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本文基于类比引力模型, 对声学黑洞的超辐射和霍金辐射现象进行了数值研究. 通过求解特定声学度规背景下标量场的径向方程, 计算了不同频率下的反射系数、透射系数和霍金辐射功率谱. 数值结果验证了超辐射的存在, 即当入射声波频率低于声学黑洞的特定频率阈值 $m \varOmega_{\rm{H}}$ 时, 反射波会被放大. 同时, 计算结果与能量守恒关系一致, 验证了数值方法的可靠性. 关于霍金辐射, 我们计算了其功率谱随频率的变化. 观察到在超辐射临界频率附近, 霍金辐射功率谱出现显著增强, 这主要是由于玻色－爱因斯坦统计分布函数在包含旋转效应的指数项趋于零时分母接近于零, 以及频率依赖的透射概率共同作用导致的. 随着频率进一步升高, 功率谱呈现出非线性变化, 这反映了频率因子、透射概率以及包含旋转效应的玻色-爱因斯坦统计分布的综合影响. 本文的研究为理解声学黑洞的量子效应提供了数值支持, 并为末来的理论和实验研究提供了参考.This paper conducts numerical studies on superradiance and Hawking radiation of a specific rotating acoustic black hole model characterized by parameters A and B, within the framework of analogue gravity. The standard radial wave equation for scalar perturbations in the effective metric of this model is solved numerically by using an adaptive Runge-Kutta method with tortoise coordinates; this approach necessitates careful numerical inversion of the coordinate transformation near the horizon via a root-finding algorithm. By imposing appropriate boundary conditions, we extract the reflection coefficient $\mathcal{R}$ and transmission coefficient $\mathcal{T}$ in a range of frequencies ω. Our results clearly demonstrate superradiance, with the reflectivity $|\mathcal{R}|^2$ exceeding unity for $\omega < m\varOmega_{\rm{H}} = 1$ (where $m=-1$ and $\varOmega_{\rm{H}}=-1$), which confirms energy extraction from the rotating background. The high accuracy of our method is validated by the flux conservation relation, $|\mathcal{R}|^2 + [(\omega - m\varOmega_{\rm{H}})/\omega]|\mathcal{T}|^2 = 1$, which typically has a numerical precision of $ 10^{-8}$. Furthermore, using the derived Hawking temperature and the rotation modified Bose-Einstein distribution, we calculate the Hawking radiation power spectrum $P_\omega$, and use the numerically obtained transmission coefficient $|\mathcal{T}|^2$ as the greybody factor of the model. A prominent feature of $P_\omega$ is its sharp enhancement (or divergence) as ω approaches the threshold $m\varOmega_{\rm{H}}$ from above, which is a characteristic directly related to the denominator of the Bose-Einstein factor. This research also reveals that superradiant amplification and Hawking spectrum characteristics are significantly dependent on the specific values of flow parameters A and B, even when the superradiant threshold $m\Omega_H$ is kept unchanged. This detailed numerical study provides quantitative results for the scattering and radiation properties of this model, and also for strong support for the analogue gravity framework.
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Keywords:
- acoustic black hole /
- superradiance /
- Hawking radiation /
- numerical calculation
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图 1 反射系数模$ |{\cal{R}}| $(实线)与透射系数模$ |{\cal{T}}| $(虚线)随入射声波频率ω的变化关系
Fig. 1. Modulus of reflection coefficient $ |{\cal{R}}| $ (solid line) and transmission coefficient $ |{\cal{T}}| $ (dashed line) as functions of incident wave frequency ω.
图 2 放大系数$ |{\cal{R}}|^{2}-1 $(黑色实线)随入射声波频率ω的变化. 该系数大于零的区域$ (0<\omega \lesssim 1.0) $代表超辐射存在的区域. 此图确定了超辐射发生的频率范围, 其上限对应理论阈值$ m \varOmega_{{\rm{H}}}=1 $
Fig. 2. Amplification factor $ |{\cal{R}}|^{2}-1 $ (black solid line) as a function of incident wave frequency ω. The region where this factor is greater than zero $ (0<\omega \lesssim 1.0) $ represents the existence of superradiance. This figure determines the frequency range for superradiance, with the upper limit corresponding to the theoretical threshold $ m \varOmega_{{\rm{H}}}=1 $.
图 3 霍金辐射功率谱$ P_{\omega} $(黑色实线)随频率ω的变化, 特别展示了超辐射临界频率($ m \varOmega_{{\rm{H}}}=1 $)附近的行为. 图中$ P_\omega $为采用自然单位制($ \hbar=k_{\rm{B}}=c=1 $)及特征长度$ r_0=1 $计算得到的无量纲功率谱密度, 代表以$ \hbar c / r_0 $为单位的能量
Fig. 3. Hawking radiation power spectrum $ P_{\omega} $ (black solid line) as a function of frequency ω, specifically showing the behavior near the superradiance critical frequency ($ m \varOmega_{{\rm{H}}}=1 $). The plotted $ P_\omega $ is the dimensionless power spectrum density calculated using natural units ($ \hbar= $$ k_{\rm{B}}=c=1 $) and characteristic length $ r_0=1 $, representing energy in units of $ \hbar c / r_0 $.
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