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].

__Main types of actions__: see first-order
metric action [including Palatini and Holst actions], and other
types of actions.

__Related issues__: see gravitational
thermodynamics; lagrangian dynamics [symmetric
variations].

__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].

@ __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].

@ __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.

@ __Finiteness, asymptotically flat__: Sorkin in(88);
Solodukhin PRD(00)ht/99 [and
AdS spacetime]; Visser PRD(09)-a0808 [4D].

@ __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].

@ __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].

@ __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 a1704; > s.a. quasilocal
general relativity.

__Related theories__: see higher-order theories; unimodular
relativity.

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