Non-extremal Reissner-Nordstrom black hole: Do asymptotic quasi-normal modes carry information about the quantum properties of the black hole?

TL;DR

This paper investigates the asymptotic quasi-normal modes of non-extremal Reissner-Nordstrom black holes, showing that their frequency gaps do not generally converge unless specific rationality conditions on surface gravities are met, impacting black hole thermodynamics theories.

Contribution

It demonstrates that the frequency gap in the asymptotic quasi-normal modes does not converge unless the ratio of surface gravities is rational with specific properties, clarifying conditions for Maggiore's conjecture.

Findings

Frequency gap does not converge in general.

Convergence occurs only when surface gravity ratio is rational with coprime integers N, M.

If convergent, the limit equals the least common multiple of the surface gravities.

Abstract

We analyze the largely accepted formulas for the asymptotic quasi-normal frequencies of the non-extremal Reissner-Nordstr\"om black hole, (for the electromagnetic-gravitational/scalar perturbations). We focus on the question of whether the gap in the spacing in the imaginary part of the QNM frequencies has a well defined limit as n goes to infinity and if so, what is the value of the limit. The existence and the value of this limit has a crucial importance from the point of view of the currently popular Maggiore's conjecture, which represents a way of connecting the asymptotic behavior of the quasi-normal frequencies to the black hole thermodynamics. With the help of previous results and insights we will prove that the gap in the imaginary part of the frequencies does not converge to any limit, unless one puts specific constraints on the ratio of the two surface gravities related to the two spacetime horizons. Specifically the constraints are that the ratio of the surface gravities must be rational and such that it is given by two relatively prime integers N, M, whose product is an even number. If the constraints are fulfilled the limit of the sequence is still not guaranteed to exist, but if it exists its value is given as the lowest common multiplier of the two surface gravities. At the end of the paper we discuss the possible implications of our results.