Gravity Theories via Algebra Gauging

TL;DR

This paper systematically derives quantized gravity theories using algebra gauging across different spacetime structures, highlighting key differences and features of Galilean and Bargmann gravity compared to relativistic gravity.

Contribution

It introduces the algebra gauging technique for constructing gravity theories in various spacetime frameworks and analyzes their unique geometric and algebraic properties.

Findings

Galilean gravity has a degenerate metric structure and additional degrees of freedom.

Explicit solutions for metric connections in Galilean spacetime are derived.

The physical implications of constraining temporal torsion are discussed.

Abstract

This work presents instructive, yet comprehensive derivation of quantized gravity theories in relativistic, classical, and semi-classical spacetime structure based on the Poincar\'e, Galilean, and Bargmann algebra, respectively. The technique of algebra gauging to construct the spacetime dynamics - inspired by the approach of notable previous works - is introduced to complement the standard vielbein formulation. The key characteristics and anomalies of Galilean gravity will then be analyzed: the degenerate metric structure, the additional degree of freedom in metric connection and the additional necessary conditions of Galilean invariance among others. General metric connection solution in Galilean spacetime differs fundamentally from that of general relativity; this will be thoroughly investigated and an explicit formula for such solution - equivalent to the parameterization by Hartong and Obers (2015) - shall be derived. Multiple derivations of the Bargmann algebra will be provided, together with both physical and algebraic motivation for the extended Bargmann frame bundle. Finally, the physical impact of constraining temporal torsion in classical spacetime will be discussed with emphasis on the geometrical interpretation of time foliations.