Extended two-dimensional characteristic framework to study nonrotating black holes

TL;DR

This paper introduces a 2D numerical solver for scalar perturbations of nonrotating black holes, enabling detailed analysis of quasinormal modes, energy exchange, and nonlinear harmonic generation in complex spacetime geometries.

Contribution

The work extends existing computational frameworks to include scalar fields in 2D, validating the solver and demonstrating its capability to analyze nonlinear effects and higher harmonic evolutions.

Findings

Successfully reproduces quasinormal modes for scalar harmonics

Calculates energy balance between black hole and surroundings

Shows nonlinear harmonic generation and complex angular structure evolution

Abstract

We develop a numerical solver, that extends the computational framework considered in [Phys. Rev. D 65, 084016 (2002)], to include scalar perturbations of nonrotating black holes. The nonlinear Einstein-Klein-Gordon equations for a massless scalar field minimally coupled to gravity are solved in two spatial dimensions (2D). The numerical procedure is based on the ingoing light cone formulation for an axially and reflection symmetric spacetime. The solver is second order accurate and was validated in different ways. We use for calibration an auxiliary 1D solver with the same initial and boundary conditions and the same evolution algorithm. We reproduce the quasinormal modes for the massless scalar field harmonics $\ell = 0$, $1$ and $2$. For these same harmonics, in the linear approximation, we calculate the balance of energy between the black hole and the world tube. As an example of nonlinear harmonic generation, we show the distortion of a marginally trapped two-surface approximated as a q-boundary and based upon the harmonic $\ell=2$. Additionally, we study the evolution of the $\ell = 8$ harmonic in order to test the solver in a spacetime with a complex angular structure. Further applications and extensions are briefly discussed.