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We consider the model of Dirac fermions coupled to gravity as proposed in\cite{VolovikBH}, in which superluminal velocities of particles are admitted.In this model an extra term is added to the conventional Hamiltonian thatoriginates from Planck physics. Due to this term a closed Fermi surface isformed in equilibrium inside the black hole. In this paper we propose thecovariant formulation of this model and analyse its classical limit. Weconsider the dynamics of gravitational collapse. It appears that the Einsteinequations admit a solution identical to that of the ordinary generalrelativity. Next, we consider motion of particles in the presence of the blackhole. Numerical solutions of the equations of motion are found whichdemonstrate that the particles are able to escape from the black hole.

Classical limit for Dirac fermions with modified action in the presence of the black hole