Free Hyperboloidal Evolution of the Einstein-Maxwell-Klein-Gordon System

TL;DR

This work develops a numerical framework for evolving the Einstein-Maxwell-Klein-Gordon system on hyperboloidal slices that reach future null infinity, enabling direct radiation extraction. It combines a conformal transformation with a BSSN/Z4-like 3+1 formulation and an adapted Lorenz gauge to handle the coupled gravity, Maxwell, and charged scalar fields, all in spherical symmetry. The authors demonstrate the approach with three cases: a charged scalar perturbation of Minkowski, a charged scalar perturbation of a Reissner-Nordström black hole, and the collapse of a charged scalar into a RN BH, including convergence tests and scri-fixing. The results establish a first step toward fully nonlinear EMKG evolutions that access null infinity, with potential applications to radiation research, cosmic censorship investigations, and extensions to more general systems and higher dimensions.

Abstract

We present simulations of the Einstein-Maxwell-Klein-Gordon system on compactified hyperboloidal slices. To the best of our knowledge, this is the first time that this setup is evolved with a common formulation like BSSN/Z4. Hyperboloidal slices smoothly reach future null infinity, the only location in spacetime where radiation (such as gravitational waves) is unambiguously defined. We are thus able to reach null infinity and extract signals there. We showcase the capabilities of our implementation in spherical symmetry with the evolution of a charged scalar field perturbing a regular spacetime and near an electrically charged black hole. We also present the collapse of a charged scalar field into a Reissner-Nördstrom black hole.

Figures (10)

• Figure 1: Variation of $\bar{\Omega}$ according to different values of the charge-to-mass ratio $Q/M$, as solution to \ref{['eq:indatagen2']} setting ${\gamma_{rr}}_0=1$. $M$ has been set to 1 and $Q$ is changing.
• Figure 2: Evolution of initial perturbations on $\bar{c}_\phi$ and $A_{3r}$, for regular initial data as given in sections \ref{['sec:regularinitialdata']} and \ref{['subsec:chargedscalarinitialdata']}. The plots show $\bar{d}_\phi$, $\bar{E}_r$, $\bar{j}_r$ and $\bar{q}_{\text{dens.}}$ evolving until the scalar field is mostly radiated out through future null infinity, at $r_{\mathscr{I}^+} = 1$.
• Figure 3: L2 norm convergence plot of the simulation shown in Fig. \ref{['fig:scalarmax']}, here plotting the quantity \ref{['convorder']}. The ideal result is a horizontal line at 4, for the expected 4th order convergence of our setup -- the orange line, which uses a series of runs with higher resolution, is generally closer to 4 than the blue one, which is a good indication.
• Figure 4: Initial data for the conformal factor $\chi$, the BH's electric field $\bar{E}^r$, \ref{['eq:electricrn']}, and the scalar potential $\Phi$, \ref{['eq:potentialindata']}. We set $Q=0.8$, $K_{\text{CMC}}=-1$ and $M=1$.
• Figure 5: Additional electric field coming from the scalar field's initial interaction with the scalar potential from the BH.