Cosserat elasticity as the weak-field limit of Einstein--Cartan relativity

TL;DR

This paper explores the weak-field limit of Einstein--Cartan relativity, revealing its connection to Cosserat elasticity and providing detailed post-Newtonian equations that incorporate spin effects and their influence on gravitational interactions.

Contribution

It demonstrates that Einstein--Cartan theory's weak-field limit corresponds to Cosserat elastic media, extending post-Newtonian theory to include spin-related fields and their gravitational coupling.

Findings

EC theory's 1PN equations resemble a micropolar elastic medium.

Spin fields evolve independently at 1PN order.

New gravitational effects due to spin include torque and dynamic mass moments.

Abstract

The weak-field limit of Einstein--Cartan (EC) relativity is studied. The equations of EC theory are rewritten such that they formally resemble those of Einstein General Relativity (EGR); this allows ideas from post-Newtonian theory to be imported without essential change. The equations of motion are then written both at first post-Newtonian (1PN) order and at 1.5PN order. EC theory's 1PN equations of motion are found to be those of a micropolar/Cosserat elastic medium, along with a decoupled evolution equation for non-classical, spin-related fields. It seems that a necessary condition for these results to hold is that one chooses the non-classical fields to scale with the speed of light in a certain empirically reasonable way. Finally, the 1.5PN equations give greater insight into the coupling between energy-momentum and spin within slowly moving, weakly gravitating matter. Specifically, the weakly relativistic modifications to Cosserat theory involve a gravitational torque and an augmentation of the gravitational force due to a `dynamic mass moment density' with an accompanying `dynamic mass moment density flux', and new forms of linear momentum density captured by a `dynamic mass density flux' and a `dynamic momentum density'.