The Maxwell Lagrangian in purely affine gravity

TL;DR

This paper demonstrates the equivalence of a purely affine gravity formulation with the Einstein-Maxwell equations, highlighting the role of conformal invariance and Legendre transformations in relating different electromagnetic Lagrangians.

Contribution

It shows that the purely affine Lagrangian for electrodynamics is dynamically equivalent to Einstein-Maxwell equations and clarifies the connection through conformal invariance and Legendre transformations.

Findings

Purely affine Lagrangian reproduces Einstein-Maxwell equations.

Conformal invariance underpins the equivalence.

Legendre transformation relates to Maxwell-Palatini Lagrangian.

Abstract

The purely affine Lagrangian for linear electrodynamics, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the Einstein-Maxwell equations in the metric-affine and metric formulation. We show that this equivalence is related to the invariance of the Maxwell Lagrangian under conformal transformations of the metric tensor. We also apply to a purely affine Lagrangian the Legendre transformation with respect to the tensor of homothetic curvature to show that the corresponding Legendre term and the new Hamiltonian density are related to the Maxwell-Palatini Lagrangian for the electromagnetic field. Therefore the purely affine picture, in addition to generating the gravitational Lagrangian that is linear in the curvature, justifies why the electromagnetic Lagrangian is quadratic in the electromagnetic field.