  We investigate light-cone structure on the world-volume of an unstable
D-brane with a tachyon decaying inhomogeneously by using a field theoretical
description. It is shown that (i) light-cones governing open strings are
narrower than those governing closed strings and will eventually collapse
inward in all directions except at kinks, where the tachyon remains at the top
of its potential; and that (ii) light-cones governing open strings at a kink
will be narrowed only in the direction perpendicular to the kink surface. It is
also shown that (iii) future-directed light-cones governing open strings near a
kink are tilted towards the kink, compared with those governing closed strings.
The result (i) implies that open strings except at kinks are redshifted,
compared with closed strings, and will eventually cease to be dynamical. On the
other hand, the result (ii) shows that open strings on a kink surface can move
freely along the kink surface and are dynamical but do not feel the existence
of the spatial dimension perpendicular to the kink surface. The result (iii)
indicates that open strings near a kink have tendency to move towards the kink.
Hence, the light-cone structure vividly illustrates how open strings behave
during the dynamical formation of a kink. We also discuss about a possibility
that the early universe has a network of various dimensional D-branes,
black-branes and tachyon matter. A problem associated with the network and a
possible solution to the problem are discussed.
