## Duke Mathematical Journal

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

### 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

$${\square}_{g}\varphi =0$$ 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 ${W}_{\mathrm{loc}}^{1,2}$. 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**

Received: 11 February 2015

Revised: 5 April 2016

First available in Project Euclid: 24 October 2016

**Permanent link to this document**

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

**Keywords**

black holes Reissner–Nordström wave equation instability of Cauchy horizon strong cosmic censorship conjecture

#### 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