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9.1 Scalar invariants and covariants 113All the considerations here concern purely local equivalence. If two metricsare given which cover disjoint regions of a single analytic space-time,the fact that each is in the analytic continuation of the other cannotbe detected by these methods. Conversely, topological identifications, ornon-trivial homotopies of the metric as a section of the bundle of symmetrictensors (called ‘kinks’; see e.g. Whiston (1981)), may give locallyequivalent but globally inequivalent metrics. A further limitation is thatwe assume that all classifying quantities, discrete or continuous, are respectivelyconstant or sufficiently smooth in the (open) neighbourhoodsconsidered. For classifications like the Petrov type this follows from thesmoothness of the invariants whose vanishing or otherwise characterizesthe type; for smooth metrics such types change only on submanifolds oflower dimension.In principle a space-time for which a Cauchy problem is well posed, andall its properties, can also be characterized by Cauchy data on a suitablehypersurface, since such data completely determine (a neighbourhood in)the space-time (see e.g. Friedrich and Rendall (2000)).The method described in §9.2 and various of the methods described in§9.3 have been implemented in computer algebra programs. We do not describesuch programs here as this information would rapidly become outdated:instead we refer interested readers to the reviews of Hartley (1996)and MacCallum (1996). However, we do discuss some efficiency considerationsin §9.3. In preparing this book, we used the system CLASSI (Åman2002), based on SHEEP and REDUCE, as described in MacCallum andSkea (1994).9.1 Scalar invariants and covariantsScalars constructed from the metric and its derivatives must be functionsof the metric itself and the Riemann tensor and its covariant derivatives(Christoffel 1869). In a manifold M of n dimensions, at most n suchscalars can be functionally independent, i.e. independent functions onM. (The term ‘functional independence’ may also, confusingly, be usedfor functional independence over the bundle of symmetric tensors or somejet bundle thereof.) The number of algebraically independent scalar invariants,i.e. invariants not satisfying any polynomial relation (called asyzygy) is rather larger: it can be calculated, e.g. by considering Taylorexpansions of the metric and of the possible coordinate transformations(Siklos 1976a). The result (Thomas 1934) is that in a general V n the numberof algebraically independent scalars constructible from the metric andits derivatives up to the pth order (the Riemann tensor and its derivatives