A quantum-regularization parameter makes the black hole geometry smoother—but in these simulations, that smoothness weakens superradiance. Yet the same parameter can also create a new mode in one black hole bomb setup.

Superradiance exhibits wave amplification of fields in a thermodynamic system, with the cost of dissipation in a viscous medium. In curved spacetime, superradiance can also occur in vacuum, especially in black hole physics, where the event horizon provides a viscous one-way membrane and an ergoregion.

Superradiant instabilities occur when the process can be carried repeatedly, because a reflective mechanism bounces the amplified wave back again. In asymptotically flat charged black holes, a reflective boundary must be imposed in a toy model to make a black hole bomb.

Quantum gravity effects are widely believed to remove the curvature singularity of a black hole and result in a regular or non-singular black hole. Because a complete theory of quantum gravity is not yet well established, various non-singular black holes have been investigated at the phenomenological level.

Recent investigations of superradiant effects and black hole bomb instability in different regular black holes have provided insights into the interplay between fundamental and phenomenological aspects. The study investigates superradiant effects in a spherically symmetric charged black-bounce spacetime, which allows regular black holes and wormhole solutions through smooth parameter tuning.

The metric is obtained by replacing the radial coordinate with the square root of the radial coordinate squared plus lambda squared, where lambda is a length scale parameter representing quantum regularization. The parameter is typically associated with the Planck length, smooths the central singularity, and preserves asymptotic Reissner-Nordström behavior.

The model provides a phenomenological framework for probing how quantum gravitational effects alter superradiant processes and black hole bombs, while allowing controlled comparison with classical results by tuning lambda. The primary focus is how this length scale parameter influences the efficiency of superradiant energy extraction and black hole bomb instabilities.

The regularizing replacement extends the natural domain of the radial coordinate from zero to positive infinity into the full range from negative infinity to positive infinity, while asymptotic flatness is preserved. Computing the Kretschmann scalar and curvatures verifies that the replacement renders the geometry globally regular, rather than simply making a coordinate transformation.

The charged complex scalar field uses a gauge-covariant derivative, with the scalar-field mass and charge represented by mu and q. The corresponding equation governing the charged scalar field is the Klein-Gordon equation. In the time domain, the scalar field admits separable solutions, with Y sub l m as a spherical harmonic function.

The temporal part is converted to the frequency domain by a Fourier transform. Substitution into the Klein-Gordon equation, together with the tortoise coordinate, gives a differential equation for the radial part in the time domain.

The effective potential generalizes Regge-Wheeler analysis by including the field charge and mass, the black-hole charge, and the quantum parameter. The usual Reissner-Nordström effective potential is recovered by setting lambda to zero.

Because the effective potential is intimately related to the effective ergo-region and affects the superradiance magnitude, the study investigates how each parameter influences the potential. The standard parameters are lambda over M equal to zero point eight, mu M equal to zero point three, q M equal to one point four, Q over M equal to zero point nine, and l equal to zero.

As lambda grows larger, the effective potential becomes smoother and the potential well becomes shallower, so the superradiant ability is expected to weaken. As mu M increases, the range where the potential is negative becomes narrower, so the amplification factor is expected to decrease.

As q M and Q over M increase, the effective potential near the black-hole horizon continuously decreases, producing a deepening effect associated with larger superradiant amplification. Figure I maps the effective potential, labeled V eff, against the scaled radius x over M in four panels, varying lambda over M, mu M, Q over M, and q M in turn.

The standard setting is lambda over M equals zero point eight, mu M equals zero point three, q M equals one point four, Q over M equals zero point nine, and angular index l equals zero. These spatial profiles provide the potential behavior needed to interpret wave propagation between the event horizon and spatial infinity.

When omega is less than q A zero at the event horizon, the reflection coefficient exceeds the incident coefficient, indicating scalar-field amplification through superradiance and energy extraction from the black hole. A dimensionless amplification factor measures the effect of superradiance for different parameter choices.

The numerical calculation imposes a purely ingoing boundary condition near the event horizon and integrates the radial equation outward to a region where the numerical solution has stabilized. The reflection and incident coefficients are identified by comparison with the asymptotic form, and the amplification factor is then obtained.

As the quantum parameter lambda over M increases, the magnitude of the amplification factor decreases over a wide frequency range because the potential well is weakened. The reduction rate of the amplification factor progressively accelerates as the dimensionless quantum parameter increases.

The amplification factor also gradually decreases in all frequency ranges as the scalar-field mass mu M increases. The critical superradiant frequency is omega sub c M equal to zero point eight seven seven five, and it is not affected by lambda over M or mu M.

Figure Two shows the scalar-field amplification factor across frequency, comparing different coupling ratios on the left and different field masses on the right. With the other parameters fixed, the curves rise through an amplification regime and return to zero at the critical superradiant frequency, omega sub c times M equals zero point eight seven seven five.

This matters because it visualizes how the amplification spectrum depends on both lambda over M and the field mass, rather than reporting a single frequency response. The black-hole charge and field charge affect the critical superradiant frequency, and increasing the charges substantially broadens the cutoff-frequency range.

The amplification factor is presented as dependent on Q over M and q M. Figure three plots the scalar-field amplification factor against frequency, varying the black-hole charge-to-mass ratio on the left and the field charge on the right.

The curves show that increasing either charge suppresses amplification at low frequency but enhances it at high frequency, as the authors note. The comparison uses fixed values of the other charge in each panel, with lambda over mass equal to zero point eight, mu times mass equal to zero point three, and angular mode l equal to zero.

When a black hole is enclosed by reflective mirrors, superradiance can amplify scalar fields confined between the event horizon and the mirror boundary. In Reissner-Nordström spacetime, stability means that only a theoretical reflective boundary can be used to confine the field.

Energy accumulation in the scalar field may eventually break the mirror, leading to the black hole bomb phenomenon. The time-domain partial differential equation is solved with the finite difference method combined with the method of lines. After spatial discretization, the calculation produces ordinary differential equations containing only time derivatives.

The fourth-order Runge-Kutta method integrates this system, with the grid spacing set to delta x over M equal to zero point two. At the mirror, the solution is set to vanish, corresponding to psi of t comma x sub m equal to zero. Because the tortoise coordinate extends from negative infinity to the mirror position, the computational domain must be truncated at the left.

Perfectly Matched Layers suppress artificial reflections by introducing an absorbing region where outgoing waves are attenuated through damping profiles. The PML boundary is evaluated by checking that no reflected waves are generated and that the absorbing layer fully attenuates incoming waves.

The truncation boundary is set at negative twenty M, with a PML layer width of two hundred M. The initial condition is a Gaussian wave packet, with peak amplitude psi zero, initial center position x in, and spatial width alpha. The calculation sets psi zero to ten to the power of negative three, alpha to five M, and the initial position to twenty M, where the effective potential is sharpest.

Figure IV validates the perfectly matched layer, or PML, used to absorb outgoing scalar waves. The left panel shows the field’s repeated reflection and transmission between the mirror and black hole, while the dashed green line marks the PML boundary.

In the right panel, the blue PML-derived amplitude is compared with the orange true amplitude, allowing the authors to assess boundary reflections and attenuation within the absorbing layer. The study considers Type I, with the mirror outside the effective ergosphere, and Type II, with the mirror inside the effective ergosphere.

The fast Fourier transform extracts the real parts of the eigenfrequencies, while a digital filter extracts their imaginary parts. The scalar-field energy gain is calculated numerically. Figure V follows a Type I black-hole bomb with the mirror at x sub m equals fifty M, across three values of lambda over M: zero, zero point eight, and one point two.

Each row shows the evolving field amplitude, its power spectrum, and the energy flux measured at x sub s equals minus forty M. The green dashed line marks the critical superradiant frequency, approximately zero point eight seven seven five in units of M, making the spectral peaks’ locations directly comparable with the superradiant threshold.

As lambda increases, the scalar-field amplitudes, energy fluxes, and imaginary parts of the eigenfrequencies all decrease monotonically. In Type II, only two eigen-modes are present, with no new modes appearing when lambda over M equals one point two, unlike Type I. A possible qualitative explanation is that the absence of a left-side mirror in Type I produces a deeper potential well that can support more eigenstates.

In contrast, the left mirror in Type II shallows the effective potential well and reduces its capacity to support eigenstates. Figure seven tracks a Type II black-hole bomb with light particles, where the mirror is at minus ten M and the rows use lambda over M equal to zero, zero point eight, and one point two.

The left column shows the field amplitude, the middle column its power spectrum, and the right column the energy flux at seventy M. Across the spectra, the green dashed line marks the critical superradiant frequency, approximately zero point eight seven seven five in units of M inverse, while the time-domain panels show growing oscillatory behavior and flux.

For mu greater than omega sub c, the calculation uses mu M equal to zero point six, Q over M equal to zero point nine, q M equal to zero point eight, and lambda equal to zero and one point two, giving omega sub c equal to zero point five zero one four. Wave amplifications are absent, although eigenfrequencies appear in the right panel.

When lambda equals one point two, a new eigenfrequency appears within the range omega less than omega sub c. Unlike the new eigenfrequency in Type I, this new eigenfrequency is not superradiantly amplified. Figure VIII tracks the scalar field at x equals zero over time, alongside its power spectrum, for the sub-case where the field mass exceeds the critical frequency in a Type II black-hole bomb.

The top row uses lambda over M equal to zero, while the bottom uses one point two; both show oscillatory amplitude and distinct spectral peaks. The dashed markers identify mu M equal to zero point six and omega c approximately zero point five, highlighting the frequencies relevant to the superradiant setup.

The study examines superradiance for a charged scalar field in a black-bounce Reissner-Nordström black hole, obtained by replacing the radial coordinate with the square root of the radial coordinate squared plus lambda squared. The quantum parameter lambda shallows the effective potential in the Regge-Wheeler equation and weakens the negativity of the energy.

The study focuses on the quantum parameter in a spherically symmetric charged black-bounce black hole; quantum parameters in nonsingular rotating black holes remain to be systematically investigated. The study uses an artificially engineered reflecting boundary beyond the event horizon to constrain the scalar field.

An alternative pathway is to use nonlinear interaction terms within the scalar-field framework to achieve analogous confinement effects. The investigation is limited to scalar-field evolution, and further studies are needed to explore black-hole mass and charge dynamics. The central result is that increasing the quantum parameter generally weakens scalar superradiant amplification by shallowing the effective potential.

The bomb configurations add a sharper twist: new eigenfrequencies can appear, but they are not always superradiantly amplified.