  We study the relations of the positive frequency mode functions of Dirac
field in 4-dimensional Minkowski spacetime covered with Rindler and Kasner
coordinates, and describe the explicit form of the Minkowski vacuum state with
the quantum states in Kasner and Rindler regions, and analytically continue the
solutions. As a result, we obtain the correspondence of the positive frequency
mode functions in Kasner region and Rindler region in a unified manner which
derives vacuum entanglement.
