# Strong cosmic censorship under quasinormal modes of non-minimally coupled massive scalar field

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## Abstract

We investigate the strong cosmic censorship conjecture in lukewarm Reissner–Nordström–de Sitter black holes (and Martínez–Troncoso–Zanelli black holes) using the quasinormal resonance of non-minimally coupled massive scalar field. The strong cosmic censorship conjecture is closely related to the stability of the Cauchy horizon governed by the decay rate of the dominant quasinormal mode. Here, dominant modes are obtained in the limits of small and large mass black holes. Then, we connect the modes by using the WKB approximation. In our analysis, the strong cosmic censorship conjecture is valid except in the range of the small-mass limit, in which the dominant mode can be assumed to be that of the de Sitter spacetime. Particularly, the coupling constant and mass of the scalar field determine the decay rate in the small mass range. Therefore, the validity of the strong cosmic censorship conjecture depends on the characteristics of the scalar field.

## 1 Introduction

The inside of a black hole is covered by its event horizon, from which no light can escape. Hence, it is not possible to detect black holes by its own radiation, classically. However, quantum theory suggests that a black hole can emit a portion of the energy from the horizon. This radiation is called Hawking radiation [1, 2]. In consideration of Hawking radiation, a black hole can be treated as a thermodynamic system with the Hawking temperature, which is proportional to the surface gravity at the horizon. Moreover, the area of the black holes horizon is irreducible in an irreversible process [3, 4, 5]. Therefore, Bekenstein–Hawking entropy of a black hole is defined as being proportional to the area of its event horizon [6, 7]. Because properties of black holes are different from any other astronomical object in the universe, the existence of the black hole evokes curiosity. However, the recent detection of gravitational wave signals, which originated from collisions between black holes, by the laser interferometer gravitational-wave observatory (LIGO) has proven that black holes are in fact stable celestial bodies spread across the universe.

The center of a black hole is the location of a curvature singularity. Physically, a visible singularity causes the breakdown of causality and loss of predictability in the theory of gravity. Hence, to avoid this unpredictability, the singularity should be invisible to the observer. This is called the cosmic censorship conjecture [8, 9, 10]. According to a given observer, the cosmic censorship conjecture is divided into two types: weak conjecture and strong conjecture. On the one hand, the weak cosmic censorship (WCC) conjecture states that the singularity should be covered by an outer horizon for an asymptotic observer. Thus, the outer horizon needs to be stable under perturbation to satisfy the WCC conjecture. The first test on the WCC conjecture was performed on the Kerr black hole [11]. Here, adding a particle into the Kerr black hole cannot overspin it beyond the extremality. Since then, the WCC conjecture has been tested in various black holes. Moreover, the validity of this conjecture depends on the state of the black hole and the method of perturbation. For example, the horizon of the near-extremal Kerr black hole becomes unstable upon adding a particle [12], but it can be still be stable when considering self-force effects [13, 14, 15, 16, 17]. This test can be extended to the Reissner–Nordström black hole where a situation similar to that of the Kerr black hole arises, as discussed in [18, 19]. Further, as there is no general proof of the validity of the WCC conjecture, the test is now extended to various black holes by adding a particle [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Particularly, when the thermodynamic pressure and volume terms are considered for the electrically charged anti-de Sitter black hole, the WCC conjecture is proven to be valid under particle absorption [33]. For the test of the WCC conjecture, adding a particle can be generalized to the scattering of the test field [34, 35, 36, 37, 38, 39, 40, 41]. Under the scattering of a scalar field, the weak cosmic censorship conjecture is shown to be valid for the Kerr-(anti-)de Sitter black holes [42].

The strong cosmic censorship (SCC) conjecture proposes that the singularity is invisible to any observer, and hence must be a spacelike singularity. It is important to note that a timelike singularity appears in well-known solutions such as Reissner–Nordström and Kerr–Newman black holes, which often leads to the notion that these black holes are counterexamples to the SCC conjecture. However, this is not true because inside the outer horizon, the timelike singularity is enclosed by a Cauchy (inner) horizon at which an inward wave undergoes an infinite blueshift. Hence, when a wave enters the black hole, the infinitely blueshifted wave makes the Cauchy horizon unstable. As a result, the singularity becomes spacelike, making the SCC conjecture valid even in this case [43, 44, 45, 46, 47, 48]. However, the issues with the SCC conjecture becomes more complicated in the Reissner–Nordström–de Sitter (RNdS) black hole. For instance, in the de Sitter (dS) spacetime, the existence of a cosmological horizon causes the redshift of an influx into the Cauchy horizon, so the redshift competes with the blueshift from the Cauchy horizon. Then, the redshift originating from the cosmological horizon becomes dominant, stabilizing the Cauchy horizon [49]. However, there can be an additional influx to the Cauchy horizon. Furthermore, this influx is predominantly blueshifted at the Cauchy horizon, which can destabilize the Cauchy horizon [50]. Recently, quasinormal modes have been physically categorized into three families based on their behaviors in an RNdS black hole, and these behaviors play an important role in the validity of the SCC conjecture [51, 52]. Particularly, the stability of the Cauchy horizon depends significantly on the competition between the surface gravity on the Cauchy horizon and the decay rate of the quasinormal mode on the outer horizon. By the analysis of the quasinormal modes in RNdS black holes, ranges have been found over which the SCC conjecture is invalid [52]. Nevertheless, the SCC conjecture in the RNdS black hole is still actively studied in [53, 54, 55, 56, 57, 58, 59, 60, 61]. The historical review can be found in [62] (and references therein).

Here, we consider a dS black hole whose metric is a solution (of the same geometry) to two theories of gravity: Einstein’s gravity coupled with the Maxwell field, and gravity theory coupled with a conformal scalar field including a quartic self-interaction potential. The solution in these two theories is known by different names. The former is called the lukewarm RNdS black hole [63] and the latter Martínez–Troncoso–Zanelli (MTZ) black hole [64]. Further, dS black holes encounter an issue with the temperature. Since dS black holes have two horizons surrounding the timelike spacetime, two temperatures for the two horizons can be obtained. It should be noted that these two temperatures are not coincident, so the system is not balanced between the input and output radiations through the horizons. Hence, the systems are thermodynamically unstable. The lukewarm RNdS black hole resolves unbalanced radiations by setting the two temperatures at a coinciding value [63, 65, 66]. In the gravity theory coupled with a conformal scalar field including a quartic self-interaction potential, the geometry becomes that of the MTZ black hole, which is a four-dimensional dS black hole with a non-singular scalar hair outside the outer horizon. Further, the MTZ black hole can satisfy the strong energy condition [64]. However, during a perturbation, an instability can be observed in the MTZ black hole [67], which is consistent with the no-hair theorem. Thermodynamically, according to the effect of the scalar field, derived from the Euclidean action, the entropy of the MTZ black hole is given by a modified form [68].

In this work, we investigate the SCC conjecture in the lukewarm RNdS (or MTZ) black hole under the quasinormal modes of non-minimally coupled massive scalar field. The decay rate of the scalar field is closely related to the investigation of the SCC conjecture. In our analysis, as the decay rate depends on non-minimal coupling and scalar field mass, we elucidate these effects in the SCC conjecture, which has not been done yet under the non-minimally coupled massive scalar field. Further, in the case of the lukewarm RNdS black hole, we will investigate the SCC conjecture for a thermally stable dS black hole and test its consistency with previous studies on non-lukewarm RNdS black holes. In the case of the MTZ black hole, the SCC conjecture for hairy black holes has not been studied much. Although the MTZ black hole is unstable, we propose it is a useful solution to extending studies on its SCC conjecture to black holes having scalar hair. It should be noted that because our analysis is based on the quasinormal resonances that are considered linear effects of the scalar field, its results depend only on the equations of motion for the scalar field rather than on the action for gravity theories. Therefore, our conclusion on the SCC conjecture are the same for both the black holes. Here, for convenience, we will call the geometry as the MTZ black hole.

The paper is organized as follows: Section 2 introduces the geometry of the MTZ black hole. Section 3 solves the non-minimally coupled massive scalar field equation at the outer horizon in the MTZ black hole. Section 4 investigates the SCC conjecture in two limits of the scalar field’s mass. Then, it approximates the quasinormal modes in the intermediate range of the mass by the WKB method. Section 5 summarizes the results.

## 2 Geometry of dS black holes

*M*and cosmological constant \(\Lambda \). The curvature singularity is located at the center of the spacetime. In the limit of the asymptotic region, the metric becomes the dS spacetime containing the cosmological horizon. The mass of the black hole is in the range of \(0<M<\frac{1}{4}\sqrt{\frac{3}{\Lambda }}\). There exists four solutions to \(g^{rr}=\Delta (r)=0\) in the spacetime

### 2.1 Lukewarm Reissner–Nordström–de Sitter Black Hole

*Q*related to

### 2.2 Martínez–Troncoso–Zanelli black hole

## 3 Non-minimally coupled massive scalar field

*m*, and

*l*are separate variables corresponding to frequency and eigenvalues with respect to rotating axis and total angular momenta. Hence, the only non-trivial equation is the radial part, which is written as

## 4 Strong cosmic censorship conjecture

Here, we investigate whether the SCC conjecture is valid with respect to the non-minimally coupled massive scalar field in the MTZ black hole. By scalar field scattering, the perturbation can be blueshifted as it comes close to the Cauchy horizon. The blueshift is given by the amplification rate, which is related to the surface gravity of the Cauchy horizon \(\kappa _\text {i}\) [50]. Note that the scalar field also undergoes exponential decay, which is given as \(|\Psi -\Psi _0|\sim e^{-\alpha t}\) with the spectral gap \(\alpha \). Then, destabilizing the Cauchy horizon depends on the competition between amplification and decay with respect to the perturbation, owing to the scalar field [52]. Further, the competition is governed by a very simple parameter, \(\beta \equiv \alpha /\kappa _\text {i}\) [52]. According to the SCC conjecture discussed in [70], the parameter \(\beta \) determines whether the energy of the scalar field at the Cauchy horizon is divergent [57]. When \(\beta <\frac{1}{2}\), amplification becomes dominant, due to which the blueshifted inward mode is able to destabilize the Cauchy horizon. In this case, the SCC conjecture becomes valid. On the contrary, if \(\beta >\frac{1}{2}\), then the quasinormal modes are damped. Further, the Cauchy horizon is still stable. In this case, the SCC conjecture is invalid. Therefore, the SCC conjecture can be elucidated from the value of \(\beta \). In the following subsections, we consider the MTZ black hole in the limits of the large mass and small mass, and these limits are interpolated by the WKB approximation as given in [53, 71, 72].

### 4.1 Large mass case: near-extremal black holes

### 4.2 Small mass case: de Sitter mode approximation

As the mass of the MTZ black hole decreases, the size of the black bole also decreases. Finally, when the mass becomes zero, the geometry becomes the dS spacetime, containing only the cosmological horizon. Thus, we expect that the quasinormal modes physically smoothly become pure dS spacetime modes in the limit of the small mass. This behavior was already found in [52], which dealt with the RNdS black hole in a massless scalar field, and was called a dS mode. We rewrite this behavior in terms of our notation and obtain the quasinormal mode of the non-minimally coupled massive scalar field in pure dS spacetime by modifying the result obtained in [75], which studied the massive scalar case.

*M*and tending to zero. Then, the potential term becomes

*k*and

*p*as

*c*. This is a suitable form of the solution with arbitrary \(\xi \) and \(\mu \) of the scalar field. Instead of the non-integer

*c*, the choice of the integer

*c*is still possible under a limited condition: \(\mu ^2+\xi {\mathcal {R}}=0\). The case of the integer

*c*is coincident with the massless scalar field without the coupling. This is already discussed in [52], and our analysis of \(\omega \) technically includes the massless case [75]. Hence, to keep the arbitrary \(\xi \) and \(\mu \) for general cases, we focus on the solution about the non-integer

*c*. According to the boundary condition for the quasinormal resonance in Eq. (22), the scalar field only has a purely outgoing mode near the cosmological horizon \(\zeta \rightarrow 0\). In addition, the scalar field is assumed to have vanished at the origin of the spacetime, \(\zeta =1\). Thus, we should take \(A_0=0\) in order to eliminate the incoming wave at the cosmological horizon. This fixes \(k=-i\ell \omega /2\). Then, the solution in Eq. (37) is reduced to

*p*in Eq. (36): \(p = -l/2\) or \(p = (1+l)/2\). (a) When we choose \(p = -l/2\), the parameters in Eq. (38) are fixed as \(c-a=-n\) or \(c-b=-n\), because the radial solution should be regular at the origin \(\zeta \rightarrow 1\). Then, the radial solution in Eq. (39) becomes

### 4.3 WKB Approximation

Each mode is represented by a pair of \(\text {Re}(\omega )\) and \(\text {Im}(\omega )\) with the same color. The real part of the quasinormal frequency is about a propagating oscillation in Fig. 3a, c. The decay rate is closely related to the imaginary part of \(\text {Im}(\omega )\) in Fig. 3b, d. Here, we need to determine the dominant mode, which is the least damping mode, so that it will have the longest life span among all the modes. In Fig. 3b, the mode with \(n=0\) is in the least damping state for various values of *n*. Further, for a fixed *n*, the least damping mode appears at the largest value of *l* in Fig. 3d. Particularly, the damping becomes smaller as *l* increases. Thus, for our analysis, we assume the eikonal limit, (\(l\gg 1\)) to \(l=100\). Therefore, the dominant mode in the quasinormal resonance is in \(n=0\) and \(l=100\). Note that the effects of \(\mu \) and \(\xi \) do not much affect our analysis; therefore, we did not introduce them in Fig. 3.

The detailed behaviors of \(\beta \) with two limits are given in Fig. 4a. The dominant modes obtained from the WKB approximation is represented by the black lines. There are two limits represented by blue and red lines. The blue line is for the \(\beta \) obtained from the Pöschl–Teller potential in the large-mass limit of the near-extremal approximation in Eq. (30). At this limit, we can clearly observe that the WKB and Pöschl–Teller potential approximations are exactly coinciding as represented by the blue point. Further, the value of \(\beta \) is much smaller than \(\frac{1}{2}\). Thus, the SCC conjecture is valid for this limit. However, in the small-mass limit of the MTZ black hole, the behavior patterns become complex. The value of \(\beta \) in the quasinormal resonance rises to the infinity in the limit of the small mass in the black hole. On the other hand, the value of \(\beta \) of the dominant quasinormal resonance in the dS spacetime is finite and much smaller than that of the black hole. As we already expected for the small-mass limit in Sect. 4.2, the dominant mode of the quasinormal resonance of the black hole, denoted by a black line, may smoothly converge on that of the dS spacetime, denoted by a red line in Fig. 4a. This is because, in the limit of the small mass, the black hole could be too small, the configuration of the quasinormal resonance may be a superposition of those resonances of black hole and dS spacetime. Then, the quasinormal resonance of the dS spacetime can be dominant as it has a smaller \(\beta \) value than the black hole. Different from the WKB approximation, in the small-mass limit, \(\beta \) has a finite value depending on \(\xi \) and \(\mu \) in Eq. (45). In Fig. 4a, there is a crossing point of red and black lines where the dominant mode transforms into that of the dS spacetime \(l=1\) rather than into the eikonal limit \(l=100\). Hence, the dominant quasinormal resonance is assumed to be the red line of the dS spacetime in the mass smaller than the crossing point. The small-mass limit shows that \(\beta \) can rise over \(\frac{1}{2}\) at the small mass. Therefore, the SCC conjecture may be invalid in the small-mass range. However, there exists a specific case where the SCC conjecture is valid for all the masses of the MTZ black hole, as shown in Fig. 4b. Here, we simplify the diagram by choosing the smaller \(\beta \) for a given mass, because the dominant mode should have the smallest damping factor, \(\text {Im}(\omega )\). The blue point is still largely coincident with the WKB approximation. Interestingly, according to the choice of \(\xi \) and \(\mu \), \(\beta \) can be lower than \(\frac{1}{2}\) at the small-mass limit as shown in Fig. 4b. This implies that the SCC conjecture is valid for all the masses of the MTZ black hole.

We have applied various methods to obtain the dominant modes of the quasinormal resonance, including approximate potentials and WKB method, in MTZ black holes. Our results are consistent with previous studies conducted on the massless scalar field. We will now check the consistency of the results with those of the previous studies that discuss RNdS black holes. As we consider the quasinormal mode of the scalar field in the MTZ black hole, our analysis needs to be consistent with [52]. Note that the detailed method is different. The MTZ black hole can be considered for the lukewarm case of \(Q/M=1\) compared with the RNdS black hole in [52]. For the near-extremal black hole of \(r_\text {o}\approx r_\text {c}\), the SCC conjecture is valid because \(\beta < \frac{1}{2}\). This is consistent with our results given in Sec. 4.1. Moreover, for the small-mass limit of \(r_\text {o},\,r_\text {i}\rightarrow 0\), the value of \(\beta \) becomes larger than \(\frac{1}{2}\), which means that the conjecture is invalid. This is also consistent with our results in Sect. 4.2, in the case of the massless scalar field of \(\xi =0\) and \(\mu =0\). Then, we can find a point \(\beta =\frac{1}{2}\) after interpolating the two limits by the WKB approximation. This is also provided in Sect. 4.3. Although MTZ and lukewarm RNdS black holes are solutions to different theories of gravity, their quasinormal frequencies are consistent with one another. Whether or not non-linear effect is subtle is still an issue in current studies. Our analysis is based on the quasinormal resonance of a linear perturbation, and does not consider non-linear effects. The non-linear effect is a subject for future study.

## 5 Summary

We investigated the validity of the SCC conjecture in the MTZ or lukewarm RNdS black hole by the quasinormal resonance of the non-minimally coupled massive scalar field. Since the instability of the Cauchy horizon depends on the amplification and decay rates of the quasinormal resonance, we obtained the overall behaviors of \(\beta \equiv \text {Im}(\omega )/\kappa _\text {i}\) with respect to the mass of the black holes. In the analysis of the SCC conjecture by quasinormal modes, the dominant mode, which is the least damping mode, plays an important role. Therefore, we first obtained the values of \(\beta \) for the small- and large-mass limits of the MTZ black hole. Then, we combined them by the WKB approximation. In the large-mass limit, for a given cosmological constant, the black hole becomes the near-extremal case where the effective potential of the scalar field reduces to the Pöschl–Teller potential. Then, we obtained the quasinormal frequency and read the value of \(\beta <\frac{1}{2}\) for the dominant mode. This implies that the amplification of the inward field is more dominant than the decay rate. Hence, the SCC conjecture is valid for large mass black holes. Note that as the mass of the black hole decreases, the value of \(\beta \) tends to increase. Particularly, at the small-mass limit, \(\beta \) rapidly diverges in the eikonal limit of the scalar field. To resolve this divergence, we carefully consider that the dominant mode of the quasinormal resonance gradually becomes similar to that of the dS spacetime, because the black hole could be too small to affect the quasinormal mode in the limit. Hence, it is possible that the least damping mode originates from that of the dS spacetime. Moreover, the dominant mode of the dS spacetime had less damping than that found by the WKB approximation. Thus, the amplification and decay rates depend on the dS spacetime mode in the small-mass limit. Instead of the mass of the black hole, the dominant mode of the dS spacetime is determined precisely from \(\xi \) and \(\mu \). In the limit of the massless scalar field without coupling, as shown in previous studies, the value of \(\beta \) is still larger than \(\frac{1}{2}\). On the contrary, we found that there is a range producing smaller \(\beta \) than \(\frac{1}{2}\) in the phase-space of \(\xi \) and \(\mu \). This implies that the SCC conjecture is invalid. Therefore, the validity of the SCC conjecture depends on the coupling constant and mass of the scalar field in the lukewarm RNdS or MTZ black hole. It should be noted that our analysis is based on the perturbations of the quasinormal mode. Therefore, it can be improved by considering backreaction or non-linear effects. Nevertheless, the results of this study extend the investigation of the SCC conjecture to the non-minimally coupled massive scalar field. Further, we found that the validity of the SCC conjecture in this case may be different from that of the massless scalar field without coupling.

## Notes

### Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (NRF-2018R1C1B6004349). B.G. would like to thank Yongwan Gim for helpful discussions. In addition, B.G. appreciates APCTP for its hospitality during completion of this work.

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