One of the recurring themes in black hole perturbation theory is that many apparently complicated dynamical questions eventually reduce to a rather geometric spectral problem1In perturbation theory, the dynamics often reduce to determining the eigenvalues of an effective differential operator.. One begins with a black hole spacetime, perturbs it slightly, separates variables, and discovers that the entire linear dynamics becomes encoded in the spectral structure of a one-dimensional differential equation. The corresponding complex frequencies are the quasinormal modes2First systematically studied in black hole physics by Vishveshwara and later developed extensively by Chandrasekhar, Leaver, and others., and these frequencies govern the characteristic ringdown behaviour of the geometry.
The paper3Feng, C., Li, S. Y., Zhang, X., Zhang, M., Zou, D. C., & Yue, R. H. (2026). Quasinormal modes of massless scalar and electromagnetic perturbations for Euler Heisenberg black holes surrounded by perfect fluid dark matter. arXiv preprint arXiv:2605.14528. under discussion studies this problem for Euler-Heisenberg black holes surrounded by perfect fluid dark matter. Although the physical setup sounds rather elaborate, the mathematical structure is quite clean. One starts with a static, spherically symmetric spacetime whose metric may be written as
$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2$$
where $d\Omega^2$ denotes the standard metric on the unit two-sphere. Thus, at this level of symmetry, essentially all of the geometry is encoded in the single function $f(r)$. In Schwarzschild spacetime this function is $f(r)=1-2M/r$, while in charged or nonlinear electrodynamic black holes additional charge-dependent terms appear. The Euler-Heisenberg corrections4These arise as effective nonlinear corrections to classical Maxwell electrodynamics due to quantum vacuum polarization effects. modify the electromagnetic sector beyond ordinary Maxwell theory, and the surrounding perfect fluid dark matter contributes an additional deformation to the radial geometry.
The perturbation problem is then introduced by placing a test field on this background. For a massless scalar field, the field equation is the curved-space wave equation
$$\Box\Phi=0$$
where $\Box$ is the d’Alembertian associated with the black hole metric. Using spherical symmetry, one separates variables by writing the field as a product of a time-dependent oscillation, an angular spherical harmonic, and a radial function:
$$\Phi(t,r,\theta,\phi)=e^{-i\omega t}Y_{\ell m}(\theta,\phi)\frac{\psi(r)}{r}$$
After substituting this ansatz into the wave equation, the problem reduces to a Schrödinger-type radial equation
$$\frac{d^2\psi}{dr_*^2}+\left(\omega^2-V(r)\right)\psi=0$$
where the tortoise coordinate5The tortoise coordinate stretches the near-horizon region infinitely, making wave propagation near the horizon easier to analyze. $r_*$ is defined by
$$\frac{dr_*}{dr}=\frac{1}{f(r)}.$$
This coordinate is useful because it sends the event horizon to $r_*\to-\infty$, making the wave boundary conditions more transparent. The function $V(r)$ is the effective potential, and it is here that the geometry of the black hole directly enters the perturbation problem.
For massless scalar perturbations, the effective potential has the form
$$V_{\rm scalar}(r)=f(r)\left(\frac{\ell(\ell+1)}{r^2}+\frac{f'(r)}{r}\right).$$
For electromagnetic perturbations, one obtains the closely related expression
$$V_{\rm EM}(r)=f(r)\frac{\ell(\ell+1)}{r^2}.$$
Thus the role of the Euler-Heisenberg and PFDM parameters is not mysterious: they alter $f(r)$, and by altering $f(r)$ they alter the height, width, and curvature of the effective potential barrier. The quasinormal frequencies are then determined by how waves scatter off this barrier.
The boundary conditions defining quasinormal modes are unusual but physically natural. Near the horizon one demands a purely ingoing wave,
$$\psi\sim e^{-i\omega r_*},\qquad r_*\to-\infty,$$
while at spatial infinity one demands a purely outgoing wave,
$$\psi\sim e^{i\omega r_*},\qquad r_*\to+\infty.$$
These two conditions make the frequency spectrum discrete and complex. Writing
$$\omega=\omega_R-i\omega_I,$$
the time dependence becomes
$$e^{-i\omega t}=e^{-i\omega_R t}e^{-\omega_I t}.$$
The real part $\omega_R$ determines the oscillation frequency, while the imaginary part $\omega_I$ determines the damping rate. A larger $\omega_I$ corresponds to faster decay of the perturbation.
Since the exact spectrum is rarely obtainable in closed form, one usually turns to approximation schemes. The standard approach in this type of problem is the WKB approximation6The WKB method is a semiclassical approximation commonly used to estimate quasinormal spectra when exact solutions are unavailable., which treats the effective potential barrier in analogy with a semiclassical tunneling problem. Near the maximum of the potential, the leading behaviour of the spectrum is controlled by the value of the potential and its curvature. Very schematically,
$$\omega^2\approx V_0-i\left(n+\frac{1}{2}\right)\sqrt{-2V_0”}.$$
Here $V_0$ denotes the maximum of the effective potential, $V_0”$ denotes its second derivative with respect to the tortoise coordinate at the maximum, and $n$ is the overtone number. Higher-order WKB corrections involve higher derivatives of the potential at the peak. This is why changes in the black hole parameters can shift the quasinormal spectrum in a systematic way: they change not only the position of the potential peak, but also its height and local curvature.
This is the main mathematical mechanism behind the paper. The Euler-Heisenberg parameter and the PFDM parameter deform the metric function $f(r)$; this deforms the effective potentials $V_{\rm scalar}$ and $V_{\rm EM}$; and those deformations propagate into the complex frequencies $\omega$. The ringdown spectrum therefore becomes a probe of the underlying spacetime geometry.
The physical interpretation is also quite elegant. A black hole is not literally a material bell, but under perturbation it behaves like a dissipative resonator7Energy is lost both through absorption at the horizon and through radiation escaping to infinity.. Its ringing frequencies are not arbitrary. They are determined by the causal structure of the horizon, the asymptotic boundary condition at infinity, and the effective potential generated by the spacetime geometry. In this sense, quasinormal modes may be regarded as spectral fingerprints of the black hole background.
For nonspecialists, the useful way to think about the calculation is this: the geometry first determines $f(r)$, the function $f(r)$ determines the wave potential $V(r)$, the potential determines the allowed complex frequencies $\omega$, and those frequencies determine how perturbations decay in time. The entire chain may be summarized as
$$\text{geometry}\longrightarrow f(r)\longrightarrow V(r)\longrightarrow \omega\longrightarrow \text{ringdown}.$$
What makes the Euler-Heisenberg plus PFDM case interesting is that both nonlinear electrodynamic effects and environmental dark matter effects enter this chain simultaneously. The resulting quasinormal spectrum therefore carries information not only about the black hole mass and charge, but also about how the surrounding matter distribution and modified electromagnetic sector reshape the effective barrier experienced by perturbing fields.
More broadly, this paper is another example of a general lesson in black hole physics: perturbations translate geometry into spectra. Once the equations are reduced to the radial wave problem, the black hole becomes something like a geometric resonator, and the quasinormal modes provide one of the cleanest ways to read off its structure.
References and Footnotes
- 1In perturbation theory, the dynamics often reduce to determining the eigenvalues of an effective differential operator.
- 2First systematically studied in black hole physics by Vishveshwara and later developed extensively by Chandrasekhar, Leaver, and others.
- 3Feng, C., Li, S. Y., Zhang, X., Zhang, M., Zou, D. C., & Yue, R. H. (2026). Quasinormal modes of massless scalar and electromagnetic perturbations for Euler Heisenberg black holes surrounded by perfect fluid dark matter. arXiv preprint arXiv:2605.14528.
- 4These arise as effective nonlinear corrections to classical Maxwell electrodynamics due to quantum vacuum polarization effects.
- 5The tortoise coordinate stretches the near-horizon region infinitely, making wave propagation near the horizon easier to analyze.
- 6The WKB method is a semiclassical approximation commonly used to estimate quasinormal spectra when exact solutions are unavailable.
- 7Energy is lost both through absorption at the horizon and through radiation escaping to infinity.