Types of Actions for General Relativity |

**In General** > s.a. gravitation;
higher-order gravity theories.

* __Possible contributions in four dimensions__:
Possible terms that can appear in the Lagrangian are [@ Zumino in(86)]

*L*_{0,4}
= *e*_{a} *e*_{b}
*e*_{c} *e*_{d}
*ε*^{abcd} : Cosmological term ,

*L*_{1,2}
= *R*_{ab} *e*_{c}
*e*_{d} *ε*^{abcd}
: Einstein-Hilbert term ,

*L*_{2,0}
= *R*_{ab} *R*_{cd}
*ε*^{abcd} : Euler invariant .

@ __Related topics__:
Robinson IJTP(98) [chiral];
Mei a0711 [with positive kinetic-energy term];
Dubois-Violette & Lagraa LMP(09)-a0907 [large classes];
Banerjee & Majhi PRD(10) [and entropy];
Kolekar & Padmanabhan PRD(10)-a1005 [thermodynamic/holographic decomposition into surface and bulk terms];
Brown PRD(11)-a1008 [for the generalized harmonic formulation];
Kol & Smolkin PRD(12)-a1009 [in terms of Newtonian fields];
Demir et al a1105 [constructed solely from the Riemann tensor];
Sengupta JPCS(12) [with Nieh-Yan, Pontryagin and Euler topological terms].

**Einstein-Hilbert Action**
> s.a. 2D gravity; noether charge.

* __Expression__: The variable is the
metric (or its inverse); With a cosmological constant Λ, the volume term is

*S*_{EH}[*g*]
= \(1\over2\kappa\)∫_{M}
d^{4}*x* |*g*|^{1/2}
(*R* − 2Λ) ,

where \(\kappa = 8\pi G/c^4\), which contains
– linearly – second derivatives of *g*.

@ __References__: Hilbert KNGWG(15);
Katanaev GRG(06)gq/05,
TMP(06)gq
[with |*g*| as a variable, polynomial];
Cheung & Remmen JHEP(17)-a1705 [as a theory of purely cubic interactions];
Gionti JPCS(19)-a1902 [Reuter-Weyer RG improved];
Takeuchi IJMPA(20)-a1811 [and Fisher information metric].

**Gibbons-Hawking-York (tr K) Action** (with all boundary terms)
> s.a. extrinsic curvature.

*

*S*_{GHY}[*g*]
= \(1\over2\kappa\){∫_{M}
d^{4}*v* (*R* −
2Λ) + 2 [∫_{Σ2}
(*K*−*K*_{0}) d^{3}*s*
− ∫_{Σ1}
(*K*−*K*_{0}) d^{3}*s*
− ∫_{τ}
(*K*−*K*_{0}) d^{3}*s*
− ∫_{B2}
*ξ* d^{2}*σ* +
∫_{B1}
*ξ* d^{2}*σ*] } ,

where *K* (*K*_{0}) is the trace of the
extrinsic curvature induced by *g* (*g*_{0})
on ∂*M*, and *ξ*:= sinh^{−1}(*u*
· *n*), with *u* the future-pointing normal to Σ, and *n* the
outward-pointing normal to *τ*.

@ __References__:
Gibbons & Hawking PRD(77);
York FP(86);
Hayward PRD(93);
Hawking & Hunter CQG(96)gq [boundaries];
Pons GRG(03)gq/01 [Lagrangian, Noether charges];
Polishchuk G&C(10) [and Hilbert-Einstein action].

**3+1 Metric Form**

* __Expression__: If *θ*
= extrinsic curvature of *τ*; *K* extrinsic curvature of \(\Sigma_t\);
*η* = *u* · *n*; *a*^{a}
= *u*^{b}∇_{b}
*u*^{a}
= acceleration of *τ*,

*S*[*q*, *N*, *N*^{a}]
= \(1\over2\kappa\)∫ d*t*
∫_{Σ}
d^{3}*x* *N* *q*^{1/2}
(^{3}*R* + *K*_{ab}
*K*^{ab} − *K*^{2}
− 2Λ) + \(1\over\kappa\)∫_{τ}
d^{3}*x* |*γ*|^{1/2}
(*θ* + *ηK* − *n*_{a}
*a*^{a})
.

@ __References__: Hawking & Hunter CQG(96)gq.

**Other Forms**
> s.a. action for general relativity [boundary terms].

* __Baierlein-Sharp-Wheeler form__:
The one obtained when the Lagrange multiplier (the lapse function) is eliminated
from the Lagrangian and one is left with a product of square roots.

@ __General references__:
Ambjørn et al NPB(97)ht/96 [Regge-calculus inspired];
Arcioni et al JHEP(01)ht [boundary action, eikonal limit];
Cremaschini & Tessarotto EPJP(15)-a1609 [synchronous Lagrangian variational principles];
Cheung & Remmen JHEP(17)-a1612 [with twofold Lorentz symmetry];
Takeuchi a1811 [in terms of Fisher information metric];
Magnano et al a1812 [dependence on the Weyl tensor];
Schmekel a2009 [action for pure gravity as a boundary term].

@ __BSW and related forms__: Carlini & Greensite PRD(95)gq;
Ó Murchadha IJMPA(02)-proc;
Shyam & Venkatesh GRG(13)-a1209 [Barbour-Foster-Ó Murchadha 3-space action].

@ __Curvature-saturated__: Kleinert & Schmidt GRG(02)gq/00
[\(\cal L\)_{cs} = \(\cal L\)_{EH}/
(1 + *l*^{ 4}
*R*^{2})^{1/2},
with *l* a length parameter].

@ __Self-dual__: Nieto & Socorro PRD(99)ht/98 [and Yang-Mills, MacDowell-Mansouri formalism];
Nieto MPLA(05)ht/04 [various versions];
> s.a. connection formulation.

@ __Bimetric reformulation__: Koivisto PRD(11)-a1103;
Jiménez et al PRD(12)-a1201 [bimetric variational principle].

__Other forms__: see action [effective classical action,
similar theories]; first-order forms; formulations
of general relativity [including embedding variables].

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