Comparing Equivalent Gravities: common features and differences (2208.03011v2)

Abstract: We discuss equivalent representations of gravity in the framework of metric-affine geometries pointing out basic concepts from where these theories stem out. In particular, we take into account tetrads and spin connection to describe the so called {\it Geometric Trinity of Gravity}. Specifically, we consider General Relativity, constructed upon the metric tensor and based on the curvature $R$; Teleparallel Equivalent of General Relativity, formulated in terms of torsion $T$ and relying on tetrads and spin connection; Symmetric Teleparallel Equivalent of General Relativity, built up on non-metricity $Q$, constructed from metric tensor and affine connection. General Relativity is formulated as a geometric theory of gravity based on metric, whereas teleparallel approaches configure as gauge theories, where gauge choices permit not only to simplify calculations, but also to give deep insight into the basic concepts of gravitational field. In particular, we point out how foundation principles of General Relativity (i.e the Equivalence Principle and the General Covariance) can be seen from the teleparallel point of view. These theories are dynamically equivalent and this feature can be demonstrated under three different standards: (1) the variational method; (2) the field equations; (3) the solutions.

• The paper shows that the actions of GR, TEGR, and STEGR differ only by boundary terms, yielding equivalent field equations.
• It employs variational methods and spherically symmetric solutions to recover the Schwarzschild metric and confirm Birkhoff’s theorem.
• The study implies that alternative geometric formulations can deepen our understanding of gravity and inspire novel approaches in cosmology and quantum gravity.

Overview of "Comparing Equivalent Gravities: Common Features and Differences"

The paper "Comparing Equivalent Gravities: Common Features and Differences," authored by Salvatore Capozziello, Vittorio De Falco, and Carmen Ferrara, presents a detailed exploration of three dynamically equivalent formulations of gravity within the framework of metric-affine geometries. This exploration is centered around the so-called "Geometric Trinity of Gravity," which includes General Relativity (GR), the Teleparallel Equivalent of General Relativity (TEGR), and the Symmetric Teleparallel Equivalent of General Relativity (STEGR). Each formulation uses different geometric invariants, namely curvature $R$, torsion $T$, and non-metricity $Q$.

Key Concepts and Formulations

• General Relativity (GR): Based on the metric tensor and the curvature scalar $R$, GR describes gravity as the curvature of spacetime. It is founded on principles like General Covariance and the Equivalence Principle. The geometric setup involves the Levi-Civita connection, ensuring metric compatibility and a torsion-free space.
• Teleparallel Equivalent of General Relativity (TEGR): This approach uses torsion $T$ instead of curvature to describe gravitational interactions. The tetrads and a Weitzenböck connection are central to TEGR. It frames gravity as a force similar to gauge theories, with the torsion tensor capturing the gravitational interactions.
• Symmetric Teleparallel Equivalent of General Relativity (STEGR): Unlike GR and TEGR, STEGR attributes gravity to non-metricity $Q$ while maintaining both zero curvature and zero torsion. This approach treats the metric tensor and the affine connection as independent entities, emphasizing the Palatini formulation where the connection emerges as a dynamic variable. The non-metricity tensor modifies the length of vectors, offering a novel geometric interpretation of gravity.

Numerical Results and Analysis

The paper rigorously demonstrates the dynamical equivalence of these gravity formulations through three standards: variational methods, field equations, and solution comparisons. At the variational level, the paper illustrates that the actions for GR, TEGR, and STEGR differ by boundary terms, yielding identical field equations upon variation. The methodology includes starting from second Bianchi identities and utilizing them to recover the field equations characteristic of each formulation.

When examining solutions, the authors focus on spherically symmetric cases, recovering the Schwarzschild metric and confirming Birkhoff's theorem across the three formulations, thereby underscoring their physical and mathematical equivalence in vacuum scenarios.

Implications and Future Directions

The implications of these equivalent formulations are profound for theoretical physics and cosmology. They challenge the uniqueness of GR as the sole geometric description of gravity under Einstein's original principles. Instead, they propose alternative geometrical and algebraic bases that could enrich our understanding of gravity and potentially highlight novel features when extending beyond classical GR.

For future developments, investigating these approaches' roles in explaining the enigmatic features of the universe, such as dark matter and dark energy, could be promising. Furthermore, these frameworks offer fertile ground for embedding quantum aspects within gravitational theories, particularly as they allow for a broader interpretation of fundamental symmetries and invariants. The potential loss of equivalence in $f(R)$, $f(T)$, and $f(Q)$ models also invites deeper analysis on how additional degrees of freedom may lead to detectable deviations from standard GR predictions.

In conclusion, the paper provides a comprehensive examination of equivalent gravity theories through the lens of metric-affine geometry, offering insights that could pave the way for future explorations in both classical and quantum gravity contexts.