Quasi Normal Modes and PV Criticallity for scalar perturbations in a class of dRGT massive gravity around Black Holes
Abstract
We investigate black holes in a class of dRGT massive gravity for their quasi normal modes (QNMs) for neutral and charged ones using Improved Asymptotic Iteration Method (Improved AIM) and their thermodynamic behavior. The QNMs are studied for different values of the massive parameter for both neutral and charged dRGT black holes under a massless scalar perturbation. As increases, the magnitude of the quasi normal frequencies are found to be increasing. The results are also compared with the Schwarzchild de Sitter (SdS) case. PV criticallity of the aforesaid black hoels under massles scalar perturbation in the de Sitter space are also studied in this paper. It is found that the thermodynamic behavior of a neutral black hole shows no physically feasible phase transition while a charged black hole shows a definite phase transition.
Keywords:
Quasi Normal Modes dRGT Massive Gravity PV Criticallity∎
1 Introduction
The existence of black holes is an outcome of Einstein’s
General Theory of Relativity (GTR). The question then is how to
realize their existence and one natural way to identify them is
to try to perturb and know their responses to the perturbation.
Regge and Wheeler1 () started way back in 1950s studying
perturbations of blackhole space times and later, serious
studies were initiated by Zerilli2 (). It was
Vishveshwara3 () who first noticed the existence of
quasinormal modes (QNMs) by studying the scattering of
gravitational waves by Schwarzschild black holes. Later, scattering
of scalar, electromagnetic and Fermi fields by different
blackhole spacetimes have been studied by
manyref7 (); ref8 (); ref9 () and references cited therein. In the frame
work of general relativity, QNMs arise as perturbations of black
hole spacetimes. QNMs are the solutions to perturbation equations
and they are distinguished from ordinary normal modes because they
decay at certain rates, having complex frequencies. The remarkable
property of the black hole QNMs(”ring down” of black holes) is that
their frequencies are uniquely determined by the mass, angular
momentum and charge(if any) of black holes. Black holes can be
detected by observing the QNMs through gravitational waves. When a
star collapses to form a black hole or when two black holes collide
or a black hole and a star collide, Gravitational Waves (GWs) are
emitted. The result of these processes is a black hole with higher
mass that absorbs the GWs 6 (). Hence the emitted GWs decay
quickly. The decay of oscillations are characterized by complex
frequencies.
The Quasi normal modes were first introduced by Vishveshwara
7 (); 8 (). Later, perturbation calculations have been done by many
to get QNM oscillations 9 (); 10 (); 11 (). To study the black hole
QNMs, the solution of the perturbed field equation are separated for
the radial and angular parts, whose radial part is the so called
ReggeWheeler equation. But this technique is time consuming and
complicated that makes it difficult to survey QNMs for a wide range
of parameter values. A semi analytic method has then been explored
12 () that has its own limitations of accuracy. Later, the
Continued Fraction Method (CFM) was proposed by Leaver. This method
is a hybrid of analytic and numerical and can calculate QNM
frequencies by making use of analytic infinite series representation
of solution 13 (). Another method is WKB approximation which is
very commonly employed and a powerful one too. However all these
methods have their own limitations. In recent years a new approach
has been introduced to study black hole QNMs called Asymptotic
Iteration Method (AIM) which is previously used to solve eigenvalue
problems 14 (). This method has been shown to be efficient and
accurate for calculating
QNMs of black holes 15 (); 16 ().
The studies of Hawking and Bekenstein made in 1970s17 (); 18 ()
helped us to view that black holes are thermal objects
possessing temperature and entropy and that laws of black hole
dynamics are analogous to the laws of classical
thermodynamics. An immediate consequence of these studies is
that they bring together quantum theory, gravity and
thermodynamics and one can hope for a quantum theory of
quantum gravity. Various methods 19 (); 20 () have been developed
to study the thermodynamics of black holes. An important fact is
that certain black holes make a transition from a stable phase to
an unstable phase and some are thermodynamically
unstable21 (). If the thermodynamic variables, pressure and
volume, are identified, then an equation of state corresponding to
the black hole can be found out and the critical points can be
determined. The PV isotherms then show their thermodynamic
behavior.
GTR helped us to have a model for our universe and the universe
can be considered as a dynamical system and most of the
cosmological and astronomical observations could find a meaningful
explanations under GTR. But there are some fundamental issues like
quantization of gravity, the initial stages of the evolution of
the Universe under BigBang theory and also certain
astronomical observations like dark matter and the late time
accelerating expansion of the universe which lacked proper
explanations under GTR 22 (); 23 (). Hence attempts are being
made for an alternative theory of gravitation.
From the perspective of the modern particle physics, GTR can be
thought of as the unique theory of a massless spin 2 particle called
graviton 26 (); 27 (); 28 (). If the assumption behind the uniqueness
theorem is broken, it can lead to alternative theories of gravity.
Theories concerning the breaking of Lorentz invariance and spin have
been explored in depth. Representing gravity as a manifestation of a
higher order spin, thereby maintaining the Lorentz invariance and
spin has also been explored largely 29 (). Yet another
possibility that has been recently explored is the so called
’Massive Gravity’(MG) theory30 (); ref12 (); ref13 (). In this model
gravity is considered to be propagated by a massive spin 2
particle. The theory gets complicated especially when the massive
spin 2 field interacts with matter. In that case, the theory goes
completely nonlinear and consequently non renormalizable. A non
self interacting massive graviton model was first suggested by Fierz
and Pauli31 () which is now called as ’linear massive gravity’.
However this model suffers from a pathology32 () thereby ruling
out the theory on the basis of solar system tests. Later,
Vainshtein33 () proposed that the linear massive gravity model
can be recovered to GTR through ’Vainshtein Mechanism’ at small
scales by including non linear terms in the hypothetical massive
gravity theory. But the Vainshtein mechanism is later found to
suffer from the so called ’BoulwareDeser’(BD) ghost34 ().
Recently it is shown by de Rham, Gabadadze and Tolly in their series
of works 35 (); 36 (); 37 () that the BD ghost can be avoided for a sub
class of massive potentials. This is called dRGT massive gravity
which includes one dynamical and one fixed metric. This
also holds true for its bi gravity extension 30 (); 32 (); ref3 ().
This paper deals with the study of quasinormal modes coming out of
massless scalar perturbations of a class of dRGT massive gravity
around both neutral and charged black holes. We use the Improved
Asymptotic Iteration Method (AIM) to calculate the QNMs. The PV
criticality condition of such black holes are also verified in the
de Sitter space. Section 2 deals with a review of the Asymptotic
Iteration Method. In section 3, the quasinormal modes of neutral and
charged black holes coming under a class of dRGT massive gravity,
proposed by Ghosh, Tannukij and Wongjun 39 (), are found out.
Section 4 deals with the PV criticality in the extended phase space
of black holes described in Section 3. Section 5 concludes the
paper.
2 Review of Asymptotic Iteration Method
Asymptotic Iteration Method(AIM) was proposed initially for finding
solutions of the second order differential equations of the form 40 (),
(1) 
where and are coefficients of the differential equation and are well defined functions and sufficiently differentiable. By differentiating (1) with respect to ,
(2) 
where the new coefficients are and . Differentiating twice with respect to leads to,
(3) 
where the new coefficients are and . This process is continued to get the derivative of with respect to as,
(4) 
where the new coefficients are related to the older ones through the following expressions,
(5)  
(6) 
where
The ratio of derivative and derivative can
be obtained from as,
By introducing the asymptotic concept that for sufficiently large values of ,
(7) 
where is a constant, we get,
from which a general expression for can be found outref10 (). From we can write,
(8) 
The roots of this equation are used to obtain the eigenvalues of
. The energy eigenvalues will be contained in the coefficients.
To get the eigenvalues, each derivative of and are
found out and expressed in terms of the previous iteration. Then by
applying the quantization condition given by , a general
expression for the eigenvalue can be arrived at. Cifti et al.
41 () first noted that this procedure has a difficulty in that,
the process of taking the derivative of and terms of
the previous iteration at each step can consume time and also affect
the numerical precision of calculations. To overcome this
difficulty, an improved version of AIM has been proposed that
bypasses the need to take derivative at each iteration. This is
shown to improve both accuracy and speed of the method. For that,
and are expanded in a Taylor series around the
point at which AIM is performed, ,
(9)  
(10) 
where and are the Taylor coefficients of and respectively. Substitution of equations and in and lead to the recursion relation for the coefficients as,
(11)  
(12) 
Applying and in , the quantization condition can be rewritten as,
(13) 
This gives a set of recursion relations that do not require any derivatives. The coefficients given by and can be computed by starting at and iterating up to until the desired number of recursions are reached. The quantization condition given by contains only term. So, only the coefficients with where is the maximum number of iterations to be performed needs to be determined. The perturbed radial wave equation of a black hole can be written in the form of a second order differential equation similar to with the coefficients containing their quasinormal frequencies. Hence the condition can be employed to extract the QNMs of a black hole15 (); 16 (). This method is used in this paper to determine the QNMs of dRGT black hole.
3 Quasinormal modes of Black Holes in dRGT massive gravity
3.1 Neutral dRGT black hole
In the standard formalism of dRGT massive gravity theory, the EinsteinHilbert action is given by 42 (); ref14 (),
(14) 
where is the metric tensor, is the Ricci scalar, represents the graviton mass and is the effective potential for the graviton and is given by 43 (),
(15) 
where and are two free parameters. These parameters are redefined by introducing two new parameters and as,
(16)  
(17) 
Varying the action given by with respect to the metric leads to the field equation,
(18) 
where,
(19) 
The constraints of this field equation can be obtained by using the Bianchi identity,
(20) 
A spherically symmetric metric has a form given by,
(21) 
with and where is a constant in terms of and ref4 (); ref5 (); ref6 (). The exact solution for this ansatz is complicated. It is simplified by choosing specific relations for the parameters. In this paper, we take . Since the fiducial metric acts like a Lagrangian multiplier to eliminate the BD ghost, to simplify the calculations, we choose the fiducial metric as, 44 (),
(22) 
where is a constant. In this paper we consider only the diagonal branch of the physical metric for simplicity ie., . Then,
By taking we get,
The nonzero components of the Einstein tensor are given by39 (),
(23)  
(24)  
(25)  
(26) 
and the tensor as,
(27)  
(28)  
(29)  
(30) 
Solving using these expressions for and gives the form of the metric as,
(31) 
where,
(32)  
(33)  
(34) 
The details of the above calculations are given by Ghosh,
Tannukij and Wagjun 39 (). When , and
will determine the nature of the solution. ie., if
we get a Schwarzschildde Sitter type
solution, if we will get a Schwarzschildanti
de Sitter type solution and when we get a Schwarzchild black hole.
In this paper, we consider a static spherically symmetric space time
with vanishing Energy momentum tensor and hence the field
perturbations in such background are not coupled to the
perturbations of the metric and therefore are equivalent to test
field in black hole background. Consider a massless scalar field
that satisfies the KleinGordon equation in curved spacetime,
(35)  
(36) 
where,
(37) 
In order to separate out the angular variables we choose the ansatz:
(38) 
where gives the frequency of the oscillations corresponding to the black hole perturbation, are the spherical harmonics and,
(39) 
Substituting in and using and we get the radial wave equation,
(40) 
By using tortoise coordinate , the above equation can be brought into the standard form45 (),
(41) 
where,
(42) 
The SdS black hole has three singularities given by the roots of , which are the event horizon, , the cosmological horizon, and at = ( + . The QNMs are defined as solutions of the above equation with boundary conditions: as and as for an time dependence that corresponds to ingoing waves at the horizon and out going waves at infinity. The surface gravity at these singular points are defined as,
(43) 
In the present study we are using improved AIM for finding the QNMs of the dRGT black hole and hence it is convenient to make a change of variable as in leading to,
(44) 
where,
(45)  
(46) 
In de Sitter space, the radial equation has got 3 singularities and these are represented as (Event horizon), (Cosmological horizon) and and hence we can write ref11 (); 15 (),
(47) 
The idea is to scale out the divergent behavior at the cosmological horizon first and then rescale at the event horizon for a convergent solution. Now to scale out the divergent behavior at cosmological horizon, we take,
(48) 
The master equation given by then takes the form,
(49) 
The correct scaling condition of QNM at the event horizon implies,
(50) 
The master equation then can be viewed of the form as,
(51) 
where and are the coefficients of the second order differential equation. It can be seen from that the coefficient of includes the frequency . Therefore the quantization condition given by can be used to find out the of by iterating to some maximum. For calculating the QNMs, we have used the MATHEMATICA NOTEBOOK given in the reference 46 (). Initially the QNMs are calculated for the SdS by making and the results are compared with referenceref1 (); ref2 () in Table . It can be seen that the results agree quite well with those found in the existing literature.
(for dRGT)  (for SdS)  

0  0.483644 – 0.0967588 i  0  0.48364  0.09677 i 
0.02  0.434585 – 0.0885944 i  0.02  0.43461  0.08858 i 
0.04  0.380784 – 0.0787610 i  0.04  0.38078  0.07876 i 
0.06  0.320021 – 0.0668449 i  0.06  0.32002  0.06685 i 
0.08  0.247470 – 0.0519043 i  0.08  0.24747  0.05197 i 
0.09  0.202960 – 0.0425584 i  0.09  0.20296  0.04256 i 
0.10  0.146610 – 0.0306869 i  0.10  0.14661  0.03069 i 
0.11  0.0461689 – 0.0063134 i  0.11  0.04617  0.00963 i 
We have executed iterations while calculating the QNMs. We have
taken while calculating the QNMs so that the
results of the calculations will correspond to that in de Sitter
space.
Table shows the quasi normal frequencies obtained through
improved AIM method. The values of and are chosen
so that remains negative. We have chosen the values
in these calculations. The table shows the quasinormal modes
calculated for and respectively for the same range
of and values. It can be seen that for the same
and , increasing the value of increases the
magnitude of the cosmological constant, which is obvious from
. Also as increases, the quasinormal frequencies are
seen to be increasing in magnitude for both and modes.
As for every , both the real and imaginary parts of the
quasinormal frequencies are seen to be continuously increasing in
magnitude as increases. Comparing these quasinormal
frequencies with Table , it can be seen that the values of the
quasinormal frequencies when takes a finite value are higher
in magnitude than when which corresponds to a Schwarzschild
case.
0.080  0.80  1.9840  1.15155 – 0.348046 i  1.62914 – 0.341517 i 
0.088  0.80  1.9904  1.15615 – 0.350418 i  1.63572 – 0.343749 i 
0.096  0.80  1.9968  1.16081 – 0.352759 i  1.64237 – 0.346001 i 
0.104  0.80  2.0032  1.16552 – 0.355121 i  1.64910 – 0.348271 i 
0.112  0.80  2.0096  1.17030 – 0.357501 i  1.65590 – 0.350560 i 
0.120  0.80  2.0160  1.17512 – 0.359902 i  1.66278 – 0.352868 i 
0.128  0.80  2.0224  1.18001 – 0.362322 i  1.66974 – 0.355195 i 
0.100  1.00  3.1000  2.81587 – 1.049800 i  3.90051 – 1.026860 i 
0.110  1.00  3.1100  2.83013 – 1.057140 i  3.91984 – 1.033950 i 
0.120  1.00  3.1200  2.84445 – 1.064510 i  3.93924 – 1.041070 i 
0.130  1.00  3.1300  2.85881 – 1.071910 i  3.95870 – 1.048210 i 
0.140  1.60  3.1400  2.87322 – 1.079340 i  3.97823 – 1.055380 i 
0.150  1.75  3.1500  2.88768 – 1.086800 i  3.99781 – 1.062580 i 
0.160  1.90  3.1600  2.90220 – 1.094280 i  4.10746 – 1.069800 i 
3.2 Charged dRGT black hole
Consider a charged black hole from the class of dRGT massive gravity with the metric,
(52) 
where39 (),
(53) 
where corresponds to the charge. Proceeding as in section , the wave equation is found as,
(54) 
where,
(55)  
(56) 
Scaling out the divergent behavior at the event horizon leads to the master equation,
(57) 
Again, the correct scaling condition of QNMs at the event horizon implies,
(58) 
where,
(59) 
The master equation is now in the form of so that the
quantization condition given by can be employed to find out
the QNMs.
Table shows the quasinormal modes calculated using the improved
AIM method for different values of and . We have
chosen the values and in these calculations. The
QNMs are studied as in the prevoius section by varying the
value while keeping the values of and the same. It
can be seen that as increases, the real part of the quasi
normal frequency deceases while the magnitude of the imaginary part
increases. For each the quasi normal frequency vary
continuously. A black hole is stable only when the imaginary part in
its Quasi normal spectrum is negative47 (). It is noted while
calculating the Quasinormal modes that the roots of the frequency,
give positive as well as negative imaginary frequencies.
Here we are interested in the stable modes and therefore considered
only the negative imaginary parts of . iterations have
been done for calculating the QNMs.
0.080  0.80  1.9840  2.43544 – 0.523799 i  1.67618 – 0.168257 i 
0.088  0.80  1.9904  2.43455 – 0.535763 i  1.67635 – 0.180489 i 
0.096  0.80  1.9968  2.43252 – 0.547233 i  1.67351 – 0.195613 i 
0.104  0.80  2.0032  2.42939 – 0.558215 i  1.67069 – 0.209057 i 
0.112  0.80  2.0096  2.42523 – 0.568718 i  1.66693 – 0.222338 i 
0.120  0.80  2.0160  2.42021 – 0.578624 i  1.66230 – 0.235427 i 
0.128  0.80  2.0224  2.41399 – 0.588313 i  1.65677 – 0.248391 i 
0.10  1.00  3.1000  0.304084 – 2.99974 i  0.9866449 – 4.93190 i 
0.11  1.00  3.1100  0.342169 – 3.05263 i  1.0195500 – 5.01834 i 
0.12  1.00  3.1200  0.378347 – 3.10442 i  1.0531600 – 5.10348 i 
0.13  1.00  3.1300  0.413140 – 3.15511 i  1.0872800 – 5.18734 i 
0.14  1.60  3.1400  0.446882 – 3.20472 i  1.1219000 – 5.26998 i 
0.15  1.75  3.1500  0.479812 – 3.25326 i  1.1570200 – 5.35141 i 
0.16  1.90  3.1600  0.512100 – 3.30072 i  1.1926600 – 5.43168 i 
4 PV Criticality of black holes
4.1 Black holes in dRGT massive gravity
In this section we look into the thermodynamic critical behavior of black holes described by the metric in the extended phase space. We intend to check whether the black hole exhibits any phase transition by showing an inflection point in the indicator diagram. Here, the cosmological constant, is treated as representing a negative pressure 48 () as,
(60) 
For the metric given by would lead to the case of a de Sitter space provided is negative. Keeping this in mind we take,
(61) 
where is the pressure. The boundary of the black hole is described by the black hole horizon, and is determined by the condition, . From this condition, the mass of the black hole can be expressed in terms of as,
(62) 
and the black hole mass is considered to be the enthalpy of the system. The thermodynamic volume, is given by,49 (); 50 ()
(63) 
Varying partially with respect to the pressure P, we get
(64) 
The temperature of the black hole, described by the metric in , given by the Hawking temperature can be written as 51 (),
(65) 
Substituting for from in the above equation and rearranging it we get an expression for the cosmological constant,
(66) 
But from , the cosmological constant can be related to the pressure as . Therefore can be written in terms of as,
(67) 
Or,
(68) 
where,
(69)  
(70) 
From , can be treated as a shifted temperature. From , thermodynamic volume is a monotonic function of the horizon radius . and hence can be considered to be corresponding to . Therefore, can be treated as an equation of state describing the black hole. The critical point is then determined by the conditions,
(71) 
and
(72) 
Substituting for P from in the above differential equation it is found that the conditions given by and are not simultaneously satisfied. The condition,
(73) 
gives the critical horizon as,
(74) 
Evaluation of gives a non zero value which can imply either a local maximum or a local minimum depending on whether the value is greater than or less than zero. The critical pressure is found out by substituting in which gives,
(75) 
This critical point corresponds to a physically feasible one if is positive 52 (). From it can be seen that this happens only if is negative irrespective of the sign of . The relation between shifted temperature, , critical pressure, and horizon radius can be found out from and as,
(76) 
This ratio is called the ‘Compressibility Ratio’. The value of compressibility ratio for a Van der Waal’s gas is . Hence, the black hole system, with the Compressibility Ratio given by , can be thought of as behaving like a near Van der Waal’s system. The diagram plotted for different shifted temperature is shown in Figure . In the first figure, the curves are plotted for , the curves are seen to show critical behavior but it likely does not correspond to a physical one because, from , for the above said values of and the critical pressure turns out to be negative for these curves. The second figure is plotted for , they show inflection point but there is no phase transition.
4.2 Charged dRGT Black Hole
Consider a charged black hole with the metric of the form . The Hawking Temperature for this metric can be found out as,
(77) 
From the above equation, the equation of state can obtained proceeding as described in Section . The mass, of the black hole can be written in terms of the horizon radius as,
(78) 
Substituting in we get,
(79) 
Writing this equation in terms of ,
(80) 
Or,
(81) 
where,
(82)  
(83)  
(84) 
describes the equation of state. The critical point is then determined by the conditions,
(85) 
and
(86) 
Unlike in the Section , it is found that and are simultaneously satisfied which gives the solutions, for critical horizon as,
(87) 
and for the critical temperature as,
(88) 
Using , and , an expression for the critical pressure can be arrived at as,
(89) 
The relation connecting shifted temperature , critical pressure, and critical horizon radius are found as,
(90) 
which is exactly the same as in the case for a Van der Waal’s system. The PV diagram plotted for different shifted temperature is shown in Figure . In the first figure, the curves are plotted for and . The second figure is plotted for and . The first figure shows an inflection point and a phase transition, but the second does not, as is obvious due to the sign change of .
5 Conclusion
In this paper, the quasinormal modes coming out of massless scalar
perturbations in black hole spacetime in a class of dRGT massive
gravity, is studied. We have used the Improved Asymptotic Iteration
Method (Improved AIM) to find out the QNMs in the de Sitter space.
We have done iterations for calculating the QNMs. The Quasi
normal modes are studied by varying the massive parameter, . It
is found that as increases the magnitude of the quasi normal
frequencies increase for neutral black hole. These QNMs are also
higher in magnitude compared to the SdS case. It is also found that
as and tend to zero, the results converge to the
SdS case. For a charged black hole, the real part of the quasi
normal frequency decreases and the magnitude of imaginary part
increases as is increased.
The criticality in the extended phase space of the aforesaid
black holes are also determined. The neutral black holes show a near
Van der Waal behavior with the compressibility ratio of . But
it does not show any physically feasible phase transition for the de
Sitter space. The charged black hole on the other hand exactly shows
a Van der Waal’s behavior and clearly exhibits a phase transition.
Acknowledgements.
The authors would like to thank the reviewers for their valuable suggestions. One of us (PP) would like to thank UGC, New Delhi for financial support through the award of a Junior Research Fellowship (JRF) during 201012 and SRF during 201213. VCK would like to acknowledge Associateship of IUCAA, Pune.References
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