The Geometrical Trinity of Gravity (1903.06830v1)

Abstract: The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the spacetime curvature, which is now the standard interpretation of gravity. However, this is not the only possibility. In these notes we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory. Based on these three alternative formulations of General Relativity, we then discuss some extensions.

• The paper presents three equivalent formulations of gravity by attributing its effects to curvature, torsion, and non-metricity.
• It contrasts the Einstein-Hilbert action with the approaches in TEGR and STEGR to demonstrate their dynamic equivalence.
• The study opens avenues for modified gravity theories and empirical tests relevant to dark energy and quantum gravity research.

The Geometrical Trinity of Gravity: An Academic Overview

The field of gravitational theory is continuously evolving, with researchers exploring different formulations and interpretations of fundamental concepts. The paper "The Geometrical Trinity of Gravity" by Jose Beltran Jimenez, Lavinia Heisenberg, and Tomi S. Koivisto contributes to this sphere by examining the framework of General Relativity (GR) and proposing three equivalent representations, each offering a unique geometrical perspective. The authors challenge the traditional approach to GR by presenting alternative formulations that ascribe gravitational interactions to curvature, torsion, and non-metricity, thus providing a broad spectrum of theoretical analysis.

Overview of the Three Formulations

1. Curvature-Based General Relativity (GR): The classical formulation of GR as envisioned by Einstein is based on the equivalence principle, which associates gravity with spacetime curvature. It entails a spacetime metric described by the metric tensor $g_{\mu\nu}$ and the Levi-Civita connection, which is symmetric and metric compatible. The dynamics in this framework are governed by the Einstein-Hilbert action, with its foundations deeply rooted in the notion that the universe is shaped by the curvature caused by mass-energy distribution.
2. Teleparallel Equivalent of General Relativity (TEGR): This formulation describes gravity through torsion instead of curvature. The absence of curvature implies a flat spacetime, with gravity encoded in the torsion tensor. The authors explore the Weitzenböck connection, a specific teleparallel geometry where torsion replaces curvature. The TEGR formulation remains dynamically equivalent to GR, offering a different perspective by employing a purely inertial connection with additional local Lorentz symmetries.
3. Symmetric Teleparallel Equivalent of General Relativity (STEGR): Proposed as another alternative, this formulation removes curvature and torsion, attributing gravity to non-metricity. The affine structure of spacetime is manipulated through the connection that retains flatness and vanishes in a suitably chosen coordinate system, the "coincident gauge." This approach is notable for its simplicity, as it describes gravity in a trivially connected spacetime, preserving the concept of non-metricity while achieving equivalency with GR.

Implications and Extensions

The paper discusses not only the formulations themselves but the potential extensions and implications of each. In revisiting GR's foundational framework, the authors open avenues for exploring modifications to GR that endeavor to address issues such as dark energy and quantum gravity. These include:

• Newer General Relativity and $f(T)$ Theories: By extending the TEGR with non-linear and higher-order derivatives, $f(T)$ theories introduce additional degrees of freedom while seeking to remain consistent with observational data.
• Newer Developments in $f(Q)$ Theories: These theories extend the symmetric teleparallel approach by adding functional dependencies on non-metricity scalars, potentially offering new insights into cosmology and the early universe.
• Matter Couplings in Various Geometries: Each formulation presents unique challenges and possibilities in coupling fermions and bosons, revealing intriguing physical effects. For instance, the non-minimal coupling of fermions to torsion in TEGR and how bosons remain unaffected by non-metricity in STEGR.

Theoretical and Practical Implications

The paper underscores the versatility and robustness of GR as it transitions through varied geometric interpretations. The consistency of physical outcomes across these frameworks reinforces the theoretical underpinnings of GR. Moreover, the potential for novel gravitational phenomena in these alternative representations offers a fertile ground for further research. By furnishing new mathematical techniques and theoretical insights, these frameworks could facilitate progress in areas such as quantum gravity and cosmological models.

The authors' work in "The Geometrical Trinity of Gravity" lays out a comprehensive groundwork for future research. It invites exploration into the empirical verifications of these frameworks and encourages cross-disciplinary dialogue that may bridge existing gaps between geometric theories and quantum field methods. As researchers build upon these foundations, further investigations may yield transformative insights into the fundamental nature of gravity, space, and time.