First-Order Tetrad Actions for General Relativity  

In General, Tetrad-Based Palatini Action > s.a. higher-dimensional gravity.
* Rem: A variational principle based on tetrads instead of metric variables is necessary if one wants to couple fermions to gravity.
* Palatini action: The Palatini action can be expressed in terms of the the metric gab and affine connection ∇a, or replacing those by a tetrad eai and a spin connection ωaij, respectively, by (\(\kappa\) = 8πG/c4)

SP[e, ω] = \(1\over2\kappa\)M d4x εabcd εijkl eai ebj Rcdkl(ω) .

@ General references: Corichi et al GRG(14)-a1312, IJMPD(16)-a1604 [rev].

Holst Action
* Idea: The Holst action is the simplest tetrad-based action producing a canonical theory without the complications of second class constraints (which appear in the simpler first-order Einstein-Palatini action); It is, furthermore, the classical starting point for loop quantum gravity and some spin-foam models; The "Holst term" was originally introduced by Hojman et al., but it was Sören Holst who first showed that its 3+1 decomposition plus partial gauge fixing gives, for compact spacetimes without boundaries, a Hamiltonian action for general relativity in terms of Ashtekar-Barbero variables; Corichi & Reyes extended this result to asymptotically flat spacetimes.
@ References: Fatibene et al IJGMP(09)-a0808 [conserved quantities and entropy]; Corichi & Wilson-Ewing CQG(10)-a1005 [symplectic structure, conserved quantities, entropy]; Pfäffle & Stephan CMP(11)-a1102 [and the spectral action principle]; Szczachor a1202-conf [supersymmetric]; Corichi & Reyes JPCS(12) [boundary terms and 3+1 decomposition]; Bodendorfer & Neiman CQG(13)-a1303 [vs Nieh-Yan density, and complex boundary term]; Geiller & Noui GRG(13) [canonical analysis]; Corichi & Reyes CQG(15)-a1505 [consistent 3+1 split for asymptotically flat spacetimes]; > s.a. connection formulation; einstein-cartan theory; isolated horizons; renormalization.

Other Forms of the Action
* Samuel-Jacobson-Smolin action: The action, in terms of a tetrad eaI and a self-dual Lorentz connection AaIJ,

SSJS[e, A] = M d4x (det e) eaI ebJ FabIJ ;

It can be shown that it is not a purely metric action [@ in Lau CQG(96)gq/95].
* Goldberg action: In terms of a tetrad eaI, the Levi-Civita connection of the tetrad ΓIJa:= eIba ebJ , and the Sparling 2-form σI ,

SG[e] = \(1\over2\kappa\)M ΓIJeJ ∧ σI ;

With some gauge fixing, this action is closely related to the "tr K" action [@ in Lau CQG(96)gq/95].
* Plebański action: The sum of a BF term and a simplicity (or metricity) constraint,

S = M (BijFij + φijkl BijBkl) .

@ Goldberg action: Goldberg PRD(88); in Lau CQG(96)gq/95, CQG(96)gq/95.
@ Ashtekar variables: Jacobson & Smolin CQG(88); Samuel Pra(87); in Ashtekar 88; Nieto MPLA(05)ht/04; Fatibene et al CQG(07)-a0706 [with Barbero-Immirzi SU(2) connection]; Ashtekar et al CQG(08)-a0802 [and covariant phase space].
@ Plebański action: Alexandrov et al CQG(07)gq/06 [and covariant canonical formulation of the Hilbert-Palatini action]; Ita AZJ-gq/07, AZJ-a0704 [instanton representation]; Noui et al GRG(09) [cosmological symmetry reduction, quantization]; Krasnov GRG(11)-a0904 [intro]; Smolin & Speziale PRD(10)-a0908 [with cosmological constant and Immirzi parameter]; Tennie & Wohlfarth PRD(10)-a1009 [matter couplings]; > s.a. BF theory.
@ BF-like formulation: Lewandowski & Okołów CQG(00)gq/99; Capovilla et al CQG(01)gq [arbitrary γ]; > s.a. BF theory.
@ Other connection-based: Jiménez-Rezende & Pérez PRD(09)-a0902 [Holst action + topological terms]; Krasnov PRL(11)-a1103 [pure connection formulation]; Robinson a1506 [generalized Chern-Simons action principles]; Celada et al PRD(15)-a1509 [derivation of Krasnov action], PRD(16)-a1605 [Plebanski-like action, and anti-self-dual gravity]; Chagoya & Sabido a1612 [self-dual gravity and the Immirzi parameter].

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