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 \(\nabla_{\!a}\), 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,
IJGMP(18)-a1707 [unified formalism];
Bernal et al PLB(17)-a1606 [non-uniqueness of the Levi-Civita solution];
Cattaneo & Schiavina a1707 [with boundaries];
Bejarano et al a1907
[inequivalence of Palatini and metric formulations];
Capriotti a1909 [Routh reduction].

@ __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,
a1805,
a1901;
Montesinos et al PRD(19)-a1903 [treatment of the second-class constraints];
> s.a. gauge symmetries.

@ __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];
Martins & Biezuner JGP(19)-a1808 [obstructions].

**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. BF theory; 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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