  We examine the influence of the dispersion relation on the Unruh effect by
Lorentz boosting the phase of Minkowski vacuum fluctuations endowed with an
arbitrary dispersion relation. We find that, unlike what happens with a linear
dispersion relation exhibited by massless fields, thermality is lost for
general dispersion relations. We show that thermality emerges with a varying
"apparent" Davies-Unruh temperature depending on the acceleration of the
observer and on the degree of departure from linearity of the dispersion
relation. The approach has the advantage of being intuitive and able to
pinpoint why such a loss of thermality occurs and when such a deviation from
thermality becomes significant. We discuss the link of our results with the
well-known fundamental difference between the thermalization theorem and the
concept of Rindler noise. We examine the possible experimental validation of
our results based on a successful setup for testing the classical analogue of
the Unruh effect recently described in the literature.
