  We perform a time-resolved optical study of Sr$_2$IrO$_4$ to understand the
influence of magnetic ordering on the low energy electronic structure of a
strongly spin-orbit coupled $J_{eff}$=1/2 Mott insulator. By studying the
recovery dynamics of photo-carriers excited across the Mott gap, we find that
upon cooling through the N\'{e}el temperature $T_N$ the system evolves
continuously from a metal-like phase with fast ($\sim$50 fs) and excitation
density independent relaxation dynamics to a gapped phase characterized by
slower ($\sim$500 fs) excitation density dependent bimolecular recombination
dynamics. The development of the insulating gap is accompanied by a transfer of
in-gap spectral weight to energies far in excess of the gap and occurs over an
unusually broad temperature window, which suggests Sr$_2$IrO$_4$ to be a
Slater- rather than Mott-Hubbard type insulator and naturally explains the
absence of anomalies at $T_N$ in transport and thermodynamic measurements.
