Quasinormal modes of non-Abelian hyperscaling violating Lifshitz black holes

Abstract

We study the quasinormal modes of scalar field perturbations in the background of non-Abelian hyperscaling violating Lifshitz black holes. We find that the quasinormal frequencies have no real part so there is no oscillatory behavior in the perturbations, only exponential decay, that is, the system is always overdamped, which guarantees the mode stability of non-Abelian hyperscaling violating Lifshitz black holes. We determine analytically the quasinormal modes for massless scalar fields for a dynamical exponent $z=2$ and hyperscaling violating exponent $\tilde{\theta }>-2$. Also, we obtain numerically the quasinormal frequencies for different values of the dynamical exponent and the hyperscaling violating exponent by using the improved asymptotic iteration method.

Appendix 1: Improved AIM

In this appendix we give a brief review of the improved AIM, which is used to solve homogeneous linear second-order differential equations subject to boundary conditions. First, it is necessary to implement the boundary conditions. For this purpose the dependent variable must be redefined in terms of a new function, say $\chi$, that satisfies the boundary conditions appropriate to the eigenvalue problem under consideration. In the study of quasinormal modes of the black holes, one solves the radial equation on the horizon and at spatial infinity, and imposes the boundary condition that on the horizon only ingoing waves exist there and at spatial infinity the appropriate boundary condition depends on the asymptotic behavior of the spacetime [in our case we imposed the scalar field to be null at spatial infinity due to the effective potential diverges there; therefore, the new radial function was defined in Eq. (48)]. Thus, in order to implement the improved AIM the differential equation must be written in the form

$$\begin{aligned} \chi ^{\prime \prime }=\lambda _{0}(y)\chi ^{\prime }+s_{0}(y)\chi . \end{aligned}$$

(52)

Then, one must differentiate Eq. (52) n times with respect to y, which yields the following equation:

$$\begin{aligned} \chi ^{n+2}=\lambda _{n}(y)\chi ^{\prime }+s_{n}(y)\chi , \end{aligned}$$

(53)

where

$$\begin{aligned} \lambda _{n}(y)= & {} \lambda _{n-1}^{\prime }(y)+s_{n-1}(y)+\lambda _{0}(y)\lambda _{n-1}(y), \end{aligned}$$

(54)

$$\begin{aligned} s_{n}(y)= & {} s_{n-1}^{\prime }(y)+s_{0}(y)\lambda _{n-1}(y). \end{aligned}$$

(55)

Then, expanding the $\lambda _{n}$ and $s_{n}$ in a Taylor series around some point $\eta$, at which the improved AIM is performed, yields

$$\begin{aligned} \lambda _{n}(\eta )= & {} \sum _{i=0}^{\infty }c_{n}^{i}(y-\eta )^{i}, \end{aligned}$$

(56)

$$\begin{aligned} s_{n}(\eta )= & {} \sum _{i=0}^{\infty }d_{n}^{i}(y-\eta )^{i}, \end{aligned}$$

(57)

where the $c_{n}^{i}$ and $d_{n}^{i}$ are the $i^{th}$ Taylor coefficients of $\lambda _{n}(\eta )$ and $s_{n}(\eta )$, respectively, and by substituting the above expansions in Eqs. (54) and (55) the following set of recursion relations for the coefficients is obtained:

$$\begin{aligned} c_{n}^{i}= & {} (i+1)c_{n-1}^{i+1}+d_{n-1}^{i}+ \sum _{k=0}^{i}c_{0}^{k}c_{n-1}^{i-k}, \end{aligned}$$

(58)

$$\begin{aligned} d_{n}^{i}= & {} (i+1)d_{n-1}^{i+1}+\sum _{k=0}^{i}d_{0}^{k}c_{n-1}^{i-k}. \end{aligned}$$

(59)

Thus, the authors of the improved AIM have avoided the derivatives that contain the AIM in [29, 33], and the quantization condition, which is equivalent to imposing a termination on the number of iterations, is given by

$$\begin{aligned} d_{n}^{0}c_{n-1}^{0}-d_{n-1}^{0}c_{n}^{0}=0. \end{aligned}$$

(60)