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This paper asks whether a divergent analytic approximation can be turned into a reliable calculator for the damped ringing of rotating (Kerr) black holes. The ringing modes, called quasinormal modes (QNMs), are usually found with numerical methods. The authors try to make a commonly used analytic expansion predictive by using two types of resummation called Padé and Borel‑Padé, and they test where these tricks work and where they fail.
The authors develop two complementary implementations. First, they make a semi‑analytic slow‑rotation expansion in the dimensionless spin a and carry the Wentzel–Kramers–Brillouin (WKB) expansion to 21st order. Second, they hold the spin fixed and implement a Padé‑resummed WKB condition solved by a fixed‑point iteration, carried out to 41st WKB order. In places they also apply a second Padé resummation to the spin series. They compare their results to Leaver’s continued‑fraction method, the standard numerical reference.
At a high level, the WKB method is a local Taylor expansion of an effective potential (the Chandrasekhar–Detweiler potential) about its peak. The ordinary WKB series is asymptotic, meaning that adding more terms eventually makes the answer worse. Padé and Borel‑Padé resummations turn the divergent series into rational or Borel‑summed forms that can converge much better. The fixed‑spin implementation finds a self‑consistent frequency by updating a trial frequency, evaluating the angular separation constant and the potential peak, and repeating until convergence.
The resummations work well in many cases. In the slow‑rotation regime the 21st‑order resummed expansion is far more accurate than the previous fourth‑order WKB results. For damped modes at larger spins, the fixed‑spin method matches Leaver’s numerical answers very closely; for example, the real part of the fundamental m=0 mode at spin a=0.99 has a fractional error below 10^−7. The authors also show order‑of‑magnitude improvements in accuracy for common l=2 modes up to moderate spins.
However, the strategy breaks down for modes that move to the so‑called zero‑damped branch as the black hole approaches extremality (spin a→1). For those modes — for example the (l,m,n)=(2,2,0) mode — the fractional error grows above 10^−3 for a>0.9. The breakdown has a physical origin: near extremality the potential develops rapid variation close to the horizon on the throat scale, with length and frequency shifts scaling like √(1−a). Nearby poles in the Chandrasekhar–Detweiler potential then produce rapid changes that a local Taylor expansion about the peak cannot capture uniformly. In short, local resummed WKB approximations are useful over a wide range of spins but are not reliable for near‑extremal zero‑damped modes.