Long-lived resonances of massive scalar fields in the Reissner-Nordström black-hole spacetime: Analytic treatment in the large-mass regime (2511.10521v1)

Abstract: The physical and mathematical properties of the composed Reissner-Nordström-black-hole-massive-scalar-field system are studied {\it analytically} in the dimensionless large-mass $Mμ\gg1$ regime [here ${M,μ}$ are respectively the mass of the central black hole and the proper mass of the scalar field]. It is proved that, for a given value ${\bar Q}\equiv Q/M$ of the dimensionless charge parameter of the central black hole, the system is characterized by the presence of quasi-resonances, linearized perturbation modes with arbitrarily long lifetimes. In particular, using analytical techniques, we determine the black-hole-field critical mass spectrum ${Mμ_{\text{crit}}({\bar Q})}$ which characterizes the long-lived resonances of the composed physical system.

• The paper derives explicit analytic expressions for the critical mass parameters that support arbitrarily long-lived quasi-resonant modes in RN black-hole spacetimes.
• It employs a WKB approximation to analyze the influence of the charge-to-mass ratio on the resonance spectrum, contrasting Schwarzschild and extremal RN limits.
• The study generalizes numerical results to closed-form formulas, offering diagnostic tools for predicting long-lived scalar field configurations with potential astrophysical implications.

Analytic Treatment of Long-Lived Resonances of Massive Scalar Fields in Reissner-Nordström Spacetimes

Introduction

The paper provides a comprehensive analytic investigation into the resonance structure of massive scalar fields in the background of Reissner-Nordström (RN) black holes, focusing on the $M\mu \gg 1$, $l \gg 1$ regime. By deriving explicit analytic expressions for the critical mass parameter supporting quasi-resonant—and thus arbitrarily long-lived—modes, the work clarifies both the physical conditions and quantitative predictions for the existence of metastable scalar field configurations in the exteriors of spherically symmetric charged black holes. The results also demonstrate that the resonance spectrum is highly sensitive to the black hole's charge-to-mass ratio, and generalize previous purely numerical findings to closed analytic formulas.

Mathematical Framework and Asymptotic Analysis

The system examined couples a minimally coupled massive scalar field ($\mu$) with an RN black hole spacetime characterized by mass $M$ and charge $Q$. The dynamics are governed by the (covariant) Klein-Gordon equation. Mode decomposition and separation of variables reduce the problem to a Schrödinger-type ODE for the radial function, with an effective potential dependent on $(M,Q,\mu,l)$.

Focusing on the double-asymptotic limit ($M\mu \gg 1$, $l\gg 1$), the effective potential simplifies and allows for a WKB-type analysis, employing the third-order Wentzel-Kramers-Brillouin method for complex resonant frequencies. Here, the imaginary part $\Im\omega_0$ dictates the decay rate of perturbations, with $\tau_{\text{relax}} = 1/\Im\omega_0$ defining the relaxation timescale.

The critical point of the analysis is the determination of conditions under which the imaginary part of the fundamental mode approaches zero, hence yielding arbitrarily long-lived quasi-bound states. These conditions reduce to finding degenerate extrema in the effective potential—a set of coupled cubic equations whose solutions determine the critical mass parameters for the field.

Explicit Analytic Results for Critical Mass

Schwarzschild (Q=0) and Extremal RN (Q=M) Limits

• Schwarzschild limit ($Q=0$):

The critical value for $\bar{\mu} \equiv M\mu/\sqrt{l(l+1)}$ for arbitrarily long-lived modes is:

$$\bar{\mu}_{\text{crit}}^{\text{Sch}} = \frac{1}{\sqrt{12}}$$

with the corresponding potential peak coinciding at $\bar{r}_{\text{crit}} = 6$.

• Extremal RN limit ($Q=M$):

The critical value is:

$$\bar{\mu}_{\text{crit}}^{\text{eRN}} = \frac{1}{\sqrt{8}}$$

with coincident peak at $\bar{r}_{\text{crit}} = 4$.

General Charge-to-Mass Ratio ($Q/M$)

For an arbitrary $Q/M \in [0,1]$, the analysis yields:

• A cubic equation for $\bar{r}_{\text{crit}}$ (dimensionless potential peak location):

$$\bar{r}^3 - 6\bar{r}^2 + 9\bar{Q}^2\bar{r} - 4\bar{Q}^4 = 0$$

with $\bar{Q} = Q/M$, solved analytically.

• The critical mass parameter for long-lived resonances:

$$M\mu_{\text{crit}}(\bar{Q}) = \sqrt{l(l+1)} \frac{ \sqrt{\bar{r}_{\text{crit}}^2 - 3\bar{r}_{\text{crit}} + 2\bar{Q}^2} }{ \bar{r}_{\text{crit}}^{3/2} - \bar{Q}^2 \bar{r}_{\text{crit}}^{1/2} }$$

where $\bar{r}_{\text{crit}}$ is obtained via the specified cubic as a (monotonically increasing) function of $\bar Q$.

WKB Regime Validity

The analytic results rely critically on the WKB approximation, valid when $$l(l+1), (M\mu)^2 \gg (\bar{\mu}_{\text{crit}}-\bar{\mu})^{-1}$$. The regime of validity is carefully characterized, and higher-order corrections are shown to be negligible in the stated limit.

Physical and Theoretical Implications

Absence of Stationary Scalar Hair:

The analysis reconfirms that static, spatially regular scalar hair is excluded for RN black holes due to no-hair theorems. However, in the large mass (and frequency) regime, the system admits quasi-stationary, long-lived resonant modes of massive scalar fields—effectively circumventing the no-hair restriction in a dynamical sense. These quasi-resonant solutions, while non-static, can persist for arbitrarily long timescales as $M\mu$ approaches the critical value.

Critical Mass Spectrum:

The explicit analytic formula for $M\mu_{\text{crit}}(\bar{Q})$ provides an immediately useful diagnostic for predicting the onset of long-lived scalar configurations about charged, nonrotating black holes. The result exhibits monotonic increase of $M\mu_{\text{crit}}$ with $\bar{Q}$.

Astrophysical Relevance:

While astrophysical black holes are expected to be nearly neutral, the analytic structure is relevant for fundamental field theory and proposed extensions involving minicharged or dark-sector fields. The analytic results may be of use in modeling late-time tails, field remnants, and potential observations of ultralight fields in gravitational wave astronomy.

Generalization of Numerical Results:

Previous studies relied extensively on numerical integration to locate quasi-bound states. This analytic treatment systematically generalizes prior work and delivers closed-form dependencies between system parameters.

Future Directions

• Extension to Rotating Black Holes: Analogy with Kerr spacetimes—where superradiant instabilities complicate the spectrum—suggests further analytic work could clarify resonant dynamics in rotating backgrounds.
• Beyond Leading WKB: Refinement of the analytic results to sub-leading corrections would enhance predictive power, especially for moderate $l$ or $M\mu$.
• Non-minimal Couplings and Nonlinearities: Incorporation of self-interactions or couplings to gauge fields may yield richer spectra and possible new regimes of quasi-resonant behavior.
• Phenomenological Applications: The analytic criteria may serve as input for models of scalar fields around primordial black holes, dark matter candidates, or as lessons for analogous phenomena in other compact object spacetimes.

Conclusion

This analysis rigorously establishes the existence and analytic dependence of critical mass parameters for long-lived scalar field resonances around RN black holes in the large-mass, high-angular-momentum regime. The work elucidates the interplay between black hole charge and scalar field parameters, emphasizing that while no true stationary configurations are possible, black holes can support field configurations with lifetimes that diverge as masses approach criticality. This closed-form approach paves the way for future analytic studies in related black hole environments, offering new tools for both theoretical and phenomenological inquiry in gravitational physics.