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

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 {\Lambda}, 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.