Scalar field solutions and energy bounds for modeling spatial oscillations in Schwarzschild black holes based on the Regge–Wheeler equation

This text discusses the behavior of solutions and the energy stability within Schwarzschild spacetimes, with a particular emphasis on the behavior of massless scalar fields under the influence of a non-rotating and spherically symmetric black hole. The stability of solutions in the proximity of the event horizon of black holes in general relativity remains an open question, especially given the difficulties introduced by minor perturbations that may resemble Kerr solutions. To address this, this work explores a simplified model, including massless scalar fields, to better understand perturbation behaviors around black holes under the Schwarzschild approach. We depart from Richard Price’s work in connection with how scalar, electromagnetic, and gravitational fields behave. The tortoise coordinate transformation is considered to set the stage for numerical solutions to the wave equations. Afterward, we explore energy estimates, which are used to gauge stability and wave behavior over time. Our analysis reveals that the time evolution of the energy does not exceed twice its initial value. Further and under the assumption of initial conditions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo></mml:math> spaces, we obtain an exponential decreasing behavior in the energy time evolution. A question to continue exploring is how perturbations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> in the initial conditions that introduce Kerr solutions as a second-order effect in the linearized equations perturb this obtained exponential decay.