Acoustic Hayward Black Hole

• Acoustic Hayward black hole is an analogue gravity model that deforms a regular Hayward spacetime by introducing fluid flow effects to create an effective acoustic metric.
• The model employs relativistic Gross–Pitaevskii theory to derive an emergent acoustic geometry and analyze phenomena such as horizon structure, shadow formation, and quasinormal modes.
• The tuning parameter ξ predominantly governs the expansion of acoustic horizons and shadows while the Hayward parameter L subtly modulates the regular core features.

An acoustic Hayward black hole is an analogue black-hole geometry constructed in the background of a regular Hayward spacetime, with phonons propagating in an effective metric rather than in the original spacetime metric directly. In the formulation based on relativistic Gross–Pitaevskii theory, phase fluctuations of a complex scalar field behave as a massless scalar on an emergent acoustic geometry whose lapse function depends both on the Hayward background and on the fluid flow. The explicit Hayward-specific construction appears in “Acoustic Black Hole in Hayward Spacetime: Shadow, Quasinormal Modes and Analogue Hawking Radiation” (Hui et al., 27 Feb 2026), where the analogue geometry is used to analyze acoustic horizons, shadow formation, quasinormal modes, grey-body factors, and analogue Hawking emission.

1. Definition and theoretical setting

The Hayward background is a regular black-hole spacetime with metric

$$ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2 d\Omega^2,\qquad f(r)=1-\frac{r_s r^2}{r^3+r_sL^2},$$

with $r_s=2M$ and Hayward length scale $L$ (Hui et al., 27 Feb 2026). The ordinary Hayward horizon is determined by

$P_3(r)=r^3-r_s r^2+r_sL^2=0,$

and its outer horizon radius is

$$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$

with

$L_0=\sqrt{\frac{2r_s}{27}}.$

The acoustic construction is obtained by starting from the relativistic Gross–Pitaevskii action

$$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$

writing $\varphi=\sqrt{\rho}\,e^{i\theta}$, and linearizing around a background $(\rho_0,\theta_0)$ in the long-wavelength limit while neglecting the quantum pressure term (Hui et al., 27 Feb 2026). This yields a covariant wave equation for the phase perturbation $\theta_1$, and the perturbation propagates in an effective acoustic metric $r_s=2M$0 determined by the background geometry and the fluid four-velocity.

In the specific Hayward realization, the fluid is chosen to be radially infalling with $r_s=2M$1 and $r_s=2M$2, and the radial flow is parameterized by

$r_s=2M$3

where $r_s=2M$4 is the tuning parameter controlling the strength of the acoustic modification (Hui et al., 27 Feb 2026). With $r_s=2M$5, the effective acoustic line element becomes

$r_s=2M$6

with

$r_s=2M$7

Using

$r_s=2M$8

this can be written as

$r_s=2M$9

The limit $L$0 recovers the ordinary Hayward black hole. This establishes the acoustic Hayward black hole as a deformation of the Hayward geometry induced by the fluid sector rather than as a separate regular-black-hole ansatz.

2. Horizon structure and parameter dependence

The acoustic horizons are determined by $L$1 (Hui et al., 27 Feb 2026). One factor reproduces the ordinary Hayward horizon, while the acoustic sector contributes the condition

$L$2

This immediately gives the existence condition

$L$3

After reducing the horizon equation to a cubic, the two acoustic horizons are

$L$4

with

$L$5

The outer acoustic horizon is $L$6, and the ordering emphasized in the analysis is

$L$7

The dependence on the two parameters is markedly asymmetric. As $L$8 increases, the outer acoustic horizon grows, and in the limit $L$9 it expands without bound, so that sound cannot escape from an ever-larger region (Hui et al., 27 Feb 2026). By contrast, the Hayward parameter $P_3(r)=r^3-r_s r^2+r_sL^2=0,$0 has a much weaker effect because it is restricted to the regular-black-hole window

$P_3(r)=r^3-r_s r^2+r_sL^2=0,$1

This suggests that, within the model, the analogue causal structure is governed predominantly by the tuning parameter $P_3(r)=r^3-r_s r^2+r_sL^2=0,$2, while the regular-core scale $P_3(r)=r^3-r_s r^2+r_sL^2=0,$3 acts as a secondary deformation.

The resulting horizon structure differs from the standard single-horizon canonical acoustic black hole discussed in GUP-corrected Abelian-Higgs constructions, where the canonical metric function is instead

$P_3(r)=r^3-r_s r^2+r_sL^2=0,$4

and no Hayward-type regular-core structure is introduced (Anacleto et al., 2021).

3. Acoustic geodesics and shadow

The acoustic shadow is obtained from critical null geodesics of the effective acoustic metric (Hui et al., 27 Feb 2026). Since phonons follow null trajectories in the analogue geometry, the construction parallels the photon-shadow calculation for an ordinary black hole, but with $P_3(r)=r^3-r_s r^2+r_sL^2=0,$5 replacing the usual lapse function.

For equatorial null geodesics, the conserved quantities are

$P_3(r)=r^3-r_s r^2+r_sL^2=0,$6

The shadow radius seen by a distant observer is

$P_3(r)=r^3-r_s r^2+r_sL^2=0,$7

where $P_3(r)=r^3-r_s r^2+r_sL^2=0,$8 is the radius of the unstable circular null orbit, referred to as the acoustic sphere. The location of that orbit follows from

$P_3(r)=r^3-r_s r^2+r_sL^2=0,$9

Numerically, a single relevant acoustic sphere is found outside the horizon, and the shadow radius grows monotonically with $$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$0 (Hui et al., 27 Feb 2026). For fixed $$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$1, increasing $$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$2 makes both the horizon and the shadow larger; dependence on $$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$3 is comparatively mild and slightly decreases the shadow size. The qualitative conclusion is therefore that the tuning parameter enlarges the acoustic shadow, whereas the Hayward parameter only weakly suppresses it.

This shadow is not a photon shadow of the underlying Hayward geometry. It is the boundary between captured and escaping phonon trajectories in the effective metric. That distinction is central: the “shadow” here is an acoustic-optics observable defined by analogue null curves, not an electromagnetic image of the spacetime itself.

4. Perturbations, effective potential, and quasinormal modes

Phonon perturbations are separated as

$$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$4

which leads to the Schrödinger-like radial equation

$$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$5

The effective potential is

$$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$6

(Hui et al., 27 Feb 2026).

The quasinormal-mode boundary conditions are standard: $$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$7 Frequencies are computed with the WKB method up to 9th order, using

$$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$8

and are cross-checked with the asymptotic iteration method (Hui et al., 27 Feb 2026).

All computed modes satisfy

$$r_{\rm H}=\frac{r_s}{3}\left(2\cos\alpha+1\right),\qquad \alpha=\frac13\arccos\!\left(1-\frac{L^2}{L_0^2}\right),$$9

so the acoustic Hayward black hole is linearly stable. Compared with the ordinary Hayward case $L_0=\sqrt{\frac{2r_s}{27}}.$0, the acoustic modes are reported to be much smaller in amplitude. As $L_0=\sqrt{\frac{2r_s}{27}}.$1 increases, both $L_0=\sqrt{\frac{2r_s}{27}}.$2 and $L_0=\sqrt{\frac{2r_s}{27}}.$3 decrease, so the oscillation frequency and damping rate both drop. The stated interpretation is that the perturbation becomes weaker and longer-lived in frequency terms because the effective barrier becomes smoother and lower (Hui et al., 27 Feb 2026).

The angular and overtone dependence follows the expected pattern within this analysis. Increasing $L_0=\sqrt{\frac{2r_s}{27}}.$4 raises the real part and lowers the damping magnitude, while increasing the overtone number $L_0=\sqrt{\frac{2r_s}{27}}.$5 lowers $L_0=\sqrt{\frac{2r_s}{27}}.$6 and increases damping. The paper characterizes the acoustic quasinormal modes as “more stable” than those of the ordinary Hayward black hole in the sense that the response exhibits weaker decay-driving structure.

A broader contextual point is that acoustic black-hole perturbations have also been related to bulk sound-channel quasinormal modes in holographic constructions based on cutoff-surface fluids and black branes, where phonon scattering is mapped to gauge-invariant sound-channel perturbations in the bulk (Ge et al., 2015). The Hayward construction is different in setting and mechanism, but it belongs to the same general program of treating acoustic excitations as probes of an effective black-hole geometry.

5. Grey-body factor and analogue Hawking radiation

The same radial wave equation is treated as a scattering problem to extract the transmission and reflection data (Hui et al., 27 Feb 2026). The asymptotic conditions are

$L_0=\sqrt{\frac{2r_s}{27}}.$7

with

$L_0=\sqrt{\frac{2r_s}{27}}.$8

The grey-body factor is the transmission probability,

$L_0=\sqrt{\frac{2r_s}{27}}.$9

The analogue Hawking temperature is defined by

$$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$0

The energy emission rate is

$$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$1

Within this model, the grey-body factor behaves in the standard way: it is near zero at low frequency and approaches unity at high frequency (Hui et al., 27 Feb 2026). The emission spectrum is blackbody-like, vanishing at very low and very high $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$2 and peaking at intermediate frequencies. Increasing $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$3 suppresses transmission because the effective barrier is higher.

The principal parameter dependence again comes from $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$4. Increasing $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$5 lowers the barrier, so both the grey-body factor and the energy emission rate are enhanced (Hui et al., 27 Feb 2026). The Hayward parameter $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$6 affects these quantities only weakly because its allowed interval is narrow and its effect on $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$7 is comparatively small. The physical interpretation offered is that the tuning parameter reshapes the effective potential and therefore the causal and scattering structure of the acoustic geometry.

This parameter-controlled enhancement is specific to the Hayward acoustic model. In other acoustic black-hole settings, the temperature relation can instead be tied directly to hydrodynamic gradients or to a parent black-brane temperature, as in cutoff-surface holographic fluids and black D3-brane constructions (Sun et al., 2017). The Hayward case does not use that holographic temperature relation; its emission properties are extracted from the effective lapse $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$8 and the associated scattering barrier.

6. Relation to the broader acoustic-black-hole literature and scope

The term “acoustic black hole” is used in multiple, technically distinct senses, and the acoustic Hayward black hole belongs to the analogue-gravity branch in which perturbations propagate on an effective curved spacetime. In this branch, acoustic horizons arise from conditions such as $$S=\int d^4x\,\sqrt{-g}\left(|\partial_\mu\varphi|^2+m^2|\varphi|^2-\frac{b}{2}|\varphi|^4\right),$$9 in flowing media, and the effective metric governs phonon propagation. Experimental and model realizations include stationary microcavity-polariton flows with a subsonic–supersonic transition across an engineered defect (Nguyen et al., 2014), analytic transonic potential flow in a thin tube (Tsuda et al., 2023), holographic superfluids with bulk acoustic horizons in AdS (Candare et al., 2024), and acoustic geometries induced on cutoff-surface fluids dual to black branes (Ge et al., 2015, Sun et al., 2017).

By contrast, some papers use “acoustic black hole” to describe tapered absorbing structures in plates or cylinders, where waves are trapped and attenuated by geometry rather than by an analogue spacetime metric. Examples include phononic thin plates with embedded acoustic black holes (Zhu et al., 2014) and rotating absorbing ABH structures used to study superradiance (Yu et al., 2024). Those systems are not Hayward black holes, and they do not employ the Hayward lapse

$\varphi=\sqrt{\rho}\,e^{i\theta}$0

The Hayward-specific literature is correspondingly narrow. The 2026 construction explicitly studies an acoustic black hole in Hayward spacetime and derives its horizon structure, shadow, quasinormal modes, grey-body factor, and energy emission rate (Hui et al., 27 Feb 2026). Earlier works on GUP-corrected acoustic black holes, holographic acoustic black holes, black D3-brane acoustic geometries, and polariton horizons do not derive a Hayward acoustic metric [(Anacleto et al., 2021); (Ge et al., 2015); (Sun et al., 2017); (Nguyen et al., 2014)]. A plausible implication is that the acoustic Hayward black hole should be understood not as the generic endpoint of analogue-gravity models, but as a specific regular-black-hole embedding of relativistic Gross–Pitaevskii phonons.

The main conceptual content of the subject is therefore threefold. First, the Hayward regular background supplies a nonsingular gravitational scaffold. Second, the fluid sector introduces a tunable acoustic deformation through $\varphi=\sqrt{\rho}\,e^{i\theta}$1. Third, observable analogue quantities—shadow radius, quasinormal spectra, transmission coefficients, and Hawking-like emission—are governed primarily by how that tuning parameter reshapes the effective potential. Within the model analyzed in (Hui et al., 27 Feb 2026), increasing $\varphi=\sqrt{\rho}\,e^{i\theta}$2 enlarges the outer acoustic horizon and the acoustic shadow, lowers the barrier, suppresses quasinormal frequencies, and enhances transmission and emission, while the Hayward parameter $\varphi=\sqrt{\rho}\,e^{i\theta}$3 plays a comparatively modest role.