The evolution of the Weyl and Maxwell fields in curved and space-times

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (U, <(phi)over tilde>) of a space-time (M, g) are considered. It is shown that the condition g(oo)(x) > O for all x is an element of U is necessary and sufficient to rewrite them in a unified manner as evolution equations partial derivative(t) phi = L((s))phi. Here L((s)) is a linear first order differential operator on the pre-Hilbert space (C-o(infinity) (U-t, C-2s+1), [., .]), where U-t subset of R(3) is the image of the coordinate map of the spacelike hypersurface t = const, and [phi, psi] = integral U-t phi(star)Q psi d((3))x with a suitable Hermitian n x n-matrix Q = Q(t, x). The total energy of the spinor field phi with respect to U-t is then simply given by E = [phi, phi]. In this way inequalities for the energy change rate with respect to time, partial derivative(t) parallel to phi parallel to(2) = 2 Re [phi, L((s))phi], are obtained. As an application, the Kerr-Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well-known superradiance of electromagnetic waves.