In the field of theoretical physics, the Holst action 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, where is the tetrad, its determinant (the space-time metric is recovered from the tetrad by the formula where the Minkowski metric), the curvature considered as a function of the connection : ,
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| - In the field of theoretical physics, the Holst action 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, where is the tetrad, its determinant (the space-time metric is recovered from the tetrad by the formula where the Minkowski metric), the curvature considered as a function of the connection : , (en)
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| - In the field of theoretical physics, the Holst action 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, where is the tetrad, its determinant (the space-time metric is recovered from the tetrad by the formula where the Minkowski metric), the curvature considered as a function of the connection : , a (complex) parameter, and where we recover the Palatini action when . It only works in 4D. To be torsion free means the covariant derivative defined by the connection when acting on the Minkowski metric vanishes, implying the connection is anti-symmetric in its internal indices . As with the first order tetradic Palatini action where and are taken to be independent variables, variation of the action with respect to the connection (assuming it to be torsion-free) implies the curvature be replaced by the usual (mixed index) curvature tensor (see article tetradic Palatini action for definitions). Variation of the first term of the action with respect to the tetrad gives the (mixed index) Einstein tensor and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the Riemann tensor (specifically the first Bianchi identity), together these imply Einstein's vacuum field equations hold. (en)
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