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.