Flow of time during energy measurements and the resulting time-energy uncertainty relations

Uncertainty relations play a crucial role in quantum mechanics. A well-defined method exists for deriving such uncertainties for pairs of observables. It does not include, however, an important family of fundamental relations: the time-energy uncertainty relations. As a result, different approaches have been used for obtaining them in diversified scenarios. The one of interest here revolves around the idea of the existence or inexistence of a minimum duration for an energy measurement with a certain precision. In our study, we use the Page and Wooters timeless framework to investigate how energy measurements modify the relative flow of time between internal and external clocks. This provides a unified framework for discussing the topic, recovering previous results and leading to new ones. We also show that the evolution of the external clock with respect to the internal one is non-unitary.