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

## TL;DR

This paper develops a covariant formulation of Dirac fermions with modified action near black holes, showing classical limits align with general relativity and particles can escape black holes due to superluminal velocities.

## Contribution

It introduces a covariant model for Dirac fermions with Planck-scale modifications and analyzes its classical limit and particle escape dynamics in black hole spacetimes.

## Key findings

- Einstein equations remain unchanged in the classical limit.
- Particles can escape black holes due to superluminal effects.
- Classical solutions match those of standard general relativity.

## Abstract

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 that originates from Planck physics. Due to this term a closed Fermi surface is formed in equilibrium inside the black hole. In this paper we propose the covariant formulation of this model and analyse its classical limit. We consider the dynamics of gravitational collapse. It appears that the Einstein equations admit a solution identical to that of the ordinary general relativity. Next, we consider motion of particles in the presence of the black hole. Numerical solutions of the equations of motion are found which demonstrate that the particles are able to escape from the black hole.