Hamiltonian mapping and quantum perturbation equations in the point matter black hole and noncommutative black hole models

The Hamiltonian mapping of the Dirac equation with a quantum perturbation in $2$D black hole models are investigated. We derive the Hamiltonians of the modified Rabi model for the point matter black hole and noncommutative black hole, and obtain perturbed expressions for the velocity and acceleration of the simulating Dirac particles. The fluctuation equations for the charge–current density in two types of black holes are derived based on the solutions of the Dirac equation. The results indicate that the Hamiltonians contain the correction terms with different powers of the creation and annihilation operators, and the accelerations have some additional Dirac Zitterbewegung terms. In addition, the coordinate operators in the fluctuation equations of the charge–current density have different powers. These characteristics can be used to distinguish different types of black holes in the analogical gravity models.