The dynamical nature of time

TL;DR

This paper explores the complex relationship between mathematical parametric time and physical dynamical time, revealing potential implications for understanding clock measurements and phenomena like the Pioneer anomaly.

Contribution

It demonstrates that the relation between mathematical and physical time in dynamics is more nuanced than traditionally assumed, with possible observable consequences.

Findings

The mathematical parametric time can be transformed by any well-behaved function without altering dynamics.

Physical dynamical time measured by clocks may differ from parametric time in significant ways.

Non-equivalence of different dynamical clock-times could explain phenomena such as the Pioneer anomaly.

Abstract

It is usually assumed that the "$t$" parameter in the equations of dynamics can be identified with the indication of the pointer of a clock. Things are not so easy, however. In fact, since the equations of motion can be written in terms of $t$ but also of $t'=f(t)$, $f$ being any well behaved function, each one of those infinite parametric times $t'$ is as good as the Newtonian one to study classical dynamics. Here we show that the relation between the mathematical parametric time $t$ in the equations of dynamics and the physical dynamical time $\sigma$ that is measured with clocks is more complex and subtle than usually assumed. These two times, therefore, must be carefully distinguished since their difference may have significant consequences. Furthermore, we show that not all the dynamical clock-times are necessarily equivalent and that the observational fingerprint of this non-equivalence has the same form as that of the Pioneer anomaly.