[Paper Review] Gauge theory of gravity: Electrically charged solutions within the metric--affine framework

Friedrich W. Hehl, J. Socorro|ArXiv.org|Mar 10, 1998

Cosmology and Gravitation Theories19 references4 citations

TL;DR

This paper presents electrically charged exact solutions in a metric-affine gravity framework, extending the Plebański–Demiański metric to include nonmetricity and torsion via three covectors representing dilation, shear, and spin charges. The key contribution is a class of solutions that generalize Einstein-Maxwell theory by incorporating gauge-theoretic gravity with non-Riemannian structures, offering a richer geometric description of charged gravitational systems.

ABSTRACT

We find a class of electrically charged exact solutions for a toy model of metric-affine gravity. Their metric is of the Plebański-Demiański type and their nonmetricity and torsion are represented by a triplet of covectors with dilation, shear, and spin charges.

Motivation & Objective

• To extend metric-affine gravity by incorporating electromagnetic coupling in a gauge-theoretic framework.
• To construct exact solutions for charged gravitational systems beyond general relativity's Riemannian geometry.
• To model nonmetricity and torsion as gauge-like fields using three covectors representing physical charges.
• To explore the interplay between electromagnetic fields and spacetime geometry in a non-Riemannian setting.
• To provide a toy model for unified gravity-electromagnetism within a metric-affine formalism.

Proposed method

• Adopt a metric-affine gravity framework where the metric and connection are independent variables.
• Introduce a gauge-theoretic formulation of gravity analogous to Yang-Mills theories, treating affine connection as a gauge potential.
• Use a triplet of covectors to represent nonmetricity and torsion, encoding dilation, shear, and spin charges.
• Solve the field equations derived from a Lagrangian that includes curvature, nonmetricity, torsion, and electromagnetic terms.
• Employ the Plebański–Demiański metric as the background geometry, generalizing Kerr-Newman-type solutions.
• Apply symmetry assumptions to reduce the field equations to a tractable system of ODEs or PDEs for the covector fields.

Experimental results

Research questions

• RQ1 How can electromagnetic charge be consistently coupled to gravity in a non-Riemannian spacetime framework?
• RQ2 What are the exact solutions for charged systems in metric-affine gravity with nonmetricity and torsion?
• RQ3 How do dilation, shear, and spin charges manifest in the geometry of charged gravitational fields?
• RQ4 Can the Plebański–Demiański metric be extended to include gauge-like gravitational degrees of freedom?
• RQ5 What is the role of the affine connection and nonmetricity in modifying the dynamics of electrically charged spacetimes?

Key findings

• The paper derives a class of exact solutions for electrically charged systems in metric-affine gravity with nonmetricity and torsion.
• The solutions are based on the Plebański–Demiański metric, which generalizes Kerr–Newman spacetimes to include cosmological and kinematic parameters.
• Nonmetricity and torsion are fully described by a triplet of covectors corresponding to dilation, shear, and spin charges.
• The field equations admit consistent solutions where the electromagnetic field is coupled to the affine connection and metric structure.
• The model provides a toy framework for unifying gravity and electromagnetism via gauge-theoretic geometry, though not yet a full quantum theory.
• The solutions are valid in a classical, effective field theory regime, with no backreaction from quantum effects considered.