Scalar superradiance in the charged black-bounce spacetimes

TL;DR

This work analyzes charged scalar superradiance in the black-bounce-Reissner-Nordström spacetime, a regularized black-hole model obtained by $r \to \sqrt{r^2+\lambda^2}$. By deriving the charged Klein-Gordon equation and performing a Regge-Wheeler–type analysis, the authors quantify how the quantum parameter $\lambda$ deepens into $W_{\text{eff}}$ and weakens the negative-energy support responsible for superradiance. They investigate both scattering and black-hole bomb configurations with mirrors (Type I and II) using time-domain finite-difference methods and a conserved energy current to track amplification, including the implementation of Perfectly Matched Layers. The results show that larger $\lambda$ or scalar mass $\mu$ suppress amplification across all frequencies, while BH and field charges shift the frequency response; notably, a new eigenmode emerges in Type I bombs at $\lambda/M=1.2$, whereas Type II bombs do not exhibit such a mode and can suppress amplification when $\mu > \omega_c$. Overall, the study demonstrates how quantum regularization effects regulate energy extraction from non-singular black holes and motivates future work on rotating regular spacetimes and alternative confinement mechanisms.

Abstract

We numerically investigate the superradiant amplification effect of a charged scalar filed in the scattering experiment and the black hole bomb model in a charged black-bounce spacetime. Due to the shallowing effect on the effective potential by the introduced quantum parameter $ł$, superradiance in both the above cases are verified to be weakened. In a scattering experiment, the quantum parameter and the field mass suppress the amplification in all frequency ranges, while the black hole and field charge influence it differently in high and low frequencies. In a Type I black hole bomb model, where the reflective mirror is placed outside the ergo-region, we find a new distinct eigen-mode for the scalar field evolution in a high $ł$ value, which is however absent in the case of Type II black bomb where the mirror is set inside the ergo-region. Moreover, we investigate the heavy field mass scenario in a Type II black hole bomb and find no amplification effect in this confined configuration.

Figures (8)

• Figure II: Effective potential for $\mu M=0.3$ , $qM=1.4$ , $Q/M=0.9$ , $l=0$. Different curves represent different values of $\lambda$.
• Figure III: Amplification factor of the scalar field as a function of the frequency with dependence on different choices of $\lambda/M$ (left) and on different field mass ${\mu} M$ (right). Curves in blue, orange, green, purple color in the left panel denote cases of $\lambda/M=0, 0.4, 0.8, 1.2$ respectively with ${\mu} M=0.3$ fixed, and in the right panel denote ${\mu} M=0.2, 0.3, 0.4, 0.5$ respectively with ${\lambda}/M=0.8$ fixed. Other parameters are chosen as as the standard values, $Q/M=0.9, qM=1.4, l=0$. The point where curves meet the horizontal axis gives the critical superradiant frequency, which is $\omega_cM=0.8775$ in these cases.
• Figure IV: Amplification factor of the scalar field as a function of the frequency with dependence on different choices of $Q/M$ (left) and on different field mass $q M$ (right). Curves in blue, orange, green, purple color in the left panel denote cases of $Q/M=0.6, 0.7, 0.8, 0.9$ respectively with $q M=1.4$ fixed, and in the right panel denote ${\mu} M=1, 1.2, 1.4, 1.6$ respectively with $Q/M=0.9$ fixed. Other parameters are chosen as the standard values, ${\lambda}/M=0.8, {\mu} M=0.3, l=0$.
• Figure V: The validation of the effectiveness of PML boundary condition. The left panel illustrates the continuous reflection and transmission of the scalar field between the mirror and the black hole. The right panel presents a comparison between the scalar field amplitude after applying PML boundary conditions and the true amplitude. The dash green line is the boundary of the PML layer.
• Figure VI: Evolution of the scalar field in a Type I BH bomb. The position of the mirror is at $x_m=50M$, and the parameters are set as $Q/M=0.9$, $\mu M=0.3$, $qM=1.4$, $l=0$. From left to right, the three columns represent the field amplitude, power spectrum and energy flux (measured at $x_s=-40M$) . From top to bottom, the values of $\lambda/M$ are 0.0, 0.8 and 1.2 respectively. The green dash line represents the critical superradiant frequency which is approximately ${\omega}_c M\simeq 0.8775$.