An alternative to the Teukolsky equation

Abstract

We conjecture a new ordinary differential equation exactly isospectral to the radial component of the homogeneous Teukolsky equation. Surprisingly our equation looks much simpler than Teukolsky's one. We find this novel relation by a hidden symmetry implied from a four-dimensional N = 2 supersymmetric quantum chromodynamics. Our proposal is powerful both in analytical and in numerical studies. As an application, we derive high-order perturbative series of quasinormal mode frequencies in the slowly rotating limit.

Exact solution of Kerr black hole perturbations via CFT$_2$ and instanton counting. Greybody factor, Quasinormal modes and Love numbers

“…This perspective has been analyzed in the context of black hole physics in [21,22,23] where it was suggested that some physical properties of black holes, such as their greybody factor and quasinormal modes, can be studied in a particular regime in terms of Painlevé equations. A decisive step forward about the quasinormal mode problem has been taken in [24], where a different approach making use of the Seiberg-Witten quantum curve of an appropriate supersymmetric gauge theory has been advocated to justify their sprectrum and whose evidence was also supported by comparison with numerical analysis of the gravitational equation (see also [25,26] for further developments). This view point has been further analysed in [27], where the context is widely generalized to D-branes and other types of gravitational backgrounds in various dimensions.…”

New Chandrasekhar transformation in Kerr spacetime

“…We have obtained the differential equation (3.21), which is also reduced to the Regge-Wheeler equation in the limit a → 0 as the (Chandrasekhar-)Detweiler and the Sasaki-Nakamura equations are. The extension to the case of different spins (the scalar waves and the electromagnetic waves) [26,27,28] and the explicit comparison of the quasi normal modes [15,16,22,29,30,31,32] would also be interesting.…”

“…As well as in the Regge-Wheeler and the Teukolsky equations, the (confluent) Heun's equation also appears in the quantization of the Seiberg-Witten curves in supersymmetric gauge theories. For example in N = 2 supersymmetric SU(2) gauge theory coupled with three matter hypermultiplets in the fundamental representation of the gauge group, the quantum Seiberg-Witten geometry gives the following differential equation [19,21,15,16,22]…”

“…By comparing (4.6) and (4.7), one can find that it is not only the exchange between m 2 and m 3 , but also 2iC is moved to m 1 , and λ is replaced with λ. Note that the Regge-Wheeler-type equation which has the parameters in (4.6) with the exchange between m 2 and m 3 is given in [16].…”

“…These correspondence is helpful to study the solution to the Regge-Wheeler and the Teukolsky equations beyond the low-frequency approximation. It is also found that the Chandrasekhar(-like) transformation in the Schwarzschild spacetime can be interpreted as the exchange of the mass parameters [15,16]. Before that it has already been known that this exchange of the parameters is regarded as a particular integral transform [23,24].…”