  The behavior of quantum speed limit time (QSLT) for a single free spin $-1/2$
particle described by Gaussian wavepackets in the framework of relativity under
dephasing noise is investigated. The dephasing noise acts only on the spin
degrees of freedom of the spin$-1/2$ particle. In particular, the effects of
initial time parameter, rapidity, average momentum and the size of the
wavepackets in the presence of the dephasing noise on the dynamics of evolution
process are studied. In general, the effects of relativity monotonically
decrease the QSLT in time. In the range of large values of average momentum,
critical values of both the rapidity and the size of the wavepackets exist at
which the QSLT has its minimum value. In the range of small values of the
average momentum, the QSLT monotonically decreases with both rapidity and the
size of the wavepackets. The decrease of QSLT in a particular range of rapidity
and with other relative parameters may be of great interest in employing fast
quantum communication and quantum computation.
