  Black-hole transients exhibit a correlation between the time lag of hard
photons with respect to softer ones and the photon index of the hard X-ray
power law. The correlation is not very tight and therefore it is necessary to
examine it source by source. The objective of the present work is to
investigate in detail the time-lag -- photon-index correlation in GX 339-4. We
have obtained RXTE energy spectra and light curves and have computed the photon
index and the time lag of the $9 - 15$ keV photons with respect to the $2 - 6$
keV ones. The observations cover the first stages of the hard state, the pure
hard state, and the hard-intermediate state. At low $\Gamma$, the correlation
is positive and it becomes negative at large $\Gamma$. By assuming that the
hard X-ray power law index $\Gamma$ is produced by inverse Compton scattering
of soft disk photons in the jet, we have reproduced the entire correlation by
varying two parameters in the jet: the radius of the jet at its base $R_0$ and
the Thomson optical depth along the jet $\tau_\parallel$. We have found that,
as the luminosity of the source increases, $R_0$ initially increases and then
decreases. This behavior is expected in the context of the Cosmic Battery. As a
further test of our model, we predict the break frequency in the radio spectrum
as a function of the photon index during the rising part of an outburst.
