Comparison of Quasinormal Modes of Black Holes in $f(\mathbb{T})$ and $f(\mathbb{Q})$ Gravity

Abstract

We investigate the quasinormal modes of static and spherically symmetric black holes in vacuum within the framework of $f(\mathbb{Q}) = \mathbb{Q} + α\mathbb{Q}^2$ gravity, and compare them with those in $f(\mathbb{T}) = \mathbb{T} + α\mathbb{T}^2$ gravity. Based on the Symmetric Teleparallel Equivalent of General Relativity, we notice that the gravitational effects arise from non-metricity (the covariant derivative of metrics) in $f(\mathbb{Q})$ gravity rather than curvature in $f(R)$ or torsion in $f(\mathbb{T})$. Using the finite difference method and the sixth-order WKB method, we compute the quasinormal modes of massless scalar field and electromagnetic field perturbations. Tables of quasinormal frequencies for various parameter configurations are provided based on the sixth-order WKB method. Our findings reveal the differences in the quasinormal modes of black holes in $f(\mathbb{Q})$ gravity compared to those in $f(R)$ and $f(\mathbb{T})$ gravity. This variation demonstrates the impact of different parameter values, offering insights into the characteristics of $f(\mathbb{Q})$ gravity. These results provide the theoretical groundwork for assessing alternative gravities' viability through gravitational wave data, and aid probably in picking out the alternative gravity theory that best aligns with the empirical reality.

Figures (16)

• Figure 1: The shape function $F(r)$ of $f(\mathbb{Q})$ BHs, where $\alpha = 0.2$, $k = 10$, and $r_0 = 0.1$ are chosen. The shape function of Schwarzschild BHs is given for comparison.
• Figure 2: The effective potential $V_\mathrm{eff}$ as a function of the tortoise coordinate $r^*$, where the orbital angular momentum $l$ is fixed to be $1$. Two sets of parameters are chosen for each type of perturbations: Set 1 ($\alpha = 0.05$, $k = 20$, $r_0 = 0.1$) and set 2 ($\alpha = 0.07$, $k = 20$, $r_0 = 0.1$). For comparison, $V_\mathrm{eff}$ of a Schwarzschild BH is also included.
• Figure 3: The integration grid is chosen for solving the wave equation. Each grid point corresponds to a value of the wave intensity $\Phi(u,v)$. The points $A$, $B$, $C$ and $D$ illustrate the relative positioning of adjacent points. Initial conditions are assigned along the $t=0$ line, and the time decay behavior for a specific $r^*$ is extracted along the $r^*=\mathrm{const.}$ line.
• Figure 4: The waveform of scalar field perturbations is shown with parameters $\alpha = 0.05$, $k = 20$, and $r_0 = 0.1$ in $f(\mathbb{Q})$ gravity. Here $s=0$ and $l=1$. The case of the Schwarzschild black hole is presented for comparison, also with $s=0$ and $l=1$.
• Figure 5: Waveform of scalar field perturbations is shown when the center of perturbation wave packets, $r_c$, is varied, where $l=1$, $\alpha=0.05$, $k=20$, and $r_0=0.1$ are set. The oscillation frequency is independent of the values of $r_c$, but the onset time of tail phases is dependent on them.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4