Linear Transformations on Affine-Connections

Abstract

We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its $\Gamma$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $\Gamma$-variation of the functional respects a certain symmetry then the functional is invariant under the associated transformation of the affine connection. We then apply our results in Metric-Affine Gravity and produce invariant actions under certain transformations of the affine connection. Finally, we derive the constraints put on the hypermomentum for such invariant Theories.

arXiv comment8 pages, Applications section added

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• Teleparallel Palatini theories
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• Parity Violating Metric-Affine Gravity Theories
• Radiatively stable ghost and tachyon freedom in Metric Affine Gravity
• Super-entropic black hole with Immirzi hair
• Generalized Ashtekar variables for Palatini f(R) models
• Projective transformations in metric-affine and Weylian geometries
• Metric-Affine Vector-Tensor Correspondence and Implications in $F(R,T,Q,\mathcal{T},\mathcal{D})$ gravity
• Quadratic Metric-Affine Gravity: Solving for the Affine-Connection
• (3+1)-Formulation for Gravity with Torsion and Non-Metricity: The Stress-Energy-Momentum Equation
• (3+1)-Formulation for Gravity with Torsion and Non-Metricity II: The Hypermomentum Equation
• The Ternary Structure of Lie Algebroid Connections
• Gravitational waves in Palatini gravity for a non-minimal geometry-matter coupling
• The role of longitudinal polarizations in Horndeski and macroscopic gravity: Introducing gravitational plasmas