Advances in High Energy Physics

Volume 2017 (2017), Article ID 5234214, 19 pages

https://doi.org/10.1155/2017/5234214

## Gravitational Quasinormal Modes of Regular Phantom Black Hole

^{1}College of Physics, Chongqing University, Chongqing 401331, China^{2}State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China^{3}Instituto de Física e Química, Universidade Federal de Itajubá, 37500-903 Itajubá, MG, Brazil^{4}Department of Astronomy, China West Normal University, Nanchong, Sichuan 637002, China^{5}Escola de Engenharia de Lorena, Universidade de São Paulo, São Paulo, SP, Brazil^{6}Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista, Guaratinguetá, SP, Brazil

Correspondence should be addressed to Jin Li; moc.liamtoh@vratsqc

Received 24 August 2016; Accepted 9 October 2016; Published 16 February 2017

Academic Editor: Christian Corda

Copyright © 2017 Jin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We investigate the gravitational quasinormal modes (QNMs) for a type of regular black hole (BH) known as phantom BH, which is a static self-gravitating 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.

#### 1. 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 magnitudes 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 model-free data analysis from 172 type Ia supernovae (SNIa) resulted in a range of for our present epoch [1]. According to the WMAP data during 7 years, [2]. By using the data from Chandra telescope, an analysis of the hot gas in 26 X-ray luminous dynamically relaxed galaxy clusters gives [3]. The data on SNIa from the SNLS3 sample estimates [4]. In fact, several DE models with a supernegative equation of state provide better fits to the above data [5–8]. And all these approaches are in favor of phantom DE scenario [9–13], in which a constant equation of state parameter is used [14, 15]. This implies that the phantom model might be meaningful for in-depth understanding of DE.

In the phantom model, the signature of the metric is , and the action of the model readswhere is the scalar curvature, is the potential of the scalar field, and 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 ago [16]. Inside the event horizon of such phantom BH there is no singularity similar to the case of regular BHs with nonlinear electrodynamics sources [17]. 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 Schwarzschild-AdS spacetime [18], which is then followed by a study on scalar wave in topological AdS spacetime [19]. There are a large number of works on regular BH’s QNMs [17, 20–25]. 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 [26] the higher dimensional gravitational perturbations can be classified into three types, namely, scalar-gravitational, vector-gravitational, and tensor-gravitational perturbations. The first two types are associated with odd (vector-gravitational) and even (scalar-gravitational) parity in accordance with the spatial inversion symmetry of the perturbations and are of great physical interest [27]. 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 4-dimensions [27–30], in higher dimensions [26], and for stationary BHs [31, 32]. In fact, gravitational perturbations of a BH may generate relatively strong gravitational waves (GWs). Recently, the GWs from a binary BH system have been detected by LIGO [33], 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 [34], which included 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 [35, 36], where the corresponding Hawking temperature can be derived from the tunneling rate [35–40]. Furthermore, a natural correspondence between Hawking radiation and QNM has been established recently [35–37, 39, 41]. Therefore, in this work, we will also investigate the Hawking radiation of regular phantom BH.

The paper is organized as follows. In Section 2, 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 3 includes two subsections. In Section 3.1, we derive the master equation for odd parity gravitational perturbation and analyze the corresponding temporal evolution of the perturbed metric; in Section 3.2, corresponding studies are carried out for the even parity gravitational perturbation. In Section 4, we calculate the Hawking radiation of the regular phantom BH. We summarize our results and draw concluding remarks in Section 5.

#### 2. 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:According to the action, (1), the field equation for a self-gravitating minimally coupled scalar field with an arbitrary potential can be expressed asBy combining the scalar field equation, a regular phantom BH solution can be obtained aswhere is the Schwarzschild mass defined in the usual way, and 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 anti-de Sitter (AdS). The corresponding parameters should be restricted in each spacetime.

*For the asymptotically flat spacetime*, in accordance with (5), one has andIn this case, the spacetime is asymptotically flat, namely, , . And is the event horizon of the phantom BH. We note when that (5) becomes Schwarzschild flat spacetime.

*For the de Sitter spacetime*, one haswhere , are the cosmological horizon and event horizon, respectively. We note when that (5) becomes Schwarzschild dS metric.

*For the anti-de Sitter spacetime*, we choose with without loss of generality. By expanding (5) around infinity, one findswhere is the event horizon of AdS spacetime. We note when that (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. Figure 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 Schwarzschild-like 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, that is, . 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.