Chaotic solutions and black hole shadow in f (R) gravity

We discuss the emergence of black hole shadow and photon-sphere in the context of f(R) gravity. It is shown that the shadow is exponentially sensitive to linear instabilities of metric coming from some f(R) solutions. Thus, the instabilities of photon circular trajectories, delimiting the black hole photon-sphere, are double exponentialized. Specifically we individuate two Lyapunov exponents, rather than only one, related to two different sources of chaos in geodesic orbits as a sort of butterfly effect. Such a result violates the black hole chaos bound proposed by Maldacena, Shenkerand Stanfordfor General Relativity. We also explore the impact of the black hole metric instabilities in f(R) gravity on the quasi-normal modes. In the framework of Extended Theories of Gravity, our analysis suggests a new paradigm to deal with black hole shadow and gravitational waves observations coming from black hole merging in the ringdown phase. (C) 2021 The Author(s). Published by Elsevier B.V.

Automated summary

In this study, the emergence of black hole shadow and photon-sphere in the context of f(R) gravity is discussed, with a focus on the sensitivity of the shadow to linear instabilities of metric from f(R) solutions. It is found that the photon circular trajectories defining the black hole photon-sphere exhibit double exponential instabilities, leading to two Lyapunov exponents related to chaos in geodesic orbits. This result challenges the black hole chaos bound proposed by Maldacena, Shenker and Stanford for General Relativity, and suggests a new paradigm for handling black hole shadow and gravitational waves observations in the ringdown phase of black hole merging in Extended Theories of Gravity.