  Every Coxeter group admits a geometric representation as a group generated by
reflections in a real vector space. In the projective representation space,
limit directions are limits of injective sequences in the orbit of some base
point. Limit roots are limit directions that can be obtained starting from
simple roots. In this article, we study the limit directions arising from any
point when the representation space is a Lorentz space. In particular, we
characterize the light-like limit directions using eigenvectors of
infinite-order elements. This provides a spectral perspective on limit roots,
allowing for efficient computations. Moreover, we describe the space-like limit
directions in terms of the projective Coxeter arrangement.
