First-Order Metric Actions for General Relativity| 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
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 ∇a is
SH[g,∇] = SP[g,∇] + (32πGγ/c4)−1 ∫M 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,
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 PLB-a1907
[inequivalence of Palatini and metric formulations];
Capriotti a1909 [Routh reduction];
Barnich et al a2004 [conserved currents].
@ 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];
Montesinos et al PRD(20)-a1911 [Holst action with first-class constraints only];
> 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];
Hansen et al a2012
[in terms of moving frames, and non-relativistic expansion].
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 = \(1\over2\kappa\)∫M d4x [gab 0Rab + Γc Γcab gab − gab Γman Γnmb + 0∇a gab Γb − (0∇m gab) Γmab ],
where \(\kappa = 8\pi G/c^4\), 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];
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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 10 dec 2020