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A Unified Approach to Geometric Modifications of Gravity

Erik Jensko

Abstract

This thesis studies modified theories of gravity from a geometric viewpoint. We review the motivations for considering alternatives to General Relativity and cover the mathematical foundations of gravitational theories in Riemannian and non-Riemannian geometries. Then, starting from the decomposition of the Einstein-Hilbert action into bulk and boundary terms, we construct new modifications of General Relativity. These modifications break diffeomorphism invariance or local Lorentz invariance, allowing one to bypass Lovelock's theorem while remaining second-order and without introducing additional fields. In the metric-affine framework, we introduce a new Einstein-Cartan-type theory with propagating torsion. Important comparisons are made with the modified teleparallel theories, and we construct a unified framework encompassing all these theories. The equivalence between theories that break fundamental symmetries in the Riemannian setting and non-Riemannian theories of gravity is explored in detail. This leads to a dual interpretation of teleparallel gravity, one in terms of geometric quantities and the other in terms of non-covariant objects. We then study the cosmological applications of these modified theories, making use of dynamical systems techniques. One key result is that the modified Einstein-Cartan theories can drive inflation in the early universe, replacing the initial cosmological singularity of General Relativity. To conclude, we discuss the viability of these modifications and possible future directions, examining their significance and relevance to the broader field of gravitational physics.

A Unified Approach to Geometric Modifications of Gravity

Abstract

This thesis studies modified theories of gravity from a geometric viewpoint. We review the motivations for considering alternatives to General Relativity and cover the mathematical foundations of gravitational theories in Riemannian and non-Riemannian geometries. Then, starting from the decomposition of the Einstein-Hilbert action into bulk and boundary terms, we construct new modifications of General Relativity. These modifications break diffeomorphism invariance or local Lorentz invariance, allowing one to bypass Lovelock's theorem while remaining second-order and without introducing additional fields. In the metric-affine framework, we introduce a new Einstein-Cartan-type theory with propagating torsion. Important comparisons are made with the modified teleparallel theories, and we construct a unified framework encompassing all these theories. The equivalence between theories that break fundamental symmetries in the Riemannian setting and non-Riemannian theories of gravity is explored in detail. This leads to a dual interpretation of teleparallel gravity, one in terms of geometric quantities and the other in terms of non-covariant objects. We then study the cosmological applications of these modified theories, making use of dynamical systems techniques. One key result is that the modified Einstein-Cartan theories can drive inflation in the early universe, replacing the initial cosmological singularity of General Relativity. To conclude, we discuss the viability of these modifications and possible future directions, examining their significance and relevance to the broader field of gravitational physics.
Paper Structure (128 sections, 1 theorem, 684 equations, 18 figures, 3 tables)

This paper contains 128 sections, 1 theorem, 684 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Why modify General Relativity?
  3. Observational challenges
  4. Theoretical challenges
  5. Quantum gravity
  6. The cosmological constant problem
  7. Theories beyond General Relativity
  8. Invariants
  9. Additional fields
  10. Non-Riemannian geometry
  11. Symmetry breaking
  12. Outline
  13. Geometric foundations of gravity
  14. Differential geometry and mathematical preliminaries
  15. Metric tensor
  16. ...and 113 more sections

Key Result

Theorem 1

The only local, divergence-free tensor constructed from the metric and its first two derivatives in four dimensions is where $G^{\mu \nu} \equiv R^{\mu \nu} - \frac{1}{2} R g^{\mu \nu}$ is the Einstein tensor and $\alpha$ and $\beta$ are constants.

Figures (18)

  • Figure 1: Torsion as the closure of infinitesimal parallelograms.
  • Figure 2: Lengths of parallel transported vectors changing due to non-metricity.
  • Figure 3: Parallel transported vectors changing around a closed loop by an angle $\alpha$ due to curvature.
  • Figure 4: Diffeomorphism from spacetime manifold $\mathcal{M}$ to itself. Transformations (a) and (b) lead to the standard coordinate transformation law $\hat{T}(\hat{x})$ (\ref{['Tensor transformation']}) whilst (c) leads to the Lie derivative $T(\hat{x})$ (\ref{['Lie derivative']}).
  • Figure 5: Relationship between different geometries. Metric-affine is the most general, from which we obtain Riemann-Cartan, Weyl or teleparallel by imposing vanishing non-metricity, torsion or curvature respectively. Riemann, symmetric teleparallel and metric teleparallel geometries possess only curvature, non-metricity or torsion respectively.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Theorem 1: Lovelock's Theorem