Canonical transformation path to gauge theories of gravity II --- Spacetime coupling of spin-0 and spin-1 particle fields

TL;DR

This paper explores how different types of spin-0 and spin-1 particles couple to gravity, revealing that massive vector fields generate spacetime torsion while massless ones do not, based on their energy-momentum tensors.

Contribution

It provides a detailed analysis of the coupling of gravitational fields with various scalar and vector fields, highlighting differences in their energy-momentum tensors and resulting spacetime properties.

Findings

Massive vector fields use the canonical energy-momentum tensor as source.

Massive vector fields generate spacetime torsion.

Massless charged vector fields are associated with the metric energy-momentum tensor and do not generate torsion.

Abstract

The generic form of spacetime dynamics as a classical gauge field theory has recently been derived, based on only the action principle and on the Principle of General Relativity. It was thus shown that Einstein's General Relativity is the special case where (i) the Hilbert Lagrangian (essentially the Ricci scalar) is supposed to describe the dynamics of the "free" (uncoupled) gravitational field, and (ii) the energy-momentum tensor is that of scalar fields representing real or complex structureless (spin-$0$) particles. It followed that all other source fields---such as vector fields representing massive and non-massive spin-$1$ particles---need careful scrutiny of the appropriate source tensor. This is the subject of our actual paper: we discuss in detail the coupling of the gravitational field with (i) a massive complex scalar field, (ii) a massive real vector field, and (iii) a massless vector field. We show that different couplings emerge for massive and non-massive vector fields. The \emph{massive} vector field has the \emph{canonical} energy-momentum tensor as the appropriate source term---which embraces also the energy density furnished by the internal spin. In this case, the vector fields are shown to generate a torsion of spacetime. In contrast, the system of a \emph{massless} and charged vector field is associated with the \emph{metric} (Hilbert) energy-momentum tensor due to its additional $\mathrm{U}(1)$ symmetry. Moreover, such vector fields do not generate a torsion of spacetime. The respective sources of gravitation apply for all models of the dynamics of the `free' (uncoupled) gravitational field---which do not follow from the gauge formalism but must be specified based on separate physical reasoning.