Photon-Fluid Rotating Black Holes

• Photon-fluid rotation black holes are analogue gravity systems where light in a nonlinear medium mimics rotating black-hole spacetimes with defined horizons and ergoregions.
• They use a hydrodynamic reformulation of the nonlinear Schrödinger equation to convert optical vortex flows into effective curved metrics analogous to Kerr and BTZ black holes.
• Experimental implementations have measured features such as event and inner horizons, superradiant instabilities, and quasinormal modes, bridging theoretical models with laboratory observations.

Searching arXiv for the cited photon-fluid and rotating black-hole analogue papers to ground the article in current arXiv records. A photon-fluid rotation black hole is an analogue-gravity configuration in which light propagating through a self-defocusing nonlinear optical medium forms a photon fluid, while small fluctuations of that fluid propagate as if they were fields on an effective rotating black-hole spacetime. In the systems studied on arXiv, the relevant geometry is typically $2+1$-dimensional, generated by draining vortex flows, and characterized by an event horizon, an ergoregion, and, in BTZ-type constructions, even an inner horizon. The subject sits at the intersection of nonlinear optics, hydrodynamic analogies, black-hole perturbation theory, and laboratory gravity, with direct links to superradiance, scalar clouds, quasinormal modes, Hawking-like emission, and black-hole spectroscopy (Vocke et al., 2017, Ciszak et al., 2021, Wu et al., 25 Apr 2025, Senjaya et al., 3 Sep 2025).

1. Conceptual and hydrodynamic foundation

The photon-fluid description begins from paraxial light propagation in a nonlinear medium. In one standard formulation, the optical field $E(\mathbf{r},z)$ obeys the nonlinear Schrödinger equation

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$

while in another frequently used form,

$$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$

After the Madelung transformation $E=\sqrt{\rho}e^{i\phi}$, the optical field is re-expressed in hydrodynamic variables, with density $\rho$ and velocity potential $\phi$, and the flow velocity becomes

$\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$

In this regime, long-wavelength fluctuations behave as sound-like excitations of a quantum fluid rather than as free photons in vacuum (Vocke et al., 2017, Chen et al., 2023).

The effective spacetime emerges when one linearizes around a stationary background. In the rotating photon-superfluid formulation, the phase perturbation $\psi_1$ satisfies a massless Klein-Gordon equation,

$$\Box \psi_1 = \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g} \, g^{\mu\nu} \partial_\nu \psi_1 \right) = 0 ,$$

with effective metric

$E(\mathbf{r},z)$0

Equivalent formulations write the line element as

$E(\mathbf{r},z)$1

Here $E(\mathbf{r},z)$2 is the local sound speed, and $E(\mathbf{r},z)$3, $E(\mathbf{r},z)$4 are the radial and azimuthal components of the photon-fluid flow. The propagation coordinate is mapped to an effective time, so the analogue spacetime is $E(\mathbf{r},z)$5-dimensional rather than $E(\mathbf{r},z)$6-dimensional (Vocke et al., 2017, Chen et al., 2023).

A central definitional point is that the “black hole” in this context is not a gravitational solution of Einstein’s equations but an emergent metric for collective excitations. The horizon and ergoregion are therefore kinematic structures of the medium: they are identified from the local flow and sound velocities, not from curvature sourced by stress-energy. This distinction is foundational for interpreting all subsequent results.

2. Rotating geometries, horizons, and ergoregions

The basic rotating configuration is the draining vortex. In one widely used realization, the background phase is

$E(\mathbf{r},z)$7

yielding azimuthal and radial velocities

$E(\mathbf{r},z)$8

The azimuthal term produces quantized circulation, while the radial term produces inward drain. This combination is the direct analogue of frame dragging plus infall in a rotating black-hole exterior (Ciszak et al., 2021).

Within this kinematic geometry, the event horizon is located where the radial flow equals the sound speed,

$E(\mathbf{r},z)$9

and the ergosphere is located where the total flow speed matches the sound speed,

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$0

These conditions reproduce, in the analogue setting, the separation between horizon and ergoregion familiar from Kerr spacetimes. In the rotating acoustic-black-hole formulation of the photon-fluid model, the metric can be written as

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$1

where $$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$2 is the horizon radius and $$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$3 is the angular velocity of the horizon. Its ergoregion boundary is

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$4

This form makes explicit the Kerr-like role of horizon rotation in the analogue system (Hod, 2021).

A second major line of development replaces the Kerr-like draining-bathtub geometry by a BTZ-like analogue. In the rotating BTZ mapping,

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$5

with horizons determined by

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$6

and ergosurface

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$7

The BTZ construction is notable because it includes an inner horizon, a feature absent in earlier optical analogues that implemented only a single horizon structure (Chen et al., 2023, Wu et al., 25 Apr 2025).

The optical beam profile required for this BTZ-like geometry is not the standard vortex $$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$8 alone. The proposed and later implemented phase structure is

$$\partial_z E = \frac{i}{2k} \nabla_\perp^2 E + i \frac{k}{n_0} \Delta n E ,$$9

which generates

$$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$0

The logarithmic and quadratic radial terms are precisely what allow $$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$1 to cross $$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$2 twice and thus realize both $$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$3 and $$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$4 (Chen et al., 2023).

3. Massive modes, scalar clouds, and superradiant instabilities

Photon-fluid rotating black holes are not restricted to massless perturbations. In the presence of suitable vortex flows, the fluctuation dynamics can be governed by a massive Klein-Gordon equation,

$$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$5

where $$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$6 is the effective mass parameter of the scalar field. In the rapidly rotating acoustic-black-hole analysis, one introduces the mode decomposition

$$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$7

which yields the radial equation

$$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$8

For bound states, the asymptotic behavior is exponentially decaying,

$$\partial_{z} E = \frac{i}{2k} \nabla_{\bot}^2 E + i \frac{k n_{2}}{n_0} E |E|^2 .$$9

and stationarity requires synchronous rotation with the horizon,

$E=\sqrt{\rho}e^{i\phi}$0

In this formulation, stationary scalar clouds exist only in the narrow dimensionless window

$E=\sqrt{\rho}e^{i\phi}$1

For $E=\sqrt{\rho}e^{i\phi}$2, the discrete resonance spectrum satisfies

$E=\sqrt{\rho}e^{i\phi}$3

These cloud solutions are the acoustic counterparts of stationary scalar clouds around rotating gravitational black holes (Hod, 2021).

The same framework also supports superradiant instabilities. In the rotating photon-fluid model with a draining vortex background, superradiant amplification occurs for

$E=\sqrt{\rho}e^{i\phi}$4

and quasi-bound states become unstable when they are simultaneously localized by the effective mass term and lie inside the superradiant band. At the threshold,

$E=\sqrt{\rho}e^{i\phi}$5

the imaginary part vanishes and one obtains stationary clouds rather than growing modes. This is the analogue-black-hole version of the “black-hole bomb” mechanism familiar from Kerr perturbation theory (Ciszak et al., 2021).

A more recent spectroscopy analysis treats relativistic massive phonon modes in a $E=\sqrt{\rho}e^{i\phi}$6-dimensional photon-fluid rotating black hole with metric

$E=\sqrt{\rho}e^{i\phi}$7

Using the ansatz

$E=\sqrt{\rho}e^{i\phi}$8

the radial equation becomes

$E=\sqrt{\rho}e^{i\phi}$9

The exact radial solution is expressed in terms of the confluent Heun function, and the quasibound-state spectrum satisfies

$\rho$0

In that analysis, the analogue black hole is superradiant in the range

$\rho$1

and the greybody factors are negative for co-rotating modes in the superradiant regime. Scalar clouds again occur at

$\rho$2

The same work also derives Hawking-radiation and amplification formulas through the Damour-Ruffini and asymptotic-matching methods (Senjaya et al., 3 Sep 2025).

4. Quasinormal modes and analogue ringdown

The ringdown sector of photon-fluid rotating black holes has been studied through quasinormal modes of the optical-field fluctuation equation. In the $\rho$3-dimensional photon-fluid model, the background vortex beam is written as

$\rho$4

and the effective rotating metric takes the form

$\rho$5

The normal-mode decomposition

$\rho$6

leads, after transformation to tortoise coordinate and field redefinition, to the Schrödinger-like equation

$\rho$7

with generalized potential

$\rho$8

Quasinormal-mode boundary conditions are ingoing at the horizon and outgoing at infinity (Liu et al., 2024).

Three numerical methods were used to compute the quasinormal frequencies: the asymptotic iteration method, Leaver’s continued fraction method, and the WKB approximation. The calculations covered the fundamental mode and overtones up to $\rho$9. The principal qualitative result is a sign-sensitive contrast between co-rotating and counter-rotating sectors. For positive winding number $\phi$0, the real part of the frequency increases sensitively and monotonically with $\phi$1, approaching

$\phi$2

for large $\phi$3, while the imaginary part becomes more negative and then saturates. For negative $\phi$4, $\phi$5 decreases with increasing $\phi$6, and the imaginary part becomes less negative, potentially approaching zero for high $\phi$7, which suggests quasi-resonances. Increasing the overtone number decreases $\phi$8 modestly and makes $\phi$9 more negative, so higher overtones decay faster (Liu et al., 2024).

The quasinormal-mode program is significant because it provides a genuine analogue of black-hole ringdown spectroscopy in an experimentally tunable optical system. A plausible implication is that the sign of $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$0 is not merely a kinematic label but an experimentally relevant discriminator between rapidly damped and long-lived analogue excitations.

5. Experimental realizations and measured horizon structure

The experimental history of photon-fluid rotation black holes proceeds from the realization of a $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$1-dimensional rotating acoustic geometry to BTZ analogues with measured inner horizons. The first direct experimental evidence of an ergosphere and a spatially separated event horizon in any system was obtained in a room-temperature photon superfluid formed by a continuous-wave laser beam propagating through a dilute methanol/graphene solution. The phase mask imposed

$\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$2

producing quantized azimuthal flow and inward radial flow. By scanning a small iris of $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$3 across the beam and analyzing the far-field momentum distribution, the experiment reconstructed the local flow velocity and sound speed and thereby identified the horizon and ergosphere directly from the crossing conditions (Vocke et al., 2017).

The BTZ line of work first established a numerically realizable optical analogue with the improved phase profile

$\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$4

arguing that the double-horizon structure can persist for a long distance, which is convenient for observing analogue Hawking or Penrose radiations. The proposal also emphasized that standard optical vortex beams do not generate an inner horizon, whereas the added logarithmic and quadratic radial phase terms do (Chen et al., 2023).

An improved photon-fluid experiment subsequently implemented this BTZ idea in a graphene/methanol thermal optical solution. The setup used a continuous-wave laser with $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$5 nm, up to $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$6 mW, a super-Gaussian profile, a spatial light modulator, a $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$7-cm glass tube, a $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$8 imaging system, a $\mathbf{v}=\frac{c}{kn_0}\nabla_\perp \phi .$9m pinhole mounted on a $\psi_1$0D translation stage, a Fourier lens of $\psi_1$1 mm, and a CCD camera. The photon-fluid velocity components were measured through Fourier-plane light-spot localization, while the phonon velocity was inferred from the optical vortex intensity distribution. The reported measured radii were

$\psi_1$2

consistent with

$\psi_1$3

This provided an experimental route to a fuller causal analogue of a rotating black hole, including the inner horizon (Wu et al., 25 Apr 2025).

Platform	Characteristic result	Source
2D photon superfluid	Ergosphere and separated horizon directly identified	(Vocke et al., 2017)
Optical rotating BTZ proposal	Inner horizon included by improved optical phase	(Chen et al., 2023)
Improved BTZ photon fluid	Measured $\psi_1$4, $\psi_1$5, and $\psi_1$6	(Wu et al., 25 Apr 2025)

These experiments establish that photon-fluid rotation black holes are not purely formal analogies. They are measurable optical systems in which both the background flow and the causal structure can be reconstructed in situ.

6. Relation to gravitational black holes and observational analogies

The photon-fluid literature is closely aligned with studies of photon dynamics in actual rotating black-hole spacetimes, but the two domains must be distinguished. In the analogue system, the relevant excitations are phonons or scalar fluctuations in a nonlinear medium. In astrophysical settings, the relevant objects are null geodesics or photons propagating through vacuum or plasma around a gravitational black hole. The similarity lies in the governing wave or Hamilton-Jacobi structure, not in ontological identity.

One related direction treats photons in plasma as effective massive particles. In the rotating Einstein-Maxwell-Dilaton black-hole analysis, the refractive index is

$\psi_1$7

so photon propagation requires

$\psi_1$8

The same study applies the Banados-Silk-West mechanism to photon collisions by treating photons as massive particles in a dispersive medium and finds that the plasma parameter has a relatively weak and unchanged effect on the center of mass energy across all cases (Raza et al., 18 Mar 2025). This is conceptually adjacent to photon-fluid analogue gravity because both rely on an effective medium, but it is a distinct construction.

Astrophysical shadow calculations supply another point of contact. Rotating black holes surrounded by perfect fluid dark matter and immersed in plasma exhibit refractive-index-modified null geodesics and shadow observables written in celestial coordinates $\psi_1$9, with the shadow size decreasing as plasma density increases in the cited model (Das et al., 2021). In Kerr-MOG, spherical photon orbits are governed by a sixth-order polynomial and remain radially unstable, with the deformation parameter $$\Box \psi_1 = \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g} \, g^{\mu\nu} \partial_\nu \psi_1 \right) = 0 ,$$0 shrinking the capture cross-section and altering the photon ring structure (Li et al., 2024). In Kerr itself, the solid angle of the shadow determines the trapped-photon ratio,

$$\Box \psi_1 = \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g} \, g^{\mu\nu} \partial_\nu \psi_1 \right) = 0 ,$$1

and rapid rotation produces strong anisotropy in the local radiation field through frame dragging (Takahashi et al., 2010).

A recurrent misconception is that black-hole-like imaging features are sufficient to demonstrate a true event horizon. That inference is not generally valid. For rotating regular no-horizon spacetimes, it has been shown that closed photon rings can exist without a horizon, and the mere existence of a closed photon ring does not prove that the compact object is necessarily a black hole (Kumar et al., 2020). By analogy, a photon-fluid rotation black hole should be interpreted as an experimentally controlled realization of black-hole kinematics—horizons, ergoregions, superradiance, and ringdown for collective modes—not as a reproduction of full gravitational dynamics.

7. Scientific significance and present boundaries

Within analogue gravity, photon-fluid rotation black holes provide a rare platform in which the metric, horizon structure, angular velocity, and perturbation spectrum are all experimentally adjustable. The platform has already been used to access Kerr-like draining vortices, BTZ-like double-horizon geometries, stationary scalar clouds, superradiant instabilities, quasibound states, Hawking-like emission, greybody factors, and quasinormal modes (Ciszak et al., 2021, Senjaya et al., 3 Sep 2025, Liu et al., 2024). The combination of exact mode analysis, WKB treatments, and direct optical measurements makes the subject unusually well integrated across theory and experiment.

The current boundary of the field is set by dimensionality, effective-medium assumptions, and the specific dispersion relations of nonlinear optics. Most realizations are $$\Box \psi_1 = \frac{1}{\sqrt{-g}}\partial_\mu \left( \sqrt{-g} \, g^{\mu\nu} \partial_\nu \psi_1 \right) = 0 ,$$2-dimensional, and the “black-hole” interpretation applies to the fluctuation sector rather than to the background optical field itself. Even so, the inclusion of a BTZ inner horizon in improved photon-fluid experiments materially expands the analogue program, because it permits laboratory access to causal structures that are closer to those of rotating relativistic black holes than earlier single-horizon optical analogues (Wu et al., 25 Apr 2025).

Taken together, the arXiv literature defines the photon-fluid rotation black hole as a technically mature analogue-gravity system: a nonlinear-optical vortex flow whose excitations obey a Klein-Gordon-type dynamics on an emergent rotating spacetime, and whose measurable signatures reproduce, in controlled laboratory form, many of the central spectral and kinematic phenomena usually associated with rotating black holes.