Chaos, Solitons and Fractals 116 (2018) 348–357 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos On the quasi-normal modes of a Schwarzschild white hole for the lower angular momentum and perturbation by non-local fractional operators Amos S. Kubeka, Emile F. Doungmo Goufo ∗ , Melusi Khumalo Department of Mathematical Sciences, University of South Africa, Florida 0003 South Africa a r t i c l e i n f o Article history: Received 22 August 2018 Revised 25 September 2018 Accepted 26 September 2018 Keywords: Schwarzschild white hole Quasi-normal mode Fractional model with non-local operator Atangana–Baleanu fractional derivative in Caputo sense Numerical scheme 26A33 35Q85 65C20 97M50 a b s t r a c t We investigate conditions for the quasi-normal modes of a Schwarzschild white hole for lower angular momentum. In determining these normal modes, we use numerical methods to solve the solution of the linearized Einstein vacuum equations in null cone coordinates. The same model is generalized to non- local fractional operator theory where the model is solved numerically thanks to a method proposed by Toufik and Atangana. In fact, approaching this kind of problem analytically seems to be an impossible task as comprehensively articulated in the literature. We show existence of quasi-normal modes of a Schwarzschild white hole for lower angular momentum l = 2. Moreover, the non-local fractional operator appears to be a perturbator factor for the system as shown by numerical simulations that compare the types of dynamics in the system. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction The linear perturbation theory applied to black hole was de- veloped many years ago. The main idea here comes from the fact that the vacuum Einstein equations are linearized about the Schwarzschild (or Kerr) geometry which is described by the stan- dard well-known coordinates, namely (t, r, θ , φ). Then, one can apply a simple separation of variables ansatz, leading to the met- ric quantities that behave like an unknown function of variable r × Y ℓm (θ , φ)exp (iσ t). It is important to mention that the angular dependence is somewhat very complicated, and the technical de- tails related to it is available in the literature. One of the common ways to obtain the quasi-normal modes is by seeking and inves- tigating the solutions to the Zerilli equation which should verify suitable boundary conditions in the event horizon’s neighborhood and infinity’s neighborhood. It was then proved that there exist so- lutions only in the case of certain special values taken by the pa- rameter σ [2,5,10]. ∗ Corresponding author. E-mail addresses: kubekas@unisa.ac.za (A.S. Kubeka), franckemile2006@yahoo.ca, dgoufef@unisa.ac.za (E.F. Doungmo Goufo), khumam@unisa.ac.za (M. Khumalo). The theory of quasi-normal mode has slowly become a cor- nerstone for the modern theory of general relativity. It appears in some kind of simulations, like the numerical relativity simulations of binary black hole coalescence. And, while it is not yet actually observed, we really expect it to be measured by the LIGO collabo- ration, and certainly by LISA, and therefore, leading to concise in- formation about the parameters that describe a black hole from some coalescence event. The approach commonly used for linear perturbations of a black hole includes the performance of linearization using well-known standard Schwarzschild (or Kerr) coordinates (t, r, θ , φ). We can also perform the same linearization making use of Bondi–Sachs coordinates, which represent a system of coordinates based on outgoing null cones. Those techniques have been done in previous works and the main objective was find analytic solutions of the linearized Einstein equations, and this, for the sole goal of testing numerical relativity codes [9,15,19]. As with the standard well-known approach, we finish by obtaining a second order ordinary differential equation involving ℓ and σ as parameters, Eq. (2.9). However, after the quasi-normal modes were found for that equation, it was pointed out that they are different from those found in Zerilli equation. The reason come from the fact that the different physical problems are considered in the two cases, as https://doi.org/10.1016/j.chaos.2018.09.047 0960-0779/© 2018 Elsevier Ltd. All rights reserved.