Revisiting environmental effects on black hole quasibound-state spectra with relativistic perturbation theory

Technical Summary: Revisiting Environmental Effects on Black Hole Quasibound-State Spectra with Relativistic Perturbation Theory

Problem Statement
Black holes (BHs) and ultralight bosons can form "gravitational atoms" via superradiance, leading to macroscopic boson clouds. While the superradiant spectrum and instability rates for isolated Kerr black holes are well-understood, realistic astrophysical environments introduce perturbations (e.g., galactic halos, accretion disks, binary companions) that may alter these dynamics. Previous studies addressing these environmental effects relied on non-relativistic treatments, modeling the system analogously to the hydrogen atom. However, this approximation fails to capture the intrinsic dissipative nature of the black hole horizon, where quasibound states possess complex frequencies (with imaginary parts encoding decay or superradiant growth). Consequently, non-relativistic approaches cannot naturally describe shifts in decay rates or the non-Hermitian nature of the spectrum, particularly when perturbations induce mode mixing between growing and decaying states.

Methodology
The authors develop and apply a fully relativistic perturbation framework to compute corrections to the eigenfrequency spectrum of massive scalar fields in perturbed black-hole spacetimes.

1. Relativistic Framework: Building on the bilinear form formalism introduced in Ref. [41], the authors utilize a conserved product based on spacetime symmetries ($t-\phi$ symmetry) to define an orthogonality relation for quasibound states. This allows for the consistent calculation of first-order frequency shifts, including imaginary components (decay/superradiant rates), which are absent in standard Hermitian quantum mechanical treatments.
2. Perturbation Orders:
   • First-Order: The authors derive first-order shifts using both a Hamiltonian formulation (semi-Newtonian approximation via an effective potential) and a fully covariant formulation (direct expansion of the Klein-Gordon operator).
   • Second-Order: The framework is extended to second order to investigate mode mixing, which arises when axisymmetry is broken (e.g., by binary companions). The authors explicitly address the "completeness assumption," testing whether quasibound states form a complete basis for reconstructing perturbations, a standard practice in non-relativistic quantum mechanics.
3. Environmental Models: Three physically motivated perturbations are analyzed:
   • Galactic Halos: Modeled as an anisotropic fluid (Einstein cluster) transitioning to a Hernquist density profile.
   • Accretion Disks: Modeled as a static, axially symmetric thin disk around a Schwarzschild BH.
   • Binary Companions: Treated via a Newtonian tidal potential approximation for large separations, inducing mode mixing.
4. Benchmarking: The perturbative predictions are rigorously benchmarked against non-perturbative numerical solutions obtained by directly integrating the Klein-Gordon equation in the full perturbed metrics.

Key Contributions and Results

• Validity of Non-Relativistic Treatments: The study confirms that non-relativistic treatments are limited. They correctly predict shifts in the real part of the frequency (energy levels) in the small coupling limit ($\alpha \ll 1$) but fail entirely to capture shifts in the imaginary part (decay/superradiant rates).
• Accuracy of Relativistic Approaches:
  • The covariant formulation provides the most accurate agreement with numerical solutions across all parameters, correctly capturing both real and imaginary shifts.
  • The Hamiltonian formulation (semi-Newtonian) performs well for galactic halos and binary companions at large separations. However, for accretion disks, it fails to reproduce decay-rate shifts in certain regimes (large disk radius $b$). This failure is attributed to the Hamiltonian approach capturing only time-redshift effects (lapse function) while missing spatial rescaling effects that significantly impact the imaginary part of the spectrum.
• Binary Companions and Mode Mixing:
  • First-order relativistic calculations for binary companions show that while energy levels shift, the companion does not, at leading order, flip the sign of the superradiant growth rate (i.e., it does not turn unstable modes stable).
  • The mechanism proposed in previous non-relativistic work (Ref. [22]) for terminating superradiance relies on mode mixing between $m>0$ and $m \le 0$ states, which is a second-order effect.
• Completeness Assumption:
  • The authors demonstrate that quasibound states do not form a complete basis for the relativistic problem.
  • When the completeness assumption is enforced in second-order calculations, the results agree with numerical solutions only when the perturbation probes the asymptotic tail of the wavefunction (large halo scales).
  • When the perturbation probes the relativistic bulk (near the horizon), the completeness assumption fails to reproduce the correct decay-rate shifts and energy shifts, highlighting the non-Hermitian nature of the spectrum.
• Second-Order Corrections: Using the galactic halo as a controlled testbed, the authors show that a fully relativistic treatment without completeness assumptions is required to accurately reproduce the spectrum, particularly the dissipative components.

Significance and Claims
The paper argues that previous estimates regarding the termination of superradiance due to environmental effects (specifically binary companions and accretion disks) must be revisited within a fully relativistic framework. The non-relativistic treatments used in prior literature (e.g., Refs. [22, 40]) miss first-order corrections to the imaginary part of the spectrum and rely on a completeness assumption that is invalid for dissipative black hole systems.

The authors conclude that while non-relativistic approaches may offer qualitative insights in specific limits, they are insufficient for quantitative predictions of superradiant dynamics in realistic environments. Future dedicated analyses of binary systems must employ the relativistic perturbative framework developed here to accurately assess the robustness of superradiance termination mechanisms, moving beyond the limitations of the hydrogenic approximation.