Overview
This research examines the applicability and consequences of Jacobson's thermodynamic approach to classical gravity when extended to non-Riemannian geometries. Jacobson's original formulation posits that gravitational field equations arise from the first law of thermodynamics as applied to a Rindler observer within Riemannian geometries. This study explores the implications of this approach in gravitational theories characterized by non-Riemannian structures.
Research Context
Three decades ago, Ted Jacobson proposed an approach to classical gravity where the gravitational field equations were presented as a consequence of the first law of thermodynamics for a Rindler observer. This formulation was developed within the framework of Riemannian geometries. The current investigation extends this conceptual framework to non-Riemannian geometries to understand the resulting gravitational theories.
Approach
The research first explored the implications of applying Jacobson's thermodynamic approach to non-Riemannian geometries. Following this, it considered the hypotheses utilized in the formulation of Lanczos-Lovelock theories of gravity. The study then sought to identify a unique alternative gravitational theory in the non-Riemannian case by combining insights from both Jacobson's approach and the Lanczos-Lovelock framework. Particular attention was paid to scenarios without non-metricity and to different identifications of the energy-momentum tensor (metric versus canonical versions).
Findings
- When Jacobson's approach was applied to non-Riemannian geometries, the theory derived from the Einstein-Hilbert action was found not to be among the available gravitational theories for Nature's selection, with the exception of the Riemannian case.
- In the search for a unique alternative in the non-Riemannian context without non-metricity, and when the energy-momentum tensor is identified as its metric version, the combination of Jacobson's approach and the hypotheses of Lanczos-Lovelock theories indicates that the theory derived from the Einstein-Hilbert action plus a term quadratic in the torsion vector is the one that would be selected by Nature.
- The strategy of combining the two approaches (Jacobson's and Lanczos-Lovelock) cannot be consistently followed in the full non-Riemannian case.
- The same inconsistency arises in the previous non-Riemannian case (without non-metricity) when the energy-momentum tensor is identified as its canonical version. This suggests a mutual inconsistency between the two approaches under these specific conditions.