Scalar quasinormal modes of black holes in Einstein-Yang-Mills gravity

Quasinormal modes of a scalar field are investigated in the background of Einstein-Yang-Mills black holes. The logarithmic term in the metric function of the five-dimensional Einstein-Yang-Mills black hole eliminates the divergence of the Hawking temperature but maintains the charge-dependent behavior of the quasinormal modes. The real and imaginary parts of quasinormal mode frequencies have the same charge-dependent behavior in different dimensions. Similar to the case of high dimensional Schwarzschild black holes, a compact approximate relation, $\omega_{\rm R}\sim D/r_{+}$, also exists between the real part of the quasinormal mode frequencies $\omega_{\rm R}$ and the dimension $D$ in the higher than five dimensional Einstein-Yang-Mills black holes.


Introduction
Quasinormal modes (QNMs) of black holes, which characterize the dynamics of perturbation fields, are usually used to describe a dissipative system that oscillates and decays at a complex frequency, ω = ω R − iω I , where the real part represents the frequency of oscillation while the imaginary part gives the damping timescale. For a stable black hole, the imaginary part is negative, i.e. ω I is positive. The fundamental mode (n = 0) is the least damped and long lived mode in a ringdown signal, which is most likely to be detected in a binary compact star merger.
The investigations of QNMs have been made in a wide range, see, for instance, some reviews [1][2][3][4]. In the Einstein theory and Einstein-Maxwell theory, the QNMs of a scalar field have been studied extensively, such as for the Reissner-Nordström-anti-de Sitter (RNAdS) black hole [5], Kerr black hole [6], and Reissner-Nordström (RN) black hole [7], respectively. In particular, it has been found [8] that the real part of QNM frequencies is proportional to the ratio of the dimension to the horizon radius, D/r + , in high dimensional Schwarzschild black holes. Furthermore, the timescale for approaching to a thermal equilibrium has been computed [9,10] numerically according to the AdS/CFT correspondence through the QNMs of a scalar field in different dimensions.
Compared with the Einstein-Maxwell theory, the Einstein-Yang-Mills (EYM) theory is more challenging in finding exact solutions due to its inherent complications. Until now, the numerical solutions have been available for the SU(N) EYM theory, where the solutions of the SU(2) EYM theory are asymptotically flat and uncharged [11,12] while that of the SU(5) EYM theory, in contrast to the case in the SU(2) EYM theory, are charged [13]. It has been shown [14][15][16] that these solutions are unstable in terms of numerical analyses under linear perturbations. Nonetheless, the black hole solutions in the four dimensional SU(N) EYM theory with a negative cosmological constant have been proved [17] to be stable. Thus, the EYM black holes in the higher than four dimensions are quite different from that in the four dimensions.
The analytical solutions have been obtained [18][19][20][21][22][23] when the symmetric group is reduced to SO(D − 1), where the order of the gauge group is less than the dimension by one. However, the stability of these solutions under linear perturbations remains to be verified. The significant difference between the SO(D− 1) EYM black hole and the Einstein-Maxwell RN black hole is the appearance of an additional logarithmic term in the metric function of the five-dimensional EYM black hole, which leads to [18] a mass-independent radius of horizon. What we are interested in is whether there are new physical effects caused by the logarithmic term, in particular, how it affects QNMs and what behaviors the QNMs exhibit in different dimensions. We investigate these issues by dealing with a scalar field around EYM black holes and applying the analytical solutions of EYM black holes with the SO(D − 1) gauge group. Our aim is to gain some insight into the stability of black holes in Einstein gravity coupled to Yang-Mills gauge fields. Specifically, we want to determine the regions of Yang-Mills charges and dimensions for stable EYM black holes. We shall focus on the EYM black holes with the SO(D − 1) gauge group in the higher than four dimensions due to their specifics.
The outline of this paper is as follows. In Sec. 2 we at first review shortly the EYM black holes, and then investigate the Hawking temperature of the EYM black holes in D = 5 and D = 6 as an example of D > 5. Sec. 3 contains the derivation of the perturbation equations of a massless scalar field for a spherically symmetric EYM black hole, and is 1 E-mail: guoy@mail.nankai.edu.cn 2 Corresponding author. 3 E-mail: miaoyg@nankai.edu.cn divided into two subsections for the cases of D = 5 and D > 5, respectively. In Sec. 3.1, we compute the fundamental QNMs under a massless scalar field perturbation and study the dependence of QNMs on gauge charges and angular quantum numbers for the case of D = 5. Next, we analyze mainly the dependence of QNMs on dimensions for the case of D > 5 in Sec. 3.2. Finally, we present our conclusions in Sec. 4.

Einstein-Yang-Mills black holes
The EYM gravity we consider here is described [18,24] by the action, where R is the Ricci scalar, F a µν = ∂ µ A a ν − ∂ ν A a µ + 1 2σ C a bc A b µ A c ν are the field strengths of SO(D − 1) Yang-Mills fields with the structure constants C a bc and coupling constant σ , and A a µ are the gauge potentials, the Latin indices, a, b, c, · · · = 1, 2, . . . , (D−1)(D−2)/2, represent the internal space of the gauge group, and the Greek indices, µ, ν, α, · · · = 0, 1, . . ., D− 1, describe the D-dimensional spacetime. The EYM field equations can be obtained 1 by the variation of the action with respect to the metric g µν , where the energy-momentum tensor can be expressed by The following intuitive result can be obtained [18,25] in terms of the D-dimensional Wu-Yang ansatz, , and the only non-zero gauge charge Q, which yields the non-zero components of the energy-momentum tensor. As a result, the D-dimensional line element takes a general form, where dΩ 2 D−2 is the line element of the unit sphere S D−2 . Correspondingly, Eq. (2) reduces to two radial equations in five dimensions and higher than five dimensions, respectively, based on which the metric functions can be expressed as where m denotes the mass of black holes. Incidentally, the two metric functions can alternatively be derived [25] in f (R) gravity when f (R) = R.
The two metric functions of the Einstein-Yang-Mills black holes are plotted in Fig. 1, where the metric functions of the Reissner-Nordström black holes are attached for comparison. It is obvious from the figure that the behavior of the metric function of the EYM black hole with respect to the radial coordinate is similar to that of the metric function of the RN black hole in D = 5, and that the EYM black hole has only one event horizon but the RN black hole has one event horizon and one Cauchy horizon in D = 6. As the dimension increases, the positions of the two horizons in the RN black holes approach each other and coincide at r = 1, while the position of the event horizon in the EYM black holes approaches one from outside. We note that the metric function of the four-dimensional Bardeen black hole converges [26] to one in the limits of r → 0 and r → ∞. However, the metric functions of the EYM black holes diverge at r → 0 for both cases of D = 5 and D > 5, and only at infinity do the two functions converge to one. That is, we want to emphasize that the singularity remains for the EYM black holes in contrast to the non-singular nature of the Bardeen black hole. For the given event horizon r + , the Hawking temperature of the Einstein-Yang-Mills black holes and Reissner-Nordström black holes can be obtained in terms of the formula, T = − 1 4π dg tt dr r=r + , as follows: The Hawking temperature is plotted in Fig. 2 with respect to the gauge charge Q and the event horizon r + , respectively. For the case of D = 5, the temperatures of both the EYM black hole and RN black hole decrease when the gauge charge Q increases, see the left diagram. Their difference is stated as follows: The temperature of the EYM black hole is higher than that of the RN black hole when Q is smaller than 0.6, but it is lower than the temperature of the RN black hole when Q is larger than 0.6. Moreover, it is also clear for the case of D = 5, see the right diagram, that the extreme temperatures exist during the evaporation of the EYM black hole and the RN black hole. Nevertheless, the most significant difference between the EYM black hole and the RN black hole is that the temperature of the EYM black hole ascends while the temperature of the RN black hole descends when Q increases in the case of D = 6, see the left diagram. In addition, also for the case of D = 6, there is one specific feature worth noting: The evaporation of the EYM black hole will cause the divergence of the Hawking temperature, see the green curve of the right diagram, because no logarithmic term appears in the metric function Eq. (9). In brief, we explicitly show the relations between the Hawking temperature and the charge in the left diagram of Fig. 2 and the relations between the Hawking temperature and the horizon in the right diagram of Fig. 2. It is interesting to note that it is actually the logarithmic term that eliminates the temperature's divergence and makes the two black holes behave similarly in the case of D = 5.

Quasinormal modes of Einstein-Yang-Mills black holes under a massless scalar field perturbation
The massless scalar field Φ propagating in a curved spacetime is governed by the Klein-Gordon wave equation, For a stationary and spherically symmetric solution, we substitute the following decomposition of variables, into Eq. (12), where Y lm (θ , ϕ) stands for the spherical harmonics of D − 2 degrees, 0 ≤ θ = (θ 1 , θ 2 , . . . , θ D−3 ) ≤ π and 0 ≤ ϕ ≤ 2π, and then derive the radial equation for ψ l (r), where f ′ ≡ d f (r)/dr, ψ ′ l ≡ dψ l (r)/dr, and ψ ′′ l ≡ d 2 ψ l (r)/dr 2 . After defining the "tortoise" coordinate by the relation, dr * = dr/ f (r), we reduce the radial equation Eq. (14) into the standard form, where the effective potentials of the Einstein-Yang-Mills black holes for the cases of D = 5 and D > 5 can be calculated from the metric functions Eqs. (6) and (7), Hence, the evolution of the linear perturbation field Φ in the black hole spacetime turns to the differential equation Eq. (15), based on which the quasinormal modes can be determined. Moreover, we have to impose the following boundary conditions in order to obtain the solution of Eq. (15), ψ l ∼ e −iω(t∓r * ) , r * → ±∞.
For an asymptotically flat spacetime, such as the EYM black hole spacetime, we see that only the outgoing modes can be present at r * → +∞. In order to find the QNMs characterizing the dynamics of a massless scalar field, we apply the improved WKB approximation [27,28]. Note that the fundamental mode (n = 0) is very long lived 2 when compared to the other modes with the overtone number n ≥ 1, so we mainly discuss it in this paper. Next, we study the fundamental modes for the EYM black holes with the SO(D − 1) gauge symmetry in D = 5 and D > 5, respectively.

The case of D = 5
In accordance with Eqs. (15) and (16) and the numerical method [27], we work out the fundamental QNM frequencies of massless scalar fields in the ranges of the gauge charges Q = 0.0, 0.1, . . ., 0.6 and of the angular quantum numbers l = 0, 1, 2, 3, 4, where the black hole mass is set to be unit. The results showing the dependence of QNMs on gauge charges and angular quantum numbers are listed in Table 1.  In order to analyze in detail the behaviors of QNMs with respect to the gauge charge Q in the EYM black hole, we compute more data for the fundamental mode (n = 0) with the angular quantum number l = 2 besides those in Table 1 and plot the frequencies with respect to the charge in Fig. 3, where the case of the RN black hole is attached for comparison.  From the left diagram of Fig. 3 we can see that the real part of frequencies decreases in the EYM black hole when the Yang-Mills charge increases, which is in contrast to the behavior of the real part of frequencies with respect to the Maxwell charge in the RN black hole. We note that the logarithmic term in the EYM black hole leads to the similar Hawking temperature profile to that of the RN black hole in five dimensions in section 2. However, the fact that the logarithmic term does not make the EYM black hole behave as the RN black hole does in five dimensions suggests that its effect on QNMs seems to be greatly weakened. In addition, from the right diagram of Fig. 3 we can see that the imaginary parts of QNM frequencies behave similarly, i.e. they decrease as Q increases in both the EYM and RN black holes. 3 For the sake of intuition, the data in Table 1 are plotted by Fig. 4. We can see from Fig. 4 that both the real and imaginary parts of the quasinormal mode frequencies at different angular quantum numbers have almost the same charge-dependent behavior, i.e., they decrease as the charge increases. However, the significant difference between the real and imaginary parts of frequencies is that the former increases while the latter decreases when the angular quantum number l increases, which is surprisingly similar [9] to that of the Schwarzschild-AdS black hole. This phenomenon is understandable in terms of the AdS/CFT correspondence because the timescale for approaching to a thermal equilibrium increases for the modes with increasing l and Q, see the right diagram of Fig. 4.

The case of D > 5
Now we pay attention to the dependence of QNM frequencies on dimensions in EYM black holes, i.e., how the dimensions of spacetime affect the quasinormal behavior of EYM black holes. By taking the same strategy as that for the case of D = 5 in Sec. 3.1, we can obtain the expected QNM frequencies that are listed in Table 2. Here we only focus on the the fundamental mode with l = 2 for different values of charges Q and dimensions D.  In order to analyze in detail the behaviors of QNMs with respect to the gauge charge Q in the EYM black hole, we compute more data for the fundamental mode (n = 0) with the angular quantum number l = 2 in D = 6 as an example besides those in Table 2, and plot the frequencies with respect to the charges in Fig. 5, where the case of the RN black hole is attached for comparison. We can see from Fig. 5 that the dependence of real parts on charge is contrary to that of the six-dimensional RN black hole but the dependence of imaginary parts on charge consists with that of the six-dimensional RN black hole. By comparing Fig. 3 with Fig. 5, we can clearly see that the dependence of QNMs on charge in the six-dimensional EYM black hole is same as that in the five-dimensional one. This shows that there is no change in dependence of quasinormal mode frequencies on charge from five to six dimensions, which further implies that the logarithmic term that appears in five dimensions does not change such a dependence.  In order to investigate the dependence of the QNMs on dimensions D in EYM black holes, we plot Fig. 6 showing the product of quasinormal frequencies and horizon radius with respect to dimensions for the values of angular quantum numbers l = 2, 3, 4, where we have taken the data on the second row in Table 2 and supplied the other data for D = 11 and D = 12. We can see from Fig. 6 that both r + ω R and r + ω I increase when l increases, where the variations of the r + ω I are very small for different values of l. It is known [8] that the real part of QNMs is approximately proportional to the ratio of the dimension to horizon radius, D/r + , in high dimensional Schwarzschild black holes. Here we conclude from the left diagram of Fig. 6 that the relation between r + ω R and D is almost linear, which is similar to the situation in high dimensional Schwarzschild black holes. As a result, we write such a relation after giving the coefficients, For the sake of intuition, the data in Table 2 are plotted by Fig. 7. We can see from Fig. 7 that both the real and the imaginary parts of quasinormal mode frequencies decrease when the charge increases in the case of D > 5, which is similar to the charge-dependent behavior in the case of D = 5, see also Fig. 4. Moreover, an increasing dimension with a fixed Q leads to an interesting effect that both the real and imaginary parts are increasing. 4 In more detail, it is quite obvious from the right diagram of Fig. 7 that the increments of the imaginary parts with respect to the dimensions decrease when the charge is increasing. In conclusion, the EYM black holes oscillate at larger frequencies but in shorter periods of damping time in higher dimensions.

Conclusions
By using the improved WKB method, we have computed the scalar QNMs of Einstein-Yang-Mills black holes in different dimensions. The logarithmic term in the metric function of the five-dimensional Einstein-Yang-Mills black hole prevents the Hawking temperature from diverging when the horizon radius shrinks, but it keeps the charge-dependent behavior of the QNMs unchanged. In any dimensions larger than four, the real and imaginary parts of quasinormal mode frequencies decrease with the increasing charge, which is different from the situation in the real parts of RN black holes. For the fundamental modes of the EYM black holes, the real parts of QNM frequencies are approximately proportional to the ratio of the dimension to the radius of horizon, D/r + , and the imaginary parts are almost same for different values of angular quantum numbers, i.e. such tiny variations of the imaginary parts imply that the Einstein-Yang-Mills black holes approach to a thermal equilibrium at a nearly same damping timescale for different angular quantum numbers. As for the tendency related to dimensions, our results show that the EYM black holes oscillate at larger frequencies but damps in shorter timescales in higher dimensional spacetimes.