On the Construction of Syzygies of the Polynomial Invariants of the Riemann Tensor

Details On the Construction of Syzygies of the Polynomial Invariants of the Riemann Tensor On the Construction of Syzygies of the Polynomial Invariants of the Riemann Tensor

Sep 2008

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008mgm..conf.2128l&link_type=abstract

"THE ELEVENTH MARCEL GROSSMANN MEETING On Recent Developments in Theoretical and Experimental General Relativity, Gravitation an

Computer Science

Scientific paper

This paper outlines our full solution to the classic problem of determining a complete set for the polynomial invariants of the Riemann tensor in a 4-D Lorentzian space. In addition to establishing a basis, we provide a constructive two-stage algorithm for expressing any invariant as a polynomial function of the basis invariants. In the first stage, a formal correspondence between the SL(2,ℂ) form of these invariants and generalized directed multigraphs is established. A novel combination of spinor algebra and elementary graph theory is used to derive an "arc-pairing" algorithm which reexpresses any invariant as a polynomial function of invariants containing maximal numbers of paired contractions. The problem is thus reduced to finding a basis for traces of products of complex 3 × 3 matrices which transform under the SO(3,ℂ) group. Techniques from matrix polynomial algebra and rotor calculus are subsequently applied to solve the reduced problem and provide the second stage of the algorithm.

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