For any static, spherically symmetric black hole obeying Einstein's equations and the dominant energy condition, the circular photon orbit's Lyapunov exponent is bounded by the photon-sphere surface gravity, the Unruh acceleration, and the inverse shadow radius.
Chaos bound and its violation in charged Kiselev black hole
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The chaos bound in the near-horizon regions has been studied through the expansions of the metric functions on the horizon. In this paper, we investigate the chaos bound in the the near-horizon region and at a certain distance from the horizon of a charged Kiselev black hole. The value of the Lyapunov exponent is accurately calculated by a Jacobian matrix. The angular momentum of a charged particle around the black hole affects not only the exponent, but also the position of the equilibrium orbit. This position gradually moves away from the horizon with the increase of the angular momentum. We find that the the bound is violated at a certain distance from the horizon and there is no violation in the near-horizon region when the charge mass ratio of the particle is fixed. The small value of the normalization factor is more likely to cause the violation.
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Bounds for Lyapunov exponent of circular light orbits in black holes
For any static, spherically symmetric black hole obeying Einstein's equations and the dominant energy condition, the circular photon orbit's Lyapunov exponent is bounded by the photon-sphere surface gravity, the Unruh acceleration, and the inverse shadow radius.