Proper acceleration, geometric tachyon and dynamics of a fundamental string near D$p$ branes

We present a detailed analysis of our recent observation that the origin of the geometric tachyon, which arises when a D$p$-brane propagates in the vicinity of a stack of coincident NS5-branes, is due to the proper acceleration generated by the background dilaton field. We show that when a fundamental string (F-string), described by the Nambu-Goto action, is moving in the background of a stack of coincident D$p$-branes, the geometric tachyon mode can also appear since the overall conformal mode of the induced metric for the string can act as a source for proper acceleration. We also studied the detailed dynamics of the F-string as well as the instability by mapping the Nambu-Goto action of the F-string to the tachyon effective action of the non-BPS D-string. We qualitatively argue that the condensation of the geometric tachyon is responsible for the (F,D$p$) bound state formation.