Coincident General Relativity (1710.03116v2)

Abstract: The metric-affine variational principle is applied to generate teleparallel and symmetric teleparallel theories of gravity. From the latter is discovered an exceptional class which is consistent with a vanishing affine connection. Based on this remarkable property, this work proposes a simpler geometrical formulation of General Relativity that is oblivious to the affine spacetime structure, thus fundamentally depriving gravity of any inertial character. The resulting theory is described by the Hilbert action purged from the boundary term and is more robustly underpinned by the spin-2 field theory, where an extra symmetry is now manifest, possibly related to the double copy structure of the gravity amplitudes. This construction also provides a novel starting point for modified gravity theories, and the paper presents new and simple generalisations where analytical self-accelerating cosmological solutions arise naturally in the early and late time universe.

• The paper introduces a new formulation of general relativity by nullifying the affine connection and focusing solely on non-metricity.
• It employs a metric-affine variational principle to reformulate the Hilbert action, revealing additional symmetries reminiscent of the double copy structure.
• Its framework yields self-accelerating cosmological solutions, offering potential new insights for modified gravity and quantum gravity research.

An Examination of "Coincident General Relativity"

The paper "Coincident General Relativity" authored by Jose Beltran Jimenez, Lavinia Heisenberg, and Tomi Koivisto presents a novel interpretation of General Relativity (GR) by leveraging the metric-affine variational principle to explore teleparallel and symmetric teleparallel theories of gravity. This work identifies an exceptional subclass where the affine connection can be nullified, offering a novel theoretical construct where GR is simplified by being framed independently of the affine spacetime structure. This paper not only simplifies the geometrical foundation of GR but also opens a path to developing modified gravity theories.

Mathematical Foundations and Theoretical Innovations

The authors propose a reformulation of GR where the affine connection is divorced from the gravitational interaction, challenging the conventional notion of gravity having a fundamentally inertial character. The resulting theory is distilled into the Hilbert action stripped of its boundary terms, grounded robustly on spin-2 field theory principles. This approach reveals an additional symmetry likely associated with the double copy structure observed in gravity amplitudes, hinting toward deeper symmetries in gravitational interactions.

The paper begins by addressing the teleparallel equivalent of GR by specifying the constraints ${R}^\alpha_{\phantom{\alpha}\beta\mu\nu} = 0$, simplifying the geometrical description by reducing the connection to the Weizenböck form. The authors then argue for a connection-free geometry, applying a purely metric paradigm where the non-metricity scalar captures the essence of gravitational dynamics.

Mathematical Formalism and Cosmological Applications

The paper extends its theoretical framework by introducing modifications in the gravitational Lagrangian through $f(Q)$ models, where $Q$ quantifies non-metricity. The implications of this theory are explored within a cosmological context, predicting naturally arising self-accelerating cosmological solutions in both early and late universe epochs. Specifically, the authors highlight the potential for novel inflationary scenarios and late-time cosmic acceleration, providing alternative mechanisms that align with observed cosmological data without invoking dark energy or exotic inflationary fields. Such solutions arise from the simplicity and symmetry restrictions imposed on the governing action.

Implications and Path Forward

The proposed "coincident GR" holds significant theoretical implications. First, it offers an alternative to the conventional teleparallel framework by eschewing torsion and curvature, achieving a dynamic theory expressed purely through non-metricity. This purification simplifies equations and potentially offers new ways to derive gravitational energy-momentum tensors.

Practically, the resulting framework could provide new insights into theories of quantum gravity and gauge theories of spacetime representations. Since the theory inherently simplifies the role of the connection, it facilitates a cleaner separation between physical gravitational fields and inertial frames of reference. This distinction could lead to advancements in our understanding of the gravitational energy quantification and related thermodynamic properties, such as entropy.

Looking forward, "coincident GR" could serve as a robust foundation for exploring modified gravity theories and addressing long-standing cosmological puzzles, such as the nature of dark energy and modifications of gravity at quantum scales. Future work should focus on detailed perturbative analyses of the $f(Q)$ models proposed, including addressing potential observational constraints and phenomenological insights that could refine or constrain model parameters.

In conclusion, the paper effectively challenges and extends GR by proposing a purified geometrical framework that hinges solely on non-metricity, offering a substantial step towards simplified and potentially unifying theories of gravity. The work serves as a catalyst for further explorations into the theoretical underpinnings of gravitation, potentially offering new avenues for research in both fundamental physics and cosmology.