Abstract
We show that the definition of proper time for Weyl-invariant space-times given by Perlick naturally extends to spaces with arbitrary non-metricity. We then discuss the relation between this generalized proper time and the Ehlers–Pirani–Schild definition of time when there is arbitrary non-metricity. Then we show how this generalized proper time suffers from a second clock effect. Assuming that muons are a device to measure this proper time, we constrain the non-metricity tensor on Earth’s surface and then elaborate on the feasibility of such assumption.
Highlights
1-form under scale transformations of the metric, i.e. it is the gauge field associated to scale transformations. This fact fostered the interest in Weyl geometries, since they provide a natural way of introducing scale transformations without changing the affine structure
Under the assumption that fundamental particles measure the generalized proper time, we show that it is possible to constrain some components of the non-metricity around Earth’s surface by considering data on the time dilation of muons accelerated by a magnetic field, which give constraints to the amount of second clock effect that we would find by considering different muon trajectories in a background with non-metricity
Since we have generalized this result for spaces with arbitrary non-metricity, it is pertinent to ask whether generalized clocks would measure a second clock effect in specific theories of modified gravity, or on the contrary, these theories give rise to spacetimes which are free of it
Summary
The dynamics of the theory is given by extremizing the corresponding action with respect to metric and connection independently (as opposed to extremizing it only with respect to the metric field). Under the assumption that fundamental particles measure the generalized proper time, we show that it is possible to constrain some components of the non-metricity around Earth’s surface by considering data on the time dilation of muons accelerated by a magnetic field, which give constraints to the amount of second clock effect that we would find by considering different muon trajectories in a background with non-metricity. We discuss these bounds in several theories featuring non-metricity.
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