First-Order Tetrad Actions for General Relativity| First-Order Tetrad Actions for General Relativity |
In General, Tetrad-Based Palatini Action
> s.a. higher-dimensional gravity.
* Rem: A variational principle based on
tetrads instead of metric variables is necessary if one wants to couple fermions to gravity.
* Palatini action: The Palatini action can
be expressed in terms of the the metric gab
and affine connection ∇a, or replacing those by a tetrad
eai and a spin
connection ωaij,
respectively, by (\(\kappa\) = 8πG/c4)
SP[e, ω] = \(1\over2\kappa\)∫M d4x εabcd εijkl eai ebj Rcdkl(ω) .
@ General references: Corichi et al GRG(14)-a1312,
IJMPD(16)-a1604 [rev].
@ Boundary terms: Corichi & Reyes JPCS(12) [and 3+1 decomposition of Holst action];
Bodendorfer & Neiman CQG(13)-a1303 [vs Nieh-Yan density, and complex boundary term];
Cattaneo & Schiavina a1707 [Holst action];
Oliveri & Speziale a1912.
Holst Action
* Idea: The Holst action is the simplest
tetrad-based action producing a canonical theory without the complications of second-class
constraints (which appear in the simpler first-order Einstein-Palatini action); It is,
furthermore, the classical starting point for loop quantum gravity and some spin-foam models;
The "Holst term" was originally introduced by Hojman et al., but it was Sören
Holst who first showed that its 3+1 decomposition plus partial gauge fixing gives, for compact
spacetimes without boundaries, a Hamiltonian action for general relativity in terms of
Ashtekar-Barbero variables; Corichi & Reyes extended this result to asymptotically
flat spacetimes.
@ References: Fatibene et al IJGMP(09)-a0808 [conserved quantities and entropy];
Corichi & Wilson-Ewing CQG(10)-a1005 [symplectic structure, conserved quantities, entropy];
Pfäffle & Stephan CMP(11)-a1102 [and the spectral action principle];
Szczachor a1202-conf [supersymmetric];
Geiller & Noui GRG(13) [canonical analysis];
Corichi & Reyes CQG(15)-a1505 [consistent 3+1 split for asymptotically flat spacetimes];
> s.a. connection formulation; einstein-cartan theory;
isolated horizons; renormalization.
Other Forms of the Action
* Samuel-Jacobson-Smolin action:
The action, in terms of a tetrad eaI and
a self-dual Lorentz connection AaIJ,
SSJS[e, A] = ∫M d4x (det e) eaI ebJ FabIJ ;
It can be shown that it is not a purely metric action
[@ in Lau CQG(96)gq/95].
* Goldberg action: In terms of a tetrad
eaI,
the Levi-Civita connection of the tetrad
ΓIJa:=
eIb
∇a
ebJ ,
and the Sparling 2-form σI ,
SG[e] = \(1\over2\kappa\)∫M ΓIJ ∧ eJ ∧ σI ;
With some gauge fixing, this action is closely related to the "tr K"
action [@ in Lau CQG(96)gq/95].
* Plebański action:
The sum of a BF term and a simplicity (or metricity) constraint,
S = ∫M (Bij ∧ Fij + φijkl Bij ∧ Bkl) .
@ 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].
@ Plebański action: Alexandrov et al CQG(07)gq/06
[and covariant canonical formulation of the Hilbert-Palatini action];
Ita AZJ-gq/07,
AZJ-a0704 [instanton representation];
Noui et al GRG(09) [cosmological symmetry reduction, quantization];
Krasnov GRG(11)-a0904 [intro];
Smolin & Speziale PRD(10)-a0908 [with cosmological constant and Immirzi parameter];
Tennie & Wohlfarth PRD(10)-a1009 [matter couplings];
> s.a. BF theory.
@ BF-like formulation: Lewandowski & Okołów CQG(00)gq/99;
Capovilla et al CQG(01)gq [arbitrary γ];
> s.a. BF theory.
@ Other connection-based:
Jiménez-Rezende & Pérez PRD(09)-a0902 [Holst action + topological terms];
Krasnov PRL(11)-a1103 [pure connection formulation];
Robinson a1506
[generalized Chern-Simons action principles];
Celada et al PRD(15)-a1509 [derivation of Krasnov action],
PRD(16)-a1605 [Plebanski-like action, and anti-self-dual gravity];
Chagoya & Sabido a1612 [self-dual gravity and the Immirzi parameter].
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