A complete set of Riemann invariants

There are at most 14 independent real algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space. In the general case, these invariants can be written in terms of four different types of quantities: R, the real curvature scalar, two complex invariants I and J formed from the Weyl spinor, three real invariants I-6, I-7 and I-8 formed from the trace-free Ricci spinor and three complex mixed invariants K, L and M. Carminati and McLenaghan [5] give some geometrical interpretations of the role played by the mixed invariants in Einstein-Maxwell and perfect fluid cases. They show that 16 invariants are needed to cover certain degenerate cases such as Einstein-Maxwell and perfect fluid and show that previously known sets fail to be complete in the perfect fluid case. In the general case, the invariants I and J essentially determine the components of the Weyl spinor in a canonical tetrad frame; likewise the invariants I-6, I-7 and I-8 essentially determine the components of the trace-free Ricci spinor in a (in general different) canonical tetrad frame. These mixed invariants then give the orientation between the frames of these two spinors. The six real pieces of information in K, L and M are precisely the information needed to do this. A table is given of invariants which give a complete set for each Petrov type of the Weyl spinor Psi(ABCD) and for each Segre type of the trace-free Ricci spinor Phi(ABC over dotD over dot). This table involves 17 real invariants, including one real invariant and one complex invariant involving Psi(ABCD), <(Psi)over bar>(A over dotB over dotC over dotD over dot) and Phi(ABC over dotD over dot) in some degenerate cases.