Production of Dirac particle in a deformed Minkowsky space-time

In this paper we study the Dirac field theory interacting with external gravitation field, described with tetrad of the form $e_b^{\mu }(x)=\epsilon ({\delta }_b^{\mu }+{\omega }_{ba}^{\mu }{x}^a)$, where $\epsilon =1$ for $\mu =0$ and $\epsilon =i$ for $\mu =1,2,3.$ The probability density of the vacuum-vacuum pair creation is given. In particular case of vanishing electromagnetic fields, we point out how this deformation modify the amplitude transition. The corresponding Dirac equation is solved.