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We study the relations of the positive frequency mode functions of Diracfield in 4-dimensional Minkowski spacetime covered with Rindler and Kasnercoordinates, and describe the explicit form of the Minkowski vacuum state withthe quantum states in Kasner and Rindler regions, and analytically continue thesolutions. As a result, we obtain the correspondence of the positive frequencymode functions in Kasner region and Rindler region in a unified manner whichderives vacuum entanglement.

Entanglement of the Vacuum between Left, Right, Future, and Past: Dirac spinor in Rindler spaces and Kasner spaces