``Weighing'' a closed system and the time-energy uncertainty principle

Abstract

A gedanken-experiment is proposed for `weighing'' the total mass of a closed system from within the system. We prove that for an internal observer the time $\tau$, required to measure the total energy with accuracy $\Delta E$, is bounded according to $\tau \Delta E >\hbar $. This time-energy uncertainty principle for a closed system follows from the measurement back-reaction on the system. We generally examine what other conserved observables are in principle measurable within a closed system and what are the corresponding uncertainty relations.