Quasilocal Energy in General Relativity  

In General > s.a. stress-energy pseudotensors.
* Motivation: The fundamental notion of energy in classical physics is quasilocal; Use in black-hole thermodynamics.
* History: A quasilocal energy had been defined for spherically symmetric solutions by Tolman and Møller, but the field expanded in the 1980s, after a more general one was proposed by Penrose, based on twistor methods.
* Criteria: (i) It must vanish for gab = ηab; (ii) It must agree with known standard definitions for spherical symmetry; (iii) The spi limit must be MADM; (iv) The scri limit must be MBondi; (v) For an apparent horizon, it must equal Mirred; (vi) It must be positive, and monotonic in a suitable sense [@ Christodoulou & Yau in(88)].
@ References: Schmekel MoG-a0708 [brief review]; Szabados LRR(09) [rev]; Anderson PRD(10)-a1008 [Hamiltonian, constraints, and initial-boundary problem]; Sun et al a1307 [optimal choice of reference, and angular momentum]; Wang a1510-ln; Chen et al GRG-a1811.

Various Expressions > s.a. bel-robinson tensor.
* Ambiguities: Bergqvist showed that there are infinitely many definitions satisfying the criteria, which differ by boundary terms for finite regions, reflecting different choices of physical processes [& Nester].
* Tolman expression: For a stationary field, if V is a region of space containing matter,

MT:= V d3x |g|1/2 gab Tab .

@ References: Tolman PR(30), 62; Papapetrou PRIA(47) [simpler]; Landau & Lifshitz v2, ch11 [simplest].
* Møller expression:

MM:= V d3x χ00i,i ,   where   χ00i:= |g|1/2 (8πG)−1 g0a gib (g0b,ag0a,b) .

@ References: Tolman & Møller; Florides GRG(94); Lessner GRG(96); Xulu MPLA(00)gq [Kerr-Newman].
* Ashtekar-Hansen mass: For a 2-sphere B of area A and induced metric σij [@ Ashtekar & Hansen JMP(78)],

MAH:= (8πG)−1 (A/16π)1/2 B d2x |σ|1/2 σij σkl Cijkl .

* Brown-York mass: If H is the trace of the extrinsic curvature of the boundary S of a compact spatial hypersurface,

MBY = (8πG)−1 S (H0H) d2s .

* Christodoulou-Ruffini black hole irreducible mass: Given by MCR = (A/16πG2)1/2.
@ Penrose twistor expression: Penrose PRS(82), in(86); Tod CQG(86); Mason CQG(89); Godazgar & Kaderli a1807 [modification, and Kerr-Schild metrics].
@ Bartnik expression: Bartnik PRL(89); Koc gq/96; Jauregui JGP(19)-a1806 [smoothing the boundary conditions].
@ Liu-Yau expression: Yu a0706 [small- and large-sphere limits]; Ó Murchadha a0706 [as energy rather than mass]; Miao et al CMP(10) [problems, also for Brown-York expression].
@ Expressions: Hawking JMP(68); Christodoulou & Yau in(88); Katz et al CQG(88); Katz & Ori CQG(90); Bergqvist & Ludvigsen CQG(91); Dougan & Mason PRL(91); Bergqvist CQG(92), CQG(93); Helfer CQG(92); Szabados CQG(93); Hayward PRD(94)gq/93; Chen & Nester CQG(99)gq/98; Beetle & Fairhurst AIP(99)gq; Epp PRD(00)gq [and angular momentum]; Hayward gq/00 [as Noether charge]; Chen et al gq/02-proc [spinor]; Zhang AMS-gq/06; So IJMPD(07)gq/06; Wang & Yau CMP(09); So & Nester PRD(09)-a0901; Zhang CQG(09)-a0905; Ó Murchadha et al a0905-wd; Liu et al CQG(11)-a1105 [and choice of reference]; Katz & Khuri MPLA(12)-a1201; Wang a1211-conf; Faraoni a1510 [Newtonian aspect of Hawking quasilocal energy]; Álvarez et al a1811 [note].

Related Topics
* Martinez conjecture: The Brown-York quasilocal energy at a black hole outer horizon is twice its irreducible mass, (A/4π)1/2.
@ Martinez conjecture: Jing & Wang PRD(02)gq/01 [and string theory].
@ Positivity: Liu & Yau PRL(03)gq, JAMS(06)m.DG/04, O'Murchadha et al PRL(04)gq/03 [Kijowski M]; Shi & Tam JDG(02)m.DG/03.
@ Bounds: Shi & Tam CMP(07)m.DG/05 [Brown-York and Bartnik M].
@ For cosmological models: Chen et al MPLA(07)-a0705-conf [Bianchi models, FLRW models]; Nester et al PRD(08)-a0803 [Bianchi models]; Afshar CQG(09) [FLRW models]; Lapierre-Leonard et al PRD(17)-a1710 [Brown-York mass].
@ For other types of solutions: Balart PLB(10) [regular black holes, and Komar charge]; Wu et al GRG(12) [spherically symmetric]; Schmekel a1807 [rotating].
@ In modified gravity theories: Faraoni CQG(16)-a1508 [scalar-tensor gravity]; Chakraborty & Dadhich JHEP(15)-a1509 [Lanczos-Lovelock gravity]; Faraoni & Coté a1907, Giusti & Faraoni CQG(20)-a2005 [scalar-tensor gravity].
@ Other topics: Wang & Yau PRL(09)-a0804 [energy-momentum surface density]; Yang & Ma PRD(09)-a0812 [in lqg]; Wang & Yau CMP(10) [limit at spatial infinity]; Chen et al CMP(11)-a1002 [limit at null infinity]; Frauendiener & Szabados CQG(11)-a1102 [post-Newtonian limit].

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