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 *g*_{ab} and the affine connection ∇_{a}, and

*S*_{P}[*g*,∇]
= (16π*G/c*^{4})^{–1} ∫_{M} d^{4}*x* |*g*|^{1/2} *R*_{ab}(∇)
*g*^{ab} .

* __Holst action__: The action, depending on the Barbero-Immirzi parameter *γ*, which expressed in terms of a metric *g*_{ab} and an affine connection ∇_{a} is

*S*_{H}[*g*,∇]
= *S*_{P}[*g*,∇] + (32π*Gγ**/c*^{4})^{–1} ∫_{M} d^{4}*x* |*g*|^{1/2} *ε*^{abcd} *g*_{ma}* R*^{m}_{bcd}(∇)
;

If ∇ is a torsion-free connection (Einstein-Palatini action principle) the theory is equivalent to general relativity and the equation obtained varying *S*_{H} with respect to it implies that *S*_{H} 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 *S*_{H} with respect to ∇ implies that *S*_{H} 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]; Cattaneo & Schiavina a1707 [with boundaries].

@ __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 CQG(17)-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]; Khatsymovsky MPLA(18)-a1705 [simplicial analog].

**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* = (16π*G*)^{–1} ∫_{M} d^{4}*x* [**g**^{ab} ^{0}*R*_{ab}
+ Γ_{c} Γ^{c}_{ab}** g**^{ab} – **g**^{ab} Γ^{m}_{an} Γ^{n}_{mb}
+ ^{0}∇_{a}** g**^{ab} Γ_{b} – (^{0}∇_{m}** g**^{ab}) Γ^{m}_{ab}
],

where **g**^{ab}:= |*g*|^{1/2}* g*^{ab}
is the densitized metric, and Γ_{c}:= Γ^{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, McKeon
IJMPA(10)-a1005 [canonical
analysis]; Tomboulis JHEP(17)-a1708 [and the self-interacting graviton].

@ __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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