Charged Scalar Field at Future Null Infinity via Nonlinear Hyperboloidal Evolution

TL;DR

The paper investigates the fully non-linear evolution of a charged scalar field on a Reissner-Nordström background using hyperboloidal slices that reach future null infinity, enabling direct extraction of quasinormal modes and late-time tails at $\mathrsfs{I}^+$. By solving the coupled Einstein–Maxwell–Klein–Gordon system with backreaction in spherical symmetry, the authors reveal how the scalar field charge $q$ and black hole charge $Q$ shape both the QNM ringing (well-fit for Schwarzschild, suppressed for charged backgrounds) and the tail decay, which transitions from gravity-dominated to electromagnetism-dominated behavior as $qQ$ increases. The analysis yields tail exponents $p_{\mathrsfs{I}^+}$ and $p_{i^+}$ tied to $\beta=\frac{-1+\sqrt{(2l+1)^2-4(qQ)^2}}{2}$, and shows that tail oscillation frequencies scale with $qQ$ but can deviate from some analytical predictions at larger $qQ$, prompting an empirical adjustment. Overall, the work demonstrates the power of non-linear, hyperboloidal evolutions to probe charged perturbations near black holes and provides insights into how EM coupling influences late-time dynamics with potential astrophysical relevance for charged spacetimes.

Abstract

Quasinormal modes and power-law late-time decay tails of a charged scalar field in a charged black hole background have been studied, but never in the fully non-linear regime, as far as we know. In this paper, we study the dependence of these properties on the charges of scalar field and black hole. For the quasinormal modes, a fit of the spherical fundamental mode is shown for the purely uncharged case and compared to the charged one. We also see for the first time the transition from gravitational decay to pure electromagnetic decay, and show disagreement with the oscillation frequency between real and imaginary parts of the scalar field prescribed in the literature. Full non-linear evolutions of hyperboloidal slices in spherical symmetry were used to obtain our results, allowing for the extraction of signals at future null infinity.

Figures (8)

• Figure 1: Numerical solution for $\psi_A$ solving \ref{['eq:psiaev']} for several values of the black hole's charge. We show a zoom-in on the region where it is easier to see the effect of the charge on $\psi_A$. $M$ is set to 1 and $K_{\text{CMC}}$ to -1.
• Figure 2: Integrating \ref{['eq:psifrompsia']} to get the numerical solution for $\psi$, assuming $Q=0$. For other values of charge, the result is similar.
• Figure 3: Comparison of quasinormal mode ringing of a scalar field perturbation of Schwarzschild and RN spacetimes (called Full GR in the plot). We also show the Cowling approximations, i.e., not evolving the background (called Cowling in the plot). We plot this and the following time-dependent graphs in terms of the logarithm of the unrescaled code time $t$ instead of $\log(t/M)$, as the total mass of the system changes depending on the contribution of the scalar field that falls into the black hole.
• Figure 4: Fitting the fundamental mode to the QNM ringing of the real scalar field, in a Schwarzschild spacetime.
• Figure 5: Late-time tails at $\mathrsfs{I}^+$ for a charged scalar field in a charged background. The scalar field's charge is set to 1 and the black hole's charge is different in each plot (from top left to bottom right: 0.0001, 0.3, 0.6 and 1.0). The black line in all graphs corresponds to the amplitude of the scalar field \ref{['eq:scalarfmodulus']}. The mass of the black hole is set to 1, making the last graph correspond to an extreme RN black hole.