The relation between non-commutative and Finsler geometry in Horava-Lifshitz black holes

TL;DR

This paper explores the connections between non-commutative and Finsler geometries in the context of Horava-Lifshitz black holes, revealing their Lagrangian equivalence under certain conditions and implications for particle behavior near black holes.

Contribution

It introduces a novel approach linking non-commutative and Finsler geometries in Horava-Lifshitz black holes and demonstrates their Lagrangian equivalence.

Findings

Lagrangians in non-commutative and Finsler geometries coincide for static black holes.

In rotating black holes, the Lagrangians differ, affecting particle dynamics.

Particles can enter black holes rapidly without singularity in the rotating case.

Abstract

In this paper we employ the Horava-Lifshitz black holes solutions and obtain the corresponding Hamiltonian. It helps us to take new variables and it will be written by harmonic oscillator form. This leads us to apply non-commutative geometry to the new Hamiltonian and obtain the corresponding Lagrangian. And then, we take some information from Finsler geometry and write the Lagrangian of the different kinds of Horava-Lifshitz black holes. We show that the corresponding Lagrangian in non-commutative and Finsler geometry for above mentioned black holes completely coincidence together with some specification of parameters. But in case of rotation, the place of center of mass energy completely different, so the particle goes to inside of black hole rapidly without falling into singularity. So in that case, two Lagrangians cover each other at $0<r<r_h.$