Summary
On the tangent bundleTM of a (pseudo-) Riemannian manifoldM only 4 metrics are induced by that ofM; they all define the same volume which in turn coincides with that induced onTM by the symplectic volume of the cotagent bundle conventionally used in the proof of Liouville’s theorem (LT) in general relativity. LT is then established for the unique volume induced by that of TM on the mass-shells of space-time. Conversely this is essentially the only volume for which LT holds.
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References
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Tzanakis, C. Liouville’s theorem and the definition of volume on the tangent bundle of space-time. Nuov Cim B 106, 781–788 (1991). https://doi.org/10.1007/BF02722546
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DOI: https://doi.org/10.1007/BF02722546