Action for General Relativity

In General > s.a. energy in general relativity.
* Remark: Here the Lagrangian must be a scalar tensor density, if we want to integrate it to define S.
* 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; Le Brun CMP(88)].
* Symmetries: Diffeomorphisms (a constant rescaling leaves the Einstein equation invariant, but not the action; there is no no 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].

Surfaces and Boundary Terms > s.a. holography; unimodular relativity.
@ 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].
@ Finiteness: Sorkin in(88) [asymptotically flat]; Solodukhin PRD(00)ht/99 [asymptotically flat and AdS spacetime]; Visser PRD(09)-a0808 [4D asymptotically flat].
@ Boundary conditions: Hawking & Horowitz CQG(96)gq/95; Avramidi & Esposito gq/99-in; 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].
@ 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; > s.a. quasilocal general relativity.
@ Non-smooth metrics: Hayward & Louko PRD(90); Mukohyama PRD(02)gq/01 [singular hypersurfaces]; Gravanis & Willison a0901 [distributional sources].

Forms of the Action and Related Topics > s.a. first-order action and other types.
* Quantum corrections: Quantum gravity corrections to the action can be modeled by a varying action, in the form of varying G and ; > 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 gq/02-PhD [binary neutron stars]; He et al IJMPD(03)gq/02 [with Killing vector fields].
@ Effective action: Elizalde & Odintsov MPLA(95) [from GUTs and renormalization group, non-local].
@ 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].
@ 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 , and renormalization]; Padmanabhan GRG(06) [semiclassical, holographic]; Ananth FdP(09)-a0902 [from square of Yang-Mills Lagrangian]; Krasnov a0910 [local actions with two propagating degrees of freedom]; > s.a. higher-order theories.
> Related issues: see gravitational thermodynamics; lagrangian dynamics [symmetric variations].

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