[Paper Review] Metric-affine f(R) theories of gravity

Thomas P. Sotiriou, Stefano Liberati|Apr 3, 2006

Cosmology and Gravitation Theories6 references3 citations

TL;DR

This paper develops metric-affine f(R) gravity theories where the metric and connection are independent, allowing for torsion and non-metricity. It shows that matter with spin sources torsion, non-metricity arises only if matter action depends on the connection, and minimal coupling fails unless modified; the theory reduces to General Relativity with a cosmological constant in vacuum or electrovacuum.

ABSTRACT

General Relativity assumes that spacetime is fully described by the metric alone. An alternative is the so called Palatini formalism where the metric and the connections are taken as independent quantities. The metric-affine theory of gravity has attracted considerable attention recently, since it was shown that within this framework some cosmological models, based on some generalized gravitational actions, can account for the current accelerated expansion of the universe. However we think that metric-affine gravity deserves much more attention than that related to cosmological applications and so we consider here metric-affine gravity theories in which the gravitational action is a general function of the scalar curvature while the matter action is allowed to depend also on the connection which is not {\em a priori} symmetric. This general treatment will allow us to address several open issues such as: the relation between metric-affine $f(R)$ gravity and General Relativity (in vacuum as well as in the presence of matter), the implications of the dependence (or independence) of the matter action on the connections, the origin and role of torsion and the viability of the minimal-coupling principle.

Motivation & Objective

• To investigate metric-affine f(R) gravity as a viable alternative to General Relativity, especially in light of cosmological observations requiring dark energy and dark matter.
• To clarify the origin and role of torsion and non-metricity in metric-affine gravity, particularly how they arise from matter coupling.
• To examine the validity and limitations of the minimal coupling principle in this framework.
• To determine under what conditions the theory reduces to General Relativity with a cosmological constant.
• To assess the phenomenological viability of metric-affine f(R) gravity for addressing cosmological puzzles without exotic matter components.

Proposed method

• Treats the metric and affine connection as independent variables in the action principle, enabling a full metric-affine formalism.
• Considers a gravitational action that is an arbitrary function of the Ricci scalar R, generalizing the Einstein–Hilbert action.
• Allows the matter action to explicitly depend on the affine connection, breaking the assumption of metric compatibility and enabling non-metricity and torsion.
• Uses variational calculus to derive field equations for both metric and connection, distinguishing between symmetric and antisymmetric parts of the matter energy-momentum tensor.
• Analyzes the electromagnetic field as a test case to demonstrate the breakdown of standard minimal coupling and to propose a modified minimal coupling principle.
• Applies the formalism to perfect fluids and spinor fields to study their induced geometric structures (torsion, non-metricity).

Experimental results

Research questions

• RQ1 How does the dependence of the matter action on the affine connection affect the emergence of torsion and non-metricity in metric-affine f(R) gravity?
• RQ2 Under what conditions does metric-affine f(R) gravity reduce to General Relativity with a cosmological constant?
• RQ3 Why does the standard minimal coupling principle fail in metric-affine gravity, and how can it be reformulated?
• RQ4 What is the physical origin of torsion in this framework, and how is it related to spinor fields?
• RQ5 Can macroscopic matter like perfect fluids induce non-metricity or torsion, and what are the implications for weak-field gravity?

Key findings

• Torsion arises only when the matter energy-momentum tensor is antisymmetric, which occurs for fields with non-zero spin, such as fermions.
• Non-metricity is generated only if the matter action explicitly depends on the affine connection, and it vanishes for standard matter like scalar fields or perfect fluids.
• The theory reduces to General Relativity with a cosmological constant in vacuum and in electrovacuum, regardless of the f(R) function.
• The standard minimal coupling principle fails in metric-affine gravity, as the electromagnetic field's coupling depends on the connection, requiring a modified formulation.
• Torsion is not introduced by perfect fluids or scalar fields, confirming that only spinor-like matter sources it, supporting a physical picture where matter 'twists' spacetime.
• The presence of higher-order curvature invariants in the action is necessary to consistently include torsion without prior constraints, aligning with quantum gravity expectations.