  Symmetries represent a fundamental constraint for physical systems and
relevant new phenomena often emerge as a consequence of their breaking. An
important example is provided by space- and time-translational invariance in
statistical systems, which hold at a coarse-grained scale in equilibrium and
are broken by spatial and temporal boundaries, the former being implemented by
surfaces - unavoidable in real samples - the latter by some initial condition
for the dynamics which causes a non-equilibrium evolution. While the separate
effects of these two boundaries are well understood, we demonstrate here that
additional, unexpected features arise upon approaching the effective edge
formed by their intersection. For this purpose, we focus on the classical
semi-infinite Ising model with spin-flip dynamics evolving out of equilibrium
at its critical point. Considering both subcritical and critical values of the
coupling among surface spins, we present numerical evidence of a scaling regime
with universal features which emerges upon approaching the spatio-temporal edge
and we rationalise these findings within a field-theoretical approach.
