Quasilocal
Energy| Quasilocal
Energy |
In General > s.a. [quasilocal general
relativity];
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 1980's, after
a more general one was proposed by Penrose, based on twistor methods.
* Criteria: (i) Must
vanish for gab = 
ab;
(ii) 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, must
equal Mirred; (vi) Must be positive,
and monotonic in a suitable sense [@ Christodoulou & Yau in(88)].
@ References: Schmekel a0708 [brief
review].
Various Expressions
* 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
75, ch11 [simplest].
* Møller expression:
MM:= V
d3x 
00i,i , where 00i:=
|g|1/2 (8
G)–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 
ij [@
Ashtekar & Hansen JMP(78)],
MAH:= (8G)–1 (A/16)1/2 B d2x ||1/2 ij kl Cijkl .
* Brown-York mass: If H is the trace of the extrinsic curvature of the boundary S of a compact spatial hypersurface,
MBY = (8G)–1
S (H0–H)
d2s .
* Christodoulou-Ruffini
black hole irreducible mass: MCR =
(A/16G2)1/2.
@ Penrose twistor expression: Penrose PRS(82), in(86); Tod CQG(86);
Mason CQG(89).
@ Bartnik expression: Bartnik PRL(89); Koc gq/96.
@ Liu-Yau expression: Yu
a0706 [small-
and large-sphere limits]; Ó Murchadha a0706 [as
energy rather than mass].
@ 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 gq/99-in;
Epp PRD(00)gq [and
angular momentum]; Hayward gq/00 [as
Noether charge]; Chen et al gq/02-in
[spinor]; Zhang gq/06;
So gq/06.
Related Topics
* Martinez conjecture:
The
Brown-York quasilocal energy at a black hole outer horizon is
twice its irreducible mass, (A/4)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 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 a0705-in
[Bianchi models, FRW]; Nester et al a0803 [Bianchi models].
@ Other topics: Wang & Yau a0804 [energy-momentum
surface density].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jun 2008