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 *g*_{ab} and affine connection ∇_{a}, or replacing those by a tetrad *e*_{a}^{i} and a spin connection *ω*_{ai}^{j}, respectively, by (\(\kappa\) = 8π*G**/c*^{4})

*S*_{P}[*e*,* ω*]
= \(1\over2\kappa\)∫_{M} d^{4}*x* *ε*^{abcd} *ε*_{ijkl}* e*_{a}^{i}* e*_{b}^{j}* R*_{cd}^{kl}(*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]; Cattaneo & Schiavina a1707 [with boundaries]; > 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 *e*_{a}^{I} and
a self-dual Lorentz connection *A*_{aI}^{J},

*S*_{SJS}[*e*,* A*] = ∫_{M} d^{4}*x* (det *e*) *e*^{a}_{I}* e*^{b}_{J}* F*_{ab}^{IJ} ;

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 *e*_{a}^{I}, the Levi-Civita connection of the tetrad Γ^{I}_{Ja}:= *e*^{I}_{b} ∇_{a}* e*^{b}_{J} , and the Sparling 2-form *σ*_{I} ,

*S*_{G}[*e*] = \(1\over2\kappa\)∫_{M} Γ^{I}_{J} ∧ *e*^{J} ∧ *σ*_{I}
;

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} (*B*^{ij} ∧ *F*_{ij}
+ *φ*^{ijkl}* B*_{ij} ∧ *B*_{kl})
.

@ __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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