Cosserat elasticity as the weak-field limit of Einstein--Cartan relativity (2405.12188v1)
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 accompanyingdynamic mass moment density flux', and new forms of linear momentum density captured by a dynamic mass density flux' and adynamic momentum density'.
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