Galileons as the Scalar Analogue of General Relativity

TL;DR

This paper draws a deep analogy between general relativity and scalar Galileon theories, showing that Galileons can serve as a scalar analogue with similar geometric structures and unique second-order dynamics.

Contribution

It establishes a correspondence between general relativity and Galileon theories, identifying geometric notions and demonstrating the uniqueness of second-order Galileon models via first-order formalism.

Findings

Galileons have geometric counterparts to Levi-Civita connection and Riemann tensor.

Galileon models are uniquely determined by first-order Palatini formalism.

Galileons are proposed as the scalar analogue of gauge theories like gravity.

Abstract

We establish a correspondence between general relativity with diffeomorphism invariance and scalar field theories with Galilean invariance: notions such as the Levi-Civita connection and the Riemann tensor have a Galilean counterpart. This suggests Galilean theories as the unique nontrivial alternative to gauge theories (including general relativity). Moreover, it is shown that the requirement of first-order Palatini formalism uniquely determines the Galileon models with second-order field equations, similar to the Lovelock gravity theories. Possible extensions are discussed.