Teleparallel Palatini theories (1803.10185v1)

Abstract: The Palatini formalism is developed for gravitational theories in flat geometries. We focus on two particularly interesting scenarios. First, we fix the connection to be metric compatible, but we follow a completely covariant approach by imposing the constraints with suitable Lagrange multipliers. For a general quadratic theory we show how torsion naturally propagates and we reproduce the Teleparallel Equivalent of General Relativity as a particular quadratic action that features an additional Lorentz symmetry. We then study the much less explored theories formulated in a geometry with neither curvature nor torsion, so that all the geometrical information is encoded in the non-metricity. We discuss how this geometrical framework leads to a purely inertial connection that can thus be completely removed by a coordinate gauge choice, the coincident gauge. From the quadratic theory we recover a simpler formulation of General Relativity in the form of the Einstein action, which enjoys an enhanced symmetry that reduces to a second linearised diffeomorphism at linear order. More general theories in both geometries can be formulated consistently by taking into account the inertial connection and the associated additional degrees of freedom. As immediate applications, the new cosmological equations and their Newtonian limit are considered, where the role of the lapse in the consistency of the equations is clarified, and the Schwarzschild black hole entropy is computed by evaluating the corresponding Euclidean action. We discuss how the boundary terms in the usual formulation of General Relativity are related to different choices of coordinates in its coincident version and show that in isotropic coordinates the Euclidean action is finite without the need to introduce boundary or normalisation terms.

• The paper presents a dual formulation of teleparallel gravity, distinguishing between metric-compatible and symmetric (non-metric) geometries using the Palatini approach.
• It employs Lagrange multipliers to enforce teleparallel conditions, reproducing the Teleparallel Equivalent of General Relativity and simplifying cosmological equations via the coincident gauge.
• The study offers new insights into black hole entropy and explores potential double-copy structures, hinting at deeper connections between gravity and gauge theories.

An Overview of Teleparallel Palatini Theories

The paper "Teleparallel Palatini Theories" presents a comprehensive analysis of gravitational theories in flat geometries under the Palatini formalism. The paper focuses on two distinct scenarios: metric compatible teleparallel geometry, akin to usual teleparallel theories, and geometries devoid of curvature and torsion, with all information encoded in non-metricity.

Teleparallelism in the Palatini Formalism

The Palatini formalism treats the metric and affine connection as independent entities, allowing a novel approach to gravitational theories. Teleparallelism under this framework is intriguing because it employs constraints, enforced through Lagrange multipliers, to maintain teleparallelism and metric compatibility.

In one scenario, the connection is set to be metric compatible, which resonates with standard teleparallel theories. Here, torsion naturally propagates, reproducing the Teleparallel Equivalent of General Relativity (TEGR) when formulated under a quadratic action featuring additional Lorentz symmetry.

The theory further extends to cases where geometrical information is conveyed solely through non-metricity. In symmetric teleparallel geometry, devoid of curvature and torsion, the connection can be entirely removed by using a coordinate gauge choice known as the coincident gauge. This approach simplifies General Relativity into a formulation akin to the Einstein action, enhanced by symmetry that reduces to linearized diffeomorphism at the linear level.

Theoretical Implications and Cosmological Applications

The research paves the way for more general theories that maintain internal consistency by considering inertial connections and associated degrees of freedom. Immediate applications of these theories are explored in cosmology, particularly in deriving new cosmological equations and understanding their Newtonian limits. The role of lapse functions in ensuring equation consistency is clarified, highlighting the dynamic nature of lapse in teleparallel cosmologies—a deviation from the standard static interpretation in GR.

Additionally, the paper considers black hole entropy, computationally evaluating it via corresponding Euclidean actions. This reveals insightful connections between boundary terms and coordinate choices, particularly in symmetric teleparallel gravity, where isotropic coordinates offer a finite Euclidean action without necessitating boundary or normalization terms.

Future Developments and Speculations

The paper explores the double-copy structure of gravity amplitudes and bootstrapping of gravity, proposing potential methods for elucidating connections between gauge theories and gravity via the double copy method. These approaches suggest new avenues for understanding gravitational interactions through analogy with Yang-Mills theories.

Conclusion

The paper contributes significantly to the theoretical landscape of gravitational physics by exploring teleparallel geometries through the Palatini formalism. It offers an innovative perspective on encoding gravitational dynamics, emphasizing the versatility and compatibility of teleparallel approaches with established theories like GR. The symmetric teleparallel formulation, particularly, opens doors to novel applications and potential refinements in cosmological modeling and fundamental physics. The robust mathematical treatment and thoughtful conclusions illuminate pathways for future research in teleparallel gravity and its possible synthesis with modern theoretical frameworks.