Exponential collapse with variable time scale driven by a scalar field

We study the dynamic collapse driven by a scalar field, when a relativistic observer falls co-moving with the collapse and cross the horizon of a Schwarzschild black-hole (BH), at $t=t_0$. During the collapse the scale of time is considered as variable. Back-reaction effects and gravitational waves produced during the exponential collapse are studied. We demonstrate that back-reaction effects act as the source of gravitational waves emitted during the collapse, and wavelengths of gravitational waves (GW) are in the range: $\lambda \ll r_s\equiv {e^{-2h_0t_0}\over 2 h_0}$, that is, smaller than the Schwarzschild radius. We demonstrate that during all the collapse the global topology of the space-time remains hyperbolic when the observer cross the horizon.