Variational formulation of Eisenhart's unified theory

TL;DR

This paper reformulates Eisenhart's classical unified field theory using a metric-affine variational principle, revealing how a Lagrange multiplier constrains torsion and acts as a source for Maxwell's equations.

Contribution

It introduces a new variational formulation of Eisenhart's theory, linking torsion constraints to electromagnetic sources within a metric-affine framework.

Findings

Reformulation of Eisenhart's theory via metric-affine variational principle

Identification of Lagrange multiplier as electromagnetic source

Derivation of sourceless field equations from torsion and Ricci tensor conditions

Abstract

Eisenhart's classical unified field theory is based on a non-Riemannian affine connection related to the covariant derivative of the electromagnetic field tensor. The sourceless field equations of this theory arise from vanishing of the torsion trace and the symmetrized Ricci tensor. We formulate Eisenhart's theory from the metric-affine variational principle. In this formulation, a Lagrange multiplier constraining the torsion becomes the source for the Maxwell equations.