Hawking Radiation and Horizon-Induced Particle Emission

• Hawking Radiation is the quantum emission of particles from horizons, generated by Bogoliubov mode-mixing in stationary or quasi-stationary backgrounds.
• It produces thermal spectra with grey-body corrections due to dispersion effects, applicable to both astrophysical black holes and laboratory analogue systems.
• Laboratory experiments using optical fibers and photonic crystals have observed both stimulated and spontaneous emissions, unifying gravitational and analogue perspectives.

Hawking radiation is the quantum emission of particles from the vicinity of horizons—regions where modes are blocked by effective geometry. Initially proposed for astrophysical black holes, it has been rigorously generalized to horizons in analogue systems (optical fibers, fluids, polaritons, etc.). The phenomenon is fundamentally kinematic: quantum fields in a stationary or quasi-stationary background with an apparent horizon produce a flux of particles with a spectrum determined by the local surface gravity. These emissions are thermal in the dispersionless limit but acquire grey-body corrections with dispersion. Laboratory implementations now achieve effective Hawking temperatures orders of magnitude exceeding those of astrophysical black holes. Stimulated and spontaneous versions are both realized, and the concept has been reformulated in statistical-mechanical and stochastic frameworks, unifying entropy counting, microstate transfer, and emergent horizon-scale noise.

1. Quantum Field-Theoretic Derivation in Curved Spacetimes

The classical starting point is a massless scalar field $\phi$ in a background with a metric in Painlevé–Gullstrand–Lemaître coordinates: $$ds^2 = -[c^2 - V(x)^2]\,dt^2 - 2 V(x)\,dx\,dt + dx^2$$ The field action,

$$L = \frac{1}{2}\sqrt{-g}\,g^{\mu\nu}\,\partial_\mu\phi \,\partial_\nu\phi$$

gives the wave equation

$$(\partial_t + \partial_x V)(\partial_t + V\partial_x)\phi - c^2\partial_x^2\phi = 0$$

In regions away from the horizon, the stationary ansatz $\phi(x,t) = e^{-i\omega t}\varphi_\omega(x)$ leads to mode decomposition: $[\omega - k V(x)]^2 = c^2 k^2$ with right- and left-moving branches %%%%2%%%%.

At the horizon $V(x_H) = -c$, the outgoing mode coordinate $u\sim t - (1/\alpha)\ln|x|$ is logarithmically singular, with $\alpha$ the surface gravity. Quantum particle creation emerges via Bogoliubov transformation between in- and out- modes: $$\varphi^{\text{out,}R}_\omega = \sum_j [\alpha_j(\omega)\varphi^{\text{in,}L}_{\omega,j} + \beta_j(\omega)\varphi^{\text{in,}R}_{\omega,j}]$$ Particle expectation: $$\langle N_\omega\rangle = \frac{\delta(0)}{e^{2\pi\omega/\alpha}-1}$$ defining the Hawking temperature

$k_B T_H = \frac{\hbar\alpha}{2\pi}$

This kinematic mechanism underpins all genuine horizon-induced radiation (Aguero-Santacruz et al., 2020).

2. Analogue Gravity: Optical Fiber Horizons and Effective Metrics

In optical analogues, a moving refractive-index perturbation $\delta n$ induces an effective metric for light: $\partial_\mu(g^{\mu\nu}\,\partial_\nu A) = 0$ with $g^{\mu\nu}$, the inverse metric, written explicitly in the co-moving frame. For non-magnetic, dispersive fibers ($\mu=1$, $\epsilon=n^2(\omega)$), a horizon exists where the pump speed matches the local phase velocity: $V = -c/n(\omega)$ The optical analogue surface gravity,

$\alpha = \frac{1}{\delta n}\frac{d\delta n}{d\tau}$

determines the Hawking temperature as in gravity. Doppler-shifted probe frequencies obey generalized scattering relations, and quantum emission occurs when a mode is blocked by the moving index front (Aguero-Santacruz et al., 2020).

3. Stimulated and Spontaneous Emission in Photonic-Crystal Fibers

Experimental realizations employ a femtosecond pump to induce the moving horizon, with a probe split into positive Hawking radiation (PHR) and negative Hawking radiation (NHR) bands at phase-matched frequencies. The UPPE equation,

$$i\partial_z E_\omega + \beta(\omega)E_\omega + \frac{\omega}{2c n(\omega)}P_{NL,\omega} = 0$$

governs field dynamics; probe light is converted into PHR and NHR via nonlinear effects, consistent with emission rate scaling: $$\text{Emission} \propto |\beta|^2 \propto \frac{1}{e^{2\pi\Delta\omega/\alpha}-1}$$ Numerical split-step Fourier integration matches observed spectra within experimental precision; signals emerge promptly after horizon formation. Stimulated emission scales linearly with probe input, while spontaneous emission remains under investigation (Aguero-Santacruz et al., 2020).

4. Spectral Deviations and Non-Planckian Emission

Numerical solutions for highly dispersive fibers demonstrate deviations from a strict Planck spectrum. The computed Bogoliubov coefficients $|\beta(\omega)|^2$ fit

$$|\beta(\omega)|^2 \approx \left[\exp\left(2\pi\omega/\alpha_\text{eff}(\omega)\right)-1\right]^{-1}$$

with effective surface gravity $\alpha_\text{eff}(\omega)$ varying up to 20% across the emission spectrum. Emission lobes in the infrared and ultraviolet are seen. Simulations using the full frequency-dependent propagation reproduce experimental amplitude and position for PHR/NHR bands within 15%. Both classical (stimulated) and quantum (spontaneous) regimes reflect the essential kinematics of horizon-induced pair creation (Aguero-Santacruz et al., 2020).

5. Generalization, Unified Criteria, and Comparative Perspectives

Authors advocate a universal definition for Hawking radiation, dispensing with “analogue” qualifiers. Necessary and sufficient physical ingredients:

• An apparent horizon inducing mode blocking
• Nonzero surface gravity $\alpha$ (gradient at the horizon)
• Adiabatic slow evolution of the background

No direct reference to Einstein equations is required; the kinematic mechanism is independent of dynamics. In both gravitational and optical systems, thermality emerges in the non-dispersive limit $k_B T_H = \hbar\alpha/2\pi$, but gravitational horizons exhibit negligible dispersion while optical media require strong corrections. For laboratory fibers, $T_\text{eff}\sim 10$–$10^3$ K is achievable, orders above astrophysical $T\sim10^{-8}$ K.

The unified concept encompasses sonic, optical, water-wave, polariton, and other horizons; both quantum spontaneous and classical stimulated emissions fall under Hawking radiation. The framework bridges horizon kinematics, Bogoliubov mode-mixing, effective metrics, nonlinear photonic experiments, and the call for a terminology broad enough to include all horizon-induced quantum emission phenomena (Aguero-Santacruz et al., 2020).