Holst action explained

See main article: tetradic Palatini action and frame field. In the field of theoretical physics, the Holst action^[1] is an equivalent formulation of the Palatini action for General Relativity (GR) in terms of vierbeins (4D space-time frame field) by adding a part of a topological term (Nieh-Yan) which does not alter the classical equations of motion as long as there is no torsion,

S = (1)/(2) \int e e ^{\alpha} _{I} e ^{\beta} _{J} (F ^{IJ} _{\alpha\beta} - \alpha \ast F ^{IJ} _{\alpha\beta} ) \equiv (1)/(2) \int e e ^{\alpha} _{I} e ^{\beta} _{J} (F ^{IJ} _{\alpha\beta} - (\alpha)/(2) \epsilon ^{IJ} _{KL} F ^{KL} _{\alpha\beta} )

where

e ^{\alpha} _{I} is the tetrad, e its determinant (the space-time metric is recovered from the tetrad by the formula g _{\alpha} = e ^{I} _{\alpha} e ^{J} _{\beta} η _{IJ} where η _{IJ} the Minkowski metric), F ^{IJ} _{\alpha\beta} the curvature considered as a function of the connection A ^{IJ} _{\alpha\beta} :F ^{IJ} _{\alpha\beta} = {A _{\alpha} }^ = 2 \partial_

Notes and References

1. Holst . Sören . Sören Holst . 15 May 1996 . Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action . Physical Review . 53 . 10 . 5966–5969 . 10.1103/PhysRevD.53.5966 . 10019884 . gr-qc/9511026 . 1996PhRvD..53.5966H . 15959938 .