Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology, and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories that include higher-order curvature invariants, and specifically the particular class of $f(R)$ theories, have a long history. In the last five years there has been a new stimulus for their study, leading to a number of interesting results. Here $f(R)$ theories of gravity are reviewed in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature. All known formalisms are presented---metric, Palatini, and metric affine---and the following topics are discussed: motivation; actions, field equations, and theoretical aspects; equivalence with other theories; cosmological aspects and constraints; viability criteria; and astrophysical applications.

f ( R ) theories of gravity

Modified Gravity and Cosmology

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

Extended Theories of Gravity

A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the ${U}_{4}$ theory arises as a local gauge theory for the Poincar\'e group in space-time. The deviations of the ${U}_{4}$ theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.

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General Relativity with Spin and Torsion: Foundations and Prospects

The field equations of general relativity with spin and torsion (${U}_{4}$ theory) are considered to describe correctly the gravitational properties of matter on a microphysical level. By an averaging procedure one arrives at a macroscopic field equation, which under normal matter densities coincides with Einstein's equation of conventional general relativity. For very high matter densities, even if the spin are randomly distributed, Einstein's equation breaks down and ${U}_{4}$ theory must be applied. It is shown how the singularity theorems of Penrose and Hawking must be modified to apply in ${U}_{4}$ theory. All known cosmological models in ${U}_{4}$ theory which prevent singularities are shown to violate an energy condition of a singularity theorem.

General relativity with spin and torsion and its deviations from einstein's theory

Within the confines of conventional general relativity, variational principles are analyzed in which the metric tensor and the asymmetric linear connection are varied independently. The constraint that space-time remain Riemannian is introduced by means of the Lagrange multiplier technique. The Lagrange multiplier which effects this constraint, the hypermomentum current, is closely related to the constraint “force” which keeps space-time Riemannian and should be a measure for the violation of the Riemannian constraint at the microscopic level.

Metric-affine variational principles in general relativity. I. Riemannian space-time

Abstract In this series of notes, we introduce a new quantity into the theory of classical matter fields. Besides the usual energy-momentum tensor, we postulate the existence of a further dynamical attribute of matter, the 3rd rank tensor ⊿ijk of hypermomentum. Subsequently, a general relativistic field theory of energy-momentum and hypermomentum is outlined. In Part I we motivate the need for hypermomentum. We split it into spin angular momentum, the dilatation hypermomentum, and traceless proper hypermomentum and discuss their physical meanings and conservation laws.

On Hypermomentum in General Relativity I. The Notion of Hypermomentum

Abstract In Part I1 of this series we presented the notion of the material hypermomentum current and motivated its introduction into general relativity. In Part II2 we showed that a general, linearly connected manifold with symmetric metric (L4, g) is the appropriate geometrical framework for such an introduction. The present paper completes the picture by giving dynamical definitions for energy-momentum and hypermomentum for a minimally coupled material Lagrangian. We derive and discuss the field equations of a new metricaffine gravitational theory which embodies these notions.

G. David Kerlick

Papers

General Relativity with Spin and Torsion: Foundations and Prospects

General relativity with spin and torsion and its deviations from einstein's theory

Metric-affine variational principles in general relativity. I. Riemannian space-time

On Hypermomentum in General Relativity I. The Notion of Hypermomentum

On hypermomentum in general relativity. III - Coupling hypermomentum to geometry