Low-order geometric actions with fields a metric and a matter field of arbitrary rank (undergraduate honors thesis)

TL;DR

This thesis classifies invariant Lagrangians involving a Lorentz metric and arbitrary rank tensor fields, providing a comprehensive understanding of their geometric actions and confirming Bray's conjecture on variational principles.

Contribution

It offers a complete classification of quadratic-in-derivatives invariant Lagrangians involving metrics and tensor fields, confirming Bray's conjecture.

Findings

Classification of invariant Lagrangians depending on metric and tensor fields.

Proof of Bray's conjecture on variational principles.

Identification of geometric actions with fields of arbitrary rank.

Abstract

We classify invariant Lagrangians of the form $L(g_{ij},g_{ij,k},g_{ij,kl},D_I,D_{I,j})$ depending at most quadratically on the variables $g_{ij,k},g_{ij,kl}$ and $D_I,D_{I,j}$, where $g$ is a Lorentz metric and $D$ is a tensor field of arbitrary rank on a smooth manifold. As a corollary, we prove a conjecture of Bray's regarding the classification of certain variational principles with variables a Lorentz metric and an affine connection.