Asymptotic silence of generic cosmological singularities

0.14 MB·English

by  Lars Andersson

Asymptotic silence of generic cosmological singularities
Lars Andersson1, Henk van Elst2, Woei Chet Lim3, and Claes Uggla4
∗ † ‡ §
1Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA
2Astronomy Unit, Queen Mary, University of London,
Mile End Road, London E1 4NS, United Kingdom
3Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 and
4Department of Physics, University of Karlstad, S-651 88 Karlstad, Sweden
(Dated: January 25, 2005)
In this letter we investigate the nature of generic cosmological singularities using the framework
developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum
G2 cosmologies, models that are expected to capture the singular behavior of generic cosmologies
with no symmetries at all. In particular, our results indicate that asymptotic silence holds, i.e.,
that particle horizons along all timelines shrink to zero for generic solutions. Moreover, we provide
evidencethatspatialderivativesbecomedynamicallyinsignificantalonggenerictimelines,andthat
5
the evolution into the past along such timelines is governed by an asymptotic dynamical system
0
which is associated with an invariant set — the silent boundary. We also identify an attracting
0
subset on the silent boundary that organizes the oscillatory dynamics of generic timelines in the
2
singular regime. In addition, we discuss the dynamicsassociated with recurring spikeformation.

n
a PACSnumbers: 04.20.-q,98.80.Jk,04.20.Dw,04.25.Dm,04.20.Ha
J

5
2 The singularity theorems of Penrose and Hawking [1] asymptotically silent for timelines along which E i 0,
α
state that generic cosmological models contain an ini- and asymptotically silent and local for timelines a→long
2 tial singularity, but do not give any information on the which E i 0 and E i∂ S 0; in the latter case
v α → α i →
nature ofthis singularity. Heuristic investigationsofthis E i = 0 defines an unphysical invariant set, the silent
1 α
5 issueledBelinskiˇı,KhalatnikovandLifshitz[2](BKL)to boundary. [In UEWE the concept of a “silent singular-
0 propose that a generic cosmological initial singularity is ity” was defined. However, the possibility of “recurring
2 spacelike, local and oscillatory. Uggla et al. [3] (UEWE) spike formation,”discussed below,motivates the present
0
reformulated Einstein’s field equations (EFEs) by intro- distinctionsanddefinitions.] Theevolutionequationsfor
4
ducing scale-invariant variables which have the property S on the silent boundary are identical to EFEs for spa-
0
/ thatallindividualtermsinEFEsbecomeasymptotically tially self-similar (SSS) and spatially homogeneous (SH)
c
bounded, for generic solutions. This made it possible to models, in a symmetry adapted Hubble-normalized or-
q
- characterize a generic cosmological initial singularity in thonormal frame [4].
gr terms of specific limits. The numerical study of the pic- Motivated by the discussion in UEWE, we conjecture
: ture proposed by UEWE was initiated in Ref. [4], spe- that , the union of the bounded vacuum SH Type–I
iv cializingtoGowdyvacuumspacetimeswhichhaveanon- (KasnUev−ra)candSHType–IIsubsetsonthesilentboundary,
X oscillatory singularity. This letter presents the results of form an attracting subset that organizes the oscillatory
r the first detailed study of the oscillatory asymptotic dy- dynamics of generic timelines approaching an asymptot-
a
namics of inhomogeneous cosmologies from the dynami- ically silent and local vacuum singularity.
cal systems point of view introduced in UEWE.
To obtain the equations for vacuum G cosmologies,
2
HerewefocusonvacuumcosmologieswithanAbelian
we introduce coordinates t,x,y ,y and an orthonor-
1 2
symmetry group G2 with two commuting spacelike mal frame: e =N 1∂ , e{ =e 1∂ }+e 2∂ +e 3∂ ,
0 − t 1 1 x 1 y1 1 y2
Killing vector fields, and the spatial topology of a 3- e = e 2∂ and e = e 2∂ + e 3∂ , cf. Ref. [7];
2 2 y1 3 3 y1 3 y2
torus. This is arguably the simplest class of inhomoge- N and e i are functions of t and x only. For compar-
α
neous models that is expected to capture the properties ison with previous work, e is aligned with the Killing
2
of a generic oscillatory singularity. Numerical investiga-
vector field ∂ . We choose 2π-periodic coordinates x,
y1
tionsofG spacetimessupportingtheBKLproposalwere
2 y and y , yielding a spatial 3-torus topology, and a
1 2
carried out by Weaver et al. in Ref. [5, 6].
temporal gauge such that the area density of the G -
2
UEWEusedanorthonormalframeformalismandfac- orbits is given by := (e 2e 3) 1 e t; this is con-
2 3 − −
A ∝
tored out the expansion of a timelike reference congru- venient since the level sets of give a global foliation
ence e0 by normalizing the dynamicalvariables with the for maximally globally hyperboAlic vacuum G2 cosmolo-
isotropic Hubble expansion rate H of e0. This yielded a gies [8], and since t + at the singularity [9]. The
dimensionlessstatevectorX =(Eαi)⊕S,whereEαi are G2 symmetry implies→0 =∞e2(f) = e3(f) for any coordi-
the Hubble-normalized components of the spatial frame nate scalarf. Thus, only N 1∂ and e 1∂ actnontriv-
− t 1 x
vectors orthogonalto e0; eα =eαi∂i, Eαi =eαi/H. iallyoncoordinatescalars,andhencetheequationsofall
Theapproachtoaninitialsingularitywillbesaidtobe spatialframevariables excepte 1 decouple; the essential
1
2
Hubble-normalized variables are thus E 1 :=e 1/H and Because of the symmetry restrictions, E 1 is the only
1 1 1
a subset of connection components, which depend on t spatial frame variable in our state space; in the present
andxonly. Insertingtheaboverestrictionsintotherela- contextasymptoticsilenceisthusassociatedwithE 1
1
→
tions in App. 5 of UEWE yields 0=Aα =N =N = 0, while E 1 = 0 is referred to as the silent boundary.
1α 33 1
Σ =U˙ =U˙ (α=1,2,3),andthe spatialframe gauge Our numerical experiments, which employ the RNPL [10]
12 2 3
R = Σ , R = Σ , R = 0. In addition, it is con- and CLAWPACK [11] packages with up to 216 spatial grid
1 23 2 31 3
− −
venient to define: Σ := 1(Σ +Σ )= 1Σ , Σ := pointsonthex-interval(0,2π),indicate thatasymptotic
1 (Σ Σ ), Σ+:= 21 Σ22 , Σ33:= −1 2Σ 11, N− := silence holds in the present G case for all timelines of
2√3 22 − 33 × √3 23 2 √3 31 − 2
1 N andN := 1 N . TheHubble-normalizedvari- a generic solution. Indeed, our numericalsimulations in-
2√3 22 × √3 23 dicate that max (E 1) decays exponentially. Moreover,
ableshavethefollowingphysicalinterpretation: Σ ,Σ , x 1
Σ , Σ2 are shear variables for e0; U˙ = U˙1 describ+es th−e theyindicatethatlimt→+∞(∆:=||E11∂xS||2)=0along
ac×celeration of e ; N and N are spatial connection generictimelinesofagenericsolution,i.e.,genericallythe
0
− × singularity is asymptotically silent and local, and hence
components that determine the spatial curvature; R
α
yields the angular velocity of the spatial frame e . in this case the asymptotic dynamics is governed by the
α
The lapse function is given by N = 1H 1(1+Σ{ ) }1. equations on the silent boundary.
−2 − + − Onthe silentboundaryE 1 =0,the integrabilitycon-
The deceleration parameter q and the spatial Hubble 1
ragnr:a=dd∂i−e1nHt:=−r1Ea∂1r11eH∂dx,e.rfieTnsephdeecsetbiyvdee(lfiyqn,+iwtiio1tnh)s∂:=y0ie:=l−dH−th2−e(11i∂n+0teHΣgr+aa)bni∂ld-t drweietdiuoacnreesahntoedre0Eqc=.on((c1ree−r)nye√die3lNdw×irt)h2Σ=t2h.e−I3gnfencNoe×nr2at,rlawcshatislteeoΣER2qe.f6=.(2[4d0]),,
ity condition ∂ r ∂ q =(q+2Σ )r (r U˙)(q+1). for which r = √3N and hence f = 1; in this case
0 − 1 + − − the equations on th×e silent boundary−are identical to
Imposing the above restrictions and gauge choices on
EFEs in vacuum yields the following evolution equations the Hubble-normalized equations of the exceptional SSS
and constraints: Type– 1VI0 models; see Wu [12, p. 635].
−
Ournumericalexperimentssuggestthat,inadditionto
∂0E11 =(q+2Σ+)E11 (1a) tEim11el→ine0s,oCf a:=ge(nU˙e,rric,Nso×l,uNti−onΣ×w)he→n 0t hol+ds fo.rOgennetrhiec
∂0(1+Σ+)=(q−2)(1+Σ+)+3Σ22 (1b) silent boundary E11 = 0, C = 0 yields→the∞Kasner and
∂ Σ =(q 2 3Σ +√3Σ )Σ (1c) SH Type–II subsets which are defined by 0 = E 1 =
0 2 + 2 1
∂ Σ +∂ N =(q−2)−Σ +(r U˙)−N N = N = U˙ = r, 1 = Σ2+ +Σ2 +Σ2 +Σ22, q = 2
0 − 1 × − − − × an−d 0 =×E 1 = Σ = N = U˙ =− r, q×= 2(1 N2),
+2√3Σ2×−2√3N−2 −√3Σ22 (1d) respectively1. × × − −
∂ N +∂ Σ =(q+2Σ )N +(r U˙)Σ (1e) Withthepresentgaugechoices,theKasnersubsetcon-
0 1 +
× − × − − tains a subset of equilibrium points: 0 = E 1 = Σ =
∂ Σ ∂ N =(q 2 2√3Σ )Σ 1 2
0 ×− 1 − − − − × N = Σ = N = U˙ = r, 1 = Σ2+ +Σ2, the Kasner
(r U˙ +2√3N )N (1f) cir×cle, ×, which−plays an essential role fo−r the asymp-
− − × − K
∂0N ∂1Σ =(q+2Σ++2√3Σ )N totic dynamics. A linear stability analysis of K shows
−− × − − that all variables are stable when t + , except for
(r U˙ 2√3N )Σ , (1g) (N ,Σ ,Σ ) which obey: → ∞
− − − × × − × 2
and N =Nˆ e−[1−k(x)]t (3a)
− −
Σ =Σˆ e k(x)t (3b)
0=(∂1 r+U˙)(1+Σ+) (2a) × × −
− Σ =Σˆ e[3 k(x)][1+k(x)]t/4 ; (3c)
0=1 (Σ2 +Σ2+Σ2 +N2 +Σ2 +N2) (2b) 2 2 −
0=(1−+Σ++)U˙ +23(N−Σ ×N Σ× ) − (2c) E11 andC decay exponentiallyanduniformly [N−Σ× ∝
× −− − × exp( t)]. Here “hatted” variables are functions of x
0=(∂1−r+√3N×)Σ2 , (2d) on√ly3.−Σˆ O(xn)/K[1,+ΣΣˆ+ (=x)].Σˆ+T,huΣs−, N=, ΣΣˆ−,, Σandarke(uxn)st:a=-
+ 2
wU˙h)eU˙re. qSin:=ce 2w(eΣ2+ar+e cΣo2−nc+ernΣe2×d +witΣh22)ge−ne13ri(c∂f1ea−turre+s, tb−ilveelwyh;−esneek(Fxi)g.>1(1a,).k(xT)he<u0n,s−ta1b−l<e mk×(oxd)e<N3,irnedsupceecs-
−
we restrict to the case Σ = 1 (Σ = 1 yields physical curvature transitions, associated with the SH
+ +
6 − −
the Minkowski spacetime). We use the gauge con- Type–II subset on E 1 = 0, while Σ and Σ induce
1 2
×
straint (2a) and the Codacci constraint (2c) to solve frame transitions that lead to rotations of the spatial
for r and U˙ and so obtain the reduced state vector frame and multiple representations of the same solution,
X = (E 1,Σ ,Σ ,Σ ,N ,Σ ,N ) = (E 1) S. Note see Fig. 1(b); nevertheless, for the present frame choice
1 + 2 1
− × × − ⊕
that the Gauß constraint (2b) implies that the compo- it is these gauge transitions that make repeated curva-
nents of S are bounded. ture transitions possible, and hence they have indirect
3
Σ Σ Σ Σ
− − − −
Σ
Q × T
2 3
Σ Σ ,Σ
× × 2
T
1 Q Σ Σ Σ Σ
1 + + + +
N
− Σ
2
Q T
3 N−,Σ2 2 (a) (b)
(a) (b) FIG. 2: Projections onto the (Σ+Σ−)-plane of a state space
orbit along the typical timeline x = 0.3 for (a) the full G2
FIG.1: (a)UnstablevariablesonK,and(b)singletransition system, and (b) its restriction to the silent boundary. In
sets associated with N− (dotted), Σ× (dash-dotted) and Σ2 both cases the orbits approach Uv−ac, i.e., ∆ → 0 in (a), and
(solid). (E11,C)→0 in both (a) and (b).
tor. However, there are indications that spiky features,
physical implications. The N , Σ and Σ transitions
2
− × closely related to the spikes in Gowdy vacuum space-
implythatk(x)changesaccordingtotherulesk 2 k,
→ − times, form along exceptional timelines; for such time-
k k and k (k+3)/(k 1), respectively.
→− → − lines neither C nor ∆ has a limit.
The variables and equations that describe on
E 1 = 0 and the exceptional SH Type–VI Ucav−asce, as Recall that for the present general G2 case, the whole
gi1ven by Hewitt et al. [13], are identical. ∗−A1s/9shown in of K is unstable with respect to at least one of N−, Σ×,
Σ ; see Eqs. (3). As in the Gowdy case, spike formation
Sec. 5 of Ref. [13], there exist two integrals that de- 2
iscausedbytheoccurrenceofazeroforoneofthesevari-
scribe the transition orbits. Although multiple transi-
ables at a point (t,x(t)) when the system is close to .
tions are possible, single transitions increasingly domi- K
It follows from Eqs. (1c) and (2d) that, for a generic
nate. However, since frame transitions constitute gauge
smooth solution, Σ cannot cross zero and thus pro-
effects we will not pursue this further. What is impor- 2
duces no spikes. Spikes in Σ are “false” (gauge)spikes,
tant physically is that the variety Uv−ac induces an in- while spikes in N are “tru×e” (physical) effects which
finite sequence of Kasner states related by SH Type– −
yieldinherentlyinhomogeneousdynamicalfeaturesinthe
II curvature transitions according to the frame invari-
Hubble-normalized Weyl curvature scalars. Linear anal-
ant BKL map: u u 1, if u 2, and 1/(u 1),
if 1 < u < 2, whe→re u i−s defined ≥frame invariantl−y by ysis at K shows E11 ∝Eˆ11e−t and
detΣαβ = 13ΣαβΣβγΣγα =2−27u2(1+u)2/(1+u+u2)3. E11∂xN Eˆ11h∂xNˆ +tNˆ ∂xk(x)ie−[2−k(x)]t .
Numerical investigations of vacuum SH Type–VI − ∝ − −
∗1/9
and SSS Type– 1VI0 models indicate that generic s−olu- The state space orbits of the spatial points outside the
−
tions asymptotically approach Uv−ac. Our investigation particle horizon of (t,x(t)), defined by N (t,x(t)) = 0,
suggests that this is also true for the evolution associ- undergocurvaturetransitionswithN = −(1)andoppo-
atedwithgenerictimelinesofthepresentinhomogeneous site signson either side of x(t) when−k(x)O>1; since x(t)
vacuum G2 cosmologies, since our numerical results in- does not go through such a transition this leads to the
dicate that (E11,C,∆)→0 when t→+∞ for a generic formation of a spike. For k(x)>2, ∂1N =E11∂xN is
timeline, see Fig. 2, and that thus the BKL map holds unstable on , and hence grows in mod−ulus at (t,x(−t)),
asymptotically for such a timeline. whichleadstKoagrowthinmodulusofΣ . Sincethepar-
Belinskiˇı[14] expressedconcernthat spatialstructure, ticle horizon size at (t,x(t)) is of order×E 1, see UEWE,
1
created by the effect of different timelines going through the above implies that ∂ N grows to order 1/E 1 and
x 1
transitions at different times, could cause problems for that ∂ N , and thus also ∆−, is then (1) at (t,x(t)).
1
theBKLscenario. Numericalexperimentsshowthatthis It therefo−re follows that the dynamicsOfails to be local.
is not the case for generic timelines. The reason is that Moreover,onecansimilarlyarguethatC becomes (1).
O
spatial structure, created by the mechanism described Since the dynamics fails to be local at (t,x(t)), it is not
above, develops on superhorizon scales; ∆ 0 within governedbythesilentboundarydynamicalsystem. Nev-
→
theshrinkingparticlehorizonsofagenerictimelinewhen ertheless,ourinvestigationsindicatethattheasymptotic
t + . dynamics is quite simple, and that it is related to that
→ ∞
Our investigations indicate that E 1 0 as t + onthesilentboundary. Numericalsimulationsshowthat
1
foralltimelines ofa genericsolution,an→dthat(C→,∆)∞ the orbit described by S(t,x(t)) during the formation
→
0 for generictimelines so that is a localpast attrac- and smoothing out of a spike is described by the map
Uv−ac
4
Σ the system spends a dominant portionof its time under-
−
going oscillations in the vicinity of the Taub points; this
can be expected to hold also in the present G case. In
2
G cosmologies, recently studied numerically by Garfin-
0
kle [18] in terms of the framework of UEWE, these is-
suesshouldcauseformidableproblems;theirresolutionis
likelytoconstituteamajorsteptowardarigorousanaly-
Σ sisofgenericspacetimesingularities,andanunderstand-
+
ing of the cosmic censorship problem.
We are grateful to Matt Choptuik for generous help
in getting started with the RNPL package. We also
thank John Wainwright for many helpful discussions.
FIG. 3: Projection onto the (Σ+Σ−)-plane of a state space
orbit undergoing a spike transition, followed by Σ2 and Σ× LA is supported in part by the NSF, contract no. DMS
inducedframetransitionsandanotherspiketransition(solid). 0407732. CU is supported by the Swedish Research
The combination of N− curvature and Σ× frame transi- Council. LAthankstheErwinSchr¨odingerInstitute,Vi-
tions corresponding to the second spike transition is shown enna, and the Albert Einstein Institute, Golm, and HvE
(dashed). See also Ref. [15, p.152].
and CU thank the Department of Mathematics, Univer-
sity of Miami, for hospitality during part of the work on
this paper.
k 4 k, equivalent to a sequence of local N –Σ –N
→ − − × −
transitions; following Ref. [15], we refer to this behavior
as a spike transition, see Fig. 3. The simplicity of this
structure suggests that there may exist an effective dy-
namical system governing the spike transitions, playing ∗ Electronic address: [email protected]
a role analogous to that of the silent boundary system. † Electronic address: [email protected]
Numerical investigations suggest that an isolated zero ‡ Electronic address: [email protected]
in N may persist as t + ; if true this yields an in- § Electronic address: [email protected]
finite−sequence of recurr→ing s∞pike transitions. Since the [1] S. W. Hawking and R. Penrose, Proc. R. Soc. Lond. A
314, 529 (1970).
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[2] V. A. Belinskiˇı, I. M. Khalatnikov, and E. M. Lifshitz,
converge exponentially to a point xspike, and since the Adv.Phys. 31, 639 (1982).
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[3] C.Uggla,H.vanElst,J.Wainwright,andG.F.R.Ellis,
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spike
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[6] B. K.Berger, J.Isenberg, and M.Weaver,Phys.Rev.D
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For the Gowdy case (Σ = 0), the sequence of spike
2 Texas at Austin (1995), Software at URL:
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laplace.physics.ubc.ca/Members/matt/Rnpl/.
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K
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[15] W. C. Lim, Ph.D. thesis, University of Waterloo (2004),
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gr-qc/0410126.
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α 1
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