  Forthcoming experiments like Project8 and Ptolemy aim at investigating with
high precision the end-point of the tritium $\beta$-decay spectrum sensitive to
the neutrino mass. In light of this, using the standard parametrization in
terms of nuclear polar form factors, we analyze the complete relativistic
expression for the spectrum of the $\beta$-electron emitted by a tritium
nucleus. Given the small parameters in the problem, we systematically discuss
the approximations that can be made, and present the first two corrections to
the standard lowest order formula. We particularly discuss the case of an
initially polarized target, and the consequences on the spectrum as a function
of the neutrino mass. We show that, while it induces an angular anisotropy that
can be measured by future experiments, such anisotropy cannot be used as an
additional handle to constrain the neutrino mass.
