Abstract
In this Letter, we construct the quasi-normal modes of three-dimensional extremal black holes in an algebraic way. We show that the infinite towers of the quasi-normal modes of scalar, vector and tensor could be constructed as the descendents of the highest weight modes. Our investigation shows that the hidden conformal symmetry suggested in Chen et al. (2010) [5] is an intrinsic property of the extremal black hole. Moreover, we notice that we need to fix the freedom in defining the local vector fields and find the right hidden conformal symmetry to obtain the physical quasi-normal modes.
Highlights
In [1], it was argued that the hidden conformal symmetry in the low-frequency scalar scattering off generic non-extremal Kerr black hole was essential to set up a Kerr/conformal field theory (CFT) correspondence
The similar analysis has been generalized to various cases, see for example [2, 3]. It was shown in [4] that the same hidden conformal symmetry may act on the solution space of the vector and tensor as well
For non-extremal black holes, there exist another set of vector fields forming hidden conformal symmetry
Summary
In [1], it was argued that the hidden conformal symmetry in the low-frequency scalar scattering off generic non-extremal Kerr black hole was essential to set up a Kerr/CFT correspondence. It was shown that for three-dimensional black holes which have two-dimensional conformal field theory (CFT) descriptions the wave equations of the vector and the rank-2 symmetric tensor could be written in terms of the Lie-induced SL(2, R) quadratic Casimir This fact shows that the hidden conformal symmetry, if it exists, is an intrinsic property of the black hole, not an artifact of the scalar equation of motion. Even though the conformal coordinate is quite different, one set of hidden conformal symmetry is very similar to the ones in the non-extremal case This makes the similar construction of the quasi-normal modes feasible.
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