  In this Letter, the dynamic behavior of Einstein-Podolsky-Rosen (EPR)
steering and the redistribution of EPR steering under a relativistic framework
are investigated. Specifically, we explore the scenario that particle A hold by
Alice is in a flat space-time and another particle B by Bob entangled with A is
in a non-inertial framework. The results show that EPR steering from Alice to
Bob is dramatically destroyed by Unruh effect caused by the acceleration of
Bob. Besides, EPR steering has an asymmetry property, and EPR steering
asymmetry increases with the growing intensity of Unruh effect, implying that
the Unruh effect can bring on EPR steering asymmetry. Furthermore, the reduced
physical accessible EPR steering from Alice to Bob is distributed to the
physical inaccessible EPR steering (from Alice to anti-Bob or from Bob to
anti-Bob). Notably, unlike entanglement and quantum discord, only one of EPR
steering from Alice to anti-Bob and Bob to anti-Bob experiences a sudden birth
with the increase of acceleration parameter, which means that they cannot
simultaneously survive. That is, the monogamy relation of EPR steering is still
tenable in such a scenario. Consequently, we believe that EPR steering could
also be served as one of important information resources within the
long-distance quantum secure communication under the relativistic framework.
