Quasinormal modes of phantom Reissner-Nordström-de Sitter black holes

One of the stunning discoveries of the modern physics is the observation of the accelerating expansion of our Universe. To understand the underlying physics of this profound phenomena, people have devoted large amounts of endeavors over the past decades. To the best of my knowledge, at preset there are two different perspectives to solve this mystery, one is to resort to modified gravity theory. This kind of viewpoint is based on the idea that the general relativity (GR) should be modified at large scale to incorporate the observed effects of Universe acceleration. One representative of such theories is the $f(R)$ gravity proposed in 1970 Buchdahl:1970ynr , where $f(R)$ introduced to the Lagrangian of the theory stands for an arbitrary function in terms of Ricci scalar $R$, and the freedom of the choices of $f(R)$ function makes it possible to explain the accelerated expansion. The another viewpoint to understand the accelerating Universe is by introducing an effective field (dark energy) which can generate repulsive gravity. A field with such repulsion feature can be formulated as a fluid with negative pressure in GR, and the most famous example of this fluid is the cosmological constant $\Lambda$ which was first proposed by Einstein and has been an essential ingredient of the $\Lambda\mathrm{CDM}$ model. Besides the cosmological constant, the so called phantom scalar field can also serve as a possible description of the dark energy Caldwell:1999ew ; Cai:2009zp . By comparing with the observational data Hannestad:2005fg ; WMAP:2008rhx , it has claimed that the phantom field endowed with negative energy density distribution can indeed be used to explain the acceleration of our Universe. The phantom field can appear in the Einstein-Maxwell-dilaton system where the sign of dilatonic kinetic term is flipped to be negative Clement:2009ai , and it also emerges in the in the study of ghost branes in the string theory Okuda2006 . Although the phantom field may lead to quantum instabilities Caldwell:1999ew ; Cai:2009zp which poses a challenge to the theory, the authors in Piazza2004 ; Nojiri:2003vn claimed that these instabilities can be avoided and it consequently makes phantom field a candidate for the dark energy model from the theoretical side. Given the usefulness of phantom field, it therefore motivates us to study the physics of phantom field, especially to probe the effects of phantom field on the background of black holes spacetime if black holes solutions can be found in this scenario, since the black holes in the Universe will be inevitably affected by dark energy.

Recently, a new spherically symmetrical black hole solution was obtained in Einstein-anti-Maxwell theory with cosmological constant, called anti-RN-(A)dS solution, or simply called phantom RN-(A)dS black holes solution Jardim:2012se . The phantom nature of the charge possessed by this kind of black holes make them drastically differ from their counterparts, namely usual RN-(A)dS black holes. Hence it is of great interest to study the characteristics of such black holes. The investigation of thermodynamical aspects of this black hole system have been performed in several works. The thermodynamics of phantom RN-AdS was discussed in Jardim:2012se , followed by a further study of geometrothermodynamics of this phantom black holes in Quevedo:2016cge . The authors in Mo:2018hav studied the phase transition and heat engine efficiency of phantom AdS black holes. While in this paper, we would like to focus on the dynamical properties of phantom RN-dS black holes under the scalar field perturbation, which means we are interested in investigating its QNMs spectrum on the background in our consideration.

QNMs has versatile applications in the black holes physics. One of its basic utilization is that we can use QNMs to examine the stability of black holes spacetime under perturbation Berti:2009kk ; Konoplya:2011qq . In the context of astrophysics, QNMs is contained in the gravitational waves (GWs) in the ringdown phase of the mergers of binary black holes system and plays an increasingly essential role in the contemporary gravitational waves astronomy LIGOScientific:2016aoc ; LIGOScientific:2017vwq , due to the fact that it can be regarded as characteristic “sound” of black holes Nollert:1999ji and serve as the basis of black holes spectroscopy. Therefore, in principle, rich informations of the GWs sources and spacetime geometry can be revealed with the successful detection of GWs. This feature of QNMs marks its great importance in the research of gravitational physics. In addition, QNMs has also been used to test GR and the validity of the famous “no-hair” theorem of black holes Dreyer:2003bv ; Berti:2005ys ; Shi:2019hqa ; Isi:2019aib , constrain modified gravity theories Liu:2020ddo ; Bao:2019kgt ; Cano:2021myl ; Blazquez-Salcedo:2016enn ; Franciolini:2018uyq ; Aragon:2020xtm ; Liu:2020qia ; Karakasis:2021tqx and examine strong cosmic censorship conjecture Cardoso:2017soq ; Liu:2019lon ; Hod:2018dpx ; Mo:2018nnu ; Dias:2018ynt ; Hod:2018lmi ; Gwak:2018rba ; Guo:2019tjy , and some other interesting discussions of QNMs in the asymptotically dS spacetime can be found in Sarkar:2023rhp ; Konoplya:2022xid ; Konoplya:2022kld ; Zhidenko:2003wq . Given the significance of QNMs introduced above, hence it is intriguing and meaningful to study QNMs when a new black hole solution is obtained.

The present work is organized as follows. In Section II, we first give a brief introduction to phantom RN-dS black holes, and then analyze the horizon structure and work out the value range of the phantom charge. In Section III, we will numerically calculate QNMs spectrum of massless neutral scalar field perturbation with two methods and analyze the numerical results. The last section is devoted to conclusions. Throughout this paper, we will work with units $G=c=1$.

In this section, we would like to briefly review phantom RN-dS black holes first, and then pin down its parameter space in which at least two horizons are present. The action of this theory is given by Jardim:2012se

where $R$ is the Ricci scalar, $\Lambda$ is cosmological constant, and $F_{\mu\nu}$ is the coupled vector field strength whose nature is characterized by constant $\eta$. For $\eta=1$, $F_{\mu\nu}$ is just the electromagnetic field strength, while for $\eta=-1$ it stands for phantom vector field strength. The terminology “phantom” is used here since the energy density of the field is negative for $\eta=-1$.

Based on action given in Eq. (1), we can get a spherically symmetrical black hole solution Jardim:2012se

where

Obviously, this is the well-known RN-(A)dS spacetime metric when $\eta=1$. In the present paper, we are interested in the less concerned case $\eta=-1$ with a positive cosmological constant $\Lambda$>0, i.e. phantom RN-dS black hole spacetime. For our convenience, we would like to define a charge parameter $Q\equiv\eta q^{2}$. With the parameter replacement, the phantom RN-dS black hole metric function $f(r)$ takes the form

To have a better understanding of this black hole spacetime, it is necessary to figure out the number of horizons, i.e. positive roots of function $f(r)$ of this spacetime. To simplify this task, we directly deal with function $h(r)\equiv r^{2}f(r)$ instead of dealing with $f(r)$. Apparently, the roots of $h(r)$ must be the roots of $f(r)$ except $r=0$, which is the location of black hole singularity so it is excluded from the possible roots. Function $h(r)$ is polynomial with the form

which is a quartic polynomial suggesting that there exists four roots for equation $h(r_{i})=0$. According to Vieta’s theorem, the roots of $h(r)$ satisfy following relations

where we have fixed black hole mass $M=1$ without loss of generality. For the physical reason, we only consider black hole parameters that yield four real roots $r_{i}\in\mathbb{R}$. Note that we have $\Lambda>0$ and $Q<0$ such that $r_{1}r_{2}r_{3}r_{4}=-3Q/\Lambda>0$. This means that the sign of $r_{i}$ has three different combinations, namely 1)four positive roots 2)two positive and two negative roots 3)four negative roots. On the other hand, we have $r_{1}+r_{2}+r_{3}+r_{4}=0$ which means that the roots can not be all positive or all negative. Finally, with Eq. (6), we are forced to get two positive roots in addition to two negative roots, so there are at most two horizons in phantom RN-dS black hole spacetime. The two positive roots are labeled as $r_{+}$ and $r_{c}$ with the relation $r_{+}<r_{c}$. The function $h(r)$ in Eq. (5) can be rewritten as

where $r_{1}<r_{2}<0<r_{+}<r_{c}$. By Eq. (7), we can see that $h(r)<0$ in the region $0<r<r_{+}$ and $r>r_{c}$, whereas $h(r)>0$ in the region $r_{+}<r<r_{c}$. This fact reveals that $r_{+}$ and $r_{c}$ is event horizon and cosmological horizon, respectively. The behavior of $h(r)$ and the location of roots are clearly demonstrated in Fig. 1, which shows the correctness of previous analysis.

Before taking a further investigation of the properties of phantom RN-dS black hole, we should figure out the parameter space in which both the event horizon and cosmological horizon can exist at the same time. Our first step is to find out the maximum value of cosmological constant $\Lambda_{max}$, at which the minimum value of charge parameter $Q$ is zero. This extreme condition requires

The solution of equations above is

This equation leads to $r=3$, which yields

For a given value of $\Lambda$, we are now supposed to calculate out the minimum and maximum value of $Q$, which is respectively denoted by $Q_{min}$ and $Q_{max}$. The similar strategy will be adopted as that used in finding $\Lambda_{max}$. After some simple calculation, we get analytical formula for $Q_{min}$ and $Q_{max}$ which are given by

where we have defined

in which $\mathrm{arg}$ is the argument function. We compare the values of $Q_{min}$ and $Q_{max}$ in terms of $\Lambda$ in Fig. 2, which shows that $Q_{min}$ is negative definite and and $Q_{max}$ is positive definite, and $Q_{max}$ is not as sensitive as $Q_{min}$ to the change of cosmological constant. This means that only $Q_{min}$ is related to charge parameter of phantom RN-dS black hole, $Q_{max}$ is RN-dS black hole relevant. Actually, $Q_{min}$ marks the charge parameter value at which event horizon coincides with cosmological horizon, and $Q_{max}$ is only relevant to RN-dS black hole, since it is a positive value where Cauchy horizon coincides with event horizon, while for phantom RN-dS black hole, it is simply required that $Q<0$ and no Cauchy horizon exists in this spacetime. When we set $\Lambda=\Lambda_{max}$, with Eq. (11a) we can directly get $Q_{min}=0$, this result is in agreement with our previous discussion about finding the value of $\Lambda_{max}$. It is well known that in RN-dS spacetime, charge parameter is limited to guarantee the existence of event horizon. In fact, the maxima and minima of $Q_{max}$ respectively is

Interestingly, on the contrary to RN-dS black hole, in phantom RN-dS black hole spacetime we find that $Q_{min}$ is not bounded from below,

which serves as a remarkable difference between phantom and usual RN-dS black hole spacetime.

In this section, we focus on the calculation of QNMs frequencies of massless neutral scalar field perturbation on the phantom RN-dS spacetime, and to this end we are required to derive the master equation of scalar field perturbation.

In this article, we have studied some properties of phantom RN-dS black hole, including its horizon structure, the value domain of the charge parameter, and the QNMs spectrum of massless neutral scalar field perturbation. One of the features of the phantom RN-dS black hole that differs from usual RN-dS black hole is that charge parameter of the phantom hole is negative. Under the negative charge condition, we find that there exists at most two horizons in this spacetime, namely event horizon and cosmological horizon. Especially, the value range of the phantom black hole charge is dependent on cosmological constant $\Lambda$ and found to be not bounded from below when $\Lambda\to 0$, which exhibits a remarkable difference from normal charged black holes whose charge value is limited in order to avoid naked singularity.

We have analyzed QNMs spectrum of scalar field perturbation obtained by AIM and confirmed by WKB approach which is greatly improved by Pade approximants, and classified the QNMs into dS modes and PS modes. When the angular number $l$ is increased, one can find that the real part of QNMs frequencies monotonously grow with $l$, as expected from the perspective of physics since a larger angular number corresponds to larger angular momentum which gives rise to a more rapid oscillation frequency. Whereas for the imaginary part related to the damping rate of the modes, its magnitude decreases implying modes with larger $l$ will live longer. When we fix angular number and cosmological constant but decreasing the charge parameter, we find that the real part of QNMs frequencies decreases while the imaginary part increases. On the other hand, when we solely increase $\Lambda$ and leave other parameters unchanged, the real part of QNMs frequencies will decrease, while the imaginary parts increase. With these results, we conclude that the increment of the magnitude of $Q$ or $\Lambda$ will result in the decrement of the magnitude of real and imaginary parts. The dS modes have the chance to become the dominant modes over PS modes, when $\Lambda<\Lambda_{cri}$. We find that $\Lambda_{cri}$ is related to the value of charge parameter $Q$, as for a larger $|Q|$, $\Lambda_{cri}$ will be smaller. Finally, we examine the deviations of dS modes from the pure dS modes. It is known that dS modes originate from pure dS modes, such that in a asymptotically dS black hole spacetime, the dominant dS modes frequencies are almost identical to the corresponding pure dS modes frequencies, just with a tiny deformation. We find that this deformation depends on charge parameter $Q$ and cosmological constant $\Lambda$. A larger $\Lambda$ or $|Q|$ can lead to a larger deviation, which can be noticeable and seem to be more sensitive to the variation of $\Lambda$.

At last, we compared the QNMs frequencies of phantom and RN-dS black holes in order to reveal more effects of phantom charge and highlight the differences between the two black holes. We find that under any combinations of parameters in our consideration, both imaginary and real part of QNMs frequencies from RN-dS black holes are never smaller than that of phantom RN-dS black holes, which means that the QNMs of phantom RN-dS black holes can live longer and oscillate less rapidly compared to RN-dS black holes. When charge is fixed and increasing cosmological constant, we find that $-\mathrm{Im}\,(\omega)$ and $\mathrm{Re}\,(\omega)$ will decrease, for both black holes. On the other hand, it was found that when the $\Lambda$ is fixed, for phantom RN-dS black holes the $-\mathrm{Im}\,(\omega)$ and $\mathrm{Re}\,(\omega)$ will become smaller with the grow of $Q$. However, for RN-dS black holes a larger $Q$ will make $\mathrm{Re}\,(\omega)$ monotonously increase but $-\mathrm{Im}\,(\omega)$ behave non-monotonically.