Eikonal quasinormal modes, greybody factors and shadow of charged accelerating black holes

Abstract: We show that the quasinormal modes, in the eikonal limit, for accelerating (non-rotating) black holes, are related to the angular velocity of the circular null geodesics and to the corresponding Lyapunov exponent, in the same way as the ones for spherically symmetric black holes are. We compute those quasinormal modes and greybody factors for neutral and charged accelerating black holes, considering massless test scalar fields, and we show that the results are universal for perturbations of any spin. We also determine the radius of the shadow cast by these black holes. Our results for charged black holes are valid for the Reissner-Nordstrom solution simply by setting the acceleration to zero.

• The paper derives analytic QNM frequencies in the eikonal limit using WKB methods, highlighting corrections from acceleration, mass, and charge.
• It establishes a universal correspondence between the real part of QNM frequencies and the angular velocity of the photon sphere, with the Lyapunov exponent controlling damping.
• The study also computes greybody factors and shadow radii, showing that acceleration narrows the ringdown spectrum while increasing the apparent shadow size.

Eikonal Quasinormal Modes, Greybody Factors, and Shadows of Charged Accelerating Black Holes

Overview and Motivation

The paper "Eikonal quasinormal modes, greybody factors and shadow of charged accelerating black holes" (2603.28557) systematically analyzes the physical properties of charged, accelerating black holes using the C-metric. Unlike the Kerr or Reissner-Nordström metrics, the C-metric accommodates acceleration and non-spherical symmetry, thereby capturing phenomena relevant to black holes in dense astrophysical environments and those experiencing post-merger recoil (superkicks). The study focuses on the eikonal limit, where the large angular momentum ($\ell \to \infty$) regime permits a WKB treatment, enabling analytic characterization of quasinormal modes (QNMs), greybody factors, and shadow radii.

Theoretical Framework: The C-metric and Perturbation Analysis

The C-metric solution, parameterized by mass ($M$), acceleration ($a$), and charge ($Q$), gives rise to spacetime geometries with acceleration horizons in addition to event horizons. This metric's lack of spherical symmetry necessitates novel separation constants in the perturbation equations. Massless scalar fields are used as probes, with the Klein-Gordon equation reducing to radial and angular components involving effective potentials $V_r$ and $V_\theta$. In the eikonal limit, the potentials simplify, and boundary conditions appropriate for QNMs (outgoing at infinity, ingoing at the horizon) make the problem non-Hermitian, resulting in complex frequencies whose imaginary part encodes damping.

Key technical contributions include deriving the large-$\ell$ behavior of the separation constant $\lambda$, showing its perturbative dependence on $a$, and establishing its universality for perturbations of any spin. The paper exploits the WKB method to obtain quasinormal mode frequencies, leveraging the equivalence between the maximization of the effective potential and the location of the circular null geodesics.

Eikonal Quasinormal Modes and Their Geodesic Correspondence

The main result is that, even for the C-metric, the real part of QNM frequencies in the eikonal limit is proportional to the angular velocity $\Omega_c$ of the photon sphere/circular null geodesics, while the imaginary part is governed by the Lyapunov exponent $M$0, as for spherically symmetric metrics. Explicit analytic expressions for $M$1 and $M$2 as functions of $M$3, $M$4, $M$5 are provided. Notably, for accelerating (but non-rotating, $M$6) black holes, expanding in $M$7 yields

• Real part: $M$8
• Lyapunov exponent (imaginary): $M$9

Both corrections in $a$0 are negative; acceleration reduces the real and imaginary frequency parts, i.e., narrows the ringdown spectrum and reduces damping.

For charged accelerating black holes ($a$1), the maximization of the eikonal potential introduces a cubic equation for the photon sphere radius $a$2, with the relevant root corresponding to $a$3 in the trigonometric solution. Analytical expressions for QNMs in terms of $a$4, $a$5, and $a$6 are developed, and the leading order $a$7 corrections to both real and imaginary parts are shown to be universally negative for $a$8.

Black Hole Shadow Radius and Gravitational Lensing

The shadow radius is computed via projection of the photon sphere using the metric function at the critical radius, generalizing prior results for spherically symmetric metrics. The identification $a$9 is confirmed, and corrections in $Q$0 are manifestly positive, meaning acceleration increases the apparent shadow size. Charged cases are handled analogously, recovering the Reissner-Nordström result when $Q$1.

The theoretical interpretation relies on integrals of motion that do not depend on the azimuthal angular momentum, ensuring the shadow remains circular for axisymmetric accelerating black holes without rotation, despite the absence of full spherical symmetry.

Greybody Factors and Their QNM Connection

Greybody factors, quantifying the transmission probability for Hawking-radiated scalars, are derived in the eikonal limit using WKB techniques. The analytic expressions relate transmission coefficient $Q$2 to the QNM frequencies and the second derivative of the potential at $Q$3, exhibiting explicit dependence on $Q$4 and $Q$5. The paper establishes the universal correspondence between QNMs and greybody factors in these spacetimes, and shows correction terms in $Q$6 are negative for transmission.

For charged black holes, lengthy algebraic expressions are provided, with polynomial checks ensuring the first order correction to the imaginary part of $Q$7 is always negative for non-extremal charges ($Q$8).

Generalization to Arbitrary Spin Perturbations

The appendices rigorously demonstrate that, in the eikonal limit, the asymptotic expansion of the separation constant $Q$9 is independent of spin. Using the Newman-Penrose formalism and Teukolsky equation for massless fields of spin $V_r$0, it is shown that the angular and radial potentials effectively decouple spin dependence in the large-$V_r$1 regime, establishing universality of the results for scalar, electromagnetic, and gravitational perturbations.

Implications and Future Directions

The results have multiple implications:

• Observational: Corrections to QNM frequencies, greybody factors, and shadow size due to acceleration can provide signatures distinguishing accelerating black holes from Kerr solutions in gravitational wave and black hole imaging data.
• Theoretical: The universality for spin perturbations and the extension to axisymmetric but non-spherically symmetric metrics deepens understanding of black hole perturbation theory and geodesic correspondence.
• Computational: The perturbative expansion in $V_r$2 provides a practical analytic framework but suggests future work exploring the super-accelerating regime ($V_r$3 not small) and highly damped QNMs.

The treatment is sufficiently general to be adapted to rotating C-metrics and non-minimal field couplings.

Conclusion

This paper provides a thorough analytic characterization of eikonal quasinormal modes, greybody factors, and shadows for charged accelerating black holes described by the C-metric, extending known correspondences from spherical symmetry to axisymmetric, accelerating solutions. Correction terms arising from acceleration and charge are derived perturbatively and shown to be universal across field spins. The results serve both as a foundational theoretical advance in black hole physics and as a reference for interpreting astrophysical observations involving non-isolated, dynamically evolving black holes.