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The relativistic framework with its symmetries offers a natural definition for the internal time of classical (non-quantum) physical systems as a Lorentz-invariant observable. The internal-time observable, measuring the system's aging or internal evolution, is identified with the proper time of the system derived from its centre-of-mass (CM) coordinate. For its definition as an observable it is required that the system be symmetric not only under Lorentz-Poincaré transformations but also under uniform scaling, with the associated existence of a dilatation function D, and yet that D be a varying-not conserved-quantity. Two alternative definitions are discussed, and it is found that in order to maintain simultaneity of the CM time with the events that define it, it is necessary to split the dilatation function into a CM part and an internal part.

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The protonic conduction of Sc-doped SrZrO 3 (SrZr 1Àx Sc x O 3) in the single crystal form is investigated. The SrZr 1Àx Sc x O 3 crystals exhibit significantly higher conductivity than the pure ones, and the crystal with x ¼ 0:05 exhibits the highest conductivity of those masured. The activation energy of the SrZr 1Àx Sc x O 3 crystal decreases rapidly with increasing Sc 3þ concentration when x 0:05, and increases when x > 0:05. The activation energy of SrZr 0:95 Sc 0:05 O 3 agrees with the energy separation between hole states at the top of the valence band and the Fermi level.