Types of Actions for General Relativity| 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)]
L0,4
= ea eb
ec ed
εabcd : Cosmological term ,
L1,2
= Rab ec
ed εabcd
: Einstein-Hilbert term ,
L2,0
= Rab Rcd
εabcd : 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
SEH[g] = \(1\over2\kappa\)∫M d4x |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.
* Idea: Obtained from the
Hilbert-Einstein action by subtracting the boundary terms containing normal
derivatives of the metric; To ensure that the induced metric is fixed on all
components of ∂M, we must include all corner terms; For a four-dimensional
M with initial and final hypersurfaces Σ1
and Σ2, and timelike boundary τ
which meets Σi
in Bi, i = 1, 2,
SGHY[g] = \(1\over2\kappa\){∫M d4v (R − 2Λ) + 2 [∫Σ2 (K−K0) d3s − ∫Σ1 (K−K0) d3s − ∫τ (K−K0) d3s − ∫B2 ξ d2σ + ∫B1 ξ d2σ] } ,
where K (K0) is the trace of the
extrinsic curvature induced by g (g0)
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; aa
= ub∇b
ua
= acceleration of τ,
S[q, N, Na] = \(1\over2\kappa\)∫ dt ∫Σ d3x N q1/2 (3R + Kab Kab − K2 − 2Λ) + \(1\over\kappa\)∫τ d3x |γ|1/2 (θ + ηK − na aa) .
@ 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
R2)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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