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

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 (WIST) as the most general structure that would be suitable to model space-time.

Experimental constraints on the second clock effect

“…In addition to that, in the limit in which the length curvature F µν goes to zero Perlick's time reduces to the Riemannian or WIST proper time. Recently, it was shown the equivalence between this definition and the one given in the well-known paper by Ehlers, Pirani, and Schild (EPS) [8,9]. The latter was entirely based on axiomatic approach which leads to a Weyl structure as the most suitable model for spacetime.…”

“…where the overdot means derivative with respect to the curve's parameter [8] . It has also been shown that Perlick's time has all the properties a good definition of proper time in a Weyl space-time should have, such as, Weyl-invariance, positive definiteness, additivity.…”

“…Another important property of Perlick's hypothesis (perhaps unexpected) concerning the proper time of a standard clock is that it also predicts the existence of the second clock effect, namely, that the clock rate of a local observer depends on its path [8]. More precisely, consider two clocks c 1 and c 2 synchronized at point A (see Fig.…”

“…It should also be noted here that any invariant scalar of this geometry must necessarily be formed by both the metric g µν and the Weyl gauge field σ µ . respect to each other at C. More precisely, let τ 1,2 ,respectively, be the proper times of clocks c 1 and c 2 after they have met at point C. Assuming Perlick's proper time hypothesis, it can be shown that [8]…”

“…In fact, the usual definition of proper time adopted in general relativity as the arc-length of a curve (the clock hypothesis) cannot be properly carried over to Weyl geometry for the simple reason that this definition is not invariant under Weyl transformations (see [5,6] and references therein). It turns out, however, that this problem has been finally settled by V. Perlick, who proposed a definition of proper time which is consistent with Weyl's principle of invariance [7,8]. Perlick's notion of proper time provides a correction to the arc-length formula, and reduces to the general relativistic proper time when the Weyl 1-form field vanishes.…”