On quasinormal modes in 4D black hole solutions in the model with anisotropic fluid

We consider a family of 4-dimensional black hole solutions from Dehnen et al. ( Grav. Cosmol. 9:153, arXiv: gr-qc/0211049, 2003) governed by natural number $q= 1, 2, 3 , \dots$, which appear in the model with anisotropic fluid and the equations of state: $p_r = -\rho (2q-1)^{-1}$, $p_t = - p_r$, where $p_r$ and $p_t$ are pressures in radial and transverse directions, respectively, and $\rho>0$ is the density. These equations of state obey weak, strong and dominant energy conditions. For $q = 1$ the metric of the solution coincides with that of the Reissner-Nordstr\"om one. The global structure of solutions is outlined, giving rise to Carter-Penrose diagram of Reissner-Nordstr\"om or Schwarzschild types for odd $q = 2k + 1$ or even $q = 2k$, respectively. Certain physical parameters corresponding to BH solutions (gravitational mass, PPN parameters, Hawking temperature and entropy) are calculated. We obtain and analyse the quasinormal modes for a test massless scalar field in the eikonal approximation. For limiting case $q = + \infty$, they coincide with the well-known results for the Schwarzschild solution. We show that the Hod conjecture which connect the Hawking temperature and the damping rate is obeyed for all $q \geq 2$ and all (allowed) values of parameters.


Introduction
The decaying oscillations such as quasinormal modes (QNMs) [2][3][4][5][6][7][8][9][10][11][12] are at presence a very interesting and popular topic of investigations. A possible application of QNMs may be related to gravitational waves [13][14][15] emitted during the ringdown (final) stage of binary black hole (BH) mergers. It is belived that the frequencies of gravitational waves may be calculated by using certain superpositions of QNMs. The importance of these experiments is following: the analysis of experimental data may clarify the nature of gravity in the regime of strong fields.
In this article we deal with 4D black hole solutions from Ref. [1]. These solutions take place in the model with anisotropic fluid with the following equations of state: (1.4) where p r and p t are pressures in radial and transverse directions, respectively, ρ > 0 is the density and q = 1, 2, 3, . . . is the natural number. (In (1.4) we put c = 1 .) It may be readily verified that these equations of state obey weak (ρ ≥ 0, ρ + p i ≥ 0), strong (ρ + j p j ≥ 0, ρ + p i ≥ 0) and dominant (ρ ≥ |p i |) energy conditions (here (p i ) = (p r , p t , p t )).
Here we obtain and analyse the QNMs for a test massless scalar field in the eikonal approximation which is the main subject of the paper. By product we present the global structure of BH solutions under consideration and calculate certain physical parameters corresponding to them (gravitational mass, PPN parameters, Hawking temperature and entropy).
The paper is organised as follows. In Section 2 we present the black hole solutions from Ref. [1]. In Section 3 we analyse the global structure of the solutions. In Section 4 we calculate certain physical parameters which correspond to the solutions under consideration. In Section 5 we find the frequences of QNMs in the eikonal approximation which correspond to massless test scalar field in the background metric of our BH solutions with anisotropic fluid for q = 1, 2, 3, . . . . Section 6 is devoted to special (integrable) cases q = 1, 2, 3 and the limiting case q = +∞. In Section 7 we verify the validity of the Hod conjecture [16] for the solutions under consideration with q > 1.

The black hole solution
Here we consider the solutions to Einstein equations where κ = 8πG/c 4 , G is Newton gravitational constant and c is speed of light.
The solutions under consideration are defined on the four-dimensional manifold with topology Here the spherical coordinate system is used: x µ = (r, θ, φ, t) with signature (+ + +, −). The energy-momentum tensor of anisotropic fluid is taken as and the equations of state read Here ρ is the mass density, p r and p t are pressures in radial and orthogonal (to radial) directions, respectively. The parameter q describes relations between the pressures and the mass density; q > 0, q = 1/2. In the present paper, the parameter q is taken to be a natural number to avoid the non-analytical behaviour of the metric at the (would be) horizon.
The solution has the following form [1]: where the function H(r) reads as follows: The metric on the sphere S 2 is denoted by dΩ 2 ; parameters P, µ > 0 are arbitrary. Originally we put r > 2µ = r h but the domain of definition of the metric will be extended below. The equations of motion (2.1) imply the following relation for the scalar curvature which will be used below for identifying the singularities of solutions for q = 2, 3, 4, . . . .

The global structure of the solution
In what follows we will use the following relation for the metric where Here A = A(r) is so-called "red shift function", C(r) > 0 is "area function". The global structure of the solutions above may be studied by analysing the behaviour of the "redshift function" (A(r)) and the "area function" C(r) (the factor at dΩ 2 ) at critical points corresponding to horizons or singularities. The Carter-Penrose diagrams can be constructed for various values of the parameters using the standard algorithm [17]. For our BH solutions it was done in Ref. [18].
In what follows we denote by r = r the maximal root of the equation H(r) = 0. We have r < 0 for odd q = 2k + 1 and r > 0 for even q = 2k.
There are three classes of important critical points of the radial coordinate r for the metric (3.1): 1) r = r h ≡ 2µ. This point corresponds to a regular external horizon.
2) r = r . This point corresponds to the singularity.
3) r = 0 for odd q = 2k + 1. This point corresponds to internal horizon. We introduce the following notations. Let Sch[r 1 , r 2 ] (r 1 < r 2 ) be a Carter-Penrose diagram of Schwarzschild type with a singularity at a point r 1 and a regular horizon at r 2 . (Fig. 1.) Similarly, we denote by RN[r 1 , r 2 , r 3 ] (r 1 < r 2 < r 3 ) the diagram of Reissner-Nordström type with singularity at r 1 , an internal horizon at r 2 , and an external horizon at r 3 . (Fig. 2.) As a result of analysis, we conclude that the structure of diagrams depends mostly on the parity of the parameter q: • For q = 2m, m ∈ N, we have a diagram of type Sch[r , r h ]. Extremal case. Let us consider an extremal case of the solution under consideration when µ → +0. By using relations (2.5), (2.6) and (2.7) we get in the limit µ → +0 ds 2 = g µν dx µ dx ν = (H e (r)) 2/q dr 2 + r 2 dΩ 2 − (H e (r)) −4/q c 2 dt 2 , where H e (r) = 1 + P q r , (3.6) with P > 0. For q > 1 the metric (3.4) describes a naked singularity corresponding to r → +0. Indeed, using relations (2.8) and (3.5) we obtain for the scalar curvature For q = 2, 3, 4, . . . we are led to relation: R[g] → +∞ as r → +0, which tells us about the singularity corresponding to r = +0. For q = 1 the metric (3.4) is coinciding with the metric of extremal Reissner-Nordström solution with "double" horizon corresponding to r = +0 and singularity (center) at r = −P + 0.

Physical parameters
In this section we deal with some physical parameters of the solutions. Here we put for simplicity c = = k B = 1.

Gravitational mass and PPN parameters
Let us consider the 4-dimensional space-time with the metric (2.5) for r > 2µ. Introducing a new radial variableR by the relation: we rewrite the metric in the following form: The parametrized post-Newtonian (Eddington) parameters are defined by the well-known relations is the Newtonian potential, M is the gravitational mass and G is the gravitational constant. From (4.2)-(4.4) we obtain: and (4.9) or, equivalently, The parameter β − 1 is proportional to the ratio of two physical parameters: the anisotropic fluid density parameter A f and the gravitational radius squared (GM ) 2 .

Hawking temperature and entropy
The Hawking temperature of the black hole may be calculated using the well-known relation [19] where here g rr = (A(r)) −1 , see (3.1). We get Here q = 1, 2, . . . . The Bekenstein-Hawking (area) entropy S = A/(4G), corresponding to the horizon at r = 2µ, where A is the horizon area, reads

Quasinormal modes
In this section we derive quasinormal modes (in eikonal approximation) for our static and spherically symmetric solution (for given q) with the metric given (initially) in the following general form where A(u), B(u), C(u) > 0 and dΩ 2 = dθ 2 + sin 2 θdφ 2 . Note that in this section and below we use the Planck units, i.e. we put = G = c = 1. We consider a test massless scalar field defined in the background given by the metric (4.2). The equation of motion in general is written in the form of the covariant Klein-Fock-Gordon equation where µ, ν = 0, 1, 2, 3. In order to solve this equation we separate variables in function Ψ as follows where Y lm are the spherical harmonics, l is the multipole quantum number, l = 0, 1, . . . and m = −l, . . . , 0, . . . , l.
Equation (5.2), after using (5.3) yields the equation describing the radial function Ψ * (u) and having a Schrödinger-like form and γ = dγ/du, γ = d 2 γ du 2 . Taking into account above expressions one can examine our black hole solution which has the following form where A(r) and C(r) according to Eq. (2.5) can be written as is the moduli function, µ > 0, P > 0, p = P/(2µ), q = 1, 2, . . . and We note that 0 < z < 1 for r > 2µ. After using the "tortoise" coordinate transformation the metric takes the following form For the choice of the tortoise coordinate as a radial one (u = r * ) we have A = B and where ω is the (cyclic) frequency of the quasinormal mode and V = V (r) = V (r(r * )) is the effective potential so that V is the eikonal part of the effective potential. Here and below we denote F = dF dr * = A dF dr .
In what follows we consider the so-called eikonal approximation when l 1. The maximum of the eikonal part of the effective potential is found from the extremum condition or, equivalently, Proposition 1. For any P > 0, µ > 0 and q ∈ N, the extremality relation (5.18) has only one solution for r > 2µ, which is the point of maximum for V(r).
The proposition is proved in Appendix. We denote this point of extremum by r 0 . In terms of variable z we get that the point z 0 = 1−2µ/r 0 is a unique solution to Eq. (5.20) for z ∈ (0, 1).
The maximum of the eikonal part of the effective potential thus becomes In Fig. 3 we plot the reduced eikonal part of the effective potential V/(l(l + 1)) (l = 0) as a function of the radial coordinate r (left panel) and the tortoise coordinate r * (right panel).
As can be seen from examples presented in figure for special fixed values of P and µ, the maximum of the effective potential is largest for q = +∞ case and smallest for q = 1 case. The case with q = 2 is in the middle. At large distances the effective potential tends to zero, as expected. Figure 3: The graphical representation of the reduced potential V/(l(l + 1)) as a function of the radial coordinate r (left panel) and the tortoise coordinate r * (right panel) for P = 2µ = 1, q = 1, 2, 3 and the limiting case q → +∞.
The second derivative with respect to the tortoise coordinate in the point of extremum is given by where A 0 = A(r 0 ), see (5.7). The calculation of second derivative gives us .

(5.24)
The last two terms in this relation may be simplified by using the relation for the third term in (5.19). We obtain Thus, by using (5.22), (5.23) and (5.25) we find The square of the cyclic frequency in the eikonal approximation reads as following [10,11] where l 1 and l n. Here n = 0, 1, . . . is the overtone number. By choosing an appropriate sign for ω we get the asymptotic relations (as l → +∞) on real and imaginary parts of complex ω in the eikonal approximation where H 0 = H(r 0 ) (see (5.9)), r 0 = 2µ/(1 − z 0 ), and z 0 ∈ (0, 1) is solution to master equation (5.20) and B 0 = B(z 0 ), where B(z) is defined in (5.27). We note that the parameters of the unstable circular null geodesics around stationary spherically symmetric and asymptotically flat black holes are in correspondence with the eikonal part of quasinormal modes of these black holes. See [20][21][22] and references therein. Due to Ref. [23] this correspondence is valid if certain restrictions on perturbations are imposed.
6 Special cases q = 1, 2, 3 and the limiting case q = +∞ In this section we consider eikonal QNM for three cases q = 1, 2, 3 when the master equation (5.20) may be solved in radicals for all values of p > 0 and also in the limiting case q = +∞.

The case q = 3
Let us consider the last case q = 3, when the master equation (5.20) of fourth power has a solution in radicals (which was obtained by Mathematica): It may be verified that z 0 = z 0 (3, p), given by relations (6.15)-(6.18), is real for all p > 0 and obey 1/3 < z 0 < 3−

The case q = +∞
In this case the relations (5.29) and (5.30) for QNM in eikonal approximation read as follows where r 0 = 3µ = 3GM corresponds the position where the black hole effective potential attains its maximum. We note that r 0 = 3µ is the radius of the photon sphere for the Schwarzschild black hole with the metric which is coinciding with limiting case of our AF metric (2.5) when q = +∞. We note that relations (6.21), (6.22) for Schwarzschild spacetime were obtained in Ref. [5].
Remark. Here we restrict our choice of a test field by a massless (spin-zero, non-charged) scalar field which is the simplest "perturbation" to study. It may be shown that the consideration of a test Maxwell field on our black hole background will lead us to two equations on functions: Ψ * ,a = a lm (r * ) and Ψ * ,b = b lm (r * ), which are certain combinations of coefficients (and their derivatives) coming from decomposing of vector potential in (vector) spherical harmonics. These equations (one of them is just an integrability condition) look like eq. (5.14) but with another potential V = V, instead of (5.15)(δV = 0 in this case). Thus, we will obtain the same spectrum of QNM in eikonal approximation for a test Maxwell field as for a massless scalar field considered here.

Hod conjecture
Here we verify the conjecture by Hod [16] on the existence of quasi-normal modes obeying the inequality |Im(ω)| ≤ πT H , where T H is Hawking temperature.
We note the Hod conjecture has been tested in theories with higher curvature corrections such as the Einstein-Dilaton-Gauss-Bonnet and Einstein-Weyl for the Dirac field (with positive result) [25]. (For negative result see Ref. [26].) Recently, we have also verified the Hod conjecture (with positive result) for a solution with dyon-like dilatonic black hole [27] for certain values of dimensionless parameter a ∈ [0, 1].
Here we verify this conjecture by using the obtained eikonal relations (5.30) for Im(ω) and the relation for the Hawking temperature (4.11). For our purpose it is sufficient to check the validity of the inequality for all p = P/µ > 0, q = 2, 3, . . . , where z 0 = z 0 (p, q) is unique solution to master equation (5.20), which obeys 0 < z 0 < 1, see Lemma in Appendix. In (7.2) we use the limiting "eikonal value" given by the first term in (5.30) for the lowest overtone number n = 0.
Proof. First we consider the case q > 2. In what follows we use the relation In what follows we use the following splitting y = y 1 y 2 y 3 , (7.5) where B(z) = 3 2 q − 2(q−2)z (1−z) 2 . For y 1 we obtain from (7.3) for all p > 0 and q > 2. Now, we use the following fact about the functioñ where 0 < u < 1 and q > 0. Namely, the functionf (q) is monotonicall decreasing in (0, +∞). This follows from the relation where x = u q and 0 < x < 1. This fact imlies for u = 0.4 the following bound for all p > 0 and q ≥ 3. Hence, we get y 1 = y 1 (p, q) < 1.0451, (7.11) for all p > 0 and q > 2. For y 2 we obtain from (7.3) for all p > 0, q > 2. The last bound is also valid for all p > 0 and q > 2. It follows from monotonical decreasing of the function B(z) in (0, 1) and 1/3 < z 0 < z * < z 1 . Here B(z) > 0 for z ∈ (0, z 1 ) and z * = z * (q), z 1 = z 1 (q) are defined in Appendix.
Remark. Let us comment also on the case q = 1 which gives us the Reissner-Nordström metric. It may be readily verified that in this case the inequality (7.2) is not satisfied for all values of p: it is valid only for 0 < p < p cr , where p cr is some critical value of parameter p [27].
As it was pointed out in [27] the violation of the Hod inequality in the eikonal regime for certain p (and n = 0) does not close the possibility for the obeying this relation for exact values of QNM for certain l = 0, 1, 2, . . . and all values of parameter p.

Conclusion
Here we have studied a non-extremal black hole solutions in a 4-dimensional gravitational model with anisotropic fluid proposed in Ref. [1]. The equations of state for the fluid (1.4) contains a parameter q which is natural number q = 1, 2, 3 . . . . We have outlined the global structure of solutions under consideration: for odd q = 2k +1 the Carter-Penrose diagram is coinciding with that of Reissner-Nordström metric (the case of time-like singularity hidden by two horizons) while for even q = 2k it is coinciding with that of Schwarzschild metric (the case of space-like singularity hidden by one horizon). For q = 1 the metric of the solution [1] coincides with the metric of the Reissner-Nordström solution while in the limit q = +∞, we get the metric of the Schwarzschild solution. We have also presented certain physical parameters corresponding to BH solutions: gravitational mass M , Hawking temperature, black hole area entropy.
We have examined the solutions to massless Klein-Fock-Gordon equation in the background of our static BH metric for given q = 1, 2, 3 . . . . . By using the tortoise coordinate we have reduced this equation to radial one governed by certain effective potential. This potential contains the parameters of solution such as P > 0, µ > 0, natural parameter q and also l which is the multipole quantum number, l = 0, 1, . . . .
Here we have studied the eikonal part of the effective potential for large l and have found a master equation for the value z 0 = 1 − 2µ/r 0 , where r 0 is the value of the radial coordinate (radius) r 0 corresponding to the maximum of the eikonal part of the effective potential. By using the maximum value of (the eikonal part) of the effective potential V 0 and r 0 , we have calculated the cyclic frequencies of the QNMs in the eikonal approximation up to solution of the master equation in z 0 . Since the master equation is an algebraic equation of order q + 1 in z 0 we were able to find analytical exact solutions for q = 1, 2, 3. For obtained values of eikonal QNMs we have also considered special cases q = 1, 2, 3 and a limiting cases q = +∞. For q = 1 our (eikonal) relations are compatible with the well-known result for Reissner-Nordström solution [24] (for n = 0), while for q = +∞ they in an agreement with the well-known result for the Schwarzschild solution [5].
We have also tested the validity of the Hod conjecture for our solutions by considering QNMs (eikonal) frequences with the lowest value of the overtone number n = 0. We have shown that the Hod conjecture is valid in the range of q > 1. This assumption is valid for these values of q > 1 since it is supported by examples of states with large enough values of the multipole number l.
We note, that the results obtained here for eikonal QMN modes of test massless (noncharged) scalar field are also valid for some other test fields, e.g. for electromagnetic one. This may be considered (by product) in a separate publication. (The results of Refs. [28,29] may be also used in future work.)

A Appendix
Here we prove the Proposition 1. Since the extremality condition (5.18) for the effective potential V (r > 2µ) is equivalent to the master equation (5.20) (z = 1 − 2µ/r), and the second derivative V 0 at the point of extremum is given by relation (5.26) with V 0 > 0 (see (5.21)), the Proposition 1 is equivalent to the following Lemma.