  We consider a particle detector model on 1+1-dimensional Minkowski space-time
that is accelerated by a constant external acceleration a. The detector is
coupled to a massless scalar test field. Due to the Unruh effect, this detector
becomes excited even in Minkowski vacuum with a probability proportional to a
thermal Bose-Einstein distribution at temperature T=a/2pi in the detector's
energy gap. This excitation is usually said to happen upon detection of a
Rindler quantum, which is defined by having positive frequency with respect to
the detector's proper time instead of inertial time. Using Unruh's fully
relativistic detector model, we show that the process involved is in fact
spontaneous excitation of the detector due to the interaction with the
accelerating background field E=ma. We explicitly solve the Klein-Gordon
equation in the presence of a constant background field E and use this result
to calculate the transition probability for the detector to become excited on
Minkowski vacuum. This transition probability agrees with the Unruh effect, but
is obtained without reverting to the concept of Rindler quantization.
