Landau levels in a gravitational field: the Levi-Civita and Kerr spacetimes case

Abstract

We have recently found that the gravitational field of a static spherical mass removes the Landau degeneracy of the energy levels of a particle moving around the mass inside a magnetic field by splitting the energy of the Landau orbitals. In this paper we present the second part of our investigation of the effect of gravity on Landau levels. We examine the effect of the gravitational fields created by an infinitely long massive cylinder and a rotating spherical mass. In both cases, we show that the degeneracy is again removed thanks to the splitting of the particle's orbitals. The first case would constitute an experimental test -which is quantum mechanical in nature -of the gravitational field of a cylinder. The approach relies on the Newtonian approximation of the gravitational potential created by a cylinder but, in view of self-consistency and for future higher-order approximations, the formalism is based on the full Levi-Civita metric. The second case opens up the possibility for a novel quantum mechanical test of the well-known rotational frame-dragging effect of general relativity. * Electronic address: fhammad@ubishops.ca † Electronic address: alexandre.landry.1@umontreal.ca

Landau Levels in a Gravitational Field: The Schwarzschild Spacetime Case

“…Furthermore, such a step allows us to clearly see how to adapt our present approach to other spacetime metrics as we show in Ref. [82] and how to extend it to include relativistic regimes as we did in Section 5.…”

“…Now, in addition to exploiting numerical methods, our next tasks-the first of which is already presented in Ref. [82] for the non-relativistic case-are to (i) first consider the effect on Landau levels of more involved spacetime metrics, and then (ii) take into account the relativistic regime into consideration and investigate closer the implications for the highly magnetized astrophysical objects. The first allow us to use this splitting to probe in novel ways the modifications brought by GR to Newtonian gravity.…”

Spin-1/2 Particles under the Influence of a Uniform Magnetic Field in the Interior Schwarzschild Solution

“…The second interesting refinement that should be discussed here is the possibility of adding to the metric components, given by Equation (18), the spacetime curvature caused by the magnetic field. The geometric effect of combining the external gravitational field sourced by a spherical mass with a uniform magnetic field B gives rise to the so-called Schwarzschild-Melvin spacetime [41].…”

“…This entails that if we had kept those higher order terms the expected correction to be brought to the quantized energy levels (29) can be obtained using time-independent perturbation theory in the manner worked out in Refs. [17,18] for the case of Landau levels in the exterior Schwazschild solution.…”

“…Such a study has recently been conducted in Refs. [17][18][19] based on the Klein-Gordon and Schrödinger equations by considering the gravitational interaction of the particle with a massive sphere as a small perturbation compared to the interaction of the particle with the magnetic field. Although relying on the Schrödinger equation and treating gravity perturbatively as done in these studies might be sufficient for laboratory applications [20][21][22][23], when seeking astrophysical applications, implementing a generalrelativistic and a non-perturbative approach and appealing to the relativistic Dirac equation are highly recommended.…”