Gravitational Quasinormal Modes of Regular Phantom Black Hole
Abstract
We investigate the gravitational quasinormal modes (QNMs) for a type of regular black hole (BH) known as phantom BH, which is a static selfgravitating solution of a minimally coupled phantom scalar field with a potential.
The studies are carried out for three different spacetimes: asymptotically flat, de Sitter (dS), and anti de Sitter (AdS).
In order to consider the standard odd parity and even parity of gravitational perturbations, the corresponding master equations are derived.
The QNMs are discussed by evaluating the temporal evolution of the perturbation field which, in turn, provides direct information on the stability of BH spacetime.
It is found that in asymptotically flat, dS and AdS spacetimes, the gravitational perturbations have similar characteristics for both odd and even parities.
The decay rate of perturbation is strongly dependent on the scale parameter , which measures the coupling strength between phantom scalar field and the gravity.
Furthermore, through the analysis of Hawking radiation, it is shown that the thermodynamics of such regular phantom BH is also influenced by .
The obtained results might shed some light on the quantum interpretation of QNM perturbation.
Keywords: regular phantom black hole; gravitational perturbation; quasinormal modes
pacs:
04.70.Bw; 04.62.+vI introduction
As a major topic in cosmology, the accelerated expansion of our universe has caused widespread concern in the scientific community. Since the effect of gravity causes the expansion speed to slow down, the accelerated expansion of the universe implies the existence of an unknown form of energy in the universe. The latter provides a repulsive force to push the expansion of the universe. Such unknown energy is called dark energy (DE). Subsequently, a large number of DE models have been proposed, among which the one with cosmological constant is the most famous. Even though the model of DE with the cosmological constant is reasonable in physical theory and consistent with most observations, two difficulties still remain unsolved, namely, how to derive “vacuum energy” from quantum field theory and why the magnitude of present DE and dark matter are of the same order.
Many modern astrophysics observations indicated the possibility of pressure to density ratio . For example, a modelfree data analysis from 172 type Ia supernovae (SNIa) resulted in a range of for our present epoch prl5 (). According to the WMAP data during 7 years, ins2 (). By using the data from Chandra telescope, an analysis of the hot gas in 26 Xray luminous dynamically relaxed galaxy clusters gives prl6 (). The data on SNIa from the SNLS3 sample estimates ins3 (). In fact, several DE models with a supernegative equation of state provide better fits to the above data r1add1 (); r1add2 (); r1add3 (); r1add5 (). And all these approaches are in favor of phantom DE scenario DEM1 (); DEM2 (); DEM3 (); DEM4 (); DEM5 (), in which a constant equation of state parameter is used r1add6 (); r1add7 (). This implies that the phantom model might be meaningful for indepth understanding of DE.
In the phantom model, the signature of the metric is , and the action of the model reads
(1) 
where is the scalar curvature, is the potential of the scalar field, corresponds to a phantom scalar field while is for a normal canonical scalar field.
Bronnikov and Fabris first investigated the properties of BH with phantom scalar field in vacuum, and derived a phantom regular BH solution 10 years agomainref2 (). Inside the event horizon of such phantom BH there is no singularity similar to the case of regular BHs with nonlinear electrodynamics sourcesregularQNMs (). Outside the event horizon, the properties of a phantom BH are similar to those of a Schwarzschild BH. Due to the absence of the singularity, such phantom regular BH solution has attracted much attention from researchers.
On the other hand, the research of BH perturbation has always been an important issue in BH physics. The first work on QNM in AdS spacetime was about scalar wave in SchwarzschildAdS spacetimeAdS0 (), which is then followed by a study on scalar wave in topological AdS spacetime AdS00 (). There are a large number of works on regular BH’s QNMsmainref (); regularQNMs (); regularQNMs2 (); regularQNMs3 (); regularQNMs4 (); finit4 (); Hawking8 (). Among various types of perturbation, gravitational perturbation is generally considered to be the most important form due to its practical significance. The intrinsic properties and the stability of a BH can be unfolded through its corresponding gravitational perturbation. In the fifties of last century, Regge and Wheeler began to study the gravitational perturbations of static spherically symmetric BHs. It was pointed out later that HD1 () the higher dimensional gravitational perturbations can be classified into three types, namely, scalargravitational, vectorgravitational, and tensorgravitational perturbations. The first two types are associated to odd (vectorgravitational) and even (scalargravitational) parity in accordance with the spatial inversion symmetry of the perturbations, and are of great physical interest wheeler (). These findings significantly simplify the study of gravitational perturbation of BH. Subsequently, people developed many new methods, and further studies on the gravitational perturbations result in a large number of master equations for various forms of BHs in 4dimensions wheeler (); Gq4D1 (); Gq4D2 (); Gq4D3 (), in higher dimensions HD1 (), and for stationary BHs Gq1 (); Gq2 (). In fact, gravitational perturbations of a BH may generate relatively strong gravitational waves (GWs). Recently, the GWs from a binary BH system has been detected by LIGO LIGO (), so BH is proven to be the most probable source of GWs by modern technology. Meanwhile since many alternative theories of gravity can produce the same GW signal within the present accuracy in far field, the reported GW detection still leaves a window for alternative gravity theories PLBnew (), which includes the theory of phantom BHs. Therefore, the properties of QNMs of gravitational perturbation near the horizon of phantom BH may provide us essential information on the underlying physics of gravity theory. This is the main purpose of the present study.
The thermodynamics of BHs is also an important subject in BH physics. Some works indicated that Hawking radiation can be considered as an effective quantum thermal radiation around the horizon Hawking1 (); Hawking2 (), where the corresponding Hawking temperature can be derived from the tunneling rate Hawking1 (); Hawking2 (); Hawking3 (); Hawking4 (); Hawking5 (); Hawking6 (). Furthermore, a natural correspondence between Hawking radiation and QNM has been established recently Hawking1 (); Hawking2 (); Hawking3 (); Hawking5 (); Hawking7 (). Therefore, in this work, we will also investigate the Hawking radiation of regular phantom BH.
The paper is organized as follows. In section II, we briefly review the regular phantom BH solutions and discuss their properties in three different spacetimes, namely, asymptotically flat, de Sitter (dS) and anti de Sitter (AdS). In this work, we focus on the odd parity and even parity gravitational perturbations. As the main component of this paper, section III includes two subsections. In subsection A, we derive the master equation for odd parity gravitational perturbation, and analyze the corresponding temporal evolution of the perturbed metric; In subsection B, corresponding studies are carried out for the even parity gravitational perturbation. In section IV, we calculate the Hawking radiation of the regular phantom BH. We summarize our results and draw concluding remarks in section V.
Ii The general metric for regular phantom black holes
In this section, we discuss the phantom () regular BH solution by considering the following static metric with spherical symmetry
(2) 
According to the action, Eq.(1), the field equation for a selfgravitating minimally coupled scalar field with an arbitrary potential can be expressed as
(3) 
By combining the scalar field equation,
(4) 
a regular phantom BH solution can be obtained as
(5) 
where
(6) 
and
(7) 
is the Schwarzschild mass defined in the usual way, and are integration constant and scale parameter respectively. Then it is necessary to determine the possible kinds of spacetime for such phantom BH, which can be classified as a regular infinity to be flat, de Sitter (dS) or Antide Sitter (AdS). The corresponding parameters should be restricted in each spacetime.
For the asymptotically flat spacetime, in accordance with Eq.(5), one has and
(8) 
In this case, the spacetime is asymptotically flat, namely, ,. And is the event horizon of the phantom BH. We note when , Eq.(5) becomes Schwarzschild flat spacetime.
For the de Sitter spacetime, one has
(9) 
(10) 
where , is the cosmological horizon and event horizon respectively. We note when , Eq.(5) becomes Schwarzschild dS metric.
For the Antide sitter spacetime, we choose with without loss of generality. By expanding the Eq.(5) around infinity, one finds
(11) 
and
(12) 
where is the event horizon of AdS spacetime. We note when , Eq.(5) can be returned to Schwarzschild AdS spacetime. In this context, the parameter measures the coupling strength between phantom scalar field and the gravity for all three spacetimes.
Since the parameters , can be expressed in terms of , and (dS), the structures of the regular phantom BH spacetime are completely determined by , and (dS) (cf. Fig.1). One can readily verify that all the spacetimes are indeed nonsingular even at . As for any asymptotically flat spacetime, such flat regular phantom BH has a Schwarzschildlike structure. However, its tendency of approaching flat spacetime at infinity becomes slower with increasing . In the dS case, the spacetime is bounded by two horizons, i.e., . There is a maximum for , and it decreases with increasing . For the AdS phantom BH, when , and for larger , the approach to the asymptotic solution becomes slower.
Iii Gravitational quasinormal frequencies for regular phantom black holes
As proposed by ReggeWheeler, two important gravitational perturbations are of odd and even parity. The perturbation gauge for each type has its own definition. In this section, we choose the ReggeWheelerZerilli gauge to discuss the master equation for each perturbation type. Here we take the magnetic quantum to make disappear completely because all values of lead to the same radial equation wheeler (). Since the total number of equations in the even parity case is bigger than that in odd parity case, the derivation of the master equation for even parity perturbation is thus more complicated. Once the master equation is derived, the effective potential and the corresponding qusinormal modes can be obtained. We will first discuss the odd parity gravitational quasinormal frequencies in asymptotically flat, dS and AdS spacetimes, then study the case for even parity. In our work, we consider the metric perturbations not only of the Ricci curvature tensor and scalar curvature (the l.h.s. of the Einstein field equation), but also of the energy momentum tensor (the r.h.s. of the Einstein field equation). On the other hand, we will not consider the perturbations of the phantom scalar field. This is because such perturbations can be canceled out through an appropriate choice of in the action (see Appendix for details).
In order to discuss gravitational perturbation, one may write down
(13) 
where the small perturbation will be divided into odd and even modes in the subsequent sections.
iii.1 Master equation and quasinormal modes for odd parity perturbation
The odd parity perturbation has the form aswheeler ()
(14) 
where ( is the Legendre function), which satisfies
(15) 
where , is the angular quantum number.
Then the separation of variables can be carried out by writing , . By substituting Eq.(5)(7), (13)(15) into the field equation Eq.(3) and only keeping the first order perturbation terms, we obtain the independent perturbation equations as follows:
(16) 
(17) 
Eq.(17) implies . Substituting into Eq.(16), we get the master equation for odd parity perturbation
(18) 
Finally, we renormalize by
(19) 
By substituting Eq.(19) into the master equation, Eq.(18), and using a tortoise coordinate , the Schrdingertype wave equation for this case can be expressed as
(20) 
where the effective potential for odd parity perturbations is
(21) 
Eq.(21) can be used to describe the effective potential of “odd”type perturbation in different spacetimes, and be utilized to discuss the relationship between the effective potential and model parameters such as the angular harmonic index and the parameter .
I) Fig.2, 3 and 4 show the potential functions, temporal evolution of the gravitational perturbation, and quasinormal frequency obtained by WKB method in asymptotically flat spacetime.

The form of the effective potential as a function of for different values of and is shown in Fig.2. From the left plot, one sees that as increases, the shape of the effective potential becomes smoother. The maximum of the effective potential decreases with increasing and the position of the peak shifts to the right. From the right plot, we see that with the increase of angular quantum number , the maximum of the effective potential also increases. is always found to be positive definite outside the event horizon, which indicates that the corresponding QNMs are likely to be stable.

We adopt the finite difference method to analyze the stability of such BH. By applying the coordinate transformation with , to Eq.(18), and integrating numerically using the finite difference method finit1 (); finit2 (); finit3 (); finit4 (), we obtain the differential equation for . Fig.3 shows the stability of a regular phantom BH in asymptotically flat spacetime with . The temporal evolution of each mode in Fig.3 corresponds to a corresponding case in Fig.2. Since decreases significantly with the growth of , the decay rate () becomes smaller and oscillation frequency (i.e., ) also drops; As increases, the values of are raised correspondingly, so that the oscillation frequency (i.e., ) slightly increases together with the decay rate ().

We employ the WKB approximation WKB1 (); WKB2 () to evaluate the quasinormal frequencies. The complex frequency is determined byWKB3 ()
(22) (23) By making use of Eq.(21), we evaluate the QNM frequencies by employing the order WKB method (see Fig.4). Figure 4 shows that the fundamental quasinormal modes () have the smallest imaginary parts, as the modes decay the slowest. As increases, the imaginary part of the corresponding quasinormal mode becomes bigger for given and . For a given principal quantum number , both the real and the imaginary parts of the frequency decrease with increasing ; on the other hand, the real part of the frequency increases significantly with the angular quantum number , while the imaginary part also increases slightly with increasing .
II) Fig. 5,6 and 7 show the potential function between the event horizon and cosmological horizon , temporal evolution of the gravitational field, and quasinormal frequency obtained by WKB method in dS spacetime.

The form of the effective potential in dS spacetime is similar to that in asymptotically flat spacetime. Fig.5 indicates that for given and , the effective potential decreases with increasing ; and for given and , it increases with increasing angular quantum number . In the range , is also positive definite, which implies that the corresponding QNM is likely to be stable.

Fig.6 studies the stability of a regular phantom with . Similar to the case in asymptotically flat spacetime, it is found with increasing , the oscillation frequency (i.e., ) decreases, while the decay rate () becomes smaller. Therefore, for smaller value of , the BH returns to its stable state more quickly as small perturbation dies out faster.

In accordance with the results of finite difference method, the frequencies presented in Fig.7 by WKB method also show that for given and , the decay rate of perturbation (i.e., ) decreases with increasing . Moreover, it illustrates that for a given , the fundamental mode can be found at . Due to the smallness of the real parts of the frequencies, these modes of oscillation persist for longer time.
III) Fig.8 and 9 show the potential function beyond the event horizon and the temporal evolution of the gravitational perturbation in AdS spacetime.

The form of the effective potential in AdS space time is quite different from that in asymptotically flat and dS spacetime. Fig.8 indicates that the value of is divergent as . The effects of and on the effective potential are similar to previous cases, for a given , smaller or larger leads to bigger .

Fig.9 studies the stability of AdS regular phantom BH spacetime with . The results are consistent with the above calculated potential function. It is again inferred that the fundamental mode of gravitational perturbation in odd parity occurs for and larger , since such kind of QNM will take a longer time to be stable.
iii.2 Master equation and quasinormal modes for even parity perturbation
Another canonical form for the gravitational perturbations is of even parity. After applying the separation of variables, it can be expressed as wheeler (); Gq4D1 ()
(24) 
where are unknown functions for the even parity perturbation. It is noted that these functions are not independent.
Now we derive the first order perturbation equations by substituting Eq.(5)(7), (13) and (24) into Eq.(3), and find the relationships among .
(25) 
(26) 
(27) 
(28) 
(29) 
where
(30) 
and
(31) 
Solving the Eq.(26), can be expressed as
(32) 
We define a function satisfying
(33) 
By observing Eqs.(25), (32) and (33), it turns out that we need to express by in order to express all perturbation functions , , and in terms of . This can be achieved by substituting Eqs.(32) and (33) into Eqs.(28),(29), and evaluating the subtraction , and one eventually obtains following expression
(34) 
where
(35) 
and
(36) 
Substituting Eq.(32)(36) into Eq.(27), can be solved as
(37) 
where , are
(38) 
(39) 
The resulting master equation can be derived by evaluating
(40) 
By substituting Eq.(32), (33) and (37) into the above equation, the corresponding master equation is given by
(41) 
where . Finally, one defines and (where is the coefficient of ), the master equation can be simplified into the following equation: