Perturbative Renormalisation Group Improved Black Hole Solution and its Quasinormal Modes

Abstract

In this work, we construct a perturbative black hole (BH) solution motivated by renormalization group (RG) improvement and investigate the quasinormal modes (QNMs) of the BH under scalar field perturbations in both Schwarzschild-de Sitter (SdS) and Schwarzschild-anti-de Sitter (SAdS) backgrounds. To compute the QNMs in the SdS spacetime, we employ the 6th-order Padé-averaged WKB approximation method, while for the SAdS background we utilize the direct shooting method. We examine the dependence of the QNM frequencies on the free parameter of the solution. Furthermore, we analyze the time evolution of a scalar field perturbation around the BH and present the corresponding time-domain profiles. The QNMs are also extracted from the time-domain data using the matrix pencil method. Using the extracted QNM frequencies, we reconstruct the waveform and compare it with the original time-domain profile, finding good agreement between the two. The QNM frequencies obtained from the 6th-order Padé-averaged WKB method and the time-domain analysis in the SdS background, as well as those obtained from the direct shooting method and time-domain analysis in the SAdS spacetime, show very good consistency.

Figures (10)

• Figure 1: Behaviour of the effective potential $V(r)$ as a function of the radial coordinate $r$ for both positive and negative values of the parameter $\zeta$, obtained by fixing $M = 1$ and $\bar{\Lambda} = 0.001$.
• Figure 2: Variation of real and imaginary parts of QNMs with respect to positive values of $\zeta$ for different values of $l$ along with $M = 1$, $n = 0$ and $\bar{\Lambda} = 0.001$.
• Figure 3: Variation of real and imaginary parts of QNMs with respect to negative values of $\zeta$ for different values of $l$ along with $M = 1$, $n = 0$ and $\bar{\Lambda} = 0.001$.
• Figure 4: Behaviour of the effective potential $V(r)$ as a function of the radial coordinate $r$ for both positive and negative values of $\zeta$, fixing $M = 1$, and $\bar{\Lambda} = -0.001$.
• Figure 5: Variation of real and imaginary parts of QNMs with respect to positive values of $\zeta$ for different values of $l$ along with $M = 1$, $n = 0$, and $\bar{\Lambda} = -0.001$.