Energy-Momentum in Gauge Gravitation Theory

TL;DR

This paper develops a method to derive energy-momentum conservation laws in gauge gravitation theory, addressing challenges posed by fermion fields and spin structures, and introduces a universal Dirac spin structure for this purpose.

Contribution

It introduces a universal Dirac spin structure and constructs canonical lifts of vector fields, enabling the derivation of energy-momentum conservation laws in gauge gravitation theory with fermions.

Findings

Constructed the universal Dirac spin structure $S$ over $ ext{Σ}$ and $X$.

Derived the stress-energy-momentum conservation law in gauge gravitation theory.

Analyzed the gravitational model with background spin structure.

Abstract

Building on the first variational formula of the calculus of variations, one can derive the energy-momentum conservation laws from the condition of the Lie derivative of gravitation Lagrangians along vector fields corresponding to generators of general covariant transformations to be equal to zero. The goal is to construct these vector fields. In gauge gravitation theory, the difficulty arises because of fermion fields. General covariant transformations fail to preserve the Dirac spin structure $S_h$ on a world manifold $X$ which is associated with a certain tetrad field $h$. We introduce the universal Dirac spin structure $S\to\Sigma\to X$ such that, given a tetrad field $h:X\to \Sigma$, the restriction of $S$ to $h(X)$ is isomorphic to $S_h$. The canonical lift of vector fields on $X$ onto $S$ is constructed. We discover the corresponding stress-energy-momentum conservation law. The gravitational model in the presence of a background spin structure also is examined.