Holst action information

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={\frac {1}{2}}\int ee_{\ I}^{\alpha }e_{\ J}^{\beta }(F_{\alpha \beta }^{\ \ \ IJ}-\alpha \ast F_{\alpha \beta }^{\ \ \ IJ})\equiv {\frac {1}{2}}\int ee_{\ I}^{\alpha }e_{\ J}^{\beta }(F_{\alpha \beta }^{\ \ \ IJ}-{\frac {\alpha }{2}}\epsilon _{\;\;\;KL}^{IJ}F_{\alpha \beta }^{\ \ \ KL})

where $e_{\ I}^{\alpha }$ is the tetrad, $e$ its determinant (the space-time metric is recovered from the tetrad by the formula $g_{\alpha \beta }=e_{\alpha }^{I}e_{\beta }^{J}\eta _{IJ}$ where $\eta _{IJ}$ the Minkowski metric), $F_{\alpha \beta }^{\ \ \ IJ}$ the curvature considered as a function of the connection $A_{\alpha \beta }^{\ \ \ IJ}$ :

F_{\alpha \beta }^{\ \ \ IJ}={A_{\alpha \beta }}^{IJ}=2\partial _{[\alpha }{A_{\beta ]}}^{IJ}+2{A_{[\alpha }}^{IK}{A_{\beta ]K}}^{J}

,

$\alpha$ a (complex) parameter, and where we recover the Palatini action when $\alpha =0$ . It only works in 4D. To be torsion free means the covariant derivative defined by the connection $A_{\alpha \beta }^{\ \ \ IJ}$ when acting on the Minkowski metric $\eta _{IJ}$ vanishes, implying the connection is anti-symmetric in its internal indices $I,J$ .

As with the first order tetradic Palatini action where $e_{\ I}^{\alpha }$ and $A_{\alpha \beta }^{\ \ \ IJ}$ are taken to be independent variables, variation of the action with respect to the connection $A_{\alpha \beta }^{\ \ \ IJ}$ (assuming it to be torsion-free) implies the curvature $F_{\alpha \beta }^{\ \ \ IJ}$ be replaced by the usual (mixed index) curvature tensor $R_{\alpha \beta }^{\ \ \ IJ}$ (see article tetradic Palatini action for definitions). Variation of the first term of the action with respect to the tetrad $e_{\ I}^{\alpha }$ 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.

^ Holst, Sören (15 May 1996). "Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action". Physical Review. 53 (10): 5966–5969. arXiv:gr-qc/9511026. Bibcode:1996PhRvD..53.5966H. doi:10.1103/PhysRevD.53.5966. PMID 10019884. S2CID 15959938.

[Holst-1] Holst, Sören (15 May 1996). "Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action". Physical Review. 53 (10): 5966–5969. arXiv:gr-qc/9511026. Bibcode:1996PhRvD..53.5966H. doi:10.1103/PhysRevD.53.5966. PMID 10019884. S2CID 15959938.

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