The dynamical properties of Dirac spinor particles in a non-static Rindler space-time is studied. The explicit representative of the four-component wave function of Dirac particles is obtained. A non-uniformly rectilinearly accelerating Rindler observer is accompanied by an event horizon depending on time. He explores some thermal radiation varying with time, whose temperature is proportional to the simultaneous acceleration. The existence of the non-static Rindler effect is confirmed.