Algebraically independent nth derivatives of the Riemannian curvature spinor in a general spacetime

Abstract

Explicit sets of spinor nth derivatives of the Riemann curvature spinor for a general spacetime are specified for each n so that they contain the minimal number of components enabling all derivatives of order m to be expressed algebraically in terms of these sets for n<or=m; this generalises an earlier results by Penrose. The minimal sets are defined recursively in a manner convenient for use in the procedures for resolving the 'equivalence problem' of the local isometry of two given spacetimes. The actual numbers of quantities to be calculated are given and the reductions in these arising in special cases such as vacuum and conformally flat spacetimes, and spacetimes with vanishing Bach tensor, are discussed, as is the embodiment of the results in the CLASSI implementation (using the computer algebra system SHEEP) of the procedure for the equivalence problem. Finally, the authors comment on the possible relevance of the results to analytic and numerical methods of solving the Einstein equations.