Is minimal coupling procedure compatible with minimal action principle?

TL;DR

This paper investigates the compatibility issues between the minimal coupling procedure and the minimal action principle in non-Riemannian space-times, showing that compatibility depends on the properties of the volume element and affecting Einstein-Cartan gravity predictions.

Contribution

It demonstrates conditions under which the minimal coupling procedure aligns with the minimal action principle in non-Riemannian geometries, impacting Einstein-Cartan theory.

Findings

Compatibility is restored if the space-time admits a connection-compatible volume element.

The compatibility affects the equations of motion in Einstein-Cartan gravity.

Qualitative predictions of Einstein-Cartan theory can change based on volume element properties.

Abstract

When space-time is assumed to be non-Riemannian the minimal coupling procedure (MCP) is not compatible, in general, with minimal action principle (MAP). This means that the equations gotten by applying MCP to the Euler-Lagrange equations of a Lagrangian $\cal L$ do not coincide with the Euler-Lagrange equations of the Lagrangian obtained by applying MCP to $\cal L$. Such compatibility can be restored if the space-time admits a connection-compatible volume element. We show how these concepts can alter qualitatively the predictions of the Einstein-Cartan theory of gravity.