Dynamical Similarity in Field Theories

TL;DR

This paper extends the concept of dynamical similarity to field theories using contact geometry, providing new formulations in both Lagrangian and Hamiltonian frameworks, with applications to general relativity.

Contribution

It introduces a novel application of dynamical similarity in field theories through contact geometry, offering a conformal-factor-free description of general relativity.

Findings

Dynamical similarity applies to field theories in both Lagrangian and Hamiltonian formalisms.

Contact geometry provides a new framework for describing field theory dynamics.

A complete, conformal-factor-independent description of general relativity is constructed.

Abstract

In previous work I have shown that Herglotz actions reproduce the dynamics of classical mechanical theories which exhibit dynamical similarities. Recent work has shown how to extend field theories in both the Lagrangian and de Donder-Weyl formalism to contact geometry. In this article I show how dynamical similarity applies in field theory. This is applied in both the Lagrangian and Hamiltonian frameworks, producing the contact equivalents. The result can be applied to general relativity where I demonstrate how to construct a complete description of the dynamics, equivalent to those derived from the Einstein-Hilbert action, without reference to the conformal factor.