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37.2 The basic formulae governing embedding 581equal footing with the classifications with respect to groups of motions orto the Petrov types, and it will give a refinement of both these schemes.Moreover, there is some hope of obtaining exact solutions by the methodof embedding, at least for some simple cases of low embedding class, solutionswhich are not readily available by other methods. We are notinterested in the embedding itself; the functions y A (x a ) will not be determinedor given here. A large number of explicit embeddings can be foundin Rosen (1965) and Collinson (1968b). Other aspects of the embeddingproblem are discussed in Goenner (1980).Nearly all work summarized in this chapter deals with local embeddingonly, i.e. the embedding of an open and simply connected neighbourhoodof a point of the given V 4 . In contrast, the global embedding of a V 4 can beconsidered. It may give a deeper insight into the geometrical properties ofspace-time. In fact, the maximal analytic extension of the Schwarzschildsolution was found by the method of embedding (Fronsdal 1959). Thenumber of extra dimensions needed for the embedding can be considerablyhigher than that for local embedding; only upper limits are known:a compact (non-compact) space-time V 4 is at most of embedding classp =46(p = 87). For theorems and results on global embeddings, seeFriedman (1965), Penrose (1965), Clarke (1970), Greene (1970). No systematicanalysis of global embedding of exact solutions has yet been done.37.2 The basic formulae governing embeddingTo get a deeper insight into the geometrical properties of the embeddingdescribed by (37.1)–(37.3), we introduce an N-leg at every point of V 4 andconsider the change of this N-leg along V 4 , i.e. we consider the covariantderivative with respect to the coordinates x n of V 4 .The N-leg in question consists of four vectors y,a A (vectors in E N , a =1,...,4) tangent to V 4 and p unit vectors n αA (α =1,...,p) orthogonalto V 4 and to each other,η AB n αA n βB = e α δ αβ , η AB n αA y B ,a =0, e α = ± 1. (37.4)(In these and the following formulae, summation over Greek indices takesplace only if explicitly indicated.)The covariant derivatives (covariant with respect to coordinates x n andmetric g ab ) of the basic vectors n αA and y B ,a are vectors and tensors (respectively)in V 4 , but again vectors in the embedding space E N and are,therefore, linear combinations of the basic vectors. Starting from the metric(37.3), we getg ab;c = η AB (y A ,a;cy B ,b + y A ,ay B ,b;c) =0, (37.5)