In General
* Idea: A spacetime is
said to be nonsingular if it is timelike and null geodesically complete, by
analogy
with Riemannian geometry, where geodesic
completeness
is equivalent to the usual metric completeness.
* Remark: When this
condition is violated we may not have what we would like to call a singularity
physically;
And, when satisfied, we might have,
e.g., timelike
lines of finite acceleration which are incomplete.
* Consequences: Naked singularities
would be a problem for predictive physics.
Trying to Avoid Singularities > s.a. quantum
cosmology;
quantum gravity, quantum-gravity
models and phenomenology.
* History: An attempt
was made by the Soviets with the mixmaster universe; Other possibilities include
cosmic censorship, violation of energy conditions,
"gravastars", perhaps quantum gravity (non-commutative? > see Kasner
Solutions).
@ General references: Einstein & Rosen BB(31), PR(35);
Einstein AM(39), AM(45);
Einstein & Straus AM(46); in Misner et al
73.
@ By violating energy conditions: Fulling & Parker PRD(73) [quantum];
Bekenstein PRD(75) [classical]; Fakir gq/98.
@ By going to a different metric: Quirós PRD(00)gq/99,
et al PRD(00)
[geometric duality in general relativity and Brans-Dicke];
Quirós gq/00,
et al gq/00/PRD
[conformal rescaling].
@ By extending the spacetime: Sniatycki in(91)
[using Jacobi metric];
Schroeter a0906 [FRW spacetime]; > s.a. schwarzschild
spacetime
@ Related topics: Heller & Sasin IJTP(95),
GRG(99)gq/98 [algebraically];
Raptis IJTP(06)gq/04
[Schwarzschild,
finitary-algebraic];
Goswami & Joshi gq/05 [by
not forming trapped surface]; Gershtein et al TMP(05)gq [in
field theory of gravitation?]; Mac Conamhna CMP(08)-a0708 [M-theory]; > s.a. modified
electromagnetic theory, non-commutative
gravity.
Singularity Theorems
* Idea: They establish
conditions under which a spacetime will develop a singularity.
* History: The existence
of singularities in general relativity has been known for a long time (e.g.,
Schwarzschild solution), but it was not until Penrose's work that it became
clear
that they are not an artifact of spherical symmetry in the collapse of
stars,
or of homogeneity and isotropy in the cosmological case, and are in
fact a
general phenomenon.
* Rigid singularity theorem:
A globally hyperbolic spacetime satisfying the strong energy condition and
containing past trapped sets, either is timelike geodesically
incomplete
or splits isometrically as
× R [@
Narita PRD(98)gq].
* Gannon's theorem:
Any localized non-trivial topology in space will develop into a singularity.
@ Gannon's theorem: Gannon JMP(75), GRG(76).
@ Versions and generalizations: Borde PRD(94)gq,
& Vilenkin IJMPD(96)gq [including
inflation]; Raychaudhuri MPLA(00)gq;
Senovilla Pra(07)gq/06-in,
a0712-in [based
on spatial averages]; Minguzzi CMP(09)
[either chronology violation or singularity].
Hawking & Penrose Theorem
* Idea: If a spacetime
satisfies the chronology condition, the Einstein equation, the condition
+
3pi > 0,
is sufficiently general, and admits a closed spacelike hypersurface, then it
cannot be geodesically complete along all timelike and null directions.
* Remark: If we want
to know where the singularity occurs, we have to add other, physically questionable,
conditions.
* And space of metrics:
The theorems imply that, if the set of Lorentz metrics on a manifold is given
a reasonable topology, those geodesically incomplete
are an open set.
@ Texts: in Hawking & Ellis 73; in Ryan & Shepley 75; in Naber
88 [II]; Joshi 93; Clarke 94 [III–IV].
@ First hint: Raychaudhuri PR(55).
@ Theorem: Hawking PRL(65);
Hawking & Ellis PL(65);
Penrose PRL(65);
Geroch PRL(66);
Hawking PRL(66),
PRS(66), PRS(66), PRS(67); Geroch AP(68),
in(70); Hawking & Penrose
PRS(70);
Khalatnikov & Lifshitz PRL(70);
Clarke CMP(76); Clarke & Schmidt GRG(77);
Tipler et al in(80); Clarke & Królak
JGP(85);
Borde CQG(87), JMP(87);
Szabados JMP(87);
Clarke pr(88); Ford IJTP(03)gq [rev,
quantum loopholes].
@ And energy conditions: Tipler PRD(78); Roman PRD(88); Parker & Wang
PRD(90).
@ And causality violation: Tipler PRL(76), AP(77);
Borde CQG(85);
Kriele
GRG(90),
PRS(90).
Other References > s.a. collapse [including
Hoop conjecture]; cosmic censorship; cosmology and models; types;
spacetime boundary.
@ Reviews: Canarutto RNC(88); Clarke in(88);
Senovilla GRG(98);
Cotsakis LNP(02)gq-in
[in cosmology]; Rendall gq/05-in;
Senovilla phy/06-in;
Cotsakis gq/07-in.
@ Philosophical: Earman 95; Lam PhSc(07).
@ General references: Geroch JMP(68),
in(68); Hájícek GRG(70);
Newman GRG(71);
Penrose in(78); Barrow & Tipler PRP(79),
PLA(81);
Fuchs et al FdP(88); Natario m.DG/06 [introduction
for mathematicians]; Joshi SA(09)feb [naked singularities].
@ And initial surfaces: Wojtkiewicz PRD(90).
@ Data at the singularity: Eardley et al JMP(72);
Tod CQG(90); & Goode & Wainwright.
@ Strength and physical properties: Kánnár & Rácz JMP(92);
Kánnár GRG(95)
[in Einstein-Cartan theory]; Kriele & Lim CQG(95);
Ori PRD(00).
@ Role, uses of singularities: Earman FP(96);
Lopez CQG(93);
Horowitz & Myers GRG(95)gq.
@ Probing singularities: Horowitz & Marolf PRD(95)gq;
Ishibashi & Hosoya PRD(99)gq;
Piechocki PLB(02);
Konkowski et al in(03)gq/04
[quantum particles]; Blau et al JHEP(06)ht [with
scalar fields].
@ In other theories: Eguchi MPLA(92)
[topological field theories]; Novello et al CQG(00)
[general relativity
+
non-linear
electrodynamics]; Ferraz Figueiro & Saa a0906 [modified-gravity
models]; > s.a. brans-dicke; 3D quantum
gravity; quantum-gravity
phenomenology.
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aug 2009