Quasinormal behavior of massless scalar field perturbation in Reissner-Nordström anti-de Sitter spacetimes

Abstract

We present a comprehensive study of the massless scalar field perturbation in the ReissnerNordström anti-de Sitter (RNAdS) spacetime and compute its quasinormal modes (QNM). For the lowest lying mode, we confirm and extend the dependence of the QNM frequencies on the black hole charge got in previous works. In near extreme limit, under the scalar perturbation we find that the imaginary part of the frequency tends to zero, which is consistent with the previous conjecture based on electromagnetic and axial gravitational perturbations. For the extreme value of the charge, the asymptotic field decay is dominated by a power-law tail, which shows that the extreme black hole can still be stable to scalar perturbations. We also study the higher overtones for the RNAdS black hole and find large variations of QNM frequencies with the overtone number and black hole charge. The nontrivial dependence of frequencies on the angular index ℓ is also discussed.

On the classification of asymptotic quasinormal frequencies for {$d$}-dimensional black holes and quantum gravity

“…The most significant finding of these authors was that purely damped RN AdS modes behave in a very peculiar way: their damping seems to go to infinity in the extremal black hole case, suggesting the possibility that extremally charged RN AdS black holes may be marginally unstable to electromagnetic and gravitational perturbations (unless, of course, the mode amplitude goes to zero in this extremal limit). This question was later solved in [107]. Indeed, [107] presents an extensive numerical study of quasinormal frequencies for massless scalar fields in d = 4 RN AdS spacetime, corresponding to tensor type perturbations in our results.…”

“…This question was later solved in [107]. Indeed, [107] presents an extensive numerical study of quasinormal frequencies for massless scalar fields in d = 4 RN AdS spacetime, corresponding to tensor type perturbations in our results. They find that as one increases the charge, at first the scalar field decays exponentially and oscillates but for θ > θ critical = 0.3895θ max the decay becomes purely exponential.…”

“…However, in the extremal limit the decay of the scalar field perturbation changes from exponential to power law, and the extremal RN AdS black hole is thus stable to scalar field perturbations (not marginally unstable as worried in [15] and mentioned above). For the higher modes both Re ω and Im ω increase with overtone number, n, although as one increases charge θ, Re ω increases slower and Im ω increases faster, with n. In the large black hole regime, with |λ| = 1 and R + H = 100, asymptotic frequencies become evenly spaced and as n → ∞ (we have translated the results adapting the conventions of [107] to our own)…”

“…This is done in a similar fashion to what we did in the case of Schwarzschild AdS black holes: one computes x 0 numerically according to its general expression (as there is no analytical solution for x 0 , and being careful with the choice of branch cuts for the several logarithms) and then plugs into our analytical formula for the frequencies in order to obtain (using the values of [107] for cosmological constant, black hole mass, charge and horizon radius) when θ = 0.3θ max . Our results are in full and complete agreement with the numerical calculations of [107], to a great degree of accuracy. However, it would be very interesting to produce further numerical data concerning higher dimensional RN AdS quasinormal modes to match against all our analytical results.…”

“…A point to mention concerns the critical charge θ critical mentioned above. Unfortunately asymptotic quasinormal modes cannot see this critical charge: as described in [107], the critical charge is found by following the frequencies of a specific mode, as one varies the charge. For the asymptotic frequencies n → +∞ and the choice of focusing on a specific mode becomes somewhat obscure.…”