Stress energy momentum in terms of geodesic accelerations and variational tensors including torsion

Abstract: General relativity and its extensions including torsion identify stress energy momentum as being proportional to the Einstein tensor, thus ensuring both symmetry and conservation. Here we visualize stress energy and momentum by identifying the associated relative fractional accelerations of geodesics encoded in the Einstein tensor. This also provides an intuitive explanation for the vanishing divergence of the Einstein tensor. In order to obtain this same energy and momentum for other actions such as that of Dirac theory including torsion, we then review the various stress energy momentum tensors resulting from the variation of different quantities derived from parallel transport, and detail their interrelationships. This provides an opportunity to revisit some classic material from a geometric point of view, including Einstein-Cartan theory, the Sciama-Kibble formalism, and the Belinfante-Rosenfeld relation, whose derivation in the mostly pluses signature would seem to not be otherwise readily available.

Overview of "Stress Energy Momentum in Terms of Geodesic Accelerations and Variational Tensors Including Torsion"

The paper by Adam Marsh delves into the geometric interpretation of stress-energy-momentum (SEM) tensors within the scope of general relativity and its extensions, including the influence of torsion. The paper aligns SEM with the fractional accelerations of geodesics, perceiving them through the lens of the Einstein tensor. It further traverses the derivation of these concepts beyond the classic continuum mechanical origins, accommodating actions like Dirac theory, where torsion plays a significant role.

Core Contributions

1. Geodesic Interpretations: The paper articulates the notion that the components of the Einstein tensor represent fractional accelerations of geodesics. This viewpoint extends beyond traditional energy density and momentum density interpretations, thus applicable in electromagnetic contexts where these mechanical analogs do not hold.

2. Visualization of Energy and Momentum: By associating geodesic accelerations with the Einstein tensor, the work lays a geometric foundation for understanding energy and momentum. This is significant for emphasizing changes in extended geodesic objects as they approach mass or pass through fields, shedding light on how energy and momentum geometrically transform spacetime.

3. Connecting Classical Concepts to Geometric Theories: The integration and reinterpretation of classical materials, including the Einstein-Cartan theory, Sciama-Kibble formalism, and the Belinfante-Rosenfeld relation, are revisited with a fresh geometric perspective. This offers clarity on tensor symmetries in frames with positive spacetime signatures.

4. Variation of SEM Tensors: The paper examines variational methods by introducing the matter action in terms of contorsion and the metric, serving as a prelude to defining generalized Einstein-Cartan SEM tensors. It confirms that Einstein’s theories maintain validity when torsion is not dynamically significant, simplifying to conventional Einstein field equations under zero torsion conditions.

Implications and Future Directions

The paper posits a significant step forward in visualizing and interpreting SEM tensors in general relativity, especially in frameworks that involve torsion. It sets the stage for addressing advanced field theories like those of Dirac within curved spacetime, anticipating further developments in quantum gravity and spin-torsion coupling. The reinterpretation of traditional concepts within this geometric framework might spur novel ways to integrate gravity with quantum mechanics.

Future investigations could extend to quantum field theory in curved spacetime, exploring how Dirac fields interact and evolve within the framework established by the paper. One potential research trajectory could entail analyzing the impact of torsion in high-energy astrophysical phenomena or in early-universe cosmology, where the effects might be more prominent. Additionally, adapting the model to computational simulations could validate its geometric interpretations against observed relativistic phenomena.

This theoretical exposition thus offers enriched insights into spacetime's geometric structure, enhancing our comprehension of gravity in a myriad of physical theories and potentially guiding future experimental verifications.