  We study special relativistic effects on the entanglement between either
spins or momenta of composite quantum systems of two spin-1/2 massive
particles, either indistinguishable or distinguishable, in inertial reference
frames in relative motion. For the case of indistinguishable particles, we
consider a balanced scenario where the momenta of the pair are well-defined but
not maximally entangled in the rest frame while the spins of the pair are
described by a one-parameter ($\eta$) family of entangled bipartite states. For
the case of distinguishable particles, we consider an unbalanced scenario where
the momenta of the pair are well-defined and maximally entangled in the rest
frame while the spins of the pair are described by a one-parameter ($\xi$)
family of non-maximally entangled bipartite states. In both cases, we show that
neither the spin-spin ($ss$) nor the momentum-momentum ($mm$) entanglements
quantified by means of Wootters' concurrence are Lorentz invariant quantities:
the total amount of entanglement regarded as the sum of these entanglements is
not the same in different inertial moving frames. In particular, for any value
of the entangling parameters, both $ss$ and $mm$-entanglements are attenuated
by Lorentz transformations and their parametric rates of change with respect to
the entanglements observed in a rest frame have the same monotonic behavior.
However, for indistinguishable (distinguishable) particles, the change in
entanglement for the momenta is (is not) the same as the change in entanglement
for spins. As a consequence, in both cases, no entanglement compensation
between spin and momentum degrees of freedom occurs.
