  We suggest a new relativity principle, which asserts the impossibility to
distinguish the state of rest and the state of motion at the constant velocity
of a system, if no work is done to the system in question during its motion. We
suggest calling this new rule as "conservative relativity principle" (CRP). In
the case of an empty space, CRP is reduced to the Einstein special relativity
principle. We also show that CRP is compatible with the general relativity
principle. One of important implications of CRP is the dependence of the proper
time of a charged particle on the electric potential at its location. In the
present paper we consider the relevant experimental facts gathered up to now,
where the latter effect can be revealed. We show that in atomic physics the
introduction of this effect furnishes a better convergence between theory and
experiment than that provided by the standard approach. Finally, we reanalyze
the Moessbauer experiments in rotating systems and show that the obtained
recently puzzling deviation of the relative energy shift between emission and
absorption lines from the relativistic prediction can be explained by the CRP.
