First-Order Metric Actions for General Relativity  

Palatini and Related Actions > s.a. 2D gravity; higher-dimensional gravity; higher-order gravity.
* Palatini action: Obtained by expressing the Einstein-Hilbert action in terms of the metric gab and the affine connection ∇a, and

SP[g,∇] = (16πG/c4)–1 M d4x |g|1/2 Rab(∇) gab .

* Holst action: The action, depending on the Barbero-Immirzi parameter γ, which expressed in terms of a metric gab and an affine connection ∇a is

SH[g,∇] = SP[g,∇] + (32π/c4)–1M d4x |g|1/2 εabcd gma Rmbcd(∇) ;

If ∇ is a torsion-free connection (Einstein-Palatini action principle) the theory is equivalent to general relativity and the equation obtained varying SH with respect to it implies that SH is metric-compatible; If ∇ is a metric-compatible connection (Einstein-Cartan action principle) in the absence of matter the theory is equivalent to general relativity and the equation obtained varying SH with respect to ∇ implies that SH is torsion-free, but in the presence of spinning matter there is torsion induced, and the matter couples to gravity differently than in general relativity; If is an arbitrary connection, in the absence of matter the theory is again equivalent to general relativity; > s.a. tetrad-based actions.
@ General references: Palatini RCMP(19); Holst PRD(96)gq/95 [for the Barbero Hamiltonian]; Burton & Mann PRD(98)gq/97 [extended S]; Liko & Sloan CQG(09)-a0810 [and Euclidean quantum gravity]; Goenner PRD(10), Koivisto PRD(11)-a1103 [alternative variational principle]; Capriotti JMP(14)-a1209, a1707 [unified formalism]; Bernal et al PLB(17)-a1606 [non-uniqueness of the Levi-Civita solution].
@ Canonical analysis: Han et al MPLA(05)gq [n-dimensional + cosmological constant + scalar]; Kiriushcheva et al IJMPA(06)ht; Chishtie & McKeon CQG(13)-a1304; Castrillón et al JMP(14); Montesinos et al a1704 [new internal gauge symmetry]; Yoon a1706.
@ Generalized forms: Rosenthal PRD(09)-a0809 [with independent covariant and contravariant metrics]; Dadhich & Pons GRG(12)-a1010 [Einstein-Hilbert and Einstein-Palatini formulations].

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

S = (16πG)–1 M d4x [gab 0Rab + Γc Γcab gabgab Γman Γnmb + 0a gab Γb – (0m gab) Γmab ],

where gab:= |g|1/2 gab is the densitized metric, and Γc:= Γaac.
@ General references: Einstein SPAW(16); Faddeev SPU(82); Lindström IJMPA(88); Ferraris & Francaviglia GRG(90) [interesting introduction]; Grigore CQG(92); Ghalati & McKeon a0711, a0712, McKeon IJMPA(10)-a1005 [canonical analysis].
@ And conserved quantities: Sorkin in(88); Fatibene et al JMP(01)gq/00 [relationships].

Other First-Order Forms of the Action > s.a. tetrad-connection based forms.
@ Other variables: Nester & Tung GRG(95)gq/94; Tung & Jacobson CQG(95)gq; Tung & Nester PRD(99) [and teleparallel]; Herfray & Krasnov a1503 [connection and Lie-algebra valued two-form field].
@ With other matter: Morales & Esposito NCB(94) [fermions]; Robinson JMP(95) [Yang-Mills fields]; > s.a. gravitating matter.

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