Special symmetries of the charged Kerr - AdS black hole of D = 5 minimal gauged supergravity

In this note we prove that the Hamilton-Jacobi equation in the background of the recently discovered charged Kerr-AdS black hole of D = 5 minimal gauged supergravity is separable, for arbitrary values of the two rotation parameters. This allows us to write down an irreducible Killing tensor for the spacetime. As a result we also show that the Klein-Gordon equation in this background is separable. We also consider the Dirac equation in this background in the special case of equal rotation parameters and show it has separable solutions. Finally we discuss the near-horizon geometry of the supersymmetric limit of the black hole. It is a curious fact that the Kerr-Newman black hole possesses a hidden symmetry which renders geodesic motion integrable [1]. This is related to the existence of a second rank Killing tensor Kμν ; by definition such a tensor satisfies ∇(μKνρ) = 0. Given a Killing tensor one may construct the quantity K = Kμν ẋ ẋ which is conserved along geodesics x(τ). Carter was the first to systematically analyse the consequence of separability of solutions to Einstein’s equations, and indeed this is how the Kerr-(A)dS black hole and its charged counterpart were first discovered [2]. Higher dimensional Kerr-(A)dS metrics have only been recently constructed [3]. The existence of a Killing tensor has been verified in five dimensions for arbitrary rotation parameters [4] and in all dimensions for equal rotation parameters [5]. This renders both the Hamilton-Jacobi (HJ) and Klein-Gordon (KG) equations separable. The charged counterparts of the Kerr-AdS black holes in D = 5 minimal gauged supergravity are far more difficult to construct. Progress was first made by tackling the special case where the rotation parameters are equal [6], and a reducible Killing tensor for this black hole was found in [7]. Further, the same Killing tensor was found for black holes with equal rotation parameters in the more general U(1)3 theory [8]. Only very recently has a charged Kerr-AdS black hole been found with arbitrary rotation parameters [9]. The purpose of this note is to show that this black hole also has a Killing tensor rendering geodesic motion integrable and the KG equation separable. We also show that the Dirac equation admits separable solutions in the special case of equal rotation parameters. Finally we discuss the near-horizon geometry of the supersymmetric limit of this black hole and show that is has an isometry group SO(2, 1) × U(1)2 as is the case for the extremal limit of the Myers-Perry metric [10]. [email protected] [email protected] [email protected]