Unruh Effect for General Trajectories

We consider two-level detectors coupled to a scalar field and moving on arbitrary trajectories in Minkowski space-time. We first derive a generic expression for the response function using a (novel) regularization procedure based on the Feynmann prescription that is explicitly causal, and we compare it to other expressions used in the literature. We then use this expression to study, analytically and numerically, the time dependence of the response function in various non-stationarity situations. We show that, generically, the response function decreases like a power in the detector's level spacing, $E$, for high $E$. It is only for stationary world-lines that the response function decays faster than any power-law, in keeping with the known exponential behavior for some stationary cases. Under some conditions the (time dependent) response function for a non-stationary world-line is well approximated by the value of the response function for a stationary world-line having the same instantaneous acceleration, torsion, and hyper-torsion. While we cannot offer general conditions for this to apply, we discuss special cases; in particular, the low energy limit for linear space trajectories.