# Elastic Li\'{e}nard-Wiechert potentials of dynamical dislocations from tensor gauge theory in 2+1 dimensions

## TL;DR

This paper develops a dual gauge theory framework to analyze the dynamics and interactions of dislocations in elastic media, deriving explicit potentials and forces, including non-reciprocal effects for moving dislocations.

## Contribution

It introduces a systematic dual gauge theory approach to dislocation dynamics, deriving explicit potentials and extending classical force results to moving dislocations.

## Key findings

- Derived explicit dual gauge potentials for moving dislocations.
- Extended Peach-Koehler force to include non-reciprocal effects.
- Provided a framework for analyzing defect interactions in elastic media.

## Abstract

The dynamics of defect excitations in crystalline solids is necessary to understand the macroscopic low-energy properties of elastic media. We use fracton-elasticity duality to systematically study the defect dynamics and interactions in the linear isotropic medium. We derive the explicit expressions for the dual gauge potentials for moving dislocations and the resulting Jefimenko equations. We also compute stresses and strains. The paper includes two physical situations: when the vacancy number is fixed and when the number is fluctuating. If defects are present we show a constraint that needs to be satisfied by them when they climb perpendicularly to their Burgers vector. Next, we extend the classic result of Peach and Koehler for the force between two dislocations and show that, similarly, to moving charges in electrodynamics, it is non-reciprocal, when one dislocation is moving. We argue that our formalism can be extended beyond Cauchy's elasticity by exploiting the simplifications provided by the dual gauge formulation of elastic stresses.