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 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.

💡 Research Summary

This paper presents a rigorous investigation into the relationship between the magnetic 2+1-dimensional space-time, specifically the magnetic BTZ solution, and its non-relativistic counterpart. The research focuses on deriving a specific non-relativistic limit, termed the Newton-Hooke (NH) limit, from the Einstein-Maxwell system in 2+1 dimensions. The authors distinguish this NH limit from the traditional Galilean limit by emphasizing the critical role of the cosmological constant ($\Lambda$). While the Galilean limit typically involves taking the speed of light ($c$) to infinity, the NH limit preserves the influence of $\Lambda$, thereby introducing a more complex dynamical structure.

The core mathematical achievement of this study is the demonstration that the geodesics of the magnetic BTZ solution, when subjected to the Newton-Hooke limit, reduce to the motion of a charged particle subjected to both a standard magnetic field and an additional harmonic potential. This specific physical configuration is known in condensed matter physics as the Fock-Darwin problem. By establishing this connection, the paper bridges the gap between the geometric properties of relativistic black hole solutions and the well-established models used to describe particles in quantum dots or two-dimensional electron gases.

Furthermore, the paper explores the profound implications of this reduction for the study of symmetries across different physical regimes. The authors analyze the generalized magnetic $NH_3$ symmetry, which serves as the fundamental symmetry group for the Fock-Darwin problem. By applying this symmetry within the context of the Virial theorem, the study provides a unified framework for understanding how dynamical systems transition from relativistic to non-relativistic scales. The significance of this work lies in its ability to show that the cosmological constant, a parameter of large-scale spacetime curvature, manifests as a restorative harmonic potential in the non-relativistic limit, thereby providing a deep link between gravitational physics and the mechanics of condensed matter systems. This research offers valuable insights into the symmetry-driven unification of seemingly disparate physical theories.