Normal Modes, Quasi-normal Modes and Super-radiant Modes for Scalar Fields in Kerr anti-de Sitter Spacetime

Normal modes, quasi-normal modes and super-radiant modes are studied to clarify the total dynamics for complex scalar fields in Kerr anti-de Sitter black hole spacetime. Orthonormal relations and quasi-orthonormal relations are obtained for normal modes and quasi-normal modes. Mode expansions are done and the conserved quantities are studied. Any modes are shown to be separated into two groups, physical modes and unphysical modes, by the zero mode line. Zero modes themselves do not exist as normalizable modes with the correct boundary condition. The allowed physical modes exclude the super-radiant instability modes in rotating black hole spacetime. The result is consistent with the co-rotating frame consideration.