Some Aspects of Holst and Nieh-Yan Terms in General Relativity with Torsion

TL;DR

This paper investigates the relationship between Holst and Nieh-Yan terms in general relativity with torsion, demonstrating the physical effects of these terms on equations of motion and torsion charge, and analyzing their holographic properties.

Contribution

It clarifies the role of Holst and Nieh-Yan terms in Einstein-Cartan theory, showing the Nieh-Yan term preserves equations of motion and affects torsion charge.

Findings

Holst term influences equations of motion with spin matter.

Nieh-Yan term does not affect equations of motion but corrects the action.

Torsion charge vanishes after horizon formation in a perfect fluid sphere.

Abstract

We explore the relation of the Holst term with the Nieh-Yan term in terms of metric variables. We show that the Holst term indeed affects the classical equations of motion in the presence of matter with spin. Therefore the correct term to add to the Einstein-Hilbert action such that the equations of motion are not affected is the Nieh-Yan term. We then calculate the torsion charge due to this term in the context of a perfect fluid sphere with torsion and show that it vanishes once a horizon is formed but not otherwise. We also show that adding on torsion to General Relativity the Einstein's equations are no longer holographic in torsion although they continue to be so for the metric.