Some aspects of Holst and Nieh-Yan terms in general relativity with torsion

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 on adding torsion to general relativity Einstein's equations are no longer holographic in torsion although they continue to be so for the metric.