Self-similar cosmological solutions with dark energy. I: formulation and asymptotic analysis

0.3 MB·English

by  Tomohiro Harada

Self-similar cosmological solutions with dark energy. I:
formulation and asymptotic analysis
1Tomohiro Harada∗, 1,2,3,4Hideki Maeda† and 5,6B. J. Carr‡
1Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
2Centro de Estudios Cient´ıficos (CECS), Arturo Prat 514, Valdivia, Chile
3Department of Physics, International Christian University,
3-10-2 Osawa, Mitaka-shi, Tokyo 181-8585, Japan
4Graduate School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan
5Astronomy Unit, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
6Research Center for the Early Universe, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan
(Dated: February 1, 2008)
Based on the asymptotic analysis of ordinary differential equations, we classify all spherically
symmetric self-similar solutions to the Einstein equations which are asymptotically Friedmann at
8 large distances and contain a perfect fluid with equation of state p = (γ −1)µ with 0 < γ <
0 2/3. This corresponds to a “dark energy” fluid and the Friedmann solution is accelerated in this
0 case due to anti-gravity. This extends the previous analysis of spherically symmetric self-similar
2
solutionsforfluidswithpositivepressure(γ >1). However,inthelattercasethereisanadditional

n parameter associated with the weak discontinuity at the sonic point and the solutions are only
a asymptotically“quasi-Friedmann”,inthesensethattheyexhibitanangledeficitatlargedistances.
J In the 0 < γ < 2/3 case, there is no sonic point and there exists a one-parameter family of

2 solutions which are genuinely asymptotically Friedmann at large distances. We find eight classes
ofasymptoticbehavior: Friedmannorquasi-Friedmannorquasi-staticorconstant-velocityatlarge

] distances, quasi-Friedmann or positive-mass singular or negative-mass singular at small distances,
c
and quasi-Kantowski-Sachs at intermediate distances. The self-similar asymptotically quasi-static
q
and quasi-Kantowski-Sachs solutions are analytically extendible and of great cosmological interest.
-
r Wealso investigate their conformal diagrams. The results of thepresent analysis are utilized in an
g accompanying paperto obtain and physically interpret numerical solutions.
[

PACSnumbers: 04.70.Bw,95.36.+x,97.60.Lf,04.40.Nr,04.25.Dm
3
v
8
I. INTRODUCTION case, for a given value of γ, the solutions are described
2
5 by two parameters and, providing one restricts atten-
0 tion to shock-free perfect fluids with positive pressure
There is great interest in spherically symmetric self-
. (1 < γ 2), they have been completely classified. This
7 similar solutions to Einstein’s equations because of their ≤
classification has been achieved using a combination of
0
numerousapplicationsinastrophysicsandcosmology[1].
7 the “comoving” approach (in which the coordinates are
Indeed there is now considerable evidence for the “simi-
0 adapted to the fluid 4-velocity) and the “homothetic”
larityhypothesis“,whichpostulatesthatsphericallysym-
: approach (in which the coordinates are adapted to the
v metricsolutionsmaynaturallyevolvetoself-similarform
homothetic vector). These approaches have been used
i
X in a variety of situations. The status of this hypothesis by Carr and Coley [4] (henceforth CC) and Goliath et
hasbeenrecentlyreviewedbyCarrandColey[2]. Inview
r al. [5], respectively, although a full understanding of the
a ofthis,itisclearlyimportanttohaveascompleteaclas-
solutions requires that one combines them [6].
sification of spherically symmetric self-similar solutions
as possible. A key step in the CC analysis is the derivation of
all possible asymptotic behaviors at large and small dis-
Spherically symmetric self-similar solutions have the
tances [7]. For positive pressure, there are at least three
featurethatalldimensionlessquantitiescanbeexpressed
kinds of behavior at large spatial distances (usually cor-
in terms of z = r/t, where r and t are suitably chosen
responding to the limit z ): (1) asymptotically
radialand time coordinates. If the sourceof the gravita- → ∞
Friedmann(1-parameter);(2)asymptoticallyKantowski-
tional field is a single perfect fluid, Cahill and Taub [3]
Sachs (1-parameter),though these are probably unphys-
have shown that the only barotropic equation of state
ical for γ > 2/3); and (3) asymptotically quasi-static
compatible with the similarity assumption has the form
(2-parameter). There are also two families of solutions
p = (γ 1)µ for some constant γ, where µ and p are
− which exist only when γ > 6/5: (4) asymptotically
the energydensityandthe pressure,respectively. Inthis
Minkowskiatinfinitez (1-parameter);and(5)asymptot-
icallyMinkowskiatfinitez but infinite physicaldistance
(2-parameter). At small spatial distances, the solutions
are of four kinds: they contain either (a) a black hole
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected] singularityor(b)anakedsingularityatfinitez (butzero
‡Electronicaddress:[email protected] physical distance) or they can be connected to z =0 via
2
a sonic point, in which case they are either (c) static or since there can be no discontinuities and solutions are
(d) represent a perturbation of the Friedmann solution. analytic everywhere. The second is that there is no ex-
The complete family of 1 < γ 2 solutions can now act static solution. although there are still asymptoti-
be found by combining the five k≤inds of large-distance cally quasi-static solutions (a point which was missed in
behavior and four kinds of small-distance behavior. The ref. [7]). On the other hand, the Kantowski-Sachs and
way in which one connects the large-distance and small- asymptotically quasi-Kantowski-Sachssolutions now be-
distance solutions depends crucially on whether or not come physical. Finally, the solutions which were asymp-
the solution passes through a sonic point. If the solu- totically Minkowski at large distances are now replaced
tions remain supersonic (or subsonic) everywhere, then withsolutionswhichareasymptoticallysingularatsmall
thesmall-z behaviorisuniquelydeterminedbythelarge- distances.
z behavior. However,ifthereisasonicpoint,thebehav- Despite these differences, many features of the CC
ior of the solutions is much more complicated because classification still apply. In particular, there is still a
the equations do not determine their behavior uniquely 1-parameter family of solutions asymptotic to the flat
there, so there can be a discontinuity. Indeed, only a Friedmann model at large distances and these are of
subset of solutions are “regular” at a sonic point in the great physical interest since they may have cosmologi-
sense that the pressure gradient is finite and they can cal applications. In the positive-pressure case, some of
be extended beyond it. Because of this feature, the fam- these solutions are supersonic everywhere and contain
ily of solutions with a regular center and a regular sonic black holes which grow at the same rate as the particle
pointhaveabandstructure,withthesolutionswhichare horizon [17, 18]. Others represent density perturbations
analytic at the center and sonic point forming a discrete in a Friedmann background which always maintain the
subset of these [8, 9, 10]. This feature is very important same formrelativeto the particle horizon[19]. However,
for naked singularity formation and critical behavior in recently it was pointed out that none of these positive-
the gravitationalcollapse of a perfect fluid [11, 12]. pressure solutions are “properly” asymptotic Friedmann
In this and the accompanying papers [13] (henceforth because they exhibit a solid angle deficit at infinity [20].
PaperII),we take afirststepin extending the classifica- They may still be relevantto the realuniverse(since ob-
tion of CC to the negative pressure case (γ < 1), using servations may not preclude such an angle deficit) but
a combination of numerical and analytical studies. In it would be more accurate to describe them as “quasi-
fact, part of the work is already done, because ref. [7] Friedmann”.
does include the asymptotic analysis for γ < 1. How- In the 0 < γ < 2/3 case, we will show that there are
ever, there are some errors in that work (see Appendix genuineasymptoticallyFriedmannsolutions. Wewillan-
A) and the full family of solutions has not yet been ana- alyzethesesolutionsnumericallyandexploittheseresults
lyzed. Thispaperwillfocusparticularlyonthecasewith inPaperII[13]tointerpretthesolutionsphysically. The
0 < γ < 2/3. Although this equation of state violates keyfeatureistheexistenceofasymptoticallyKantowski-
the strong energy condition and was little emphasized Sachs and static solutions, both of which are extendible
by CC, it may be very relevant for cosmology – both in analytically. This leads to the possibility of cosmological
the early universe (when inflation occurred [14]) and at black hole, wormhole and white hole solutions.
the current epoch [15] (when accelerationmay be driven The plan of this paper is as follows. In Section II,
by some form of “dark energy”). In both situations the we present the basic field equations for spherically sym-
matter model exhibits “anti-gravity” but the dominant, metric self-similar spacetimes. We describe some exact
null and weak energy conditions still hold. solutions in Section III and solutions which are asymp-
As in the positive pressure case, it should be stressed totic to these In Section IV. We summarize our results
and discuss their implications in Section V.
that our classificationdoes not cover imperfect or multi-
ple fluids or solutions with shocks. The accelerated ex-
pansion of the universe can also be realized by a scalar
field with a flat potential (i.e. quintessence). The non- II. BASIC EQUATIONS
existenceofself-similarblackholesolutionsembeddedin
an exact Friedmann universe for this case was already We consider a spherically symmetric spacetime with
demonstrated in ref. [16] and this is clearly complemen- the line element
tary to the present work.
There are several key differences between the 0<γ < ds2 = e2Φ(t,r)dt2+e2Ψ(t,r)dr2+R(t,r)2dΩ2, (2.1)
−
2/3 and 2/3 < γ < 2 cases. A purely formal difference
is that the limiting values of z for large and small spa- where dΩ2 dθ2+sin2θdϕ2. We take the matter field
≡
tialdistancesintheflatFriedmannsolutionarereversed: to be a perfect fluid with energy-momentum tensor
large spatial distances now correspond to z 0 and
→
small ones to z . The other differences have more T =pg +(µ+p)u u , (2.2)
µν µν µ ν
→ ∞
physicalsignificance. Thefirstisthattherearenosound-
waves, since the sound-speed c = dp/dµ = c√γ 1 where p and µ are the pressure and the energy density,
s
−
is imaginary. This considerably simplifies the analysis, respectively, u is the 4-velocity of the fluid, and we use
µ
p
3
units with c = 1. We will adopt comoving coordinates, latter case, ξµ is also null and the similarity surface is
so that the 4-velocity is called a similarity horizon.
It is helpful to rewrite Eq. (2.12) in the conformally
∂ ∂
uµ =e−Φ . (2.3) static form:
∂xµ ∂t
V V
The field equations can then be written in the following ds2 = e2τ e2Φ (1+V)dτ + dz (1 V)dτ dz
− z − − z
form: (cid:20) (cid:26) (cid:27)(cid:26) (cid:27)
+z2S2dΩ2 , (2.14)
p = (µ+p)Φ , (2.4)
,r − ,r wheretheconforma(cid:3)lfactordependsonlyonτ ln t. To
R ≡ | |
µ = (µ+p) Ψ +2 ,t , (2.5) obtain the causal structure of the self-similar spacetime,
,t ,t
− R we can then use the analogy with the static spacetime.
(cid:18) (cid:19)
m = 4πµR R2, (2.6) This is because the homothetic Killing vector, similarity
,r ,r
surfaces and similarity horizons are the counterparts of
m = 4πpR R2, (2.7)
,t − ,t the Killing vector, Killing surfaces and Killing horizons
0 = R,rt+Φ,rR,t+Ψ,tR,r, (2.8) in the static case (see e.g. ref. [22]).
−
R The Einstein equations imply that p, µ and m must
m = (1+e−2ΦR 2 e−2ΨR 2), (2.9)
2G ,t − ,r have the form
where a comma denotes partial differentiation and G is W(z)
8πGµ = , (2.15)
the gravitational constant. The first two equations cor- r2
respond to the energy-momentum conservation and the P(z)
8πGp = , (2.16)
next two specify the Misner-Sharp mass m. Five of the r2
above six equations are independent. Throughout this 2Gm = rM(z), (2.17)
paper, we call the direction of increasing (decreasing) t
the future (past) direction. whereweassumethattheenergydensityisnon-negative
A spacetime is self-similar if it admits a homothetic (W 0). ThentheEinsteinandenergy-momentumcon-
≥
Killing vector ξµ, which is defined by servation equations reduce to ordinary differential equa-
tions for the non-dimensional functions with respect to
ξgµν =2gµν, (2.10) the self-similar variable z:
L
where ξ denotes the Lie derivative along ξµ. Cahill 2P +P′ = (W +P)Φ′, (2.18)
L − −
andTaub[3]firstinvestigatedsphericallysymmetricself- S′
similar solutions in which the homothetic Killing vector W′ = (W +P) Ψ′+2 , (2.19)
− S
isneitherparallelnororthogonaltothe fluidflowvector. (cid:18) (cid:19)
They showed that – by a suitable coordinate transfor- M +M′ = WS2(S+S′), (2.20)
mation – such solutions can be put into a form in which M′ = PS2S′, (2.21)
−
all dimensionless quantities are functions only of the di- 0 = S′′+S′ Φ′S′ Ψ′(S+S′), (2.22)
mensionless variable z r/t. In this case, ξµ is given − −
by ≡ M = S[1+z2e−2ΦS′2 e−2Ψ(S+S′)(22].,23)
−
∂ ∂ ∂ wherea prime denotes a derivativewith respectto ln z .
ξµ =t +r . (2.11) | |
∂xµ ∂t ∂r We assume the equation of state has the form p =
(γ 1)µ, which is the only barotropic one compatible
Thelineelementinasphericallysymmetricself-similar wit−h the homothetic assumption. The dominant energy
spacetime can be written as condition requires 0 γ 2. In this paper we exclude
≤ ≤
the value of 1 (corresponding to dust) since this needs
ds2 = e2Φ(z)dt2+e2Ψ(z)dr2+r2S2(z)dΩ2. (2.12)
− special treatment and has been analyzed in ref. [23]. We
also exclude γ = 0, corresponding to a cosmological
A hypersurface Σ of constantz is called a similarity sur-
constant, since this is incompatible with self-similarity.
face and is generated by the homothetic Killing vector.
Equations (2.18) and (2.19) can then be integrated to
The induced metric on Σ is
give
ds2 = (1 V2)e2Φdt2+r2S2dΩ2, (2.13)
Σ − − eΦ = c0z2(γ−1)/γW−(γ−1)/γ, (2.24)
where V z eΨ−Φ is the speed of the fluid relative to eΨ = c S−2W−1/γ, (2.25)
≡ | | 1
the surfacesof constantz. The arearadiusR=rS must
be positive but S and r need not be. (Note that CC where c0 and c1 are integration constants. The velocity
adopt a convention in which r is always positive.) A function V can be shown to be
similaritysurfaceisspacelikefor(V2 1)e2Φ >0,timelike c
for (V2 1)e2Φ <0 and null for (1 −V2)e2Φ =0. In the V = c1z(2−γ)/γS−2W(γ−2)/γ. (2.26)
− − 0
4
Equations(2.18)–(2.23) reduceto ordinarydifferential These equations are consistent provided γ is not 2/3 or
equations for S, S′ and W: 0. Since S andW aredetermined in terms ofγ, c and
0 0 0
c , where c and c are just gauge constants, there is no
1 0 1
V2−(γ−1)W′ = γc21W(γ−2)/γ 2(γ−1) free parameter in this solution for a given value of γ.
γ W 2S4 − γ Introducingconstantsa andb inplaceofS andW ,
0 0 0 0
S′ we obtain:
2V2 , (2.27)
− S eΦ = eΦF a , (3.5)
γ 2 (γ 1)W′ ≡ 0
S′′ =S′ − − eΨ = eΨF b0z−q, (3.6)
γ − γW ≡
(cid:18) (cid:19) b
(S+S′) 2S′ + W′ , (2.28) S = SF ≡ 1 0q z−q, (3.7)
− S γW | − |
(cid:18) (cid:19) 4
M =WS2(γS′+S), (2.29) W = WF ≡ 3γ2a2z2, (3.8)
0
M =S(cid:20)1+c−02z2(2−γ)/γW2(γ−1)/γS′2 M = MF ≡ 9γ2a204|b130−q|3z2−3q. (3.9)
c−2S4W2/γ(S+S′)2 . (2.30) Here the constants a and b can be chosen arbitrarily
− 1 0 0
(cid:21) and
From Eqs. (2.29) and (2.30), we obtain the following re- 2
q . (3.10)
lation between S, S′ and W: ≡ 3γ
The constants c and c in Eqs. (2.24) and (2.25) are
WS2(γS′+S)=S 1+c−2z2(2−γ)/γW2(γ−1)/γS′2 0 1
0 given in terms of a and b as
0 0
(cid:20)
c−2S4W2/γ(S+S′)2 . (2.31) 4 (γ−1)/γ
− 1 (cid:21) c0 = a0 3γ2a2 , (3.11)
(cid:18) 0(cid:19)
Using this and Eq. (2.29), we obtain another constraint b3 4 1/γ
c = 0 , (3.12)
between M, S and W: 1 (1 q)2 3γ2a2
− (cid:18) 0(cid:19)
2 2
M M whereEqs.(3.1)–(3.4),(3.7)and(3.8)areused. Onecan
1 V2 γ 1+
− WS3 − − WS3 put the metric into a more familiar form,
(cid:18) (cid:19) (cid:18) (cid:19)
+γ2c2W−2/γS−6 1 M =0. (2.32) ds2 = dt˜2+t˜2q(dr˜2+r˜2dΩ2), (3.13)
1 − S −
(cid:18) (cid:19) by introducing new coordinates
t˜=a t, r˜=a−qb r1−q/1 q . (3.14)
III. EXACT SOLUTIONS 0 0 0 | − |
It should be noted that r˜= correspondsto r = for
A. Friedmann solution the decelerating case (2/3<∞γ 2) but to r =0 fo∞r the
≤
accelerating one (0<γ <2/3). In the γ =2/3 case, the
The flatFriedmannsolutioninself-similarcoordinates homotheticKillingvectorisparalleltothefluidflowand
corresponds to these equations no longer apply [24].
There exists a finite non-zero z where V crosses 1.
1
S =S0z−2/(3γ), W =W0z2, (3.1) Figure 1 shows the conformal diagram of the flat Fried-
mann solution for 0 < γ < 2/3, including the similarity
where S and W are constants. V is determined by
0 0 surfaces. We can see that the initial singularity is null.
Eq. (2.26) as
The similarity horizon z = z corresponds to the cos-
1
c mological event horizon, while z = and r = is a
V =V0z1−2/(3γ), V0 = c1S0−2W0(γ−2)/γ. (3.2) regular center. The dotted and dot-d∞ashed lines∞denote
0
the singularity and infinity, respectively.
and Eq. (2.28) holds trivially. From Eqs. (2.27) and
(2.31), S and W must satisfy
0 0
B. Kantowski-Sachs solution
2 γc2W(γ−2)/γ
V2 = 1 0 , (3.3)
3γ 0 2S4 The Kantowski-Sachs solution in self-similar coordi-
0
nates corresponds to
2 2
1 = c−2S6W2/γ 1 . (3.4)
1 0 0 − 3γ S =S0z−1, W =W0z2. (3.15)
(cid:18) (cid:19)
5
z=+0,t= γ z−(2−3γ)/γ
∞ V = | | , (3.23)
(2 3γ)(2 γ)
− −
where r¯ is a radpial coordinate. As expected, V is only
z real for 0 < γ < 2/3. In the Kantowski-Sachs solution,
= the area of the 2-sphere with constant t and r does not
z
z 1 0 depend on r but only on t, so it expands with time. The
= =
∞,r = z,1t tRo.pologyof the constantt spacelikehypersurface is S2×
= z Thereagainexistsanon-zerofinitez whereV crosses
1
∞ 1. r = or z = for fixed t(> 0) can be analytically
∞ ∞
extended to negative r or negative z beyond z = .
±∞
Figure 2 shows the conformal diagramof the (extended)
Kantowski-Sachs solution for 0 < γ < 2/3. The simi-
FIG. 1: The conformal diagram of the flat Friedmann solu- larity surfaces are also shown. We can see that the ini-
tion for 0<γ <2/3. There is a similarity horizon at z =z1 tial singularity is null. There are two similarity horizons
(> 0), corresponding to the cosmological event horizon, and z = z , corresponding to two cosmological event hori-
1
a regular centerat z =∞ andr=∞. Theinitial singularity zons.±The correspondingconformaldiagram(Fig. 15)in
att=0isnull,whilenullinfinityisgivenbyt=∞. Thedot- ref. [22] is incorrect.
ted and dot-dashed lines denote the singularity and infinity,
respectively. Thethinsolidcurvesandlinesdenotesimilarity z = 0,t= z =+0,t=
surfaces, i.e. orbits of z =constant. − ∞ ∞
z1
= − z z
= =
Then Eq. (2.28) holds trivially and Eq. (2.31) yields z ± z1
0
∞ =
(1−γ)W0S03 =S0[1+c−02W02−2/γS02]. (3.16) z= ,r= = z,1t
From Eq. (2.26), we have −z1, ± z
t
= ∞
V =V z3−2/γ, V = c1S−2W1−2/γ, (3.17) 0
0 0 c 0 0
0
FIG.2: TheconformaldiagramoftheKantowski-Sachssolu-
and hence V 0 as z . Eq. (2.27) is trivially
→ → ∞ tion for 0<γ <2/3. The solution is analytically extendible
satisfied to lowest order but determines W to the next
0 beyond z = ±∞ and r = ±∞. There are two similarity
lowest order:
horizons z = ±z1, both corresponding to cosmological event
horizons. The initial singularity at t = 0 is null and future
1 γ
1 c−2 = W−1+2/γ. (3.18) nullinfinity is given byt=∞.
γ − 0 4 0
(cid:18) (cid:19)
Equations (3.16) and (3.18) then give
(2 3γ)(2 γ)
−4(1 γ−) W0S02 =1, (3.19) C. Absence of static solution
−
sowecanobtainaphysicalsolutiononlyfor0<γ <2/3 The static solution would need to have
andthereremainsnofreeparameterinthiscase. Thisso-
lutionwasfirstobtainedinref.[25],althoughthatpaper S =S0, W =W0,
introduceddifferentvariablestodealwiththeunphysical
in which case Eq. (2.26) gives
solutions for the 2/3<γ <2 case.
The Kantowski-Sachs solution in more standard coor- c
dinates can be written as [21] V =V0z(2−γ)/γ, V0 = c1S0−2W0(γ−2)/γ. (3.24)
0
(2 3γ)(2 γ)
ds2 = − − dt2+t4(1−γ)/γdr¯2 Equations (2.27) and (2.31) then reduce to
− γ2
+t2(dθ2+sin2θdφ2), (3.20) γc2W(γ−2)/γ 1 γ
0= 1 0 +2 − (3.25)
4(1 γ) S4 γ
8πGµ = − , (3.21) 0 (cid:18) (cid:19)
(2 3γ)(2 γ)t2
− − and
4(1 γ)2t
2Gm = (2−3γ−)(2−γ), (3.22) W0S03 =S0[1−c−12S0W02/γ], (3.26)
6
respectively. For γ > 1, Eq. (3.25) can be satisfied and does not need to be small.) From Eqs. (4.3) and (4.4),
there isa staticself-similarsolution,butthere isno such the linearized equations at spacelike infinity become
solution for 0 γ <1 However, as we will see later, the
≤ 2
absence of an exactstatic solution does not preclude the 0 = 3q2(γ 2)A+4B′+ A′, (4.7)
− γ
possibility of an asymptotically static solution.
1 γq
0 = B′′+3(1 q)B′+ − A′ (4.8)
− γ
(cid:18) (cid:19)
IV. ASYMPTOTIC BEHAVIORS
in both cases. These equations lead to two independent
solutions and the general solution is given by a linear
A. Friedmann asymptote for z→0 combination of these.
The first solution is
We now focus on self-similar solutions which are
A(z) = A z2(2−3γ)/(3γ)+ , (4.9)
asymptotic to the flat Friedmann model at large spatial 1 ···
distances, i.e. in which Φ, Ψ, S, W and M approach B(z) = β+B z2(2−3γ)/(3γ)+ , (4.10)
1
···
the form given by Eqs. (3.5)-(3.9) as r˜ . In this
→ ∞ where
case, it is convenient to introduce new functions A and
B, defined by
A (2 3γ)B , (4.11)
1 1
≡ −
W = WF(z)eA(z), (4.1) B1 ≡ −ab022γ4((33γγ+−22))(e−2β −e4β). (4.12)
S = S (z)eB(z), (4.2) 0
F
β is a constant and the dots denote higher order terms.
which describe the deviations from the flat Friedmann A and B converge at large distances in both the decel-
solution. Equations(2.27),(2.28)and(2.26)canthenbe erating (2/3 < γ 2) and accelerating (0 < γ < 2/3)
≤
written as cases. This solutionwas first obtainedin ref.[17] for the
radiation case (γ = 4/3) and in ref. [19] for more gen-
A′ = γV2[3γq2e−(γ−2)A/γ −2(q+2B′)], (4.3) eralγ. However,it is notproperlyasymptotic to the flat
2[V2 (γ 1)] FriedmannsolutionbecauseEq.(4.2)showsthatthereis
− − γ 1 a residual eβ term in S at infinity, so Eq. (2.12) implies
B′′ = (B′ q) 1 q+B′+ − A′ that there is a solid angle deficit [20]. We describe so-
− − − γ
(cid:18) (cid:19) lutions with this asymptotic behavior as asymptotically
1
q+2B′+ A′ (1 q+B′), (4.4) quasi-Friedmann.
− γ − The second solution is
(cid:18) (cid:19)
b
V = 0z1−qe−2B+(1−2/γ)A, (4.5) A A z(2−γ)/γ, (4.13)
a ≈ 0
0 1
B A z(2−γ)/γ, (4.14)
0
respectively. Equation (2.31) yields another relation: ≈ −6γ
where A is a constant and x y means (x/y) 1 in
4b3 0 ≈ →
0 z2−3qeA+3B(1+3γB′) the relevant limit. The above solution was also found in
9γ2a2 1 q 3
0| − | ref. [26]. The decelerating solution must be discarded,
b b2 since this diverges at large distances (z ). On
= 0 z−qeB 1+ 0 z2−2q → ∞
1 q a2(1 q)2 the other hand, the accelerating solution is “properly”
| − | (cid:20) 0 − asymptotic to Friedmann since A and B both converge
e2(γ−1)A/γ+2B( q+B′)2
× − tozeroatlargedistances(z 0),sothereisnosolidan-
1 e2A/γ+6B(1 q+B′)2 . (4.6) gle deficit. In the following,→we generally omit the term
− (1 q)2 − “properly”andsimplydescribethesesolutionsasasymp-
− (cid:21)
totically Friedmann.
The flatFriedmannsolutionisgivenbyA=B =0for We conclude that there is a 1-parameter family of
allz. Spacelikeinfinitycorrespondstoz forthede- asymptoticallyFriedmannsolutionsatlargedistancesfor
→∞
celerating case (2/3<γ 2) and z 0 for the acceler- 0 < γ < 2/3. Note that, in the positive pressure case
ating case (0<γ <2/3).≤Note that→V z1−2/(3γ) (γ >1),itiswellknownthatthereisnophysicalsolution
∝ →∞
at largedistances if A and B arefinite. This means that whichisexactlyFriedmannatsufficientlylargebutfinite
z = (2/3 < γ 2) and z = 0 (0 < γ < 2/3) are distances [17]. Although one might envisage attaching
∞ ≤
horizontal lines in a conformal diagram. an interior black hole solution to an exterior Friedmann
Theasymptoticformofself-similarsolutionswhichap- solutionatasonicpoint(sincetherecanbeadiscontinu-
proachtheflatFriedmannatlargedistancescanbefound ity there), no such solution is possible. This conclusion
byneglectingtheV−2 termandlinearizingtheequations trivially applies when the pressure is negative (γ < 1)
with respect to A, A′, B′ and B′′. (It is noted that B because there are no sound-waves in this case.
7
B. Friedmann asymptote for z→∞ asz . OnlyifS0andW0arethesameasinthestatic
→∞
solution do we describe such solutions as asymptotically
We are also interested in solutions which are asymp- static. Equation (2.26) implies
totically Friedmann at small distances from the origin.
c
We therefore seek solutions which have V ≈V0z(2−γ)/γ, V0 = c1z(2−γ)/γS0−2W0(γ−2)/γ.
0
(4.22)
S S z−2/(3γ), W W z2 (4.15)
≈ 0 ≈ 0 This solution can be compatible with Eq. (2.27) if and
only if
for z and0<γ <2/3,where the constants S and
0
→∞
W may be different from the exactflat Friedmann case.
0 1W′ S′
Equation (2.26) implies +2 z−2(2−γ)/γ, (4.23)
γ W S ∝
c
V V z1−2/(3γ), V = 1S−2W(γ−2)/γ. (4.16)
≈ 0 0 c 0 0 asz . This combinationappearsinthe secondterm
0 →∞
on the right-hand side of Eq. (2.28) and this can be re-
The exact Friedmann relation (3.4) still applies but garded as a higher order term. It then follows that
Eq. (2.27) now yields to lowest order
S =S +S z1−2/γ +S z2(1−2/γ)+ . (4.24)
0 1 2
···
2 1 γ W′ γc2W(γ−2)/γ
V2+ − 2 z−2(1−2/(3γ)) = 1 0 . From Eq. (4.23), W must be of the form
3γ 0 γ W − 2S4
(cid:18) (cid:19) 0
(4.17)
W =W +W z1−2/γ +W z2(1−2/γ)+ , (4.25)
Hence (W′/W) 2 is either proportional to z2−4/(3γ) 0 1 2 ···
−
or falls off even faster. Therefore Eq. (2.28) implies the
where
following asymptotic behavior:
1W S
1 1
+2 =0. (4.26)
S = S z−2/(3γ) 1+S z2(1−2/(3γ))+ ,(4.18) γW S
0 1 0 0
···
h i
W = W z2 1+W z2(1−2/(3γ))+ , (4.19) From Eq. (2.31), we obtain the relation
0 1
···
h i
where W S3 =S 1+ 1 2 2c−2W2(1−2/γ)S2 c−2S6W2/γ .
2 1 γ γc2W(γ−2)/γ 0 0 0" (cid:18) − γ(cid:19) 0 0 1 − 1 0 0 #
V2+ − W = 1 0 , (4.20) (4.27)
3γ 0 γ 1 2S4
0 To lowest order Eq. (2.27) becomes
1
S = W . (4.21)
1 −5γ(3 2γ) 1 2 1W S γc2W1−2/γ 2(1 γ)
− 2V2 1 2 +2 2 = 1 0 + −
0 − γ γW S 2S4 γ
Allcoefficientsofhigherordertermsaredeterminedfrom (cid:18) (cid:19)(cid:20) 0 0(cid:21) 0
(4.28)
S and W . Since the value for S can be different from
0 0 0 and to the next lowest order Eq. (2.28) becomes
the exact Friedmann case, there is a 1-parameter family
ofasymptoticallyquasi-Friedmannsolutionsatsmalldis- 2 2 1 2 2
tances. SinceV 0asz ,z = isaverticallinein 2 1 S = 1 1 S W
→ →∞ ∞ − γ 2 − − γ − γ 1 1
aconformaldiagram. Thesesolutionscanbe interpreted (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
as self-similar models with a regular center as r 0 for 1 2S2 W2
fixed t = 0. On the other hand, in the limit t →0 for − 2 1− γ S0 S + γW (4..29)
6 → (cid:18) (cid:19) (cid:18) 0 0(cid:19)
fixed r = 0, they correspond to simultaneous big bang
6
models in the comoving time-slicing. If S had the ex- S and W are determined from Eqs. (4.26) and (4.27)
0 1 1
actFriedmannvalue, wewouldhaveFriedmannitself, so in terms of S and W , while S and W are determined
0 0 2 2
this istheonlysolutionwhichisproperlyasymptotically byEqs.(4.28)and(4.29)intermsofS ,S ,W andW .
0 1 0 1
Friedmann. AllhigherordertermsaredeterminedintermsofS and
0
W . Hence there is a 2-parameter family of asymptoti-
0
callyquasi-staticsolutions. Since there isno exactstatic
C. Static asymptote for z→∞ solution, there are no asymptotically static ones. Also
V as z for 0 < γ < 2/3, so z = is a
→ ∞ → ∞ ∞
As we have seen, there exists no static self-similar so- horizontal line in a conformal diagram.
lutionfor0<γ <2/3. However,wewillshowthatthere Ifweconsidertheanalyticcontinuationbeyondz =
∞
are still solutions which can be described as asymptoti- intothenegativez region,themetricshouldremainana-
cally quasi-static. In this case, we assume lytic in terms of the local inertial Cartesian coordinates.
Forasymptoticallyquasi-staticsolutions,oneshouldcon-
W W >0, S S >0 sider the analytic continuation beyond t = +0 because
0 0
≈ ≈
8
S = R/r and W = 8πGµr2 are finite. For this purpose, while Eq. (2.27) gives
it is convenientto see how the propertime τ behavesfor
fixedr>0. Forthis classofsolutions,Eq.(2.24)implies 2(1 γ) 1 γ 2 γc2W1−2/γ
− V2+2 − 3 W = 1 0 .
eΦ =c0z−2(1−γ)/γW0(1−γ)/γ, (4.30) γ 0 γ (cid:18) − γ(cid:19) 2 2S04 (4.39)
so τ t−1+2/γ. When we continue the solution analyti- At second order Eq. (2.28) becomes
∝
cally to the negative t regionbeyond t= 0, this means
that τ C t−1+2/γ, where C is a po±sitive constant 3 1 2
depend≈ing±onrr|a|ndtheupper(lowrer)signcorrespondsto 4− γ S2 =− 1− γ 3− γ W2+S12 .
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
thepositive(negative)t. ToexpressW andS asanalytic (cid:0) (cid:1)(4.40)
functions of τ, we use the following unique continuation: Hence the coefficients of all higher order terms are de-
termined in terms of S and W . This means there is a
0 0
S = S0 S1 z 1−2/γ +S2 z 2(1−2/γ)+ , (4.31) 2-parameter family of asymptotically quasi-Kantowski-
± | | | | ···
W = W0 W1 z 1−2/γ +W2 z 2(1−2/γ)+ (.4.32) Sachs solutions. If the values for S0 and W0 are the
± | | | | ··· same as those for the exact Kantowski-Sachs solution,
This expression gives a Taylor series expansion around all coefficients of higher order terms vanish, so the solu-
τ = 0 in terms of τ. Because of the presence of the odd tionisalsoexactlyKantowski-Sachs. Inotherwords,the
powersofτ, thisis notreflection-symmetricaboutt=0. onlysolutionwhichisasymptoticallyKantowski-Sachsis
Onthe other hand, in the limit r for fixedt=0, Kantowski-Sachs itself. On the other hand, even if the
→∞ 6
one has a vacuum of infinite radius which is not asymp- first order terms vanish, i.e. S =0, we can see that W
1 2
totically flat because m/R approaches a non-zero con- may not vanish, so S and W are different from their
0 0
stant. We call this quasi-static spacelike infinity. In Kantowski-Sachs values. Since V 0 as z for
→ → ∞
Paper II, we will see that there are a class of solutions 0 < γ < 2/3, z = is a vertical line in a conformal
∞
describing a Friedmann universe emergent from a white diagram.
hole, where a Friedmann spacelike infinity and a quasi- For asymptotically quasi-Kantowski-Sachs solutions,
static spacelike infinity are connected. onecanconsidertheanalyticcontinuationbeyondr=
becausezS =R/tandW/z2 =8πGµt2 arenon-zeroan∞d
finite for fixed t = 0. It is useful to see how the proper
D. Kantowski-Sachs asymptote for z→∞ length λ changes6 around r = for fixed t = 0. Equa-
∞ 6
tion (2.25) implies
For the asymptotically quasi-Kantowski-Sachs solu-
tions, we assume eΨ =c z2−2/γS−2W−1/γ, (4.41)
1 0 0
S S z−1, W W z2 (4.33)
0 0
≈ ≈ so λ r3−2/γ. This means that, when we continue
as z . From Eq. (2.26), we have the so∝lution analytically to the negativer regionbeyond
→∞
r = , one has λ C r3−2/γ, where C is a posi-
c t t
V ≈V0z3−2/γ, V0 = c1S0−2W01−2/γz3−2/γ, (4.34) tive c∞onstant depend≈ing∓on|t|and the upper (lower) sign
0 corresponds to positive (negative) r. To obtain analytic
and hence V 0 as z . Equation (2.27) is trivially expressions for W and S in terms of λ, we use the fol-
satisfied to lo→west orde→r b∞ut to the next order requires lowing continuation:
either
S = S z−1(1 S z 3−2/γ +S z 2(3−2/γ)+ (4).,42)
W′ 0 ± 1| | 2| | ···
2 z2(3−2/γ) (4.35) W = W z2(1+W z 2(3−2/γ)+ ). (4.43)
W − ∝ 0 2
| | ···
or that the right-handside falls offeven faster than this.
The above expression gives a Taylor series expansion
Equations (2.28) and (4.35) imply
around λ = 0. Since W = 0 in the above expression,
1
W/z2 =8πGt2 genericallyhasanextremumatz = ,
S = S0z−1(1+S1z3−2/γ +S2z2(3−2/γ)+···)(,4.36) while zS = R/t does not. So this expression is±n∞ot
W = W0z2(1+W2z2(3−2/γ)+ ) (4.37) reflection-symmetric with respect to r = . On the
··· ±∞
other hand, in the limit t 0 for fixed r = 0, we have
Equation (2.31) yields → 6
R 0 and µ , which corresponds to an initial
→ → ∞
singularity. In Paper II, we will see that this analytical
(1−γ)W0S03 = S0 1+c−02W02(γ−1)/γS02 continuation is crucial for obtaining black hole solutions
h 2 2 embeddedinaFriedmannbackgroundaswellasaclassof
3 c−2S6W2/γS2 (,4.38) wormhole solutions connecting one Friedmann spacelike
− − γ 1 0 0 1# infinity to another quasi-Friedmann spacelike infinity.
(cid:18) (cid:19)
9
1
E. Constant-velocity asymptote for z→∞
Here we seek solutions in which V tends to a finite
positive value V as z . Differentiating Eq. (2.26) 0.5
∞
and noting that V′/V →0∞as z , we have
→ →∞
W′ 2γ S′
1 . (4.44) V∞ 0
W ≈ − 2 γ S
(cid:18) − (cid:19)
Substituting this relationinto Eq.(2.28), we findto low-
est order -0.5
S′′ 1 4(1 γ) S′ 2 2(2 γ2)S′
− − . (4.45)
S ≈−γ − (2 γ) S − γ(2 γ) S -1
− (cid:18) (cid:19) −
0 0.5 1 1.5 2
To lowest order this ordinary differential equation has a γ
general solution of the form
S S+zp+ +S−zp−, (4.46) FIG. 3: V∞ for theasymptotically constant-velocity solution
≈ 0 0 as a function of γ. Although only the part of the solid curve
with 0 < γ < 2/3 is relevant to the present analysis, it is
where S0,± are arbitrary constants and extended to 0 < γ < 2 for completeness. The other root
for V∞ is also plotted with a dashed line but this may be
(2 γ2) (1 γ)(4 8γ+4γ2 γ3) unphysical.
p = − − ± − − − .
±
(6 5γ)γ
p −
(4.47)
One can show that the square root is real for 0 < γ <
the energy density µ is very slow, the solution is far
2/3. Also Eq. (2.31) is consistent only if the first term
from asymptotically flat. In fact, both m and m/R di-
dominates the second term in Eq. (4.46), so S+ = 0 is
0 verge to infinity. This solution cannot be analytically
excluded.
extended beyond z = because R diverges to infinity
One can easily show that the first term on the right- ∞
as r for fixed t = 0, while µ diverges to infinity as
handside ofEq.(2.27)convergestozerofor0<γ <2/3 → ∞ 6
t 0 for fixed r = 0. We describe these as asymptoti-
and so → 6
cally constant-velocity solutions. The infinity reached as
γ(1 γ)+ (1 γ)(4 8γ+4γ2 γ3) r forfixedt=0will be calledthe constant-velocity
→∞ 6
V∞ = − − − − . (4.48) spacelike infinity (cf. the asymptotically Minkowski so-
2 γ
p − lutions for γ >6/5 described in CC).
This is plotted in Fig. 3. Note that the solution with
similarasymptoticbehaviorforγ >6/5 [4]hastheother
branch of the square root. For this solution, we have F. Singular asymptote for z→z∗
S S zp+, W W z1−2γp+/(2−γ). (4.49) We assume that ln S diverges as z z , while V
0 0 ∗
≈ ≈ | | →
tends to a finite value V . Then Eq. (2.26) implies
∗
The condition V V yields a relationbetween S and
∞ 0
W : → W′ 2γ S′
0
(4.51)
c W ≈−2 γ S
V = 1S−2W(γ−2)/γ. (4.50) −
∞ c 0 0
0 and Eq. (2.28) yields to lowest order
There is no other relation between them, so there is a
S S Z (2−γ)/(6−5γ), (4.52)
1-parameterfamily of solutions with this asymptotic be- ≈ 0| |
havior. Since V V < 1 as z for 0 < γ < 2/3, W W Z −2γ/(6−5γ), (4.53)
∞ 0
→ → ∞ ≈ | |
z = is a vertical line in a conformal diagram.
∞
Wenowdiscussthephysicalsignificanceofthisasymp- where
toticsolution. Asγ increasesfrom0to2/3,p+ decreases z
from 1/2 to 3/4, while V decreases from 1 to 1/3. Z ln . (4.54)
Becau−se M z−1+(6−5γ)p+/(2−∞γ), we have R = rS , ≡ z∗
µ = W/(8π∝Gr2) 0, m = rM/(2G) →a∞nd Equation (2.27) implies V2 =1, while Eq. (2.29) gives
→ → ∞ ∗
2m/R = M/S as r for fixed t = 0. There-
→ ∞ → ∞ 6
fore, this solution approaches a vacuum region with in- γ(2 γ)
M M = − W S3, (4.55)
finite physical radius. However, because the fall-off of → 0 ± 6 5γ 0 0
−
10
where the upper (lower) sign corresponds to positive which tends to . Hence the similarity surface be-
±∞
(negative) Z. From Eq. (2.26), we have comes spacelike (timelike) for the positive (negative)
mass case. We will see in Paper II that the positive-
c21 z 2(2−γ)/γS−4W2(γ−2)/γ =1 (4.56) mass singular behavior is associatedwith black hole and
c2| ∗| 0 0 white hole singularities, while the negative-mass one is
0
associated with naked singularities.
and hence z is determined from S and W . To next
∗ 0 0 It should be noted, however, that the mass of the sin-
| |
lowest order, Eq. (2.28) implies
gularity is not constant but is proportional to the time
coordinate t. There is a 2-parameter family of solutions
S =S0 Z (2−γ)/(6−5γ) 1+S1 Z (2−3γ)/(6−5γ)+ (4.,57) belonging to this class.
| | | | ···
h i
W =W Z −2γ/(6−5γ) 1+W Z (2−3γ)/(6−5γ)+ (4.,58)
0 1
| | | | ···
h i V. SUMMARY AND DISCUSSION
where
(2 γ) In this paper, we have analyzedspherically symmetric
S = − W . (4.59)
1 1
−2(4 3γ) self-similar spacetimes. We have shown that the metric
−
of these spacetimes is conformally static, with the con-
Through Eq. (2.26), we obtain
formal structure being determined by the form of the
velocity function V. The homothetic Killing vector here
V =1+V Z (2−3γ)/(6−5γ)+ , (4.60)
1| | ··· playsananalogousroletotheKillingvectorinthestatic
case. We have expressed the Einstein field equations in
where
thecomovingapproachasordinarydifferentialequations
2 γ andanalyticallyinvestigatedthe possibleasymptoticbe-
V = 2S − W , (4.61)
1 − 1− γ 1 havior of perfect fluid solutions with p = (γ 1)µ and
(cid:18) (cid:19) −
0<γ <2/3. InPaperII,wefocusonsolutionswhichare
and hence asymptotic to the flat Friedmann universe at large dis-
γ tances because we are interested in solutions embedded
S = V , (4.62)
1 8(1 γ) 1 in an accelerating Friedmann background [13].
− These solutions can be understood in terms of the
γ(4 3γ)
W = − V . (4.63) “completeclassification”providedbyCarrandColey[4].
1 1
−4(2 γ)(1 γ)
− − However,thisclassificationmainlyappliesforperfectflu-
ids with positive pressure and we have extended it to
To next lowest order Eq. (2.27) then gives
0<γ <2/3. Wefindthereareeightpossibleasymptotic
6 5γ γc2W−(2−γ)/γ behaviors at small, intermediate and large distances in
V1 =± 2− γ 1 20S4 , (4.64) thisclassofsolutions. However,itshouldbestressedthat
(cid:18) − (cid:19) 0 thereareotherasymptoticbehaviors,suchasKantowski-
which is consistent with Eq. (2.31). Sachs asymptotes at z 0, and these are discussed
→ ±
We describethese solutionsasasymptotically singular. elsewhere [27].
Forfixedr =0,wehaveR=rS 0,µ=W/(8πGr2) Tables I and II summarize the results. Table I lists
and m 6= rM/(2G) rM /→(2G) as Z 0. Fro→m two exact solutions: Friedmann (F) and Kantowski-
0
E∞q. (4.55), we have ≈ → Sachs (KS). Table II lists eight asymptotic behav-
iors: asymptotically Friedmann (F) and asymptoti-
m≈ r2MG0 =±γ6(2−5γγ)W0S02R (4.65) ccaallllyy qquuaassi-i-FFrrieieddmmaannnn(Q(QFF),)asfoyrmpztot→ical0ly, qausaysmi-psttaottiic-
−
(QS), asymptotically quasi-Kantowski-Sachs (QKS) and
and so the mass is positive (negative) for Z 0+ asymptotically constant-velocity (CV) for z ,
(Z 0−). (Note that S can be negative but R→= rS asymptoticallypositive-masssingular(PMS)and→asym∞p-
→
is positive even after the extension to the negative r re- totically negative-mass singular (NMS) for z z .
∗
gion.) Since the physical properties of these two lim- There are three key differences from the po→sitive pres-
its are very different, we distinguish these two cases as sure case. First, there is no sonic point in the negative-
asymptotically positive-mass singular and asymptotically pressure case and hence no additional ambiguity associ-
negative-mass singular. For the positive-mass singular atedwiththesonicpoint. Second,thesesolutionscanbe
case,Z 0+,V 1+ andm>0; forthe negativecase, properly asymptotic to the flat Friedmann model, in the
Z 0−→, V 1−→and m < 0. Although V 1 in the sense that there is no solid angle deficit. This contrasts
→ → →
limit Z 0, the first component of the metric (2.13) is with the positive pressure case, where the solutions are
→
only asymptotically quasi-Friedmann. Third, the exis-
(1 V2)e2Φ
− − tence of exact and asymptotically Kantowski-Sachs so-
2c2 z 4(γ−1)/γW2(1−γ)/γV Z −(2−γ)/(6−5γ()4,.66) lutions, which can be extended beyond the timelike hy-
≈ 0| ∗| 0 1| |