  We study the evolution of neutrinos in a background matter moving with a
linear acceleration. The Dirac equation for a massive neutrino electroweakly
interacting with background fermions is obtained in a comoving frame where
matter is at rest. We solve this Dirac equation for ultrarelativistic
neutrinos. The neutrino quantum states in matter moving with a linear
acceleration are obtained. We demonstrate that the neutrino electroweak
interaction with an accelerated matter leads to the vacuum instability which
results in the neutrino-antineutrino pairs creation. We rederive the
temperature of the Unruh radiation and find the correction to the Unruh effect
due to the specific neutrino interaction with background fermions. As a
possible application of the obtained results we discuss the neutrino pairs
creation in a core collapsing supernova. The astrophysical upper limit on the
neutrino masses is obtained.
