On the effective action of a space-like brane

Starting from the non-BPS D(p+1)-brane action, we derive an effective action in (p+1) space dimensions by studying the fluctuations of various bosonic fields around the time-like tachyonic kink solution (obtained by Wick rotation of the space-like tachyonic kink solution) of the non-BPS brane. In real time this describes the dynamics of a space-like or Euclidean brane in (p+1) dimensions containing a Dirac-Born-Infeld (DBI) part and an Wess-Zumino (WZ) part. The WZ part is purely imaginary and so the action is complex if it represents the source of the time-dependent background of type II string theory, i.e., S-brane. On the other hand, the WZ part as well as the action is real if it represents the source in type II* string theory, i.e., E-brane. The DBI part is the same as obtained before using different method. This is then further illustrated by considering brane probe in space-like brane background