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;
Feng & Matzner GRG(18) [Weiss variational principle].

@ __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];
Aghapour et al a1808 [and double-foliation formalism].

@ __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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