  The uncertainty principle bounds our ability to simultaneously predict two
incompatible observables of a quantum particle. Assisted by a quantum memory to
store the particle, this uncertainty could be reduced and quantified by a new
Entropic Uncertainty Relation (EUR). In this Letter, we explore how the
relativistic motion of the system would affect the EUR in two sample scenarios.
First, we show that the Unruh effect of an accelerating particle would surely
increase the uncertainty if the system and particle entangled initially. On the
other hand, the entanglement could be generated from nonuniform motion once the
Unruh decoherence is prevented by utilizing the cavity. We show that, in a
uncertainty game between an inertial cavity and a nonuniformly accelerated one,
the uncertainty evolves periodically with respect to the duration of
acceleration segment. Therefore, with properly chosen cavity parameters, the
uncertainty bound could be protected. Implications of our results for
gravitation are also discussed.
