The Scattering of Dirac Spinors in Rotating Spheroids

TL;DR

This paper investigates how Dirac spinors scatter when interacting with rotating spheroid stars, deriving a relation between scattering cross-section and stellar matter density applicable to weak-field stars like white dwarfs.

Contribution

It provides a novel analysis of Dirac spinor scattering in the gravitational field of rotating spheroids, specifically deriving the scattering solution and its dependence on stellar matter density.

Findings

Scattering cross-section is proportional to stellar matter density.

Results are valid for weak gravitational fields and incompressible fluid stars.

Applicable to white dwarfs and similar stars, not compact objects like neutron stars.

Abstract

There are many stars that are rotating spheroids in the Universe, and studying them is of very important significance. Since the times of Newton, many astronomers and physicists have researched gravitational properties of stars by considering the moment equations derived from Eulerian hydrodynamic equations. In this paper we study the scattering of spinors of the Dirac equation, and in particular investigate the scattering issue in the limit case of rotating Maclaurin spheroids. Firstly we give the metric of a rotating ellipsoid star, then write the Dirac equation under this metric, and finally derive the scattering solution to the Dirac equation and establish a relation between differential scattering cross-section, $\sigma$, and stellar matter density, $\mu$. It is found that the sensitivity of $\sigma$ to the change in $\mu$ is proportional to the density $\mu$. Because of weak gravitational field and constant mass density, our results are reasonable. The results can be applied to white dwarfs, main sequence stars, red giants, supergiant stars and so on, as long as their gravitational fields are so weak that they can be treated in the Newtonan approximations, and the fluid is assumed to be incompressible. Notice that we take the star's matter density to be its average density and the star is not taken to be compact. Obviously our results cannot be used to study neutron stars and black holes. In particular, our results are suitable for white dwarfs, which have average densities of about $10^{5}-10^{6}$\,g~cm$^{-3}$, corresponding to a range of mass of about $0.21-0.61 M_{\bigodot}$ and a range of radius of about $6000-10000$\,km.