Back-reaction spacetime waves from a radially varying mass in a Hayward black hole

In this work, we describe back-reaction spacetime waves from a radially varying mass in a Hayward geometry. We develop a method to find boundary conditions relying on the simple form of the metric near the center of the black hole. This solution is evolved in the radial coordinate r, after a Fourier transform in time t is applied with periodic boundary conditions, combining a high order Runge–Kuta scheme with a novel implementation of the Frobenius method to avoid singular points. We present three distinct examples of the evolution of the norm |RNL| in terms of t and r, with values of the parameter L = 1, 2, 3; the mass m = 1; the Hayward’s characteristic length ℓ = 0.001; and the cosmological parameter Λ = 1000. The numerical simulation starts at r0 = ℓ/10. The solutions suggest that oscillations, immediately after passing through the Schwarzschild radius, grow stronger and become more pronounced as the parameter L increases. Additionally, in the radial coordinate, the further |RNL| progresses, the less it oscillates, indicating a decaying behavior for larger r. Whereas the time evolution shows the anticipated periodic behavior and similarly for bigger L these ripples are more prominent.