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

Jonathan Luk, Sung-Jin Oh

TL;DR

The paper proves that linear scalar perturbations on subextremal Reissner–Nordström spacetimes generically induce blow-up of the nondegenerate energy at the future Cauchy horizon, aligning with blue-shift instability and supporting a linearized analogue of strong cosmic censorship. The authors reduce to the spherically symmetric mode, introduce the L-condition computed from null infinity, and establish a chain of conditional lower bounds that force divergence along the Cauchy horizon. A key output is that Price's law decay along the event horizon is sharp for generic data, and they demonstrate the existence of blow-up-inducing perturbations via a concrete construction with $\mathfrak{L}\neq 0$. Collectively, these results illuminate a linear instability mechanism with potential implications for the nonlinear Einstein–Maxwell system and for understanding the generic structure of Cauchy horizons in charged black holes.

Abstract

It has long been suggested that solutions to linear scalar wave equation $$\Box_gφ=0$$ on a fixed subextremal Reissner-Nordström spacetime with non-vanishing 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 $W^{1,2}_{loc}$. 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.

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

TL;DR

The paper proves that linear scalar perturbations on subextremal Reissner–Nordström spacetimes generically induce blow-up of the nondegenerate energy at the future Cauchy horizon, aligning with blue-shift instability and supporting a linearized analogue of strong cosmic censorship. The authors reduce to the spherically symmetric mode, introduce the L-condition computed from null infinity, and establish a chain of conditional lower bounds that force divergence along the Cauchy horizon. A key output is that Price's law decay along the event horizon is sharp for generic data, and they demonstrate the existence of blow-up-inducing perturbations via a concrete construction with . Collectively, these results illuminate a linear instability mechanism with potential implications for the nonlinear Einstein–Maxwell system and for understanding the generic structure of Cauchy horizons in charged black holes.

Abstract

It has long been suggested that solutions to linear scalar wave equation on a fixed subextremal Reissner-Nordström spacetime with non-vanishing 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 . 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.

Paper Structure

This paper contains 18 sections, 35 theorems, 174 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Geometry of Reissner-Nordström
  3. Coordinates in the exterior region
  4. Coordinates in the interior region
  5. The wave equation
  6. Nondegenerate energy
  7. Notation
  8. Statement of main theorem and outline of the proof
  9. Outline of the proof
  10. Wave equation in the exterior of the black hole
  11. The strong cosmic censorship conjecture and previous works
  12. "Poor man's" linearized problem
  13. Spherically symmetric Einstein-Maxwell-scalar field system
  14. Outline of the paper
  15. Lower bound on $\{r=R\}$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Generic smooth and compactly supported initial data to wave.eqn on $\Sigma_{0}$ give rise to solutions that are not in $W^{1,2}_{loc}$ in a neighborhood of any point on the future Cauchy horizon $\mathcal{C} \mathcal{H}^{+}$.

Figures (7)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 2 more figures

Theorems & Definitions (57)

  • Theorem 1.1: Main theorem, first version
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6: Main theorem, second version
  • proof
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • ...and 47 more