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The speciality index and the Lifshitz–Khalatnikov Kasner index parametrization

The speciality index function for any Petrov type I, II or D spacetime is shown to be a natural function of a single complex scalar quantity μ (natural modulo permutation symmetries). For the family of Kasner spacetimes, this quantity is a function of the Kasner indices alone which coincides with the real Lifshitz–Khalatnikov parameter u for those indices.