  Black hole X-ray binaries in their hard and hard-intermediate states display
hard and soft time lags between broadband noise variations (high-energy
emission lagging low-energy and vice versa), which could be used to constrain
the geometry of the disk and Comptonising corona in these systems.
Comptonisation and reverberation lag models, which are based on light-travel
delays, can imply coronae which are very large (hundreds to thousands of
gravitational radii, $R_{g}$) and in conflict with constraints from X-ray
spectral modelling and polarimetry. Here we show that the observed large and
complex X-ray time lags can be explained by a model where fluctuations are
generated in and propagate through the blackbody-emitting disk to a relatively
compact ($\sim$10 $R_{g}$) inner corona. The model naturally explains why the
disk variations lead coronal variations with a Fourier-frequency dependent lag
at frequencies $<1$ Hz, since longer variability time-scales originate from
larger disk radii. The propagating fluctuations also modulate successively the
coronal seed photons from the disk, heating of the corona via viscous
dissipation and the resulting reverberation signal. The interplay of these
different effects leads to the observed complex pattern of lag behaviour
between disk and power-law emission and different power-law energy bands, the
energy-dependence of power-spectral shape and a strong dependence of
spectral-timing properties on coronal geometry. The observed spectral-timing
complexity is thus a natural consequence of the response of the disk-corona
system to mass-accretion fluctuations propagating through the disk.
