[Paper Review] Teleparallel Gravity, Covariance and Their Geometrical Meaning

Martin Krššák|arXiv (Cornell University)|Jan 16, 2024

Relativity and Gravitational Theory4 citations

TL;DR

This paper argues that pure gauge connections are essential for defining teleparallel geometries and enabling covariance in gravity theories based on torsion and non-metricity. It shows that teleparallel frameworks naturally provide a covariant formulation for the Einstein and Møller Lagrangians, and that modified teleparallel theories in the Weitzenböck and coincident gauges are equivalent to Riemannian-based $f(\mathbf{\Omega})$ and $f(\mathbf{G})$ theories, with no dynamical difference in degrees of freedom when covariance is properly maintained.

ABSTRACT

We explore the geometrical meaning of teleparallel geometries and the role of covariance in their definition. We argue that pure gauge connections are a necessary ingredient for describing geometry and gravity in terms of torsion and non-metricity. We show the other viable alternative is using the Einstein and Moller Lagrangians, but these are defined through the Riemannian connection coefficients and hence do not involve torsion nor non-metricity. We argue that the teleparallel geometries can be defined on the manifold without introducing any additional structures and that they naturally provide the covariant framework for the Einstein and Moller Lagrangians. We explore some consequences of this viewpoint for the modified theories of gravity as well.

Motivation & Objective

• To resolve the foundational debate on whether teleparallel connections are fundamental or merely gauge artifacts in gravity theories.
• To clarify the role of pure gauge connections in attributing non-trivial geometric meaning to torsion and non-metricity.
• To demonstrate that teleparallel geometries provide a natural covariant framework for the Einstein and Møller Lagrangians.
• To show that modified teleparallel theories in Weitzenböck and coincident gauges are equivalent to Riemannian $f(\mathbf{\Omega})$ and $f(\mathbf{G})$ theories.
• To argue that the number of physical degrees of freedom remains unchanged in the covariant formulation of teleparallel gravity.

Proposed method

• Using an extended notation to distinguish between geometric quantities associated with general metric-affine, Riemannian, teleparallel, and symmetric teleparallel connections.
• Analyzing the geometric meaning of teleparallelism through the lens of metric-affine geometry, identifying pure gauge connections as fundamental for non-trivial torsion and non-metricity.
• Comparing two views: (1) teleparallel connections as fundamental gauge fields, and (2) gauge-fixed geometries using anholonomy coefficients and partial derivatives.
• Demonstrating that the Einstein and Møller Lagrangians, though defined via Riemannian connection coefficients, are fully consistent and equivalent when framed in teleparallel geometry.
• Applying the Stückelberg trick to restore covariance and showing that field equations in the covariant formulation determine a preferred basis.
• Using perturbative analysis to compare degrees of freedom in $f(\mathbf{\Omega})$, $f(\mathbf{G})$, $f(\mathbf{\tilde{T}})$, and $f(\mathbf{\tilde{Q}})$ theories, showing equivalence in propagating modes.

Experimental results

Research questions

• RQ1 Can teleparallel geometries be defined independently of additional structures, and do they naturally support covariance?
• RQ2 Why is the pure gauge nature of teleparallel connections central to attributing geometric meaning to torsion and non-metricity?
• RQ3 Are the Einstein and Møller Lagrangians truly distinct from teleparallel formulations, or are they equivalent under covariance?
• RQ4 Do modified teleparallel theories in Weitzenböck and coincident gauges differ dynamically from their Riemannian counterparts?
• RQ5 Is the number of physical degrees of freedom preserved when moving from gauge-fixed to covariant formulations of teleparallel gravity?

Key findings

• Pure gauge connections are necessary to assign non-trivial geometric meaning to torsion and non-metricity, as these quantities are not tensors without them.
• Teleparallel geometries provide a natural covariant framework for the Einstein and Møller Lagrangians, which are otherwise defined using Riemannian connection coefficients.
• The Lagrangians $f(\mathbf{\tilde{T}})$ and $f(\mathbf{\tilde{Q}})$ in Weitzenböck and coincident gauges are equivalent to $f(\mathbf{\Omega})$ and $f(\mathbf{G})$ theories in Riemannian geometry, respectively.
• The field equations in the covariant formulation determine a preferred basis, and once fixed, the theory reduces to its Riemannian analog.
• The number of propagating degrees of freedom in $f(\mathbf{\tilde{T}})$ gravity remains five, consistent with the standard result, and is unchanged in the covariant formulation.
• The antisymmetric tetrad perturbations and spin connection contributions combine into a single effective degree of freedom, preserving the physical content of the theory.