185:
Now we look at light rays that are directed outward, along these normal vectors. The rays will either be diverging (the usual case one would expect) or converging. Intuitively, if the light rays are converging, this means that the light is moving backwards inside of the ball. If all the rays
90:âinwardsâ. The set of all such vectors engenders one outgoing and one ingoing null congruence. The surface is designated trapped if the cross sections of both congruences decrease in area as they exit the surface; and this is apparent in the mean curvature vector, which is:
59:
surfaces (topological spheres, tubes, etc.) with restricted bounds, their area tending to decrease locally along any possible future direction and with a dual definition with respect to the past. The trapped surface is a spacelike surface of co-dimension 2, in a
31:
defined the notion of closed trapped surfaces in 1965. A trapped surface is one where light is not moving away from the black hole. The boundary of the union of all trapped surfaces around a black hole is called an
501:
688:
775:
47:, trapped null surfaces are defined as only null vector fields giving rise to null surfaces. But marginally trapped surfaces may be spacelike, timelike or null.
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are negative, signifying that the mean curvature vector is timelike and future directed. The surface is marginally trapped if the outer expansion θ
535:
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More formally, if every null congruence orthogonal to a spacelike two-surface has negative expansion, then such surface is said to be trapped.
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vectors. The basic picture to think of here is a ball with pins sticking out of it; the pins are the normal vectors.
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This is the gold standard in black holes because of its place in history. It is also quite thorough.
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can be expressed as a linear combination of two future directed null vectors, normalised by:
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Bengtsson, Ingemar (December 22, 2011). "Some
Examples of Trapped Surfaces".
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Closed trapped surfaces are a concept used in black hole solutions of
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around the entire surface are converging, we say that there is a
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Senovilla, Jose M. M. (September 15, 2011). "Trapped
Surfaces".
427:
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Fashion, Faith, and
Fantasy in the New Physics of the Universe
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Trapped null surfaces are used in the definition of the
43:
is often used interchangeably. However, when discussing
282:"Gravitational collapse and space-time singularities"
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The surface is trapped if both the null expansions θ
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567:
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143:as a closed surface on which outward-pointing
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8:
615:Penrose interpretation of quantum mechanics
382:S. W. Hawking & G. F. R. Ellis (1975).
776:Mathematical methods in general relativity
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432:
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147:are actually converging (moving inwards).
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305:
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229:International Journal of Modern Physics D
178:) surface, and find its outward pointing
536:The Large, the Small and the Human Mind
384:The large scale structure of space-time
219:
420:This book is somewhat more up-to-date.
321:Nielsen, Alex B. (February 10, 2014).
23:which describe the inner region of an
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635:PenroseâHawking singularity theorems
139:of points defined in the context of
16:Concept used in black hole solutions
86:vector is directed âoutwardsâ and k
14:
680:Orchestrated objective reduction
553:White Mars or, The Mind Set Free
280:Penrose, Roger (January 1965).
1:
640:Riemannian Penrose inequality
119:= 0 and the inner expansion θ
547:and Stephen Hawking) (1997)
522:The Nature of Space and Time
323:"Revisiting Vaidya Horizons"
154:which typically surrounds a
406:University of Chicago Press
797:
704:Conformal cyclic cosmology
665:Penrose graphical notation
388:Cambridge University Press
605:Weyl curvature hypothesis
307:10.1103/PhysRevLett.14.57
259:10.1142/S0218271811020354
625:NewmanâPenrose formalism
585:Abstract index notation
396:Robert M. Wald (1984).
348:10.3390/galaxies2010062
744:John Beresford Leathes
684:PenroseâLucas argument
675:PenroseâTerrell effect
470:The Emperor's New Mind
64:. It follows that any
620:MooreâPenrose inverse
595:Geometry of spacetime
208:Raychaudhuri equation
750:Illumination problem
610:Penrose inequalities
188:trapped null surface
133:trapped null surface
127:Trapped null surface
62:Lorentzian spacetime
41:trapped null surface
486:The Road to Reality
478:Shadows of the Mind
339:2014Galax...2...62N
298:1965PhRvL..14...57P
251:2011IJMPD..20.2139S
400:General Relativity
141:general relativity
21:general relativity
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699:Andromeda paradox
670:Penrose transform
600:Cosmic censorship
235:(11): 2139â2168.
203:Null hypersurface
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732:Jonathan Penrose
689:FELIX experiment
655:Penrose triangle
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545:Nancy Cartwright
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513:Coauthored books
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39:A related term
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29:Roger Penrose
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25:event horizon
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755:Quantum mind
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580:Spin network
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333:(1): 62â71.
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781:Black holes
166:We take a (
770:Categories
214:References
172:orientable
162:Definition
156:black hole
145:light rays
51:Definition
734:(brother)
728:(brother)
559:) (1999)
529:) (1996)
369:1112.5318
267:119249809
242:1107.1344
176:spacelike
57:spacelike
55:They are
740:(sister)
722:(father)
568:Concepts
327:Galaxies
197:See also
713:Related
335:Bibcode
294:Bibcode
247:Bibcode
168:compact
555:(with
539:(with
525:(with
505:(2016)
497:(2010)
489:(2004)
481:(1994)
473:(1989)
412:
265:
180:normal
462:Books
364:arXiv
263:S2CID
237:arXiv
135:is a
123:⤠0.
93:H= âθ
82:The k
79:= â2
410:ISBN
343:doi
302:doi
255:doi
137:set
101:â θ
75:¡ k
772::
543:,
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290:14
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233:20
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174:,
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131:A
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27:.
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447:e
440:t
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331:2
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257::
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121:â
117:+
113:Âą
107:+
105:k
103:â
99:â
97:k
95:+
88:â
84:+
77:â
73:+
71:k
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