Zero, normal and super-radiant modes for scalar and spinor fields in Kerr–anti de Sitter spacetime

We study zero and normal modes for scalar and spinor fields in Kerr–anti de Sitter spacetime as a bound state problem with Dirichlet and Neumann boundary conditions. Zero mode is defined as its momentum near the horizon to be zero: $p_H=\omega -{\Omega }_{H}m=0$, and is shown not to exist as a physical state for both scalar and spinor fields. Physical normal modes must satisfy the spectrum condition $p_H>0$ as results of (i) non-existence of zero mode and (ii) the analyticity with respect to the rotation parameter of Kerr–anti de Sitter spacetime. Under the spectrum condition, the super-radiant modes are shown to be type 2 super-radiance, which is characterized by negative frequency $\omega <0$ and positive momentum on the horizon $p_H=\omega -{\Omega }_{H}m>0$. The type 2 super-radiant modes are necessary for the completeness relation of normal modes and are stable. Applying the spectrum condition to black hole thermodynamics, the brick wall model is shown to be well-defined. (paper)