  This article is motivated by the observation, that calculations of the Unruh
effect based on idealized particle detectors are usually made in a way that
involves integrations along the {\em entire} detector trajectory up to the
infinitely remote {\em future}. We derive an expression which allows
time-dependence of the detector response in the case of a non-stationary
trajectory and conforms more explicitely to the principle of causality, namely
that the response at a given instant of time depends only on the detectors {\em
past} movements. On trying to reproduce the thermal Unruh spectrum we are led
to an unphysical result, which we trace down to the use of the standard
regularization $t\to t-i\eps$ of the correlation function. By consistently
employing a rigid detector of finite extension, we are led to a different
regularization which works fine with our causal response function.
