333:
214:
745:
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150:
1059:
Global theory of minimal surfaces: proceedings of the Clay
Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001
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328:{\displaystyle m_{H}(\Sigma ):={\sqrt {\frac {{\text{Area}}\,\Sigma }{16\pi }}}\left(1-{\frac {1}{16\pi }}\int _{\Sigma }H^{2}da\right),}
1062:. Clay mathematics proceedings. Providence, RI : Cambridge, MA: American Mathematical Society ; Clay Mathematics Institute.
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921:
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118:
1086:
779:
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919:
Finster, Felix; Smoller, Joel; Yau, Shing-Tung (2000-06-01). "Some Recent
Progress in Classical General Relativity".
774:
762:
758:
39:
812:
65:
860:
447:
444:
A result of Geroch implies that
Hawking mass satisfies an important monotonicity condition. Namely, if
986:
940:
824:
391:
155:
1042:
1030:
956:
930:
898:
848:
636:
518:
1063:
1022:
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994:
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mass, depending on whether the surface is asymptotic to spatial infinity or null infinity.
397:
663:
990:
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flows outward at a speed equal to the inverse of the mean curvature. In particular, if
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960:
1057:
980:
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On the behavior of quasi-local mass at the infinity along nearly round surfaces
47:
1008:
844:
17:
1026:
757:
Hawking mass is not necessarily positive. However, it is asymptotic to the
999:
982:
Quasi-Local Energy-Momentum and
Angular Momentum in GR: A Review Article
935:
952:
115:
be a 3-dimensional sub-manifold of a relativistic spacetime, and let
51:
903:
680:
is the unit vector opposite of the mean curvature direction, then
43:
42:. It is a measure of the bending of ingoing and outgoing rays of
54:
surrounding the region of space whose mass is to be defined.
27:
One of the possible definitions of mass in general relativity
454:
131:
75:
475:
has nonnegative scalar curvature, then the
Hawking mass of
740:{\displaystyle {\frac {d}{dt}}m_{H}(\Sigma _{t})\geq 0.}
542:
is a family of connected surfaces evolving according to
603:{\displaystyle {\frac {dx}{dt}}={\frac {1}{H}}\nu (x),}
689:
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750:Said otherwise, Hawking mass is increasing for the
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507:
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467:
433:
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350:
327:
200:
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107:
145:{\displaystyle \Sigma \subset {\mathcal {M}}^{3}}
893:Shi, Yuguang; Wang, Guofang; Wu, Jie (2008).
421:about the central mass is equal to the value
152:be a closed 2-surface. Then the Hawking mass
8:
108:{\displaystyle ({\mathcal {M}}^{3},g_{ab})}
817:Annals of the New York Academy of Sciences
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38:is one of the possible definitions of
979:Szabados, László B. (December 2004).
7:
837:10.1111/j.1749-6632.1973.tb41445.x
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641:
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468:{\displaystyle {\mathcal {M}}^{3}}
369:
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195:
172:
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25:
495:is non-decreasing as the surface
394:, the Hawking mass of any sphere
811:Geroch, Robert (December 1973).
1056:Hoffman, David A., ed. (2005).
922:Journal of Mathematical Physics
728:
715:
594:
588:
234:
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181:{\displaystyle m_{H}(\Sigma )}
175:
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102:
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1:
1047:: CS1 maint: date and year (
865:: CS1 maint: date and year (
780:Inverse mean curvature flow
752:inverse mean curvature flow
653:{\displaystyle \Sigma _{t}}
535:{\displaystyle \Sigma _{t}}
1103:
985:. Vol. 7. p. 4.
775:Mass in general relativity
40:mass in general relativity
633:is the mean curvature of
508:{\displaystyle \Sigma }
488:{\displaystyle \Sigma }
375:{\displaystyle \Sigma }
201:{\displaystyle \Sigma }
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441:of the central mass.
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414:{\displaystyle S_{r}}
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110:
813:"Energy extraction*"
687:
673:{\displaystyle \nu }
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392:Schwarzschild metric
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1000:10.12942/lrr-2004-4
991:2004LRR.....7....4S
945:2000JMP....41.3943F
829:1973NYASA.224..108G
1087:General relativity
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703:
626:{\displaystyle H}
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434:{\displaystyle m}
351:{\displaystyle H}
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208:is defined to be
16:(Redirected from
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953:10.1063/1.533332
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929:(6): 3943–3963.
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972:Further reading
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823:(1): 108–117.
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360:mean curvature
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32:Hawking energy
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936:gr-qc/0001064
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917:Section 2 of
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891:Section 4 of
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880:Hoffman 2005
875:
861:cite journal
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816:
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801:, p. 21
799:Hoffman 2005
794:
756:
749:
612:
443:
389:
337:
61:
36:Hawking mass
35:
31:
29:
18:Hawking mass
882:, Lemma 9.6
786:References
386:Properties
58:Definition
48:orthogonal
1043:cite book
1009:2367-3613
904:0806.0678
853:222086296
845:0077-8923
732:≥
720:Σ
668:ν
642:Σ
586:ν
524:Σ
503:Σ
483:Σ
370:Σ
297:Σ
293:∫
286:π
274:−
259:π
251:Σ
232:Σ
196:Σ
173:Σ
126:⊂
123:Σ
46:that are
1081:Category
1035:40602589
1027:28179865
961:18904339
769:See also
1018:5255888
987:Bibcode
941:Bibcode
825:Bibcode
761:or the
390:In the
358:is the
50:to a 2-
1066:
1033:
1025:
1015:
1007:
959:
851:
843:
613:where
338:where
52:sphere
1031:S2CID
957:S2CID
931:arXiv
899:arXiv
849:S2CID
763:Bondi
44:light
1064:ISBN
1049:link
1023:PMID
1005:ISSN
867:link
841:ISSN
660:and
246:Area
62:Let
30:The
1013:PMC
995:doi
949:doi
833:doi
821:224
759:ADM
362:of
188:of
34:or
1083::
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1041:{{
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951::
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835::
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724:t
716:(
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700:t
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693:d
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592:x
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581:H
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567:t
564:d
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556:d
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455:M
429:m
407:r
403:S
346:H
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315:a
312:d
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279:1
271:1
267:(
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229:(
224:H
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70:(
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