Equivalence Problems in General Relativity: A Symmetry Approach

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Ian Anderson


Four-dimensional space-times with symmetry play a central role in the theory of general relativity. In 1961 in his book Einstein Spaces, A.Z. Petrov gave a complete classification of four-dimensional space-times with symmetry according their local isometry group. The infinitesimal isometries generated by the group actions as classified and given by Petrov form finite-dimensional Lie algebras. One would like to be able to compare any other Lorentz metric and its Lie algebra of Killing vectors with those in Petrov's classification. A classifier program created for Maple takes as input a Lorentz metric, from which is computed its Lie algebra of Killing vectors. It then computes a list of invariant properties defined for any abstract Lie algebra as well as invariant properties specific to Lie algebras of Killing vectors and their associated metric tensors. As it computes these properties, it compares them to an internal database of these same properties that has been compiled for Petrov's metrics and their Lie algebras of Killing vectors. From this comparison it's determined which of Petrov's Lie Algebras of Killing vectors have the same properties in common with the given metric and its Killing vectors supplied by the user to the program. However, the database contains enough properties such that Petrov's metrics and Lie algebras of Killing vectors have been distinguished one from another, up to an equivalence. Therefore, up to isomorphism or diffeomorphism, the results of the classifier are unique, according to Petrov's classification.

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