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116 9 Invariants and the characterization of geometriesand then use the remaining non-zero components of those tensors in thechosen tetrads, which are scalar contractions between the curvature or itsderivatives and the tetrad vectors, as the scalar invariants required. Theseare the so-called Cartan invariants or Cartan scalars. That they are notequivalent to the scalar polynomial invariants can be seen by consideringplane waves again; in that case the surviving Cartan invariant of theRiemann tensor is Ψ 4 = 1, whereas all the scalar polynomial invariantsare zero. We describe in the next section how the Cartan invariants canbe used to locally characterize space-times.One may also consider covariants of the Riemann tensor and its derivatives.In general, for a given tensor or spinor and a scalar polynomialinvariant obtained by contracting it with arbitrary vectors or spinors, acovariant is defined to be any other scalar polynomial in the same variables;for example, a covariant of φ AB A˙ζA ζ B ¯ζȦ is a scalar expression inthe coefficients φ AB Ȧ and the variables ζB , ¯ζ A ˙ . Covariants differ from invariantsin that they do not depend solely on the (metric and the) tensoror spinor itself. The degree of a covariant is the degree in the tensor orspinor, e.g. in φ AB Ȧ. Covariants and invariants together are called concomitants.For example, the covariantsQ =Ψ AB EF Ψ CDEF ζ A ζ B ζ C ζ D ,R =Ψ ABC K Ψ DE LM Ψ FKLM ζ A ζ B ζ C ζ D ζ E ζ F ,(9.4)of the expression (4.28) involving the Weyl spinor are used in some methodsof Petrov classification; see §9.3.9.2 The Cartan equivalence method for space-timesThe method to be outlined here is a specialized form of a more generalmethod, due to Cartan, applicable to the equivalence of sets of differentialforms on manifolds under appropriate transformation groups (Gardner1989, Olver 1995). For sufficiently smooth metrics, it gives sets of scalarsproviding a unique local characterization, and thus leads to a procedurefor comparing metrics.To relate two apparently different metrics, we need to consider coordinateor basis transformations, and therefore to consider a frame bundleas defined in §2.11. The basis of the method is that if the metrics areequivalent, the frame bundles they define are identical (locally). Moreover,the frame bundles possess uniquely-defined bases of 1-form fields,{ω a , Γ a b}, as described in §2.11, which would therefore also be identicalfor equivalent metrics. The same is more generally true for any (sufficientlysmooth) manifolds equipped with uniquely-defined 1-form bases