On the magnetic 2+1- D space-time and its non-relativistic counterpart

TL;DR

This paper studies the Newton–Hooke limit of a purely magnetic BTZ spacetime in 2+1 dimensions by starting from the Einstein–Maxwell system. In the NH limit, the relativistic geodesic motion maps to the nonrelativistic dynamics of a charged particle in a plane under a uniform magnetic field and an additional harmonic potential (the Fock–Darwin problem), establishing a concrete link between relativistic geometry and condensed-matter analogues. The work clarifies the physical nature of the magnetic field in the magnetically charged BTZ solution and demonstrates how a generalized NH3 symmetry governs the resulting dynamics, with implications for the Virial theorem and possibly nonrelativistic cosmology. It sets the stage for future exploration of symmetry generators and information-theoretic aspects in this relativistic-to-nonrelativistic correspondence.

Abstract

We present here an interesting non-relativistic limit, referred to as the Newton-Hooke (NH) limit, of the purely magnetic BTZ solution by starting from the Einstein-Maxwell system in the 2+1 dimensions. The Newton-Hooke limit is different from the Galilean limit in the sense that the former contains an additional parameter Λ, the cosmological constant, over and above the speed of light, c. We show that under this limit, the geodesics of the magnetic BTZ solution reduce to the two-dimensional motion of a charged particle in a normal magnetic field together with the presence of an extra harmonic potential, sometimes called the Fock-Darwin problem, which serves as a precursor to model certain condensed matter theories. Our present study has significance in analyzing the symmetries of different dynamical systems, from relativistic and/to nonrelativistic theories. Also, we discuss here one of the applications of the generalized (magnetic) NH_3 symmetry in the context of the Virial theorem, where this symmetry is the symmetry group of the Fock-Darwin problem mentioned above.