Long-lived quasinormal modes, shadows and particle motion in four-dimensional quasi-topological gravity

Abstract

We investigate massive scalar perturbations and several characteristics of particle motion in the spacetime of regular black holes arising in four-dimensional quasi-topological gravity. Quasinormal modes are computed using high-order WKB approximations with Padé resummation and verified through time-domain integration. For moderate values of the scalar-field mass, the time-domain profiles confirm the WKB results with excellent accuracy. As the mass increases, the damping rate decreases substantially, indicating the approach to the quasi-resonant regime of long-lived modes. For sufficiently large masses, the late-time signal becomes dominated by oscillatory power-law tails, which mask the quasi-resonant mode in the time-domain profile. In addition, we analyze photon motion and circular geodesics, including the photon-sphere radius, shadow size, Lyapunov exponent, and ISCO characteristics. These quantities exhibit only moderate deviations from their Schwarzschild values, unlike the Hawking temperature of the black hole.

Figures (4)

• Figure 1: Black hole model I. Effective potential as a function of the tortoise coordinate $r^{*}$ for $\ell=0$ (left panel: $\mu=0$ (blue), $\mu=0.2$ (black), and $\mu=0.4$ (red)) and $\ell=1$ (right panel: $\mu=0$ (blue), $\mu=0.3$ (black), and $\mu=0.6$ (red)). The parameters are $4l^{4}=0.94$ and $M=1$.
• Figure 2: Black hole model II. Effective potential as a function of the tortoise coordinate $r^{*}$ for $\ell=0$ (left panel: $\mu=0$ (blue), $\mu=0.2$ (black), and $\mu=0.6$ (red)) and $\ell=1$ (right panel: $\mu=0$ (blue), $\mu=0.3$ (black), and $\mu=0.9$ (red)). The parameters are $l=0.51$ and $M=1$.
• Figure 3: Right Panel: Black hole model I. Time-domain profile for $\ell=1$ perturbations. The parameters are $l=0.51$ and $M=1$, $\mu=0.1$. The time-domain integration dominant mode is $\omega =0.29641 - 0.087796 i$, while the WKB data is $\omega = 0.296401 - 0.0877942 i$, confirming great accuracy of the WKB method. Left Panel: Black hole model II. Time-domain profile for $\ell=1$ perturbations. The parameters are $l=0.51$ and $M=1$, $\mu=0.1$. The time-domain integration dominant mode is $\omega =0.612748 - 0.166874 i$, while the WKB data is $\omega = 0.612667 -0.166952 i$.
• Figure 4: Black hole model I. Logarithmic time-domain profile for $\ell=1$ perturbations. The parameters are $l=0.51$, $M=1$, $\mu=1$. The intermediate oscillatory late time tails with power-law envelope dominate even in the early phase, so that the quasi-resonant modes cannot be extracted from the signal.