Scalar field perturbations in asymptotically Lifshitz black holes

Abstract

We consider scalar field perturbations about asymptotically Lifshitz black holes with dynamical exponent z in D dimensions. We show that, for suitable boundary conditions, these Lifshitz black holes are stable under scalar field perturbations. For z=2, we explicitly compute the quasinormal mode frecuencies, which result to be purely imaginary, and then obtain the damping-off of the scalar field perturbation in these backgrounds. The general analysis includes, in particular, the z=3 black hole solution of three-dimensional massive gravity.Comment: 13 pages, no figures. v2: 14 pages, references added. To appear on PR

Lifshitz quasinormal modes and relaxation from holography

“…This result is interesting when taking into account the conjectures of [12,16], which state that for (most) Lifshitz black holes the quasinormal modes are purely imaginary. Moreover, finding underdamped cases for d > z + 1 contrasts the overdamped cases found for d ≥ 4, z = 2 in a R 2 gravity setting [16] and d ≥ 2, z = 2 in a R 3 gravity setting [17]. We present a cartoon of the qualitative structure of the overtones of the quasinormal modes in the complex plane in figure 4.…”

“…For reviews on this topic we recommend [13,14]. A summary of previously obtained analytic and numerical solutions of quasinormal modes of spin-zero fields, making usage of various bulk actions, entails: d = 2, z = 3 in New Massive Gravity (NMG) [15], d ≥ 4, z = 2 in a R 2 gravity setting [16] and d ≥ 2, z = 2 in a R 3 gravity setting [17]. The case for d = 3, z = 2 has been studied in the EinsteinProca-Scalar (EPS) background [6,18], in the EMD setup [12] and in a topological black hole in a Einstein-Maxwell-Proca (EMP) background [19][20][21].…”

Nonlinearly charged Lifshitz black holes for any exponent z > 1

“…We are also convinced that nonlinear electrodynamics will still be useful to explore Lifshitz black hole solutions and its generalizations in contexts beyond standard gravity, as for example for the Lovelock gravity [17] or even in higher-order gravity theories [15,16,18,19].…”

“…From the very beginning of the Lifshitz advent it becomes clear that pure Einstein gravity with eventually a cosmological constant resists to support such configurations, since it is clear from Birkhoff's theorem and its generalizations that in highly symmetric vacuum situations there is no freedom to choose the boundary conditions and they become fixed by the dynamics. Consequently, Lifshitz asymptotic requires the introduction of matter sources [5][6][7][8][9][10][11][12][13][14] or/and to consider higher-curvature gravity theories [15][16][17][18][19]. One of the first and simplest examples that has been used for supporting the Lifshitz spacetimes at any dimension D is a Proca field coupled to Einstein gravity with a negative cosmological constant [5].…”

“…The second case is for Notice that using the limit procedure of subsection 3.1 we obtain exactly this inequality from the one of (5.6) in the limit z → (D − 2) + . When the inequality is saturated we have an extremal black hole with horizon 19) and the gravitational potential is rewritten as …”