Action-angle variables for spherical mechanics related to near horizon extremal Myers–Perry black hole

Abstract

We provide a systematic account of integrability of the spherical mechanics associated with the near horizon extremal Myers-Perry black hole in arbitrary dimension for the special case that all rotation parameters are equal. The integrability is established both in the original coordinates and in action-angle variables. It is demonstrated that the spherical mechanics associated with the black hole in d = 2n + 1 is maximally superintegrable, while its counterpart related to the black hole in d = 2n lacks for only one integral of motion to be maximally superintegrable.

Integrable models associated with Myers–Perry–AdS–dS black hole in diverse dimensions

“…We presented both the Hamiltonians and the integrals of motion as well as performed a reduction to a spherical mechanics which is governed by the Casimir invariant of the conformal group SO(2, 1). These models provide one-parameter deformations of the systems constructed recently in [20,21,22]. It was demonstrated that they are superintegrable but not maximally superintegrable, lacking one integral of motion in the odd-dimensional case and two integrals of motion in the even-dimensional case.…”

“…In this regard the Myers-Perry black hole with all rotation parameters being equal to each other is of particular interest because its symmetry is enlarged to the unitary algebra (in direct sum with extra so(2, 1) algebra in the near horizon case) which is the largest finite-dimensional symmetry algebra possible. In particular, this gives a clue to the construction of new maximally superintegrable models in [20,21,22].…”

“…It is an extension of the metric constructed in [20] which now includes a cosmological constant λ. A salient feature of the near horizon metric (22) is that it exhibits extra symmetries generated by the Killing vectors…”

“…Since the µ i sector in (22) does not mix with other coordinates, the corresponding piece in the metric can be inverted separately.…”

Separability of Klein-Gordon Equation on Near Horizon Extremal Myers-Perry Black Hole

“…One can construct another solution to Einstein equations in the Near Horizon extremal limit [9][10][11][12] of MP (NHEMP) black hole [13] (see [14][15][16][17][18][19] for recent studies). The extremal limit of the parameters describes a black hole with the biggest allowed angular momentum for a given BH mass.…”