Scalar quasinormal modes of anti–de Sitter static spacetime in Horava-Lifshitz gravity with $U(1)$ symmetry

In this paper, we investigate the scalar quasinormal modes of Hořava-Lifshitz theory with $U(1)$ symmetry in static anti–de Sitter spacetime. The static planar and spherical black hole solutions in lower energy limit are derived in nonprojectable Hořava-Lifshitz gravity. The equation of motion of a scalar field is obtained, and is utilized to study the quasinormal modes of massless scalar particles. We find that the effect of Hořava-Lifshitz correction is to increase the quasinormal period as well as to slow down the decay of the oscillation magnitude. Besides, the scalar field could be unstable when the correction becomes too large.

Figure 1

Scalar quasinormal modes vs temperature of Hořava-Lifshitz planar black hole spacetime, where $T=\frac{3r_0}{4\pi }$.

Figure 2

Scalar quasinormal modes vs temperature of Hořava-Lifshitz spherical black hole spacetime, where $T=\frac{1+3r_0}{4\pi }$.

Figure 3

The time evolutions of scalar perturbations in Hořava-Lifshitz planar black hole spacetime for $r_0=1$.

Figure 4

The time evolutions of scalar perturbations in Hořava-Lifshitz spherical black hole spacetime for $r_0=1$.

Figure 5

Instability in planar and spherical black hole spacetime, for given $r_0=1$ and $L=0$.

Figure 6

The relation between the number of terms used in the expansion $n_N$ and the value of critical ${\rho }_c$ in planar (left-hand side) and spherical (right-hand side) black hole spacetime, for given $r_0=1$. The calculations are carried out by assuming $\omega =0$ for critical case.

Figure 7

The relation between the critical value ${\rho }_c$ and the temperature $T$ for planar (left hand side, $T=\frac{3r_0}{4\pi }$) and spherical (right hand side, $T=\frac{1+3r_0}{4\pi }$) black hole spacetime. For critical mode, the frequency $\omega$ vanishes. The value $n_N=50$ is used in the calculations.