Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I

Abstract: We study the problem of stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we consider solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We obtain boundedness, decay and non-decay results. Our estimates hold up to and including the horizon. The fundamental new aspect of this problem is the degeneracy of the redshift on the event horizon. Several new analytical features of degenerate horizons are also presented.

• The paper demonstrates that energy boundedness persists in extreme Reissner-Nordström spacetimes despite the absence of redshift, highlighting conserved quantities along the horizon.
• It employs advanced vector field techniques and modified currents to overcome degeneracies at the event horizon and control perturbation dynamics.
• Findings reveal a unique instability mechanism where certain low-frequency modes fail to decay, challenging traditional notions of black hole stability in general relativity.

Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations

The study presented in "Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations" addresses the profound question of the behavior of perturbations in the context of extreme black holes, particularly those described by the Reissner-Nordström metric with equal mass and charge parameters. The central focus is the analysis of the linear wave equation $\Box_{g}\psi = 0$ and the determination of stability properties for solutions of this equation within the unique geometry of extreme black holes.

Analytical Framework

Extreme Reissner-Nordström spacetimes are characterized by a degenerate event horizon where the surface gravity vanishes. This peculiar feature impacts the dynamics of wave equations dramatically, as traditional methods relying on redshift effects fail. Instead, the problem requires innovative approaches to overcome the lack of dispersion typically provided by redshift.

The paper explores these dynamics by deploying vector field methods, robust analytical techniques suited to the Lorentzian geometry, relying heavily on spacetime and hypersurface energy estimates. Through a meticulous derivation of integral identities and continuity relations, the paper establishes several pivotal results:

1. Boundedness, decay, and non-decay: Solutions exhibit bounded energy but specific geometric quantities along the horizon are conserved, prohibiting decay, which represents a form of instability unique to extreme configurations.
2. Degeneracy and trapping at the horizon: The analytical structure reveals a trapping phenomenon at degenerate horizons, distinct from trapping at photon spheres, which in the non-extreme case are mediated by redshift effects.
3. Vector fields and currents: The construction of vector fields such as $N$ (generalized redshift vector field) and the modification of currents through novel techniques enables control over energies and reveals subtleties in the analytical behavior of perturbation solutions.

Main Findings

The paper's primary results demonstrate:

• Energy boundedness: Despite the degenerate redshift, it is possible to bound energies away from the horizon, indicating stability in specific senses.
• Spacetime decay for certain frequencies: While spacetime estimates confirm integrated decay for specific harmonic components, a subset of solutions associated with lower frequencies persist due to conserved quantities along the horizon.
• Trapping and nonlinear instability: Non-decay results elucidate that extremal black holes possess inherent instability characteristics evidenced by conserved wave elements along the event horizon.

Implications and Future Directions

These findings have compelling implications for both mathematical physics and the understanding of black hole stability. Extreme Reissner-Nordström black holes challenge prevailing notions of cosmic censorship and stability, indicating that extremal conditions foster both stable and unstable dynamics. The paper suggests significant theoretical conjectures regarding the uniqueness and behavior of waves in black hole spacetimes that possess degenerate horizons.

Looking ahead, further research could explore the nonlinear stability and dynamics within more general types of perturbations, extending findings to broader classes of extremal spacetimes. Moreover, understanding these behaviors in rotating black holes could illuminate the complexities underscored by the intricate interaction between angular momentum and electromagnetic fields.

In conclusion, this research presents foundational insights into the perturbative dynamics of extreme Reissner-Nordström spacetimes, enriching the depth of theoretical strategies in dealing with degenerate horizon phenomena. The implications, particularly those that relate to conserved quantities and the broader stability problem in general relativity, form a robust platform for advancing our understanding of extreme astrophysical objects.