[Paper Review] Metric-Affine Gravity and Cosmology/Aspects of Torsion and non-Metricity in Gravity Theories

Damianos Iosifidis|arXiv (Cornell University)|Feb 25, 2019

Cosmology and Gravitation TheoriesPhysics and Astronomy105 references37 citations

TL;DR

This PhD thesis investigates metric-affine gravity, focusing on torsion and non-metricity, and develops methods to solve for the affine connection, explores cosmological implications, and analyzes scale transformations and invariances.

ABSTRACT

This Thesis is devoted to the study of Metric-Affine Theories of Gravity and Applications to Cosmology. The thesis is organized as follows. In the first Chapter we define the various geometrical quantities that characterize a non-Riemannian geometry. In the second Chapter we explore the MAG model building. In Chapter 3 we use a well known procedure to excite torsional degrees of freedom by coupling surface terms to scalars. Then, in Chapter 4 which seems to be the most important Chapter of the thesis, at least with regards to its use in applications, we present a step by step way to solve for the affine connection in non-Riemannian geometries, for the first time in the literature. A peculiar f(R) case is studied in Chapter 5. This is the conformally (as well as projective invariant) invariant theory f(R)=a R^{2} which contains an undetermined scalar degree of freedom. We then turn our attention to Cosmology with torsion and non-metricity (Chapter 6). In Chapter 7, we formulate the necessary setup for the $1+3$ splitting of the generalized spacetime. Having clarified the subtle points (that generally stem from non-metricity) in the aforementioned formulation we carefully derive the generalized Raychaudhuri equation in the presence of both torsion and non-metricity (along with curvature). This, as it stands, is the most general form of the Raychaudhuri equation that exists in the literature. We close this Thesis by considering three possible scale transformations that one can consider in Metric-Affine Geometry.

Motivation & Objective

• Introduce and motivate generalized non-Riemannian geometry in Metric-Affine Gravity (MAG) and its physical significance.
• Provide a systematic method to determine the affine connection for MAG theories and classify when such theories reproduce Einstein gravity in vacuum.
• Investigate cosmological implications of torsion and non-metricity, including modified Friedmann equations and Raychaudhuri equation.
• Extend MAG to include scale transformations and construct invariant quadratic actions incorporating torsion and non-metricity.

Proposed method

• Present a comprehensive review of MAG geometry and variational formalisms using coordinate and differential-form languages.
• Propose a novel approach to breaking projective invariance in metric-affine f(R) theories by treating torsion and non-metricity vectors on equal footing.
• Derive a general procedure (three theorems) to solve for the affine connection in MAG and illustrate with explicit examples.
• Study the duality between torsion and non-metricity in f(R) gravity, and analyze cosmological solutions in specially chosen MAG models.
• Derive the most general Raychaudhuri equation in spaces with torsion and non-metricity, and apply it to cosmology.
• Examine scale transformations (conformal, projective, frame rescaling) and construct invariant theories under these transformations.

Experimental results

Research questions

• RQ1 How can the affine connection be solved for in general Metric-Affine Gravity theories?
• RQ2 Under what conditions do MAG theories reduce to Einstein gravity in vacuum or reproduce familiar GR results?
• RQ3 How do torsion and non-metricity modify cosmological dynamics, such as Friedmann-like equations and Raychaudhuri equation?
• RQ4 What are the effects and classifications of scale transformations (conformal, projective, frame rescaling) on MAG actions?
• RQ5 Can torsion and non-metricity be excited or mapped onto each other in specific MAG models (duality)?

Key findings

• A detailed exposition of generalized geometry clarifies the roles of torsion and non-metricity with illustrative examples.
• A three-theorem framework is developed to exactly solve for the affine connection in MAG for selected theories.
• A classification is provided for theories that yield Einstein gravity in vacuum within MAG.
• A cosmological analysis yields generalized Friedmann equations and shows torsion/non-metricity influence on cosmology, including dualities in specific models.
• The generalized Raychaudhuri equation in spaces with torsion and non-metricity is derived and applied to cosmology.
• Scale transformations are analyzed, and conditions for conformal, projective, and frame-rescaling invariances are established for quadratic MAG actions.