  In this paper we will review some facts, both classical and recent,
concerning the geometry and analysis of the Kerr and related black hole
spacetimes. This includes the analysis of test fields on these spacetimes.
Central to our analysis is the existence of a valence $(2,0)$ Killing spinor,
which we use to construct symmetry operators and conserved currents as well as
a new energy momentum tensor for the Maxwell test fields on a class of
spacetimes containing the Kerr spacetime. We then outline how this new energy
momentum tensor can be used to obtain decay estimated for Maxwell test fields.
An important motivation for this work is the black hole stability problem,
where fields with non-zero spin present interesting new challenges. The main
tool in the analysis is the 2-spinor calculus, and for completeness we
introduce its main features.
