Some effects on relativistic quantum systems due to a weak gravitational field

Abstract

We study the behaviour of relativistic quantum particles in the space-time generated by a moving mass current, in the weak field approximation. We solve the Dirac equation in this gravitational field and calculate the current associated with the particles.The study of the behaviour of quantum systems under the influence of curved space-times goes back to the end of the 1920s and to the beginning of the 1930s[1], when the generalization of the Schrödinger and Dirac equations to curved spaces has been discussed, motivated by the idea of constructing a theory combining quantum physics and general relativity. Along this line of research the hydrogen atom has been studied in particular curved space-times [2,3]. These investigations showed that the energy levels of an atom placed in a gravitational field is shifted as a result of the interaction of the atom with the space-time curvature [3]-[5]. This shift in the energy of each atomic level would depend on the features of the space-time.The general theory of relativity, as a metric theory, predicts that gravitation is manifested as the curvature of space-time. Therefore, it is of interest to know how the curvature of spacetime at the position of the atom affects its spectrum. On the other hand, we know that there are situations in which particles are constrained to move in a region where the Riemann curvature tensor vanishes and even in this case they exhibit gravitational effects arising from a region of non-zero curvature from which they are excluded [6]. In a more general sense, we have the case in which particles are constrained to move in a region where the Riemann curvature tensor does not vanish but does depend on certain parameter of the metric such as the velocity or the angular momentum of the source. In this case we have effects on the system associated with parameters which do not influence the curvature of the space-time as we will see.In what follows we present the study concerning the behaviour of a relativistic particle placed in the gravitational field generated by a cylindrical distribution of matter with uniform density along the z-axis moving slowly, whose metric reads [7] where Φ(ρ) represents the Newtonian potential produced by this source and satisfies the condition Φ(ρ) 2 ≈ 0, in the weak field approximation and v is the velocity of the distribution of matter. This quantity also satisfies the condition v 2 ≈ 0. This metric is characterized by two parameters, namely, the mass of the source and its velocity. It is interesting to call attention to the fact that in the weak field approximation, the Riemann curvature tensor outside the cylindrical source is completely determined by the Newtonian potential. For this space-time, the curvature outside the distribution of matter does not depend on its velocity, in the weak field approximation. This means that for the weak gravitational field associated with slowly moving mass currents, the local effects of the curvature associated with the velocity of the source are absent outside it.The covariant Dirac...

Casimir energy and gravitomagnetism

“…Gravitomagnetic effects on quantum fields have been studied in some detail by several authors, especially taking into account topological effects, as in the so-called gravitational Aharonov-Bohm effect (see, e.g., [18][19][20][21][22]).…”