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Black Hole Superradiance in Dynamical Spacetime

William E. East, Fethi M. Ramazanoğlu, Frans Pretorius

TL;DR

This paper investigates black hole superradiance beyond the linear regime by solving the full Einstein equations for a nearly extremal Kerr BH ($a=0.99$) interacting with gravitational wave packets up to about $0.1M$. The authors use a generalized harmonic evolution and Kerr-Schild initial data to track energy and angular momentum transfer via gravitational waves and to monitor the apparent horizon. They find that linear amplification, about 40%, persists at small amplitudes but decreases with increasing wave energy, with nonlinear effects including higher-mode coupling to $ (l,m)=(4,4) $ and pronounced apparent-horizon distortions; backreaction reduces the rotational energy extracted, consistent with horizon area constraints. The work demonstrates the significance of dynamical backreaction in BH superradiance, clarifies energy-extraction limits in a nonlinear regime, and highlights horizon geometry as a key factor in highly dynamical spacetimes, suggesting directions for future horizon studies.

Abstract

We study the superradiant scattering of gravitational waves by a nearly extremal black hole (dimensionless spin $a=0.99$) by numerically solving the full Einstein field equations, thus including backreaction effects. This allows us to study the dynamics of the black hole as it loses energy and angular momentum during the scattering process. To explore the nonlinear phase of the interaction, we consider gravitational wave packets with initial energies up to $10%$ of the mass of the black hole. We find that as the incident wave energy increases, the amplification of the scattered waves, as well as the energy extraction efficiency from the black hole, is reduced. During the interaction the apparent horizon geometry undergoes sizable nonaxisymmetric oscillations. The largest amplitude excitations occur when the peak frequency of the incident wave packet is above where superradiance occurs, but close to the dominant quasinormal mode frequency of the black hole.

Black Hole Superradiance in Dynamical Spacetime

TL;DR

This paper investigates black hole superradiance beyond the linear regime by solving the full Einstein equations for a nearly extremal Kerr BH () interacting with gravitational wave packets up to about . The authors use a generalized harmonic evolution and Kerr-Schild initial data to track energy and angular momentum transfer via gravitational waves and to monitor the apparent horizon. They find that linear amplification, about 40%, persists at small amplitudes but decreases with increasing wave energy, with nonlinear effects including higher-mode coupling to and pronounced apparent-horizon distortions; backreaction reduces the rotational energy extracted, consistent with horizon area constraints. The work demonstrates the significance of dynamical backreaction in BH superradiance, clarifies energy-extraction limits in a nonlinear regime, and highlights horizon geometry as a key factor in highly dynamical spacetimes, suggesting directions for future horizon studies.

Abstract

We study the superradiant scattering of gravitational waves by a nearly extremal black hole (dimensionless spin ) by numerically solving the full Einstein field equations, thus including backreaction effects. This allows us to study the dynamics of the black hole as it loses energy and angular momentum during the scattering process. To explore the nonlinear phase of the interaction, we consider gravitational wave packets with initial energies up to of the mass of the black hole. We find that as the incident wave energy increases, the amplification of the scattered waves, as well as the energy extraction efficiency from the black hole, is reduced. During the interaction the apparent horizon geometry undergoes sizable nonaxisymmetric oscillations. The largest amplitude excitations occur when the peak frequency of the incident wave packet is above where superradiance occurs, but close to the dominant quasinormal mode frequency of the black hole.

Paper Structure

This paper contains 4 sections, 4 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Results
  4. Conclusions

Figures (4)

  • Figure 1: Top: Energy spectrum of outgoing GWs after the scattering of an ingoing GW packet with central frequency $\omega_0 M=0.75$ and various amplitudes. The lower amplitude waves have been rescaled according to their leading-order dependence on ingoing amplitude. We also show a case with $\omega_0 M=0.87$ and the same initial energy as the $A=12$ case. Bottom: Truncation error for the $A=12$ case, consistent with between third- and fourth-order convergence (the error has been scaled assuming the latter).
  • Figure 2: Top: The energy in outgoing GWs for different values of the ingoing amplitude $A$, relative to the value obtained by scaling the $E_{\rm GW}$ value at $A=4$ as $A^2$, as predicted by linear theory. Below that is shown the ratio of the energy to angular momentum of the outgoing GWs. The curves are fits of the form $a+bA^2$. We indicate the numerical error by showing the results from the low and medium resolution runs at $A=4$ and $12$. Bottom: The changes in the BH quantities of total mass, angular momentum times the initial BH rotational frequency (upper panel), as well as irreducible mass, and dimensionless spin (lower panel) implied by the above fits. We also show the changes in the BH that would occur if the outgoing GWs were always proportional to the ingoing GWs (labeled linear, i.e., ignoring the $A^2$ term of the fit shown in the top two panels).
  • Figure 3: The efficiency of BH energy extraction $\eta$, as a function of the energy extracted from the BH, that is implied by the fit to simulation results in Fig. \ref{['egw_fig']} (lower amplitude branch). We also show the efficiency expected for a process where the change in BH angular momentum to BH mass is given by $dJ=(m/\omega_0)dM$.
  • Figure 4: AH quantities during interaction with different frequency GW packets, each with initial mass $\approx 0.1M$. Shown (in units where $M=1$) are the mass, irreducible mass, and angular momentum of the BH (top panel), the ratio of proper polar to equatorial circumference (middle panel), and the maximum Ricci curvature of the horizon surface (bottom panel). For the latter two quantities, we also show the values of an unperturbed BH with extremal spin.