Arbitrariness and Usefulness of the Expressions of Elastic wave's Energy, Momentum and Angular Momentum

TL;DR

This paper clarifies the correct expressions for elastic wave energy, momentum, and angular momentum using Noether's theorem, demonstrating their practical usefulness and addressing recent controversies in the field.

Contribution

It systematically analyzes elastic wave conserved currents, explains the validity of various expressions, and introduces a new energy density for earthquake measurement.

Findings

Different conserved current expressions are all correct for elastic waves.

A new energy density expression improves earthquake energy measurement.

Decomposition of wave components is meaningful and observable.

Abstract

Elastic angular momentum is an emerging field, with some controversies on the correct field-theory expressions and the decomposition of longitudinal and transverse components. Motivated by the recent two papers [Phys.Rev.Lett. 128, 064301(2022), Phys.Rev.Lett. 129, 204303(2022)] on this issue, we systematically analyze by Noether's theorem the canonical and Belinfante energy-momentem and angular momentum, then explain why the two familiar expressions, together with other various conservered currents, are all correct for elastic wave. Remarkbly, to illustrate the usefullness of different expressions, we give an example on earthquake energy measurement with a new energy density expression which is more advantageous in practical measurement. Moreover, since the elastic wave is distinct from a quantum one, we suggest that the decomposition of longitudinal and transverse components, in fact, makes sense and can be clearly expressed by the observable displacement field. We hope that this paper would clarify the controversies, and finally we give prospect of future work on the Geometric Spin Hall Effect of elastic wave, where the various expressions of conserved currents would exhibit different applications.