Massive Scalar Perturbations and Quasi-Resonance of Rotating Black Hole in Analog Gravity

Black holes, predicted as fundamental “particles” within the framework of general relativity, play the role as the crucial objects for understanding the evolution of the universe and probing the theories of gravity Yagi:2016jml . With the detection of gravitational waves produced by binary black hole mergers via ground-based gravitational wave detectors LIGOScientific:2016lio ; LIGOScientific:2016vlm ; LIGOScientific:2018mvr , gravitational wave astronomy has emerged as a significant tool for systematically investigating black hole properties and near-horizon physics Barausse:2016eii ; Barack:2018yly ; Baibhav:2019rsa . However, space-based gravitational wave detectors like LISA LISA:2017pwj , Taiji Hu:2017mde ; Gong:2021gvw and TianQin Gong:2021gvw ; TianQin:2015yph , which are capable of detecting low-frequency gravitational waves, will not be operational for several years. Meanwhile, the analysis of gravitational wave data still faces challenges such as immense computational demands and the need for a vast template database Christensen:2022bxb ; Thrane:2018qnx . On the other hand, we also face huge challenges of observing some intriguing features of black holes in astrophysical environments, such as Hawking radiation, which is rather difficult to be observed, and such a difficulty prevents us from examining the validity of quantum field theory in curved spacetime. These obstacles delay the fulfillment of the growing demand for exploring black holes in the near term.

Due to the challenges in the detections of black holes and their relevant important characteristics arising from fundamental physics, it is fantastic if we can study black hole physics in laboratory by contemporary technology via the language of analogy, i.e. analog gravity. The theoretical side of this idea had been realized by Unruh in 1981, whose pioneering work Unruh:1980cg laid the basis of the analog gravity research over the past decades. Since then, people have made significant strides in this field. The analog studies of properties associated with astrophysical black holes have been extensively discussed, including mechanisms such as black hole superradiance Basak:2002aw ; Richartz:2009mi ; Anacleto:2011tr ; Patrick:2020baa , and ongoing discussions on black hole thermodynamics Zhang:2016pqx , particularly topics like black hole entropy Zhao:2012zz ; Anacleto:2014apa ; Anacleto:2015awa ; Anacleto:2016qll . In recent years, relativistic acoustic black holes have also been constructed in Minkowski spacetime using the Abel mechanism Ge:2010wx ; Anacleto:2010cr ; Anacleto:2011bv ; Anacleto:2013esa and other methods Bilic:1999sq ; Fagnocchi:2010sn ; Visser:2010xv ; Giacomelli:2017eze . Furthermore, researchers have explored acoustic black holes in curved spacetimes, including Schwarzschild spacetime Guo:2020blq ; Qiao:2021trw ; Vieira:2021ozg ; Ditta:2023lny and Reissner-Nordström (RN) spacetime Ling:2021vgk ; Molla:2023hou , as detailed in Ge:2019our . On another front, the first acoustic black hole was successfully created in a rubidium Bose-Einstein condensate as early as 2009 Lahav:2009wx . Following this, several key experiments MunozdeNova:2018fxv ; Isoard:2019buh reported observations of thermal Hawking radiation in analog black holes and corresponding temperature measurements. For the latest research on analog Hawking radiation, see Anacleto:2019rfn ; Balbinot:2019mei ; eskin2021hawking . A series of foundational works PhysRevA.70.063615 ; PhysRevA.69.033602 ; PhysRevLett.91.240407 laid the groundwork for studying analog gravity in ultracold quantum gases, which has seen further progress in recent years Tian:2020bze ; PhysRevA.106.053319 ; PhysRevD.105.124066 ; PhysRevD.107.L121502 . Additionally, an exciting recent breakthrough Svancara:2023yrf reported the observation of rotating curved spacetime characteristics within giant quantum vortices, further fueling interest in the study of analog gravity.

In modern black hole physics, quasinormal modes (QNMs) serve as a vital tool for probing information of black holes Berti:2009kk ; Konoplya:2011qq . Linear perturbations around a perturbed black hole typically manifest as damped oscillations, where the real part of the QNM corresponds to the oscillation frequency, and the imaginary part determines the lifetime of the perturbation. During the ringdown phase of binary black hole mergers Ma:2022wpv ; Ghosh:2021mrv and in extreme mass-ratio inspiral gravitational wave systems Thornburg:2019ukt ; Nasipak:2019hxh , the QNM spectrum carries vital information about the black hole. Notably, QNMs are independent of the initial conditions of the perturbation, which makes them essential in gravitational wave observations and the study of astrophysical black holes Franchini:2023eda . Due to the importance of QNMs, it is natural and significant to study QNMs in analog gravity. Actually, QNMs have garnered significant attention in the study of analog gravity models. The theoretical studies on quasinormal modes (QNMs) can be traced back to early works on 2+1 and 3+1 dimensional acoustic black hole models Visser:1997ux ; Berti:2004ju ; Cardoso:2004fi , and recent advances in this area have also garnered widespread attention Daghigh:2014mwa ; Hegde:2018xub ; Patrick:2020yyy ; Liu:2024vde . In 2018, Torres demonstrated that vortices in the scattering of water waves by rotating analog black holes could be fully characterized by QNMs Torres:2017vaz . This finding was further supported by experimental validation, where the QNMs of a 2+1 dimensional analog rotating black hole was studied Torres:2020tzs . Additionally, the excitation of QNMs in the Hawking radiation spectrum of analog black holes has been explored within quantum fluid analog models Jacquet:2021scv , highlighting the broader relevance of QNMs in both classical and quantum analog systems. The study of QNMs in analog black holes allows for mutual validation through both theoretical and experimental approaches, thereby establishing a key research method for simulating and analyzing astrophysical black holes within the framework of analog gravity using laboratory platforms.

Recently, it was realized that rotating acoustic black holes can be generated within a self-defocusing optical cavity based on the fact that the fluid dynamics are also applicable to nonlinear optics PhysRevA.78.063804 , thereby establishing a new platform which is called photon-fluid for analog black hole research. The spin of the current analog black hole can be achieved by introducing a driving light beam with a vortex profile into the cavity. Ever since the seminal work of PhysRevA.78.063804 , this new analog gravity system has attracted much attentions in community, including research works on acoustic superradiance and superradiant instability PhysRevA.80.065802 ; Ciszak:2021xlw , experimental construction of this analog rotating black hole Vocke2018 , measurement of superradiance in laboratory Braidotti:2021nhw , and the potential applications of the fluids system in the analog simulations of quantum gravity Braunstein:2023jpo . In addition, a recent paper Krauss:2024vst provided exciting insights about employing photon-fluid system to help resolve long-standing problems related to quantum gravity, including the black hole information-loss paradox and the removal of spacetime singularities.

These advancements have demonstrated that the photon-fluid system is a versatile and promising platform with significant potential to enhance our understanding of gravity through analogy. In this paper, we focus on investigating the massive QNMs of an analog rotating black hole under the massive scalar field perturbations in a two-dimensional photon-fluid model. As a natural extension of the recent work on the massless QNMs in photon-fluid Liu:2024vde , we examine the properties of fundamental QNM spectra, calculated using the Continued Fraction Method and WKB approximation. Additionally, we explore the possibility of quasi-resonance within this analog gravity model, noting its potential significance. The presence of quasi-resonance is particularly important for studying QNMs in analog gravity experiments, as its slow damping rate and extended duration provide a valuable opportunity for detailed investigation in a laboratory setting. With the experimental advancements mentioned above, we are optimistic that the QNMs we have investigated here will soon be observable in apparatus of this photon-fluid model by which provides a novel testbed to black hole physics and theories of gravitation.

This work is organized as follows. In Sec. II, we derive the master equation of the fluctuation field. In Sec. III, we introduce the numerical methods employed in our calculations of QNF. In Sec. IV, the numerical results of QNF are demonstrated and analyzed. In Sec. V, we investigate the existence of quasi-resonance. The Sec. VI, as the last section, is devoted to conclusions and discussions.

In this section, we briefly illustrate the basic parts of this photon-fluid system. The dynamics of the photon-fluid in our consideration is governed by the nonlinear Schrödinger equation (NSE) describing the slowly varying envelope of the optical field

where $z$ is the coordinate of propagation direction, $|E|^{2}$ stands for optical field intensity and $\nabla^{2}E$ is defined by the transverse coordinates $(x,y)$, $k=2\pi n_{0}/\lambda$ is the wave number, $\lambda$ is the laser wavelength in vacuum, $n_{0}$ is the linear refractive index, $n$ is the intensity dependent refractive index defined by $n=n_{0}-\Delta n=n_{0}-n_{2}|E|^{2}$ where $n_{2}>0$ is the material local nonlinear coefficient. Since the dynamics of this system lives on the transverse plane $(x,y)$, the propagation coordinate $z$ acts as effective time by relation $t=\frac{n_{0}}{c}z$. In addition to the local nonlinearity, a non-local thermo-optical nonlinearities can be introduced to some medium such that Marino:2019flp ; Ciszak:2021xlw

Based on NSE, the following effective metric of an analog rotating black hole can be obtained PhysRevA.78.063804 ; PhysRevA.80.065802 ; Ciszak:2021xlw

where $r_{H}$ is the event horizon, $\Omega_{H}$ represents the angular velocity of the black hole, $\xi$ describes the healing length, $j$ is an integer representing the topological charge of the optical vortices, and they are given by

In this analog black hole back ground, the propagation of density perturbation $\rho_{1}$ of optical field is governed by a massive Klein-Gordon equation Ciszak:2021xlw

where the mass term $\mu^{2}$ comes from the introduction of the non-local thermo-optical nonlinearities $n_{th}$ introduced by Eq. (2), and $g$ is the determinant of the effective metric. With the massive Klein-Gordon equation, we can obtain the master equation of the perturbation field

where we have set $r_{H}=1$ which means that $r$ is measured in units of $r_{H}$, and both $\omega$ and $\Omega_{H}$ are measured in units of $r_{H}^{-1}$. The derivation of this equation can be found in Appendix A. Then for our asymptotically flat analog black hole spacetime, the QNF, i.e. $\omega$, can be determined by solving Eq. (6) associated with the following boundary conditions

which means that the perturbation field is ingoing at the event horizon and outgoing at infinity.

In this section, we would like to introduce two different numerical methods for calculating QNF of this analog black hole. As a standard procedure, the QNF is usually calculated by one method and then confirmed by another one, since this process can not only enhance the reliability of our numerical results but also examine the validity of the numerical methods.

In this section we demonstrate and analyze the properties of the numerical results of QNF of massive scalar perturbation. To obtain the QNF, we adopt previously introduced CFM and WKB method. In the cases of winding number $m>0$, both CFM and WKB are used and they give consistent results, while in the $m<0$ case we only use CFM due to the poor performance of WKB for massive QNF.

In Table 1 we show the numerical results of the fundamental QNF at winding number $m=1$ for different values of mass $\mu$ and angular velocity $\Omega_{H}$. The data displayed in this table are obtained by CFM and WKB whose results are in good agreement with each other, which confirms the validity of the numerical methods we adopted. For the QNF’s real part $\omega_{R}$ which stands for the oscillation frequency of QNMs, it can be observed that $\omega_{R}$ will monotonically increase with the grow of $\Omega_{H}$, and this behavior holds for all the mass values considered in this table. On the hand, one can find that for any given $\Omega_{H}$, the $\omega_{R}$ can be also improved by increasing the value of mass $\mu$. These results suggest that QNMs with higher oscillation frequency can be observed either by choosing a background driving optical field whose profile can induce faster rotating analog black hole, or choosing the medium of which non-local thermo-optical nonlinearities can bring us bigger effective mass of perturbation field. For the QNF’s imaginary part $\omega_{I}$ which is negative and reflects the damping rate of QNMs, we find that its magnitude becomes larger when increasing $\Omega_{H}$, and this property holds for all scalar perturbations with different mass in our current consideration. This result indicates that QNMs will have a shorter life in this analog rotating black hole spacetime with higher angular velocity $\Omega_{H}$, since a negative $\omega_{I}$ with larger magnitude clearly suggests a faster damping rate of QNMs. In contrast, it is found that the magnitude of $\omega_{I}$ can be decreased by increasing the mass value $\mu$ of the perturbation field, such that for a given $\Omega_{H}$ the QNMs can live longer if it has a bigger effective mass. Furthermore, the behavior of $\omega_{R}$ and $\omega_{I}$ under the change of $\Omega_{H}$ manifests a remarkable contrast, as $\omega_{R}$ monotonically grows with the increment of $\Omega_{H}$ and seems to be unbounded, while for $\omega_{I}$ whose magnitude can be also monotonically improved by higher $\Omega_{H}$ but all $\omega_{I}$ seems to approach a constant around $\omega_{I}\approx-0.25$ when $\Omega_{H}$ is large enough. Additionally, the differences between QNF, both real and imaginary part, induced by different mass values, will be suppressed by sufficient large $\Omega_{H}$, as a natural consequence of the effects of $\mu$ on QNF can be ignored when compared with the effects of large $\Omega_{H}$.

The dependence of real part and imaginary part of QNF given in Table 1 on the angular velocity are separately plotted in Fig. 1 in order to have a more perspicuous demonstration of the features of QNF we have discussed above. In the following Table 2 and 3 we respectively illustrate the QNF for $m=5$ and $m=10$, and the corresponding figures are shown in Fig. 2 from which we find that the QNF with higher positive winding number $m$ behaves qualitatively the same as the $m=1$ case, especially all the $\omega_{I}$ for the $m$ ranges from $m=1$ to $m=10$ approaches $\omega_{I}\approx-0.25$ when $\Omega_{H}$ is increasing to large value.

Based on the $\omega_{R}$ in Fig. 1 and 2 are shown by log-log plots, the slope corresponding to any given mass value $\mu$ at high $\Omega_{H}$ can be determined by linear regression

which is dependent neither on winding number nor perturbation field mass, and directly leads to

By taking $\Omega_{H}=1$, the $m$ dependent constant $C_{m}$ is determined to be $C_{m}\approx m$, such that when $\Omega_{H}\to+\infty$ we have

Now we focus our attention on the QNF for negative winding number $m<0$. In the negative $m$ case, our numerical methods are not capable of dealing with large $\Omega_{H}$, which is in contrast to $m>0$ case where the numerical methods work well for large $\Omega_{H}$. On the other hand, the capability of WKB method to calculate precision massive QNF in this case is depressed. As a consequence, we can only rely on CFM to calculate QNF for $\Omega_{H}$ which is not large.

In Table 4, we present QNF data for $m=-5$ under different mass and angular velocity values. One can observe that the QNF in current case shows a remarkably contrasting features compared to positive $m$ case, as both $\omega_{R}$ and the magnitude of $\omega_{I}$ decreases with the grow of $\Omega_{H}$. However, the effects of perturbation field mass values $\mu$ on QNF are remained qualitatively identical to $m>0$ case, as when increasing $\mu$ the $\omega_{R}$ will become bigger while $\omega_{I}$ will be smaller. Hence we may conclude that the characteristics of the effects of $\Omega_{H}$ on QNF depend on the sign of winding number which will not change the way of how mass $\mu$ affects QNF. The data of QNF for $m=-10$ are listed in Table 5 from which a qualitatively identical behaviors of QNF can be found. This result may suggest that the fundamental QNF under all the negative $m$ has the same properties. The corresponding plots of QNF in current two cases are demonstrated in Fig. 3 which gives clear illustrations of features of QNF for negative $m$. With the help of the plots, we find that the differences between QNF induced by different mass values will increase when $\Omega_{H}$ grows, this is another behavior differs from the $m>0$ case.

As the last part of this section, we would like to compare the QNF among different wingding numbers in order to disclose how the QNF is affected by $m$. The comparisons are made in Fig. 4 where we plot the QNF among $m=-10$ to $m=10$ by separately fixing $\mu=0$ and $\mu=0.3$ in each plot. The plots of $\omega_{R}$ and $\omega_{I}$ are placed on the left side and right side of Fig. 4, respectively. For the $\omega_{R}$, it can be found that different mass values $\mu$ do not have noticeable influences on the behaviors of $\omega_{R}$. Regarding the influences of the winding number on $\omega_{R}$, it is clearly to see that among the positive $m$, the QNF with a larger winding number always has a higher $\omega_{R}$. When it comes to negative $m$, the larger $\omega_{R}$ comes from the modes with higher magnitude $|m|$. The plots for $\omega_{I}$ under the two different mass values manifest obvious unique features. In the $\mu=0$ plot, the $\omega_{I}$ for all $m>0$ coincides and forms a single branch, while another branch is formed by $\omega_{I}$ for all wingding numbers $m<0$. This result means that the $\omega_{I}$ of fundamental QNF with mass $\mu=0$ depends on the sign of winding number whose magnitude can just mildly affect it. However, for the $\omega_{I}$ of massive perturbation field, as shown in our plot in which $\mu=0.3$, this kind of degeneracy is broken. When we change $\mu=0$ into $\mu=0.3$, the original two branches start to diverge. This divergency in $m<0$ branch happens and becomes more pronounced when increasing $\Omega_{H}$ to larger values. In contrast, in the $m>0$ branch, the divergence occurs at small $\Omega_{H}$ and it starts to disappear if we keep increasing $\Omega_{H}$ to large values.

In this section we discuss an interesting phenomenon uncovered in the spectrum of massive QNMs. In the calculation of QNF of massive perturbation field, people have found that the $\omega_{I}$ of QNF will increase and gradually approaches zero when we improve the mass $\mu$ to some value, while the $\omega_{R}$ remains to be nonvanishing. As a consequence, the QNMs can be arbitrarily long-lived due to the $\omega_{I}$ which represents damping rate of QNMs can be sufficiently small. This phenomenon is called quasi-resonance which was first observed in Ohashi:2004wr where the authors studied the massive scalar field perturbation in the Reissner-Nordström black hole spacetime. Subsequently, it was realized in Konoplya:2004wg that quasi-resonance can exist only if the effective potential is nonzero at infinity. Note that the effective potential in the master equation of the massive scalar perturbation in present work, therefore we would like to learn whether the quasi-resonance can exist in our analog rotating black hole system. To this end, we need to observe how the QNF behaves when we change the value of perturbation field mass.

In Fig. 5 we demonstrate the dependence of real and imaginary part of fundamental QNF for $m=\pm 5$ at $\Omega_{H}=1$ on the perturbation field mass $\mu$. We should point out that we calculated QNF for $m=-5$ in a small range of $\mu$ compared to $m=5$, since our numerical method can yield convergent results only for comparatively small $\mu$ in the context of negative winding number. For both positive and negative $m$, this figure shows that when increasing $\mu$, the $\omega_{R}$ will grow while the magnitude of $\omega_{I}$ decreases meanwhile with a trend to get close to zero. The zero-approaching tendency of $\omega_{I}$ suggests that a sufficient small damping rate can be achieved if we further increase $\mu$ to some certain value, but the numerical method start to work poorly which stops us here. In this sense, we may claim that the quasi-resonance can exist in present analog rotating black hole system. Another property reflected by the plots is that the QNF with negative wingding number seem to be more sensitive to the change of $\mu$, as the slope of the curves for $m=-5$ seems to be higher than that of $m=5$. On the other hand, we combine the two plots together in Fig. 5 to get the behaviors of QNF on the complex plane, as shown in Fig. 6 which gives a more clear and straightforward illustration of how QNF behaves under the variation of mass $\mu$.

In this paper, we have investigated the fundamental QNF of massive scalar field perturbation in the analog rotating black hole spacetime which is established by photon-fluid system. The QNF with opposite sign of winding number is found to behave rather differently, while the modes with the same sign of winding number have qualitatively similar properties. If winding number $m>0$, both the $\omega_{R}$ and the magnitude of $\omega_{I}$ which is negative increase with the $\omega_{H}$. When $\omega_{H}$ is large enough, the effects of the mass values of perturbation field on the QNF will be suppressed and all the QNF behaves identically, as the $\omega_{R}$ will asymptotically behave as $\omega_{R}\approx m\Omega_{H}$, and the $\omega_{I}$ will approach a constant $\omega_{I}\approx-0.25$. In winding number $m<0$ case, the increase of $\Omega_{H}$ will lead to the decrease of both $\omega_{R}$ and magnitude of $\omega_{I}$, which is contrary to $m>0$ case. The influences of winding number on QNF are also analyzed. We find that among the positive winding numbers, the QNMs with bigger $m$ always has a larger $\omega_{R}$, and this result is reversed among the QNMs with $m<0$. We also note that the perturbation field mass seems not to change the behaviors of $\omega_{R}$ among different winding numbers, while it has been found to have noticeable influences on $\omega_{I}$. In the $\mu=0$ case, we find a kind of degeneracy, which manifests as all the $\omega_{I}$ with the same sign of winding number almost coincide with each other. As a consequence, we can only find two branches of curves standing for positive and negative winding numbers, respectively. This phenomenon indicates that for the massless QNF, only the sign of winding number can influence $\omega_{I}$ which is almost independent of the magnitude of $m$. However, when the perturbation field mass $\mu$ becomes nonzero, we find that the degeneracy is broken, as the original single branch will diverge. Note that the divergence will be enhanced when increasing $\Omega_{H}$ for $m<0$ QNMs, and for $m>0$ it will be suppressed and eventually converge. Despite the QNMs with opposite sign of winding number usually have very contrasting characteristics, the perturbation field mass $\mu$ can influence the QNF with all different $m$ in the same way: it increases the $\omega_{R}$ and reduces the magnitude of $\omega_{I}$.

Besides the conventional properties of QNMs, we also studied the quasi-resonance which is an interesting feature of the massive QNMs. The quasi-resonance is a kind of QNF whose real part $\omega_{R}$ remains nonvanishing while the imaginary part $\omega_{I}$ can approach zero which suggests the existence of arbitrarily long-lived QNMs. We find that the QNF in complex plane will quickly move toward the real axis as we have shown in Fig. 6, which may indicate the existence of quasi-resonance in current analog rotating black hole system. It is worth pointing out that the quasi-resonance is favored in the study of black hole physics through analog gravity, since its long damping time will make it less hard for us to successfully detect QNMs in experiment. As an aside, note that this analog rotating black hole has been experimentally constructed in Vocke2018 , and the recent work in Solidoro:2024yxi has demonstrated that the QNMs can indeed be easily excited in finite-sized experimental settings of analog gravity. In light of these two developments, we hopefully expect that the successful detection of the QNMs in analog gravity model by photon-fluid system can be realized, and thus bringing us a more promising future of studying the physics in the black holes spacetime by analogy.