Ruoff J., Andersson N.: w-mode instability of ultracompact relativistic stars. Phys. Rev. D 70(4), 043003 (2004)ADSCrossRefGoogle Scholar
  27. 27.
    Kokkotas K., Schutz B.: Normal modes of a model radiating system. Gen. Relativ. Gravit. 18(9), 913–921 (1986)ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Lee John M.: Introduction to Smooth Manifolds. Springer, New York (2012)CrossRefGoogle Scholar
  29. 29.
    Metcalfe, J., Sterbenz, J., Tataru, D.: Local energy decay for scalar fields on time dependent non-trapping backgrounds (2017). arXiv preprint arXiv:1703.08064
  30. 30.
    Moschidis G.: Logarithmic local energy decay for scalar waves on a general class of asymptotically flat spacetimes. Ann. PDE 2, 5 (2016). doi: 10.1007/s40818-016-0010-8 CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Moschidis G.: The r p-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications. Ann. PDE 2, 6 (2016). doi: 10.1007/s40818-016-0011-7 CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Moschidis, G.: Superradiant instabilities for short-range non-negative potentials on kerr spacetimes and applications (2016). arXiv preprint arXiv:1608.02041
  33. 33.
    Oliveira L., Cardoso V., Crispino L.: Ergoregion instability: the hydrodynamic vortex. Phys. Rev. D 89(12), 124008 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    Rodnianski I., Tao T.: Effective limiting absorption principles, and applications. Commun. Math. Phys. 333, 1–95 (2015)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Shlapentokh-Rothman Y.: Exponentially growing finite energy solutions for the Klein–Gordon equation on sub-extremal Kerr spacetimes. Commun. Math. Phys. 329(3), 859–891 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Shlapentokh-Rothman, Y.: Quantitative mode stability for the wave equation on the Kerr spacetime. Annales Henri Poincaré 16(1), 289–345 (2015)Google Scholar
  37. 37.
    Tataru D.: Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem. Commun. Partial Differ. Equ. 20(5-6), 855–884 (1995)zbMATHGoogle Scholar
  38. 38.
    Tataru D., Tohaneanu M.: A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. 2011(2), 248–292 (2011)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Yoshida S., Eriguchi Y.: Ergoregion instability revisited—a new and general method for numerical analysis of stability. Mon. Not. R. Astron. Soc. 282(2), 580–586 (1996)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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