## Duke Mathematical Journal

### Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations

#### Abstract

It has long been suggested that solutions to the linear scalar wave equation

$$\Box_{g}\phi=0$$ on a fixed subextremal Reissner–Nordström spacetime with nonvanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{\mathrm{loc}}$. This instability is related to the celebrated blue-shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein–Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner–Nordström spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price’s law decay is generically sharp along the event horizon.

#### Article information

Source
Duke Math. J., Volume 166, Number 3 (2017), 437-493.

Dates
Revised: 5 April 2016
First available in Project Euclid: 24 October 2016

https://projecteuclid.org/euclid.dmj/1477321009

Digital Object Identifier
doi:10.1215/00127094-3715189

Mathematical Reviews number (MathSciNet)
MR3606723

Zentralblatt MATH identifier
1373.35306

Subjects
Primary: 35Q75: PDEs in connection with relativity and gravitational theory
Secondary: 83C57: Black holes

#### Citation

Luk, Jonathan; Oh, Sung-Jin. Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations. Duke Math. J. 166 (2017), no. 3, 437--493. doi:10.1215/00127094-3715189. https://projecteuclid.org/euclid.dmj/1477321009

#### References

• [1] L. Andersson and P. Blue, Hidden symmetries and decay for the wave equation on the Kerr spacetime, Ann. of Math. (2) 182 (2015), 787–853.
• [2] P. Blue and J. Sterbenz, Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space, Comm. Math. Phys. 268 (2006), 481–504.
• [3] S. Chandrasekhar and J. B. Hartle, On crossing the Cauchy horizon of a Reissner-Nördstrom black-hole, Proc. Roy. Soc. London Ser. A 384 (1962), 301–315.
• [4] D. Christodoulou, The Formation of Black Holes in General Relativity, EMS Monogr. Math., EMS, Zurich, 2009.
• [5] D. Civin, Stability of charged rotating black holes for linear scalar perturbations, Ph.D. dissertation, University of Cambridge, Cambridge, 2014, http://www.repository.cam.ac.uk/handle/1810/247397.
• [6] J. L. Costa, P. M. Girão, J. Natário, and J. D. Silva, On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant, Part 2: Structure of the solutions and stability of the Cauchy horizon, Comm. Math. Phys. 339 (2015), 903–947.
• [7] J. L. Costa, P. M. Girão, J. Natário, and J. D. Silva, On the global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant, Part 3: Mass inflation and extendibility of the solutions, preprint, arXiv:1406.7261v3 [gr-qc].
• [8] M. Dafermos, Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations, Ann. of Math. (2) 158 (2003), 875–928.
• [9] M. Dafermos, The interior of charged black holes and the problem of uniqueness in general relativity, Comm. Pure Appl. Math. 58 (2005), 445–504.
• [10] M. Dafermos, Black holes without spacelike singularities, Comm. Math. Phys. 332 (2014), 729–757.
• [11] M. Dafermos and J. Luk, Stability of the Kerr Cauchy horizon, in preparation.
• [12] M. Dafermos and I. Rodnianski, A proof of Price’s law for the collapse of a self-gravitating scalar field, Invent. Math. 162 (2005), 381–457.
• [13] M. Dafermos and I. Rodnianski, The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math. 62 (2009), 859–919.
• [14] M. Dafermos and I. Rodnianski, “A new physical-space approach to decay for the wave equation with applications to black hole spacetimes” in XVIth International Congress on Mathematical Physics (Prague, 2009), World Sci., Hackensack, N.J., 2010, 421–432.
• [15] M. Dafermos and I. Rodnianski, A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math. 185 (2011), 467–559.
• [16] M. Dafermos and I. Rodnianski, The wave equation on Schwarzschild-de Sitter spacetimes, preprint, arXiv:0709.2766v1 [gr-qc].
• [17] M. Dafermos and I. Rodnianski, Lectures on black holes and linear waves, preprint, arXiv:0811.0354v1 [gr-qc].
• [18] M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, preprint, arXiv:1010.5137v1 [gr-qc].
• [19] M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes, I–II: The cases $\vert a\vert \ll M$ or axisymmetry, preprint, arXiv:1010.5132v1 [gr-qc].
• [20] M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes, III: The full subextremal case $\vert a\vert <M$, Ann. of Math. (2) 183 (2016), 787–913.
• [21] M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman, A scattering theory for the wave equation on kerr black hole exteriors, preprint, arXiv:1412.8379v1 [gr-qc].
• [22] R. Donninger and W. Schlag, Decay estimates for the one-dimensional wave equation with an inverse power potential, Int. Math. Res. Not. IMRN 2010, no. 22, 4276–4300.
• [23] R. Donninger, W. Schlag, and A. Soffer, A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta, Adv. Math. 226 (2011), 484–540.
• [24] R. Donninger, W. Schlag, and A. Soffer, On pointwise decay of linear waves on a Schwarzschild black hole background, Comm. Math. Phys. 309 (2012), 51–86.
• [25] A. Franzen, Boundedness of massless scalar waves on Reissner-Nordström interior backgrounds, Comm. Math. Phys. 343 (2016), 601–650.
• [26] R. Geroch and J. Traschen, Strings and other distributional sources in general relativity, Phys. Rev. D (3) 36 (1987), 1017–1031.
• [27] Y. Gursel, V. Sandberg, I. Novikov, and A. Starobinsky, Evolution of scalar perturbations near the Cauchy horizon of a charged black hole, Phys. Rev. D 19 (1979), 413–420.
• [28] W. A. Hiscock, Evolution of the interior of a charged black hole, Phys. Lett. A 83 (1981), no. 3, 110–112.
• [29] B. S. Kay and R. M. Wald, Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation $2$-sphere, Classical Quantum Gravity 4 (1987), no. 4, 893–898.
• [30] J. Luk and S.-J. Oh, Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry, Anal. PDE 8 (2015), 1603–1674.
• [31] J. Marzuola, J. Metcalfe, D. Tataru, and M. Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), 37–83.
• [32] J. M. McNamara, Instability of black hole inner horizons, Proc. Roy. Soc. London Ser. A 358 (1978), 499–517.
• [33] J. Metcalfe, D. Tataru, and M. Tohaneanu, Price’s law on nonstationary spacetimes, Adv. Math. 230 (2012), 995–1028.
• [34] E. Poisson and W. Israel, Inner-horizon instability and mass inflation in black holes, Phys. Rev. Lett. 63 (1989), no. 16, 1663–1666.
• [35] E. Poisson and W. Israel, Internal structure of black holes, Phys. Rev. D (3) 41 (1990), 1796–1809.
• [36] R. H. Price, Nonspherical perturbations of relativistic gravitational collapse, I: Scalar and gravitational perturbations, Phys. Rev. D (3) 5 (1972), 2419–2439.
• [37] J. Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds: With applications to black hole spacetimes, Anal. PDE 8 (2015), 1379–1420.
• [38] J. Sbierski, On the initial value problem in general relativity and wave propagation in black-hole spacetimes, PhD dissertation, University of Cambridge, Cambridge, 2014, http://www.repository.cam.ac.uk/handle/1810/248837.
• [39] M. Simpson and R. Penrose, Internal instability in a Reissner-Nordström black hole, Internat. J. Theoret. Phys. 7 (1973), 183–197.
• [40] D. Tataru, Local decay of waves on asymptotically flat stationary space-times, Amer. J. Math. 135 (2013), 361–401.
• [41] D. Tataru and M. Tohaneanu, A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN 2011, no. 2, 248–292.
• [42] S. Yang, Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal. 209 (2013), 683–728.