A black hole does not have to be described by the vacuum solutions of general relativity for its ringdown to retain a recognizable connection to the spacetime around it. In a new perturbative framework, Ariadna Uxue Palomino Ylla and colleagues treat deviations from Schwarzschild and Kerr black holes as an anisotropic fluid and show how those deviations can shift the frequency and damping of the black hole’s quasinormal modes. The key result is that, in the static case, the difference between those two shifts is tied directly to the local combination of energy density and tangential pressure at the unstable photon orbit.
When a black hole is disturbed, the resulting gravitational radiation includes a ringdown phase characterized by damped oscillations. The complex frequencies of those oscillations are called quasinormal modes, or QNMs. The authors investigate whether changes in those frequencies can reveal information about the effective matter source responsible for “hair” around a black hole.
The challenge is that calculating QNMs directly can become difficult when the black-hole geometry is more complicated than the standard vacuum solutions. Conventional approaches require the appropriate perturbation equations and effective wave potential for each particular black-hole model and perturbation sector. The authors instead develop an analytic approach that connects the leading changes in the ringdown spectrum to changes in the unstable circular orbits followed by photons.
That connection is made in the eikonal, or high-angular-momentum, regime. In a static, spherically symmetric spacetime, the real part of the QNM frequency is related to the orbital frequency of the unstable photon orbit, while its imaginary part is controlled by the orbit’s Lyapunov exponent, which measures the instability of the orbit. The authors use this established correspondence to translate changes in the photon orbit into expected changes in the QNM frequency and damping.
There is an important limitation. The correspondence is not an exact description of the complete QNM spectrum. The eikonal approximation assumes large angular quantum number and comparatively small overtone number. The authors explicitly note that it is therefore not expected to give precise values for the dominant gravitational-wave mode, conventionally the fundamental mode with (ℓ, m, n) = (2, 2, 0). In Schwarzschild spacetime, for example, the leading eikonal estimate gives Mω ≃ 0.3849 − 0.0962i for ℓ = 2 and n = 0, compared with the more accurate gravitational value Mω ≃ 0.3737 − 0.0890i.
The authors therefore describe their results as first-order, leading-eikonal predictions rather than full calculations of the gravitational QNM spectrum for individual hairy black-hole models.
The matter source leaves two different fingerprints
The framework starts with a general static, spherically symmetric black-hole geometry and introduces an anisotropic fluid characterized by an energy density ρ, radial pressure Pᵣ, and tangential pressure Pθ. The pressures are written in terms of state parameters, Pᵣ = wᵣρ and Pθ = wθρ. The matter variables are treated as small first-order perturbations of a Schwarzschild background.
For this perturbative system, the authors derive fractional shifts in the orbital frequency and Lyapunov exponent. The oscillation-frequency shift is
δΩ/Ω₀ = 3δf(r₀)/2,
while the damping-related shift is
δλ/λ₀ = 3δf(r₀)/2 − 4πr₀²ρ(1 + wθ),
evaluated at the Schwarzschild photon-sphere radius r₀ = 3M.
Both quantities therefore contain the same geometric correction. But the Lyapunov exponent has an additional term involving the tangential pressure. That distinction becomes the central feature of the static analysis.
The authors make the connection to the matter distribution explicit for the class of geometries satisfying h(r) = f(r)⁻¹. They introduce a mass function through f(r) = 1 − 2m(r)/r, choose the asymptotic Schwarzschild mass as the reference mass, and obtain the matter-dependent relation between the mass correction and the energy density. For positive energy density, the resulting frequency shift is positive, meaning the hairy black hole’s QNM oscillates faster than its vacuum counterpart within the perturbative approximation.
More strikingly, subtracting the oscillation shift from the damping shift removes the integrated mass contribution:
δλ/λ₀ − δΩ/Ω₀ ≃ −4πr₀²ρ(1 + wθ).
The remaining quantity depends on the local combination ρ + Pθ at the photon orbit.
This gives the authors a direct way to connect a ringdown difference with an energy condition on the effective matter. If the energy density is positive and the tangential null-energy condition is satisfied, then ρ + Pθ ≥ 0, making the difference between the fractional damping and oscillation shifts non-positive. A positive value would instead require 1 + wθ < 0, corresponding to a violation of the tangential null-energy condition in the effective matter description.
The same matter distribution also shifts the unstable circular photon orbit itself. For regular matter with positive energy density, the authors find that the photon-orbit radius moves inward relative to the Schwarzschild value.
Three static black-hole models test the framework
The authors apply the formalism to three specific geometries: the Bardeen, Hayward, and Kiselev black holes. The Bardeen and Hayward models are regular black holes that approach Schwarzschild at large distances, while the Kiselev model describes a non-vacuum black hole surrounded by an anisotropic matter distribution.
For the Bardeen black hole, the additional parameter q is treated as small compared with the black-hole mass. Keeping the leading correction in q², the orbital frequency becomes
Ω⋆ ≃ [1/(3√3M)](1 + q²/(6M²)),
while the Lyapunov exponent becomes
λ⋆ ≃ [1/(3√3M)](1 − q²/(9M²)).
The unstable photon orbit is shifted inward by approximately 5q²/(6M). The resulting leading-eikonal oscillation-frequency shift is positive while the damping-rate shift is negative.
The Bardeen model also provides a clear example of the relationship between the perturbative regime and the energy conditions. The null and weak energy conditions are satisfied, while the strong and dominant conditions impose additional restrictions on q/M. The authors note that the dominant energy condition is not compatible with the small-|q|/M regime used for the perturbative calculation.
The Hayward model produces a similar qualitative pattern. For small q, the leading corrections are
Ω⋆ ≃ [1/(3√3M)](1 + q³/(27M³))
and
λ⋆ ≃ [1/(3√3M)](1 − 2q³/(27M³)).
The photon-orbit radius again shifts inward. The authors find that the oscillation-frequency shift is positive and the damping-rate shift negative. For the standard Hayward branch, the null and weak energy conditions are satisfied for positive q, while the dominant condition is again incompatible with the small-deviation regime.
The Kiselev model behaves differently because its matter distribution is controlled by both a strength parameter k and an effective equation-of-state parameter wq. The metric correction is proportional to −k/r^(1+3wq), so changing wq changes how the hair falls off, or grows, with radius. The authors examine a range of wq values and find that the signs of the frequency and damping shifts depend on the parameters.
In the range associated in the model with quintessence, −1 < wq < −1/3, the behavior differs from the Bardeen and Hayward cases. The paper reports that the two shifts have the same sign in this regime, with the sign depending on k. For other parameter ranges, Kiselev configurations can produce the same pattern seen in the Bardeen and Hayward examples, in which the oscillation-frequency shift is positive and the damping-rate shift negative.
The illustrative ringdown waveforms make these differences visible in the model calculations. For the Bardeen and Hayward examples, the signals have similar decay rates to the vacuum case but noticeably different frequencies. In the Kiselev example, changing wq changes the damping behavior. In particular, the case wq = −1 has a smaller decay rate and a slower oscillation than the other displayed cases.
Those waveforms are illustrative rather than full low-angular-momentum gravitational-wave calculations. They use n = 0 and ℓ = 4, and the authors emphasize that they are constructed from the geodesic/eikonal estimates.
Rotation separates the ringdown into two branches
The same strategy is then extended to stationary rotating hairy black holes. Here the geometry is more complicated because rotation breaks the degeneracy associated with the azimuthal quantum number m. The authors therefore restrict their analysis to null rays trapped in the equatorial plane, corresponding to the eikonal sector with ℓ = |m|. This produces co-rotating and counter-rotating branches.
The rotating geometries are constructed from a mass function m(r) and include rotating versions of the Bardeen, Hayward, and Kiselev models. For the rotating geometries generated through the Newman–Janis procedure, however, the authors caution that the effective stress-energy tensor need not correspond to the same underlying matter model as the static seed solution. The density and pressures in this part of the analysis are therefore treated as effective source variables of the resulting geometry.
The rotation changes more than the numerical value of a single static shift. The two photon-orbit branches respond differently to the hair, producing different changes in their orbital frequencies and Lyapunov exponents. The authors find that both branches continuously approach the static result as the spin parameter tends toward zero.
For the rotating Bardeen model, the calculated shifts depend on both the hair parameter and the spin. The counter-rotating and co-rotating branches have different magnitudes of the Lyapunov-exponent shift, with the counter-rotating shift larger in magnitude and the co-rotating shift smaller in the displayed calculations. All of the curves approach zero hair-induced shift as q/M approaches zero, recovering the Kerr solution continuously.
The rotating Hayward model is treated in the same perturbative way, using the corresponding mass function and expanding in small q. The resulting photon-orbit and QNM shifts again separate into co-rotating and counter-rotating branches.
The rotating Kiselev model shows the same branch structure while retaining its dependence on k and wq. In the quintessence range, the authors find that the Lyapunov-exponent shift is positive. Outside that range, some values of wq give the same signs of the frequency and damping shifts as the Bardeen and Hayward models.
What the ringdown calculation can and cannot establish
The framework gives a unified way to express leading changes in QNM frequencies through the effective matter distribution and its equation of state. In the static case, the distinction between the frequency shift and damping shift isolates the local tangential combination ρ + Pθ at the photon orbit. The authors use this relationship to connect a property of the calculated ringdown with the tangential null-energy condition of the effective source.
For the Bardeen and Hayward examples, the authors find that the perturbative approximation is incompatible with the dominant energy condition. They state that if the corresponding deviation were confirmed through QNM observations, the effective matter description would therefore require a violation of the dominant energy condition or some other modification of the gravitational-wave emission mechanism. For Kiselev, by contrast, the compatibility with the standard energy conditions depends on k and wq, and all of those conditions can be satisfied within suitable ranges even when |k| is small.
The authors emphasize that these conclusions concern the effective sources used to construct the geometries. In particular, for rotating models produced through the Newman–Janis procedure, the reconstructed stress-energy tensor is a diagnostic of the effective geometry-source pair rather than proof of a unique underlying matter model.
The connection to an observable ringdown is also explicitly qualified. Within the eikonal approximation, the real part of the QNM frequency determines the ringdown frequency and the imaginary part determines the damping time. The authors therefore identify the fractional orbital-frequency shift with the corresponding leading fractional ringdown-frequency shift, while the negative fractional Lyapunov shift corresponds approximately to the fractional damping-time shift.
But applying the formulas quantitatively to observed gravitational-wave modes would require more. The authors state that a dedicated perturbation equation for each hairy metric, calibration against numerical QNM calculations, or a parametrized ringdown treatment would be needed, particularly for the dominant (ℓ, m, n) = (2, 2, 0) mode. Those calculations are beyond the scope of the work.
The study instead establishes a first-order framework for connecting the geometry of a hairy black hole to the matter variables that support it, and then connecting those quantities to leading-eikonal changes in its ringdown. The authors also note that their formulas could be extended beyond the simple first-order equations of state used here, including higher-order perturbative calculations and more general equations of state such as polytropic models.
The study was posted on arXiv.
