Quasilocal Energy in General Relativity| 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,a − g0a,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 (H0−H) 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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