Behavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity
De-Cheng Zou Yunqi Liu Ruihong Yue







 ^1Center for Gravitation and Cosmology,
College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China

^2School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China







In this work, we utilize the quasinormal modes (QNMs)
of a massless scalar perturbation to probe the Van der Waals-like
small and large black holes (SBH/LBH) phase transition of
charged topological Anti-de Sitter (AdS) black holes in four-dimensional
massive gravity. We find that the signature of this SBH/LBH phase transition
is detected in the isobaric as well as in the isothermal process.
This further supports the idea that the QNMs can be an efficient tool to
investigate the thermodynamical phase transition.




04.50.Kd, 04.70.-s, 04.25.D-


Behavior of quasinormal modes and Van der Waals-like phase
transition of charged AdS black holes in massive gravity
    De-Cheng Zou^1[Email:dczou@yzu.edu.cn],
Yunqi Liu^2[Email:liuyunqi@hust.edu.cn]
and Ruihong Yue^1 [Email:rhyue@yzu.edu.cn]
    December 30, 2023
==============================================================================================================================





§ INTRODUCTION



Einstein's general relativity introduces gravitons as massless spin-2
particles <cit.>.
However, understanding the quantum behavior of gravity could be related to the possible
mass of the graviton. This Einstein theory, modified at large distances in massive
gravity provides a possible explanation for the accelerated expansion of the Universe
that does not require any dark energy. Actually, the massive gravity and its extensions, such as
bimetric gravity, can yield cosmological solutions which do display late-time
acceleration in agreement with observations <cit.>.
Very recently, the LIGO collaboration reporting the discovery of gravitational wave asserted
that <cit.> “assuming a modified dispersion relation for gravitational
waves, our observations constrain the Compton wavelength of the graviton to
be λ_g>10^13 km, which could be interpreted
as a bound on the graviton mass m_g<1.2×10^-22eV/c^2”.
In order to have massive graviton,
the first attempt for constructing a massive theory
was in the work of Fierz and Pauli <cit.> which was done in
the context of linear theory. Unfortunately this theory possesses so-called van Dam,
Veltman and Zakharov discontinuity problem. The resolution to this problem was
Vainshtein's mechanism, which requires the
system to be considered in nonlinear framework.
As is now well known, it usually brings about Boulware-Deser ghost <cit.>
by adding generic mass terms for the graviton on the nonlinear level.
Subsequently, a nonlinear massive gravity theory was proposed by de Rham, Gabadadze
and Tolley (dRGT) <cit.>,
where the mass terms are added in a specific way to ensure that
the corresponding equations of motion are at most second order differential equations so
that the Boulware-Deser ghost is eliminated.
Later, spherically symmetric black hole solutions were
constructed in the dRGT massive gravity <cit.>,
including its extension in terms of electric
charge  <cit.>,
black string <cit.>, BTZ like black holes <cit.>
and some other solutions with higher curvature correction terms
<cit.>. This program goes beyond solution constructions
in the dRGT massive gravity and focuses on the investigation holographic implications <cit.>,
discussing the thermodynamical properties <cit.>
and calculating QNMs under massless scalar perturbations for the BTZ like
black hole <cit.>.

The thermodynamical phase transition of a black hole is always a hot topic
in black hole physics. It may shed light on
the understanding of the relation between gravity and thermodynamics.
Recently, thermodynamics of AdS black holes has been generalized
to the extended phase space where the cosmological constant is treated
as the pressure of the black hole <cit.>.
A particular emphasis has been put on the study of the black hole phase
transitions in AdS spacetime in Ref.<cit.>, which asserted the
analogy between the Van der Waals liquid-gas system
behavior and the charged AdS black hole.
Subsequently a broad range of thermodynamic behaviors have been discovered,
including reentrant phase transitions and more general Van der Waals behavior  <cit.>.
Recently, some investigations od the thermodynamics of AdS
black holes in the massive gravity showed generalization to the extended phase
space <cit.>,
including the higher curvature terms <cit.>.


For a long time, thermodynamical phase transitions of the black hole are supposed
to detect by some observational signatures. Considering
that QNMs of dynamical perturbations are characteristic
issues of black holes <cit.>,
it is expected that black hole phase transitions can be reflected in the
dynamical perturbations in the surrounding geometries of black holes through
frequencies and damping times of the oscillations. Moreover,
the QNM frequencies of AdS black holes have direct interpretation
in terms of the dual conformal field theory CFT <cit.>.
A lot of discussions have been focused on this topic and more and
more evidence has been found between thermodynamical phase transitions
and dynamical perturbations. See for examples <cit.>.
In the extended phase space, we have recovered the deep relation between the
dynamical perturbation and the Van der Waals-like SBH/LBH phase transition
in the four-dimensional Reissner-Nordström-Anti de Sitter (RN-AdS) black holes
with spherical horizon (k=1) <cit.>. Later, matters have been
generalized to higher-dimensional RN-AdS black holes <cit.> ,
including time-domain profiles <cit.>, and higher-dimensional
charged black holes in the presence of Weyl coupling <cit.>.

It is necessary to point out that in four-dimensional dRGT massive gravity,
there always exists so-called Van der Waals-like SBH/LBH phase transition
for the charged AdS black holes when the horizon topology is spherical (k=1),
Ricci flat (k=0) or hyperbolic (k=-1) <cit.>.
In particular, this phenomenon rarely occurs, since this Van der Waals-like
 SBH/LBH phase transition was usually recovered in a
variety of spherical horizon black hole backgrounds.
Motivated by these results,  in this paper we find crucial and well justified to
reconsider the charged topological AdS black hole in
four-dimensional dRGT massive gravity.
We further use the QNM frequencies of a massless scalar perturbation to
probe the Van der Waals-like SBH/LBH phase transitions of charged topological
black holes (k=0, k=±1), respectively.

This paper is organized as follows. In Sect. <ref>, we will review
the Van der Waals-like SBH/LBH phase transition of charged topological
AdS black holes in four-dimensional massive gravity. In Sect. <ref>,
we will disclose numerically that the phase transition can be reflected
by the QNM frequencies of dynamical perturbations. We end the paper with
conclusions and discussions in Sect. <ref>.



§ PHASE TRANSITION OF CHARGED TOPOLOGICAL ADS BLACK HOLE IN MASSIVE GRAVITY


We start with the action of four-dimensional massive gravity in the
presence of a negative cosmological constant <cit.>

    I=1/16π∫d^4x√(-g)[R-2Λ
    -1/4F_μνF^μν+m^2∑_i^4c_i U_i(g,f)],

where f is a fixed symmetric tensor usually called the reference metric,
c_i are constants, m is the mass parameter related to the graviton mass,
and F_μν is the Maxwell field strength defined as
F_μν=∂_μA_ν-∂_νA_μ with vector potential A_μ.
Moreover, U_i are symmetric polynomials of the eigenvalues
of the 4× 4 matrix K^μ_ν≡√(g^μαf_αν)

    U_1   =   [ K],
     U_2   =   [ K]^2-[ K^2],
     U_3   =   [ K]^3-3[ K][ K^2]+2[ K^3],
     U_4   =   [ K]^4-6[ K^2][ K]^2
    +8[ K^3][ K]+3[ K^2]^2-6[ K^4].

The square root in K is understood as the matrix square root, ie.,
(√(A))^μ_ ν(√(A))^ ν_λ=A^μ_ λ, and the
rectangular brackets denote traces [ K]= K^μ_ μ.

The action admits a static black hole solution with metric

    ds^2=-f(r)dt^2+1/f(r)dr^2+r^2h_ijdx^idx^j,

where the coordinates are labeled x^μ=(t, r, x^1, x^2) and h_ij describes
the two-dimensional hypersurface with constant scalar curvature 2k.
The constant k characterizes the geometric property of a
hypersurface, which takes values k=0 for the flat case, k=-1 for
negative curvature and k=1 for positive curvature, respectively.
In a four-dimensional situation, we have U_3= U_4=0.
Then the solution of charged topological AdS
black hole is given by <cit.>

    f(r)=k+8π P r^2/3-m_0/r+q^2/4r^2
    +c_0c_1m^2r/2+c_0^2c_2m^2,

where P equals to -Λ/8π.
Moreover, the parameters m_0 and q
are related to mass and charge of black hole

    M=V_2/8πm_0,   Q=V_2/16πq.

Here V_2 is the volume of space spanned by coordinates x^i.
When m→0, the solution (<ref>)
reduces to the RN-AdS black hole.

The reference metric now can have a special choice

    f_μν=diag(0,0,c_0^2h_ij).

Without loss of generality
we have set c_0=1 in our following discussions.

In terms of the radius of the horizon r_+, the mass M,
Hawking temperature T, entropy S and electromagnetic potential Φ
of the black holes can be written as

    M   =   V_2 r_+/8π(k+c_2m^2+q^2/4r_+^2+8π P r_+^2/3
    +c_1m^2 r_+/2),
    
    T   =   -q^2/16π r_+^3+k+c_2m^2/4π r_++2 Pr_+
    +c_1m^2/4π,
    
    S   =   V_2/4 r_+^2,    Φ=q/r_+

In the extended phase space, the black hole mass M is considered as the
enthalpy rather than the internal energy of the
gravitational system <cit.>.

From Eq. (<ref>), the equation of state of the black hole can be obtained

    P=T/2r_+-c_1m^2/8π r_+-k+c_2m^2/8π r_+^2
    +q^2/32π r_+^4.

To compare with Van der Waals fluid equation in four dimensions, we can translate
the “geometric” equation of state to a physical one by identifying the specific
volume v of the fluid with horizon radius of black hole as v=2r_+.

As usual, a critical point occurs when P has an inflection point

    ∂ P/∂ r_+|_T=T_c, r_+=r_c
    =∂^2 P/∂ r_+^2|_T=T_c, r_+=r_c=0,

which leads to

    r_c=√(6)q/2√(k+c_2m^2),   
    T_c=2(k+c_2m^2)^3/2/3√(6)π q
    +c_1m^2/4π,  
    P_c=(k+c_2m^2)^2/24π q^2.

Evidently the critical behavior occurs when
k+c_2m^2>0, which is a joint effect of horizon topology k and c_2m^2.
Previous thermodynamical discussions for RN-AdS black holes
show that the Van der Waals-like SBH/LBH only occurs for spherical horizon
topology k=1. The graviton mass significantly modifies this behavior and
a non-zero m admits that possibility of critical behavior for k≠1.
In addition, it has been shown <cit.> that when k+c_2m^2>0,
the small and large black hole phases are both
locally thermodynamically stable because the corresponding heat capacities
are always positive[We thank Hai-Qing Zhang for pointing this out.].

The equilibrium thermodynamics is governed by the
Gibbs free energy, G=G(T, P, q), which obeys the thermodynamic
relation G=M-TS. For later discussions, it is convenient to rescale the Gibbs free
energy in the following way: g=4π/V_2G. Then g reads

    g=3q^2/16r_++(k+c_2m^2)r_+/4
    -2π P r_+^3/3.

Here r_+ is understood as a function of pressure and temperature, r_+=r_+(P,T), via
the equation of state (<ref>).




§ PERTURBATIONS OF CHARGED TOPOLOGICAL ADS BLACK HOLE IN MASSIVE GRAVITY



Now we study the evolution of a massless scalar field perturbation in the surrounding
geometry of these charged topological AdS black holes.

A massless scalar field Ψ(r,t,Ω)=ϕ(r)e^-iω tY_lm(Ω),
obeys the Klein-Gordon equation

    ∇_μ^2Ψ(r,t,Ω)=1/√(-g)∂_μ(√(
    -g)g^μν∂_νΨ(r,t,Ω))=0,

where Y_lm(Ω) is a normalizable harmonic function on the 2-dimensional hypersurface.
In particular, the Laplace operator on Ω yields

    ∇^2_ΩY_lm(Ω)=-κ^2Y_lm(Ω),


It is necessary to point out that the eigenvalue κ^2 usually gets
different values in consideration of different horizon topologies.
For the spherical (k=1) and flat (k=0) topology,
the eigenvalue κ^2 can be zero. Then
the radial function ϕ(r) obeys

    ϕ”(r)+(f'(r)/f(r)+2/r)ϕ'(r)
    +ω^2ϕ(r)/f(r)^2=0,

where ω are complex numbers ω=ω_r + iω_im, corresponding to
the QNM frequencies of the oscillations describing the perturbation.
For the hyperbolic horizon topology(k=-1), the
eigenvalue κ^2 of the Laplace operator on Ω cannot be zero <cit.>, and
is given by 1/4+ξ^2, where ξ=L_Ω(L_Ω+1),
L_Ω=0,1,2,...<cit.>.
Then the radial function ϕ(r) obeys the following differential
equation:

    ϕ”(r)+(f'(r)/f(r)+2/r)ϕ'(r)
    +(ω^2/f(r)-κ^2/r^2)ϕ(r)/f(r)=0.


Here we define ϕ(r) as φ(r)exp[-i∫ω/f(r)dr],
where the exp[-i∫ω/f(r)dr] asymptotically approaches to ingoing wave
near horizon, then Eqs. (<ref>)(<ref>) become

    φ”(r)+φ'(r)(f'(r)/f(r)-2iω/f(r)+2/r)
    -2iω/r f(r)φ(r)=0,             k=0,1

and

    φ”(r)+φ'(r)(f'(r)/f(r)-2iω/f(r)+2/r)
    -(2iω+κ^2/r)φ(r)/r f(r)=0,     k=-1.

In this paper, we only consider ξ=0, namely κ^2=1/4 for k=-1.

We are going to study whether the signature of Van der Waals-like SBH/LBH
phase transition of charged topological AdS black holes can be
reflected by the dynamical QNMs behavior in the massless scalar perturbation.
For Eqs. (<ref>) and (<ref>), we have φ(r)=1 in the limit of r→ r_+.
At the AdS boundary (r→∞), we need φ(r)=0.
Under these boundary conditions, we will numerically solve Eqs. (<ref>) and (<ref>)
separately to find QNM frequencies by adopting the shooting method.
In the context of the Van der Waals phase transition picture, the dynamical
perturbations in the isobaric process and isothermal process will be discussed.
In our following numerical computations we will set q=2, m=1, c_1=0.05
and c_2=2.




 §.§ Isobaric phase transition



Due to the pressure P (or l) being fixed in this case, the black hole horizon r_+
is the only variable in the system. The behavior of an isobar with different horizon topologies
are plotted in Fig. <ref>. For P<P_c, the oscillating part displays
the occurrence of an SBH/LBH phase transition in the system and the Gibbs free energy depicts
a swallow tail behavior, also signaling a first-order SBH/LBH phase transition.
Here the intersection point indicates the coexistence of two phases in equilibrium.
The critical pressure P_c is
obtained by ∂ T/∂ r_+=∂^2 T/∂ r_+^2=0.





In Table. <ref>(see appendix), we further list the QNM frequencies of massless scalar
perturbation around small and large black holes for a first order SBH/LBH phase transition.
Fixing the pressure with P=0.003,
we obtain the phase transition temperature T_*≃0.04567,
T_*≃0.06945 and T_*≃0.08664 in the cases of k=-1, 0 and 1, respectively,
where the small and large black hole phases can coexist.
With regard to a small black hole phase, the radius of black hole becomes
smaller and smaller when the temperature decreases from the phase
transition temperature T_*. In this process the absolute values of the imaginary part
of the QNM frequencies decrease, while the real part frequencies change very little.
On the other hand, when the temperature for the large black hole phase increases
from the phase transition temperature T_*, the black hole gets bigger.
The QNM frequencies increase in the real and absolute value of imaginary parts.
Consequently, the massless scalar perturbation
outside the black hole gets more oscillations but it decays faster. These results
are consistent with the overall discussions reported in <cit.>.
Figure <ref> illustrates the QNM frequencies for small and large black hole phases.
Increase in the black hole size is indicated by the arrows.




In addition, at the critical position P=P_c, with P_c≃0.0033157 for k=-1,
P_c≃0.0132629 for k=0 and P_c≃0.0298416 for k=1,
a second-order phase transition occurs. The QNM frequencies
of the small and large black hole phases are plotted in Fig. <ref>.
We see that QNM frequencies of two black hole phases
show the same behavior as the black hole horizon increases at the critical point.







 §.§ Isothermal phase transition


Fixing the black hole temperature T, the associated P-r_+ diagram
of charged topological AdS black holes is displayed in the right part of Fig. <ref>.
For T<T_c there is an inflection point and
behavior is reminiscent of the Van der Waals liquid-gas system. Moreover,
the behavior of Gibbs free energy is plotted in the left panel of
Fig. <ref>. Similarly to Fig. <ref>, characteristic first order
SBH/LBH phase transition behavior shows up.

Table <ref> (see appendix) displays the QNM frequencies of
small and large black hole phases at temperature T=0.79T_c
for different horizon topologies (k=0,±1) in the isothermal precess.
Then the first order SBH/LBH phase transition happens
at P_*≈0.001728 for k=-1,
at P_*≈0.007189 for k=0 and at P_*≈0.016215 for k=1,
where the small and large black holes possess the same Gibbs free energy
and same pressure. The data above (below) the horizontal line are for the small (large)
black hole phase, respectively. The drastically different QNM
frequencies for small and large black hole phases are plotted in Fig. <ref>.
From the figure we see different slopes of the QNM frequencies in
the massless scalar perturbations revealing that small and large black holes
are in different phases.









In the isothermal transition, the QNMs can be affected by the value of the
pressure P(l) and the horizon radius r_+, which are related by a fixed temperature.
To illustrate the effects of the two parameters, we list the influence of r_+ on the
frequencies for small and large black holes by fixing P(l) in Table <ref> (see appendix)
and QNM frequencies by fixing the black hole size r_+ in Table <ref> (see appendix).
From Tables <ref> and <ref> one can see that there is competition
between the pressure P and horizon radius r_+. Each of these parameters aims
to overwhelm the other which affects the decay rate of the field.


In order to further discuss how these two factors affect the QNM frequencies,
we perform a double-series expansion of the frequency ω(r_++Δ r_+,P+Δ P)

    ω(r_++Δ r_+,P+Δ P)   =   ω(r_+,P)+∂ω/∂ r_+Δ r_+
    +∂ω/∂ PΔ P
          +𝒪(Δ r_+^2,Δ P^2,Δ r_+·Δ P).

Obviously, the changes of the QNM frequency are under two influences, one is from
the change of the black hole size r_+ and the other is from the change of the pressure P
(or AdS radius l). For simple discussions, we define
Δ_1≡∂ω/∂ r_+Δ r_+ and
Δ_2≡∂ω/∂ PΔ P.

Note that the choice of the step of pressure Δ P in linear approximation is
related to Δ r_+, which is brought about by

    dP=(-T/2r_+^2+c_1m^2/8π r_++k+c_2m^2/4π r_+^3
    -q^2/8π r_+^5)dr_+

from the equation of state (<ref>). In Table <ref> (see appendix), we list
the QNM frequencies from the linear approximation for small and large black hole
phase. One can see that the behavior of ω̃ is in good agreement
with the numerical computation results listed in Table <ref>.
Comparing Δ_1 and Δ_2 in Table <ref>, the change of P (or l)
in small black hole phase clearly wins over the change of the black hole size,
which dominantly contributes to the behavior of QNM frequencies for small black
hole phase. For the large black hole phase,
the contributions of Δ_1 and Δ_2 on the real part of the QNM
frequency are comparable. But the change of P (or l) wins out a little.


In addition, for the isothermal phase transition at T=T_c, the QNM frequencies
for the small black hole and large black hole are plotted in Fig. <ref>,
which shows the same behavior as the horizon radius increases.







§ CONCLUSIONS AND DISCUSSIONS



We have calculated the QNMs of massless scalar field perturbation around small
and large charged topological AdS black holes in four-dimensional dRGT massive gravity.
When the Van der Waals-like SBH/LBH phase transition
happens in the extended space, no matter whether in the isobaric process by
fixing the pressure P or in the isothermal process by
fixing the temperature T of the system, the slopes of the QNM frequencies
change drastically being different in the small and large black hole phases
as the horizon radius r_+ is increasing. This
clearly shows the signature of the phase transition
between small and large black holes. Moreover, we have also found that,
at the critical isothermal and isobaric phase transitions,
QNM frequencies for both small and large black holes have the
same behavior, suggesting that QNMs are not appropriate to
probing the black hole second order phase transition.

Comparing with the action of Eq. (<ref>), Ref. <cit.> recently asserted
the existence of a Van der Waals-like SBH/LBH phase transition with
the massive potential U_3≠0 in the five dimensional case.
Moreover, the charged black hole <cit.>,
the Born-Infeld black hole <cit.> and black hole in the
Maxwell and Yang-Mills fields <cit.>
have recently been  constructed in Gauss-Bonnet massive gravity.
The Van der Waals-like SBH/LBH phase transition also appears in these models.
It would be interesting to extend our discussion to these
black hole solutions.



The work is supported by the National Natural Science Foundation
of China (NNSFC) (Grant No.11605152), and Natural Science Foundation
of Jiangsu Province (Grant No.BK20160452).
D.C.Z. are extremely grateful to
Hai-Qing Zhang and Hua-Bi Zeng for useful discussions.





§ APPENDIX SECTION


Here we present the related QNM frequencies of the massless scalar perturbation
around small and large black holes in the isobaric as well as in the
isothermal process.














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