Action for General Relativity  

In General
* Idea: The bulk term of the Lagrangian density for general relativity (without a cosmological constant) is \(\cal L\)EH = |g|1/2 R.
* Arbitrariness: There are several possible prescriptions, differing by boundary terms which depend on the variables one fixes at the boundary when varying the action; The interpretation is that one changes the reference points for quasilocal densities (but does not affect, e.g., gravitational thermodynamics).
* With a null boundary: The action cannot be defined, since (1) Given two regions I and II of spacetime, separated by a null boundary, there is no unique way of reconstructing the total metric on I ∪ II; If we had the action for I and II separately, we would want their sum to be equal to the action for I ∪ II; But this is not uniquely determined, so it must be that the action for I and II cannot be defined; and (2) The extrinsic curvature cannot be defined for a null surface, and it comes into the proper boundary term for the action.
* Positive action conjecture: The action of any non-singular, non-flat asymptotically Euclidean metric with R = 0 is positive [@ Page PRD(78); Gibbons, Hawking & Perry NPB(78); Le Brun CMP(88)].
* Symmetries: Diffeomorphisms; A constant rescaling leaves the Einstein equation invariant, but not the action, and there is no corresponding conserved current.
@ References: Charap & Nelson JPA(83); in Wald 84; York FP(86); Soh PLB(91); Dragon PLB(92) [regularity]; Peldán CQG(94)gq/93; László gq/04 [global approach].
blue bullet Main types of actions: see first-order metric action [including Palatini and Holst actions], and other types of actions.
blue bullet Related issues: see gravitational thermodynamics; lagrangian dynamics [symmetric variations].
blue bullet Related topics: see energy in general relativity; history of relativistic gravity.

Types of Metrics and Generalizations
* Quantum corrections: Quantum gravity corrections to the action can be modeled by a varying action, with varying G and Λ; The effective classical action of quantum gravity contains infinitely many independent couplings, determined by renormalization; > s.a. quantum-gravity renormalization.
@ Specific type of spacetimes: Grigoryan Ast(89) [star, boundary conditions]; Frolov & Martinez CQG(96)gq/94 [black holes]; Gladush JMP(01)gq/00, GRG(04)gq/03 [dust shell]; Baker PhD-gq/02 [binary neutron stars]; He et al IJMPD(03)gq/02 [with Killing vector fields].
@ Euclideanized: Schoen & Yau PRL(79) [positivity]; Soo PRD(95)gq; Esposito CQG(99)gq/98.
@ Complexified: Hayward PRD(96)gq/95; Louko & Sorkin CQG(97)gq/95 [and 2D topology change]; > s.a. types of action [self-dual].
@ Non-smooth metrics: Hayward & Louko PRD(90); Mukohyama PRD(02)gq/01 [singular hypersurfaces]; Gravanis & Willison JMP(09)-a0901 [distributional sources]; Huber a2001 [distributional Kerr-Schild geometries].
@ Quantum corrections and effective action: Elizalde & Odintsov MPLA(95) [from GUTs and renormalization group, non-local]; Fabris et al IJMPA(12) [cosmology and astrophysics]; Anselmi JHEP(13)-a1302 [the classical action of quantum gravity and its properties]; Codello & Jain IJMPD(16)-a1605-GRF [cosmology]; Padmanabhan PLB(21)-a2011 [from the quantum microstructure of spacetime].
@ Other generalized and similar theories: Saa JGP(95) [with torsion]; Burton & Mann PRD(98)gq/97; Bimonte et al PLB(98)gq [non-commutative]; Bonanno et al IJMPA(05)ht [varying G and Λ, and renormalization]; Padmanabhan GRG(06) [semiclassical, holographic]; Ananth FdP(09)-a0902 [from square of Yang-Mills Lagrangian]; Krasnov EPL(10)-a0910 [local actions with two propagating degrees of freedom]; Faddeev a1003; Freedman a1008 [on a formal chain of combinatorial spacetimes]; > s.a. higher-order theories.

Surfaces and Boundary Terms > s.a. gravitational entropy; holography; types of actions [Gibbons-Hawking-York boundary term].
* Idea: The most commonly used boundary term for the action of general relativity is the Gibbons-Hawking-York term, which for a spacelike or timelike piece of the boundary is (up to a constant) the integral of the trace of the extrinsic curvature of the boundary; For null pieces of the boundary, it appears to be zero.
@ General references: Brill & Hayward PRD(94)gq [additivity]; Mann & Marolf CQG(06) [holographic renormalization]; Mukhopadhyay & Padmanabhan PRD(06)ht [bulk-surface "holography"]; Mann et al CQG(06) [and conserved quantities]; Chamseddine & Connes PRL(07)-a0705 [from spectral action]; Bianchi & Wieland a1205 [horizon boundary term and near-horizon energy]; Neiman a1212, JHEP(13)-a1301, PRD(13)-a1305, Bodendorfer & Neiman CQG(13)-a1303 [complex boundary term and null boundaries, black-hole entropy, Holst modification vs Nieh-Yan density, etc]; Padmanabhan MPLA(14); Coley a1503-talk [global topology and geometric invariants]; York a1512 [bundary terms for true stationary point]; Krishnan & Raju MPLA(17)-a1605 [Neumann boundary term]; Chakraborty ch(17)-a1607; Feng & Matzner GRG(18) [Weiss variational principle]; Oliveri & Speziale GRG(20)-a1912 [with tetrad variables]; Schmekel a2009 [action for pure gravity as a boundary term].
@ Finiteness, asymptotically flat: Sorkin in(88); Solodukhin PRD(00)ht/99 [and AdS spacetime]; Visser PRD(09)-a0808 [4D]; Miyashita a2007 [and AdS].
@ With null boundaries: Parattu et al GRG(16)-a1501, EPJC(16)-a1602; Lehner et al PRD(16)-a1609; Wang a1609 [for 1st-order, connection form]; Aghapour et al CQG(18)-a1808 [and double-foliation formalism]; Chakraborty & Padmanabhan a1909 [surface term as heat density].
@ Boundary conditions: Hawking & Horowitz CQG(96)gq/95; Avramidi & Esposito gq/99-conf; Kraus et al NPB(99)ht [asymptotically flat and AdS]; Pons GRG(03)gq/01 [Noether charges, presymplectic]; Padmanabhan GRG(02)gq, GRG(03), ASS(03)gq/02 [and entropy, horizons, holography]; Smoot PhD-a1205 [differential bulk-surface relation]; Park JHEP(19)-a1811 [boundary dynamics].
@ Bounded region, corners: Hayward & Wong PRD(92); Hayward PRD(93); Fabbrichesi et al NPB(94) [and Planck-scale scattering]; Lau CQG(96)gq/95; Hawking & Hunter CQG(96)gq; Momen PLB(97)ht/96; Brown et al AP(02)gq/00; Jubb et al CQG(17)-a1612 & CQG+(17); Ruan & Yang CQG(17)-a1704; Freidel et al a2104 [corner symmetry algebra, charge bracket]; > s.a. quasilocal general relativity.
blue bullet Related theories: see higher-order theories; unimodular relativity.

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