Computer Science
Scientific paper
May 1997
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1997gregr..29..539z&link_type=abstract
General Relativity and Gravitation, Volume 29, Issue 5, p.539-581
Computer Science
21
Scientific paper
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 r\^ole 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 {\it complete\/} set for each Petrov type of the Weyl spinor $\PsiABCD$ and for each Segre type of the trace-free Ricci spinor $\Phi_{AB\dot C \dot D}$. This table involves 17 real invariants, including one real invariant and one complex invariant involving $\PsiABCD$, $\bar{\Psi}_{\dot A\dot B\dot C\dot D}$ and $\Phi_{AB\dot C \dot D}$ in some degenerate cases.
McIntosh Colin B. G.
Zakhary E.
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