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Integration methods within existing tetrad formalisms in general relativity

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Abstract

The NP and GHP formalisms are reviewed in order to understand and demonstrate the important role played by the commutator equations in the structure of the system of equations in each formalism, and also in the associated integration procedures. Particular attention is focused on how the commutator equations are to be satisfied (or checked for consistency) in each of the formalisms. In particular, it is shown that in Held's integration method in the GHP formalism, it is usually sufficient—alongside the GHP Ricci and Bianchi equations—to apply the GHP commutator equations to two complex, zero-weighted quantities which consist of four real, functionally independent scalars. This result is used, first of all, to suggest an additional step in Held's method, which ensures that there is no possibility of ambiguity in the procedure; secondly a restatement/ modification of Held's integration method is suggested, which enables the integration procedure to be completely self-contained and fully co-ordinate- and gauge-independent. An example of its use for a subclass of Petrov type D vacuum spaces is given.

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Edgar, B. Integration methods within existing tetrad formalisms in general relativity. Gen Relat Gravit 24, 1267–1295 (1992). https://doi.org/10.1007/BF02418213

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