  In this work we analyze the amount of entanglement associated with the spin
and momentum degrees of freedom of a single massive spin-$\frac{1}{2}$ particle
from a relativistic perspective. The effect of a Lorentz boost introduces a
Wigner rotation that correlates the spin and momentum degrees of freedom. We
show that the natural basis to discuss the geometrical effects of the boost are
the helicity eigenstates in the rest frame. In the mid-relativistic regime
(where the Wigner rotation angle is limited by $\delta < \frac{\pi}{2}$) we
prove for states with equal helicity that the entanglement with respect to the
Wigner rotation angle is monotonically decreasing, however, in the
ultra-relativistic regime ($\delta > \frac{\pi}{2}$) the entanglement is
increasing. If the states are prepared as a superposition of unequal helicity
eigenstates, the monotonic behaviour is inverted. This implies that in the
ultra-relativistic regime a geometrical setup can be found such that the amount
of entanglement exhibits local maxima or minima. This shows a counter-intuitive
behaviour of the relative amount of entanglement, an effect due to the internal
and external geometrical configuration space, and points towards the
difficulties in achieving a Lorentz invariant formulation of entanglement in
general.
