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Extended Theories of Gravity

Salvatore Capozziello, Mariafelicia De Laurentis

TL;DR

This paper surveys Extended Theories of Gravity (ETGs) as gauge-theoretic extensions of GR designed to address infrared and ultraviolet shortcomings, including quantum gravity and cosmological puzzles. It develops the theoretical foundation by reviewing gauge symmetries, gravity as a gauge theory with curvature and torsion, and space-time deformations, then moves to concrete ETG classes such as f(R) gravity and scalar-tensor theories in both metric and Palatini formalisms. The work also discusses the quantum-field-theory perspective in curved space-time, conformal transformations, torsion, and the initial-value problem, highlighting how ETGs generate new gravitational modes, modified Newtonian limits, and potential cosmological applications. It concludes with a roadmap for future ETG research, emphasizing principled, non-trial-and-error advances over simple phenomenological fitting.

Abstract

Extended Theories of Gravity can be considered a new paradigm to cure shortcomings of General Relativity at infrared and ultraviolet scales. They are an approach that, by preserving the undoubtedly positive results of Einstein's Theory, is aimed to address conceptual and experimental problems recently emerged in Astrophysics, Cosmology and High Energy Physics. In particular, the goal is to encompass, in a self-consistent scheme, problems like Inflation, Dark Energy, Dark Matter, Large Scale Structure and, first of all, to give at least an effective description of Quantum Gravity. We review the basic principles that any gravitational theory has to follow. The geometrical interpretation is discussed in a broad perspective in order to highlight the basic assumptions of General Relativity and its possible extensions in the general framework of gauge theories. Principles of such modifications are presented, focusing on specific classes of theories like f (R)-gravity and scalar-tensor gravity in the metric and Palatini approaches. The special role of torsion is also discussed. The conceptual features of these theories are fully explored and attention is payed to the issues of dynamical and conformal equivalence between them considering also the initial value problem. A number of viability criteria are presented considering the post-Newtonian and the post-Minkowskian limits. In particular, we discuss the problems of neutrino oscillations and gravitational waves in Extended Gravity. Finally, future perspectives of Extended Gravity are considered with possibility to go beyond a trial and error approach.

Extended Theories of Gravity

TL;DR

This paper surveys Extended Theories of Gravity (ETGs) as gauge-theoretic extensions of GR designed to address infrared and ultraviolet shortcomings, including quantum gravity and cosmological puzzles. It develops the theoretical foundation by reviewing gauge symmetries, gravity as a gauge theory with curvature and torsion, and space-time deformations, then moves to concrete ETG classes such as f(R) gravity and scalar-tensor theories in both metric and Palatini formalisms. The work also discusses the quantum-field-theory perspective in curved space-time, conformal transformations, torsion, and the initial-value problem, highlighting how ETGs generate new gravitational modes, modified Newtonian limits, and potential cosmological applications. It concludes with a roadmap for future ETG research, emphasizing principled, non-trial-and-error advances over simple phenomenological fitting.

Abstract

Extended Theories of Gravity can be considered a new paradigm to cure shortcomings of General Relativity at infrared and ultraviolet scales. They are an approach that, by preserving the undoubtedly positive results of Einstein's Theory, is aimed to address conceptual and experimental problems recently emerged in Astrophysics, Cosmology and High Energy Physics. In particular, the goal is to encompass, in a self-consistent scheme, problems like Inflation, Dark Energy, Dark Matter, Large Scale Structure and, first of all, to give at least an effective description of Quantum Gravity. We review the basic principles that any gravitational theory has to follow. The geometrical interpretation is discussed in a broad perspective in order to highlight the basic assumptions of General Relativity and its possible extensions in the general framework of gauge theories. Principles of such modifications are presented, focusing on specific classes of theories like f (R)-gravity and scalar-tensor gravity in the metric and Palatini approaches. The special role of torsion is also discussed. The conceptual features of these theories are fully explored and attention is payed to the issues of dynamical and conformal equivalence between them considering also the initial value problem. A number of viability criteria are presented considering the post-Newtonian and the post-Minkowskian limits. In particular, we discuss the problems of neutrino oscillations and gravitational waves in Extended Gravity. Finally, future perspectives of Extended Gravity are considered with possibility to go beyond a trial and error approach.

Paper Structure

This paper contains 106 sections, 2 theorems, 1114 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A short history of Theories of Gravity
  3. What we mean for a "Good Theory of Gravity"
  4. General Relativity and its shortcomings
  5. UV scales: the Quantum Gravity Problem
  6. Issues and shortcomings in Quantum Gravity
  7. IR scales: Dark Energy and Dark Matter
  8. A summary of gauge symmetries
  9. What is a gauge symmetry?
  10. The Einstein connection
  11. The Weyl Gauge
  12. Electromagnetism as a Gauge Theory
  13. The Yang-Mills Gauge
  14. Geometry and Gauge
  15. Local gauge transformations
  16. ...and 91 more sections

Key Result

Proposition 12.1

If a generalized infinitesimal Lagrangian symmetry $Z$ is a $\cal J$-prolongation, then it is a Noether vector field.

Figures (3)

  • Figure 1: The six polarization modes of gravitational waves. We illustrate the displacement induced at phases spaced by $\pi$ radians by each mode on a circle of test particles at rest before the wave impinges upon them. The wave propagates out of the plane of the page in (a), (b), and (c) and into this plane in (d), (e), and (f). While (a) and (b) describe the "plus" and "cross" modes, respectively, (c) corresponds to the scalar mode, and (d), (e), and (f) to the D, B, and C modes.
  • Figure 2: The coordinate systems used to calculate the polarization tensors and a view of the coordinate transformation.
  • Figure 3: Angular pattern functions of an interferometric detector for the various polarizations. From left to right and from top to bottom, one sees constant level surfaces corresponding to the "plus", "cross", B, C, D, and scalar modes.

Theorems & Definitions (4)

  • Definition 12.1
  • Definition 12.2
  • Proposition 12.1
  • Proposition 12.2