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 *κ* = 8π*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].

**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].

@ __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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