Analog Hawking Temperatures

Updated 25 July 2025

Analog Hawking temperatures are defined as effective temperatures quantifying horizon-induced quantum emission in systems with engineered analog event horizons.
Exact solutions using fourth-order Fuchsian ODEs and generalized hypergeometric functions yield scattering coefficients that precisely determine the analog Hawking temperature.
Combined perturbative and nonperturbative methods demonstrate that mode mixing induces a thermal spectrum, offering a universal framework for lab-based studies of quantum horizon effects.

Analog Hawking temperatures quantify the effective temperature associated with particle or wave emission—mimicking black hole Hawking radiation—in laboratory or theoretical systems that possess analog event horizons. These settings exploit the mathematical and physical analogy between quantum field theory in curved spacetime and the propagation of perturbations in dispersive or inhomogeneous media, enabling controlled investigations of horizon-induced thermality and quantum effects outside astrophysics.

1. Mathematical Foundations of Analog Hawking Temperatures

The foundational principle underlying analog Hawking temperatures is the kinematic equivalence between wave propagation in a curved spacetime and in certain engineered media. For example, shallow water surface waves on moving water can be mapped onto fields in a curved geometry by comparing their dispersion relations and identifying an effective "horizon" where the background flow speed matches wave characteristics (1008.1911). The dispersion relation for surface waves in moving water,

$(f + v(x)k)^2 = \frac{gk}{2\pi} \tanh(2\pi kh)$

mirrors the role of gravitational redshift near a black hole. The analog surface gravity $g_H$ at the horizon sets a Boltzmann-like ratio of converted mode amplitudes: $\frac{|\beta_f|^2}{|\alpha_f|^2} = \exp\left(-\frac{4\pi^2 f}{g_H}\right)$ whose exponent defines the analog Hawking temperature.

Similar correspondences are found in Bose–Einstein condensates (BECs), optical fibers, dielectric media, and plasma systems, where the local flow or refractive index profile creates an effective horizon and determines the analog "surface gravity" (e.g., via the spatial derivatives of flow velocity and sound speed or refractive index) (Linder et al., 2015, Boiron et al., 2014, Moreno-Ruiz et al., 2019, Nova et al., 2018, Bera et al., 2020, Fiedler et al., 2021). Often, the analog Hawking temperature $T_H$ is obtained through a semiclassical analysis of mode mixing induced by the background inhomogeneity: $k_B T_H \propto \hbar \left.\frac{d(v + c)}{dx}\right|_{\rm horizon}$ where $v$ is the flow velocity, $c$ the characteristic signal speed, and the derivative is evaluated at the analog horizon.

2. Exact and Perturbative Solution Methodologies

Analytical treatments of analog Hawking radiation often require solving wave equations incorporating high-order dispersion in nonuniform backgrounds. Recent developments employ exact techniques based on the theory of Fuchsian ordinary differential equations (ODEs). For a realistic monotonic refractive index profile in a dispersive dielectric, the linearized field equation transforms into a fourth-order Fuchsian ODE with three regular singular points (Trevisan et al., 6 Jun 2024). Its solution,

$f(z) = z^{\cdots} \; {_4F_3}(\alpha_1, \alpha_2, \alpha_3, \alpha_4;\beta_1, \beta_2, \beta_3; z)$

gives a global, analytic description of the scattering dynamics. The explicit $\alpha_i, \beta_j$ parameters are set by the dispersion relation roots in the left and right asymptotic regions.

An essential feature is the derivation of connection formulas—relations between local solutions near different singularities—that yield the exact transmission and reflection coefficients governing the conversion between positive- and negative-norm modes. The mode mixing responsible for analog Hawking radiation is then rigorously captured by these connection coefficients, enabling a precise, nonperturbative computation of the emission spectrum and the extraction of the analog Hawking temperature: $T_H = \frac{\omega}{\log(|P|/|N|)}$ where $|P|, |N|$ are the relevant scattering coefficients.

Complementary to this is the perturbative approach, which expands the solution in a small parameter characterizing the background inhomogeneity amplitude (e.g., for a soliton-like profile in a dielectric, the parameter $\eta$ ). This allows analytical control even in situations without a true event horizon (subcritical regime) and leads to an "effective temperature" governing the emission spectrum. In near-critical limits, this temperature converges to the standard analog Hawking temperature (Trevisan et al., 2023).

3. Thermality, Spectra, and the Stokes Phenomenon

A central issue in analog models is the emergence—and degree—of spectral thermality akin to true Hawking radiation. Exact solutions via generalized hypergeometric functions reveal that, under suitable conditions (such as in the transcritical regime where a real horizon forms), the ratio of scattering coefficients approaches an exponential law at low frequencies,

$\log \left(\frac{|P|}{|N|}\right) \sim \beta \omega$

implying a thermal spectrum characterized by $T_H$ (Trevisan et al., 6 Jun 2024).

Furthermore, the mathematical paper of the Stokes phenomenon—how different exponential asymptotics switch dominance as one traverses sectors in the complex plane—provides a rigorous account of mode conversion as wavepackets cross the horizon and explains the generation of analog Hawking quanta. The sectors and associated Stokes lines align with the physical process by which incident modes pick up outgoing and amplified partners, establishing the foundation for horizon-induced thermality.

The analysis also distinguishes between subcritical (no true horizon) and transcritical regimes. In subcritical cases, spontaneous pair creation still occurs and may display nearly thermal features, but the effective temperature is reduced compared to the critical value. In the transcritical regime, the extracted temperature from the exact connection coefficients agrees with previous semiclassical or Orr–Sommerfeld approximations.

4. Universality and Robustness Across Physical Platforms

Exact and perturbative results in dispersive dielectrics mirror, and generalize, findings in other analog settings. Water channel experiments directly measure the exponential relation of outgoing mode amplitudes as a function of frequency, confirming a Planck-like law and demonstrating the independence of thermal emission from the details of high-frequency ("ultraviolet") physics (1008.1911). In BECs and nonlinear optics, controlled manipulation of background inhomogeneity (e.g., soliton amplitudes or velocity gradients) tunes the effective temperature, and the spectra reveal both thermal and nonthermal features depending on the degree of reflection, resonance, or dispersion (1103.2994, Linder et al., 2015, Boiron et al., 2014).

Experimental results further support the universality of analog Hawking temperatures. In regimes with sufficiently gradual changes in the effective metric or refractive index, the analog temperature matches predictions from surface gravity or gradient computations. In contrast, when dispersion or resonant structures dominate, nonthermal features and resonant peaks appear, requiring the full machinery of scattering theory for interpretation.

5. Physical Implications and Experimental Significance

The existence of exact solutions and the underlying Fuchsian ODE structure enables researchers to predict quantitatively the particle (or photon) emission spectrum, the degree of deviation from pure thermality, and the sensitivity of the temperature to background profile details (Trevisan et al., 6 Jun 2024). This framework permits rigorous benchmarking of analog models against Hawking's original predictions and informs the design of laboratory experiments.

A notable implication is the agreement between the analog temperature obtained from exact mathematical connection formulae and from physical considerations (such as the surface gravity analogy), reinforcing the principle that horizon-induced quantum emission is a kinematics-driven effect, governed fundamentally by local wave propagation in an inhomogeneous medium.

Moreover, the mathematical technique elucidates the mechanism of mode mixing responsible for universality of Hawking temperatures and provides the means to distinguish between genuinely thermal emission and artifacts of background geometry or dispersion.

6. Open Directions and Theoretical Extensions

Future research aims to generalize these results to more complex or higher-dimensional analog models, broaden the class of refractive index or flow profiles amenable to exact solution, and rigorously incorporate band-limited or lossy media. The interplay between local and global monodromy data, the role of the Stokes phenomenon, and the classification of spectral and non-spectral sectors in higher-order Fuchsian ODEs will deepen understanding of analog horizon physics.

Additionally, connecting exact results for analog models to unresolved questions in gravitational Hawking radiation—such as the robustness of thermality in extreme dispersion, the fate of information, and the nature of backreaction—remains a significant objective. The established mathematical machinery, leveraging generalized hypergeometric functions and rigid ODE theory, provides a solid foundation for these explorations.

Table: Key Features of the Exact Solution Framework for Analog Hawking Temperatures in Dielectric Media

Feature	Description	Reference
Governing Equation	Fourth-order Fuchsian ODE with three regular singularities for a monotonic index profile	(Trevisan et al., 6 Jun 2024)
General Solution	Expressed via $_4F_3(\alpha_1,\dots;\beta_1,\dots; z)$ , parameters set by dispersion relation roots	(Trevisan et al., 6 Jun 2024)
Scattering Coefficients	Connection formulas (via Gamma functions) yield the transition amplitudes between incoming and outgoing mode branches	(Trevisan et al., 6 Jun 2024)
Thermality Criterion	Thermal spectrum emerges when $\log(\|P\|/\|N\|) \sim \beta\omega$ , with $T_H = \omega/\log(\|P\|/\|N\|)$ at low frequencies	(Trevisan et al., 6 Jun 2024)
Regimes	Subcritical (no horizon, suppressed temperature), transcritical (with horizon, standard $T_H$ recovered)	(Trevisan et al., 6 Jun 2024)
Role of Stokes Phenomenon	Explains asymptotic mode conversion and connects mathematical sector jumps to physical emission processes	(Trevisan et al., 6 Jun 2024)

In conclusion, analog Hawking temperatures, computed via rigorous mathematical solutions such as generalized hypergeometric functions and their connection formulas, provide a universal, model-independent framework for quantifying horizon-induced quantum emission in a wide array of physical systems. These exact methods unify the understanding of thermality, spectral features, and the emergence of Hawking-like effects, regardless of the physical origin of the analog horizon.