Quasinormal frequencies of self-dual black holes

One simplified black hole model constructed from a semiclassical analysis of loop quantum gravity (LQG) is called the self-dual black hole. This black hole solution depends on a free dimensionless parameter $P$ known as the polymeric parameter and also on the $a_0$ area related to the minimum area gap of LQG. In the limit of $P$ and $a_0$ going to zero, the usual Schwarzschild solution is recovered. Here we investigate the quasinormal modes (QNMs) of massless scalar perturbations in the self-dual black hole background. We compute the QN frequencies using the sixth-order WKB approximation method and compare them with numerical solutions of the Regge-Wheeler equation. Our results show that, as the parameter $P$ grows, the real part of the QN frequencies suffers an initial increase and then starts to decrease while the magnitude of the imaginary one decreases for fixed area gap $a_0$. This particular feature means that the damping of scalar perturbations in the self-dual black hole spacetimes is slower, and the oscillations are faster or slower according to the value of $P$.

Figure 1

Scalar field normal modes. The markers denote the multipole number as $\ell =0$ (circle), $\ell =1$ (square), $\ell =2$ (triangle), while the colors denote the value of the polymerization parameter: $P=0.1$ (blue), $P=0.2$ (green), $P=0.3$ (orange), $P=0.6$ (red). The black markers denote the frequencies for the Schwarzschild black hole.

Figure 2

Real (blue/top) and imaginary (red/bottom) parts of the quasinormal frequencies as a function of the WKB order for the $\ell =0$, $n=0$ mode for some values of the polymerization parameter $P$.

Figure 3

Log-log plots of the time domain profiles for the scalar perturbations at $x=10M$ for several values of the polymerization parameter $P$.

Figure 4

Numerical evolution of waveform (solid line) for $\ell =0$ and the fit from the sixth-order WKB frequencies, for several values of the polymerization parameter. For the fit, we took the fundamental mode and the first overtone.

Figure 5

Numerical evolution of waveform (solid line) for $\ell =1$ and the fit from the sixth-order WKB frequencies for several values of the polymerization parameter. For the fit, we took the fundamental mode and the first overtone.