First-Order Actions for General Relativity

Palatini Action > s.a. 2D gravity; higher-dimensional gravity; higher-order gravity.
* Expression: The variables are the metric g or vierbein e, and the connection or spin connection , and

S[g,] = (16G)^–1 _M d⁴x |g|^1/2 R_ab() g^ab = (16G)^–1 _M d⁴x ^abcd _ijkl e_aⁱ e_b^j R_cd^kl() .

@ General references: Palatini RCMP(19); Holst PRD(96)gq/95 [for Barbero Hamiltonian]; Burton & Mann PRD(98)gq/97 [extended S].
@ Canonical analysis: Han et al MPLA(05)gq [n-dimensional + cosmological constant + scalar]; Kiriushcheva et al IJMPA(06)ht.

First Order Metric Form > s.a. 2D gravity.
* Expression: One chooses a background connection ⁰, in order to identify the part of R to subtract, and

S = (16G)^–1 _M d⁴x [g^{ab 0}R_ab + _c ^c_ab g^ab – g^{ab m}_an ⁿ_mb + ⁰_a g^ab _b – (⁰_m g^ab) ^m_ab ],

where g^ab:= |g|^1/2 g^ab is the densitized metric, and _c:= ^a_ac.
@ General references: Einstein SPAW(16); Faddeev SPU(82); Lindström IJMPA(88); Ferraris & Francaviglia GRG(90) [interesting introduction]; Grigore CQG(92); Ghalati & McKeon a0711, a0712 [canonical analysis].
@ And conserved quantities: Sorkin in(88); Fatibene et al JMP(01)gq/00 [relationships].

Connection and Self-Dual Forms
* Samuel-Jacobson-Smolin action: In terms of a tetrad e_a^I and a self-dual Lorentz connection A_aI^J,

S[e,A] = _M d⁴x (det e) e^a_I e^b_J F_ab^IJ .

It can be seen not to be a purely metric action [@ in Lau CQG(96)gq/95].
* Goldberg action: In terms of a tetrad e_a^I,

S[e] = (2)^–1 _M ^I_J e^J  _I ,

where ^I_Ja:= e^I_{b a} e^b_J is the Levi-Civita connection of the tetrad, and _I the Sparling 2-form; With some gauge fixing, this action is closely related to the "tr K" action [@ in Lau CQG(96)gq/95].
* Plebanski action: The sum of a BF term and a constraint,

S = _M (B^ij F_ij + ^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].
@ BF-like formulation: Lewandowski & Okolów CQG(00)gq/99; Capovilla et al CQG(01)gq [arbitrary ]; > s.a. BF.
@ Other variables: Nester & Tung GRG(95)gq/94; Tung & Jacobson CQG(95)gq; Tung & Nester PRD(99) [and teleparallel].
@ Plebanski action: Alexandrov et al CQG(07)gq/06 [and covariant canonical formulation of Hilbert-Palatini action].
@ With other matter: Morales & Esposito NCB(94) [fermions]; Robinson JMP(95) [Yang-Mills fields]; > s.a. gravitating matter.

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