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A Note on the Problem of Proper Time in Weyl Space–Time

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Abstract

We discuss the question of whether or not a general Weyl structure is a suitable mathematical model of space–time. This is an issue that has been in debate since Weyl formulated his unified field theory for the first time. We do not present the discussion from the point of view of a particular unification theory, but instead from a more general standpoint, in which the viability of such a structure as a model of space–time is investigated. Our starting point is the well known axiomatic approach to space–time given by Elhers, Pirani and Schild (EPS). In this framework, we carry out an exhaustive analysis of what is required for a consistent definition for proper time and show that such a definition leads to the prediction of the so-called “second clock effect”. We take the view that if, based on experience, we were to reject space–time models predicting this effect, this could be incorporated as the last axiom in the EPS approach. Finally, we provide a proof that, in this case, we are led to a Weyl integrable space–time as the most general structure that would be suitable to model space–time.

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Notes

  1. Actually, after showing that Weyl’s curvature F has to vanish, they conclude that space–time geometry has a Riemannian representation. This is true, since the vanishing of F implies (in a simply connected domain) that the Weyl structure is integrable, and in any Weyl integrable structure there is a Riemannian representative of the class.

  2. See, for instance, the footnote in page 68 of [1].

  3. In [1], in order to reduce the Weyl structure to a Riemannian one, two possible additional axioms, regarding the behaviour of clocks, are introduced. The first one is stated as follows. Given two freely falling, infinitesimally proximate clocks \(C_{1}\) and \(C_{2}\), if we consider a regular sequence of events \((p_{1},p_{2},\dots )\) in the world line of \(C_{1}\) determined by the ticking of this clock, and the Einstein-simultaneous sequence of events \((q_{1},q_{2},\dots )\) in the world line of \(C_{2}\), then \((q_{1},q_{2},\dots )\) should also be a regular sequence of events. With the help of the geodesic deviation equation, EPS show that this hypothesis reduces the Weyl structure to a Riemannian one. Another way to achieve the same the same goal, is to consider as an axiom that the “norm” of parallel transported vector fields at a point can not depend on the curve chosen to make the transport, associating this “norm”, for the case of time-like curves, to the ticking rate of clocks.

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Acknowledgements

R. Avalos and C. Romero would like to thank CNPq and CLAF for financial support. Funding was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico.

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  1. Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, PB, 58059-970, Brazil

    R. Avalos, F. Dahia & C. Romero

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  1. R. Avalos
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Avalos, R., Dahia, F. & Romero, C. A Note on the Problem of Proper Time in Weyl Space–Time. Found Phys 48, 253–270 (2018). https://doi.org/10.1007/s10701-017-0134-z

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