Quasinormal modes and greybody factors of a four-dimensional Lifshitz black hole with
We study scalar perturbations for a four-dimensional asymptotically Lifshitz black hole in conformal gravity with dynamical exponent , and spherical topology for the transverse section, and we find analytically and numerically the quasinormal modes for scalar fields for some special cases. Then, we study the stability of these black holes under scalar field perturbations and the greybody factors.
Lifshitz spacetimes have received considerable attention from the condensed matter point of view due to the AdS/CFT correspondence, i.e., searching for gravity duals of Lifshitz fixed points for condensed matter physics and quantum chromodynamics Kachru:2008yh (). From the quantum field theory point of view, there are many invariant scale theories of interest when studying such critical points. Such theories exhibit the anisotropic scale invariance , , with , where is the relative scale dimension of time and space, and these are of particular interest in studies of critical exponent theory and phase transitions. Systems with such behavior appear, for instance, in the description of strongly correlated electrons. The importance of possessing a tool to study strongly correlated condensed matter systems is beyond question, and consequently much attention has focused on this area in recent years.
One of the most well studied systems in the context of gauge/gravity duality, is the holographic superconductor. In its simplest form, the gravity sector is a gravitating system with a cosmological constant, a gauge field and a charged scalar field with a potential. The dynamics of the system defines a critical temperature above which the system finds itself in its normal phase and the scalar field does not have any dynamics. Below the critical temperature the system undergoes a phase transition to a new configuration. From the gravity side this is interpretated as the black hole to acquire hair while from boundary conformal field theory site this is interpretated as a condensation of the scalar field and the system enters a superconducting phase. In this sense, Lifshitz holographic superconductivity has been a topic of numerous studies and interesting properties are found when one generalizes the gauge/gravity duality to non-relativistic situations Hartnoll:2009ns (); Brynjolfsson:2009ct (); Sin:2009wi (); Schaposnik:2012cr (); Momeni:2012tw (); Bu:2012zzb (); Keranen:2012mx (); Zhao:2013pva (); Lu:2013tza (); Tallarita:2014bga ().
The Lifshitz spacetimes are described by the metrics
where represents a dimensional spatial vector, is the spacetime dimension and denotes the length scale in the geometry. If , the spacetime is the usual anti-de Sitter metric in Poincaré coordinates. Furthermore, all scalar curvature invariants are constant and these spacetimes have a null curvature singularity at for , which can be seen by computing the tidal forces between infalling particles. This singularity is reached in finite proper time by infalling observers, so the spacetime is geodesically incomplete Horowitz:2011gh (). The metrics of Lifshitz black holes asymptotically have the form (1); however, obtaining analytical solutions does not seem to be a trivial task, and therefore constructing finite temperature gravity duals requires the introduction of strange matter content with a theoretical motivation that is not clear. Another way of finding such a Lifshitz black hole solution is by considering carefully-tuned higher-curvature modifications to the Hilbert-Einstein action, as in new massive gravity (NMG) in 3-dimensions or corrections to general relativity. This has been done, for instance, in AyonBeato:2009nh (); Cai:2009ac (); AyonBeato:2010tm (); Dehghani:2010kd (). A 4-dimensional topological black hole with was found in Mann:2009yx (); Balasubramanian:2009rx () and a set of analytic Lifshitz black holes in higher dimensions for arbitrary in Bertoldi:2009vn (). Lifshitz black holes with arbitrary dynamical exponent in Horndeski theory were found in Bravo-Gaete:2013dca () and non-linearly charged Lifshitz black holes for any exponent in Alvarez:2014pra (). Thermodynamically, it is difficult to compute conserved quantities for Lifshitz black holes; however, progress was made on the computation of mass and related thermodynamic quantities by using the ADT method Devecioglu:2010sf (); Devecioglu:2011yi () as well as the Euclidean action approach Gonzalez:2011nz (); Myung:2012cb (). Also, phase transitions between Lifshitz black holes and other configurations with different asymptotes have been studied in Myung:2012xc (). However, due to their different asymptotes these phases transitions do not occur.
Conformal gravity is a four-derivative theory and is perturbatively renormalizable Stelle:1976gc (); Stelle:1977ry (). Also, it contains ghost-like modes in the form of massive spin-2 excitations. However, a solution to the ghost problem in fourth order derivative theories was shown in Mannheim:2006rd () by using the method of Dirac constraints Dirac () to quantize the Pais-Uhlenbeck fourth order oscillator model Pais:1950za (). In this work, we consider a matter distribution outside the event horizon of the Lifshitz black hole in -dimensions in conformal gravity with a spherical transverse section and dynamical exponent . It is worth mentioning that for the previously mentioned anisotropic scale invariance corresponds to space-like scale invariance with no transformation of time. The matter is parameterized by scalar fields minimally and conformally coupled to gravity. Then, we obtain analytically and numerically the quasinormal frequencies (QNFs) Regge:1957td (); Zerilli:1971wd (); Zerilli:1970se (); Kokkotas:1999bd (); Nollert:1999ji (); Konoplya:2011qq () for scalar fields, after which we study their stability under scalar perturbations. Also, we compute the reflection and transmission coefficients and the absorption cross section.
The study of the QNFs gives information about the stability of black holes under matter fields that evolve perturbatively in their exterior region, without backreacting on the metric. In general, the oscillation frequencies are complex, where the real part represents the oscillation frequency and the imaginary part describes the rate at which this oscillation is damped, with the stability of the black hole being guaranteed if the imaginary part is negative. The QNFs are independent of the initial conditions and depend only on the parameters of the black hole (mass, charge and angular momentum) and the fundamental constants (Newton constant and cosmological constant) that describe a black hole, just like the parameters that define the test field. On the other hand, the QNFs determine how fast a thermal state in the boundary theory will reach thermal equilibrium according to the AdS/CFT correspondence Maldacena:1997re (), where the relaxation time of a thermal state is proportional to the inverse of the imaginary part of the QNFs of the dual gravity background, which was established due to the QNFs of the black hole being related to the poles of the retarded correlation function of the corresponding perturbations of the dual conformal field theory Birmingham:2001pj (). Fermions on a Lifshitz background were studied in Alishahiha:2012nm () by using the fermionic Green’s function in 4-dimensional Lifshitz spacetime with ; the authors considered a non-relativistic (mixed) boundary condition for fermions and showed that the spectrum has a flat band. Also, the Dirac quasinormal modes (QNMs) for a 4-dimensional Lifshitz black hole were studied in Catalan:2013eza (). Generally, the Lifshitz black holes are stable under scalar perturbations, and the QNFs show the absence of a real part CuadrosMelgar:2011up (); Gonzalez:2012de (); Gonzalez:2012xc (); Myung:2012cb (); Becar:2012bj (); Giacomini:2012hg (). The QNFs have been calculated by means of numerical and analytical techniques, some remarkably numerical methods are: the Mashhoon method, Chandrasekhar-Detweiler, WKB method, Frobenius method, method of continued fractions, Nollert, asymptotic iteration method (AIM) and improved AIM among others. In the context of black hole thermodynamics, QNMs allow the quantum area spectrum of the black hole horizon to be studied CuadrosMelgar:2011up () as well as the mass and the entropy spectrum.
On the other hand, knowledge of black holes perturbations is also useful for studying the Hawking radiation, which is a semiclassical effect and gives the thermal radiation emitted by a black hole. At the event horizon, the Hawking radiation is in fact blackbody radiation. However, this radiation still has to traverse a non-trivial curved spacetime geometry before reaching a distant observer that can detect it. The surrounding spacetime thus works as a potential barrier for the radiation, giving a deviation from the blackbody radiation spectrum, seen by an asymptotic observer Maldacena:1996ix (). Thus the total flux observed at infinity is that of a -dimensional greybody at the Hawking temperature. The factors that modify the spectrum emitted by a black hole are known as greybody factors and can be obtained through the classical scattering (for a review see Harmark:2007jy ()). In this sense, the scalar greybody factors for an asymptotically Lifshitz black hole were studied in Gonzalez:2012xc (); Lepe:2012zf (), and particle motion on these geometries in Olivares:2013zta (); Olivares:2013uha (); Villanueva:2013gra ().
The paper is organized as follows. In Sec. II we give a brief review of the -dimensional Lifshitz black hole in conformal gravity. In Sec. III we calculate the QNFs of scalar perturbations for the -dimensional Lifshitz black hole with spherical topology and for some special cases analytically and numerically by using the improved AIM. Then, in Sec. IV, we study the reflection and transmission coefficients and the absorption cross section. Finally, our conclusions are in Sec. V.
Ii -dimensional asymptotically Lifshitz black hole in conformal gravity
In this work we consider a matter distribution described by a scalar field outside the event horizon of a four-dimensional asymptotically Lifshitz black hole in conformal gravity with and spherical topology for the transverse section Lu:2012xu (). Conformal gravity is a limit case of Einstein-Weyl gravity. The action of Einstein-Weyl gravity is given by
is the Ricci scalar and is the cosmological constant. When goes to infinity we have the special case of conformal gravity, and the field equations in vacuum are given by , where is the Bach tensor defined by:
where is the Weyl tensor. The following metric solves the field equations Lu:2012xu ()
For , there is an event horizon at the largest root of , given by
and for the singularity is naked. Note that the requirement implies that . When the solution becomes extremal, and for the entropy vanishes in this case. The Kretschmann scalar (for ) is given by
therefore, there is a curvature singularity at . In the next section, we determine the QNFs by considering the Klein-Gordon equation in this background and by establishing the boundary conditions on the scalar field at the horizon and at spatial infinity.
Iii Quasinormal modes of a -dimensional Lifshitz black hole
The QNMs of scalar perturbations in the background of a four-dimensional asymptotically Lifshitz black hole in conformal gravity with dynamical exponent are given by the scalar field solution of Klein-Gordon equation with suitable boundary conditions. This means there are only ingoing waves on the event horizon and we consider that the scalar field vanishes at spatial infinity, known as Dirichlet boundary conditions. These fields are considered as mere test fields, without backreaction over the spacetime itself. Therefore, it is not necessary for such fields to have the same symmetries as the background spacetime. On the other hand, if one considers the backreaction of the matter fields over the spacetime, in order to look for exact solutions to the field equations, the relation between symmetries of the spacetime and the matter fields is not trivial, for a recent study about symmetry inheritance of scalar fields see Smolic:2015txa () and references therein. In the case considered here, the gravitational field equations imply that the trace of the stress-energy tensor must vanish, due to the Bach tensor is traceless, therefore if one go beyond the probe-field approximation, this implies that the stress-energy tensor of the matter fields must be traceless. Based on these arguments, first we will consider a test scalar field minimally coupled to curvature, then we will consider a test scalar field conformally coupled to curvature, which have a traceless stress-energy tensor, and we find analytically and numerically the quasinormal frequencies for scalar fields for some special cases.
iii.1 Scalar field minimally coupled to gravity
In this section we calculate the QNMs of the Lifshitz black hole for a test scalar field minimally coupled to gravity. The Klein-Gordon equation in curved spacetime is
where is the mass of the scalar field , which is minimally coupled to curvature. By means of the following ansatz
where is a normalizable harmonic function on the two-sphere which satisfies
being the eigenvalues for the spheric manifold, with , the Klein-Gordon equation reduces to
Now, by considering and by introducing the tortoise coordinate , given by , the latter equation can be rewritten as a one-dimensional Schrödinger equation
where the effective potential is given by
In order to find analytical solutions to the radial equation (12), we perform the change of variables and get the following equation:
where the prime denotes the derivative with respect to , and and are the roots of
and are given by
Additionally, performing another change of variable and noting that , we arrive at the following expression
where we have defined and now a prime means derivative with respect to . In Fig. (3), we plot as a function of and we observe that can be positive or negative depending on the values of the parameter :
On the other hand, equation (18) can be manipulated and put into the following form
We note that for this equation corresponds to a Riemann differential equation, whose general form is M. Abramowitz ()
where and are the singular points, and the exponents and are subject to the condition
The complete solution is denoted by the symbol
and the Riemann function can be reduced to the hypergeometric function through
and the exponents are given by
Therefore, the solution to equation (21) can be written as
where we have defined the constants and as
In the near-horizon limit, the above expression behaves as
Now, we impose as a boundary condition that classically nothing can escape from the event horizon. So, choosing the exponent as
implies that we must take in order to have only ingoing waves at the horizon. Therefore, our solution simplifies to
Now, we implement boundary conditions at spatial infinity. In order to do so we employ the Kummer relations M. Abramowitz (), and write the solution as
At the limit , the above solution becomes
Now, we choose the exponents and as follows
So, imposing the condition that the scalar field be null at spatial infinity, we can determine the QNFs. The second term of equation (36) blows up when unless we impose the condition or ; therefore, we obtain the following set of QNFs:
These QNFs are purely imaginary and negative for , which guarantees that the Lifshitz black hole is stable under massless scalar field perturbations for the mode with the lowest angular momentum. For there are QNFs with imaginary and positive value, and the Lifshitz black hole is unstable under scalar field perturbations. Also, we note that if we interchange the values of the exponents in equation (37) the same QNFs are obtained. It is worth mentioning that Eq. (15) with can be written as
under the change of variable , and if we define , the above equation leads to the hypergeometric equation
and the constants are given by
The general solution of the hypergeometric equation (50) is
and it has three regular singular points at , , and . is a hypergeometric function and and are integration constants. Note that the above QNFs could be computed using the solution (46).
In this case it is possible to obtain an analytical solution for all values of the angular momentum . Thus, for or equivalently , the radial equation (12) can be written as
where we have considered . Using the decomposition , with
we can write (47) as a hypergeometric equation for K
where the coefficients are given by
The general solution of the hypergeometric equation (50) is
and it has three regular singular points at , , and . is a hypergeometric function and and are constants. Thus, the solution for the radial function is
So, in the vicinity of the horizon, and using the property , the function behaves as
and the scalar field , for , can be written as follows:
in which, the first term represents an ingoing wave and the second an outgoing wave in the black hole. So, by imposing that only ingoing waves existing at the horizon, this fixes . The radial solution then becomes
To implement boundary conditions at infinity (), we apply Kummer’s formula for the hypergeometric function M. Abramowitz (),
With this expression, the radial function (58) reads
and at infinity it can be written as
So, for , the field at infinity vanishes if or for , and for , the field at infinity vanishes if or . Therefore, the QNFs are given by
where . This expression can be written as 111The same QNFs can be obtained by imposing that only outgoing waves exist at spatial infinity.
Because not all the QNFs have a negative imaginary part we conclude that this black hole is not stable under scalar field perturbations for the case when .
In the extremal case , or equivalently , the radial equation (18) reads
and its solution is given by
where is the confluent Heun function. Thus, when , and in order to have a regular scalar field at spatial infinity, we must set ; therefore, the solution reduces to
where the property was used Fiziev (). However, we observe that when , the scalar field is null ; therefore, there are no QNMs in this case.
iii.2 Scalar field conformally coupled to gravity
In this section we calculate the QNMs of the Lifshitz black hole for a test scalar field conformally coupled to gravity. The Klein-Gordon equation for a scalar field non-minimally coupled to curvature is
where is the mass of the scalar field , is the non-mininal coupling parameter and is the Ricci scalar, which reads
For a conformally coupled scalar field case we must take and . Now, by means of the following ansatz
where is a normalizable harmonic function on the two-sphere which satisfies Eq. (11), the Klein-Gordon equation reduces to
which can be written as a one-dimensional Schrödinger equation with an effective potential that vanishes at spatial infinity. Therefore, we will consider only outgoing waves at the asymptotic region as boundary condition. It is worth to mention that Eq. (72) only has analytical solution for as we will show below. Therefore, we will perform numerical studies for by using the improved AIM Cho:2009cj (), which is an improved version of the method proposed in Refs. Ciftci (); Ciftci:2005xn () and it has been applied successful in the context of QNMs for different black holes geometries, see for instance Cho:2009cj (); Cho:2011sf (); Catalan:2013eza (); Zhang:2015jda (); Barakat:2006ki ().
iii.2.1 Numerical analysis
In order to implement the improved AIM we make the following consecutive change of variables and to Eq. (69), as we do in the previous sections. Then, the Klein-Gordon equation yields
Now, we must consider the behavior of the scalar field on the event horizon and at spatial infinity. Accordingly, on the horizon, , the behavior of the scalar field is given by
So, if we consider only ingoing waves on the horizon, we must impose . Asymptotically, from Eq. (73), the scalar field behaves as
So, in order to have only outgoing waves at infinity we must impose . Therefore, taking into account these behaviors we define
Then, by inserting these fields in Eq. (73) we obtain the homogeneous linear second-order differential equation for the function
Then, in order to implement the improved AIM it is necessary to differentiate Eq. (77) times with respect to , which yields the following equation: