"Truth is a valid concept. Unfortunately people who use the word often violate logic"
I used the word "ultimately" to clearly distance it from our current staus as thinking beings.
It is a valid concept for human beings at this point. My wording ("or even that "Truth" is ultimately a valid concept") was intentional and in keeping with the topic--a topic that conceptually goes beyond our universe and possibly into space branes. Perhaps humanity and everything else will someday resolve into a universal energy soup which is not only beyond our current concepts but indeed beyond "dark matter" or "dark energy" on which topics we know less than a cave man knew about constructing a Dell computer. Perhaps everything is "god" as Spinoza believed and all concepts are meaningless because EVERYTHING is TRUTH.
The following analysis might put us both on the same page. ;-)
This is the html version of the file intlpress.com.
Page 1
c 2004 International P ress
Adv. T heor. M ath. P hys. 8 (2004) 319–343
Transition from b ig crunch to b ig
b ang in b rane cosm ology
C laus G erhardt
Institut für Angew andte M athem atik
R uprecht-K arls-U niversität
69120 H eidelb erg, G erm any
gerhardt@ m ath.uni-heidelb erg.de
A bstract
W e consider branes N = I × S0, w here S0 is an n–dim ensional
space form , not necessarily com pact, in a S chw arzschild-AdS (n+ 2) b ulk
N. T he branes have a big crunch singularity. If a brane is an AR W
space, then, under certain conditions, there exists a sm ooth natural
transition flow through the singularity to a reflected brane ˆN, w hich
has a big bang singularity and w hich can be view ed as a brane in a
refl ected S chw a rzschild-AdS (n+ 2) b ulk ˆN. T he joint b ranes N ? ˆN ca n
thus be naturally em bedded in R2 × S0, hence there exists a second
possibility of defining a sm ooth transition from big crunch to big bang
by requiring that N ? ˆN form s a C8-hy pe rsurfa ce in R2 × S0. T his
last notion of a sm ooth transition also applies to branes that are not
AR W spaces, allow ing a w ide range of possible equations of state.
0 Introduction
T he problem of finding a sm ooth transition from a spacetim e N w ith a b ig
crunch to a spacetim e ˆN w ith a big bang singularity has b een the focus of
som e recent w orks in general relativity, see e.g., [6, 8] and the references
therein. F or ab stract spacetim es, i.e., for spacetim es that are not em b edded
in a bulk space, it is even a non-trivial question how to define a sm ooth
tra nsitio n.
e -print a rchiv e : lanl.arxiv.org
Page 2
320
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
In tw o recent papers [2, 4] w e studied this problem for a special class of
spacetim es, so-called A R W spaces, and used the inverse m ean curvature flow
to prove that, by reflecting the spaces, a sm ooth transition from big crunch
to big bang is possible.
In this paper w e look at branes in a Schw arzschild-A dS (n+ 2) b u lk N,
w here the branes are assum ed to lie in the black hole region, i.e., the radial
coordinate is the tim e function. For those branes that are A R W spaces the
transition results from [4]can be applied to conclude that a sm ooth transition
flow from a brane N to a properly reflected brane ˆN exists. H ow ever, the
assum ption that the branes are A R W spaces reduces the num ber of possible
branes drastically, cf. T heorem 0.4 and T heorem 0.5. Fortunately, in the
case of em bedded spacetim es, it is possible to define a transition through
the singularity w ithout using the inverse m ean curvature flow .
L et N b e a S ch w arzsch ild -A d S (n+ 2) bulk space w ith a black hole singu-
larity in r = 0. W e assum e that the radial coordinate r is negative, r < 0.
T hen, by sw itching the light cone and changing r to -r w e obtain a reflected
S ch w a rzsch ild -A d S (n+ 2) bulk space ˆN w ith a w hite hole singularity in r = 0.
T hese tw o bulk spacetim es can be pasted together such that N ?ˆN is a
sm ooth m anifold, nam ely, R2 × S 0, w hich has a m etric singularity in r = 0.
In th e b lack (w h ite) h ole region r is the tim e function and it is sm ooth across
th e sin g u la rity.
N ow , let us consider branes N in the black hole region of N. These
branes need not to be A RW spaces, they are only supposed to satisfy the
first of the five assum ptions im posed for A R W spaces. W e call those branes
to b e of class (B), cf. D efi n ition 0.1.
T he relation betw een the geom etry ofthe branes and physics is governed
by the Israeljunction condition.
W e shall prove existence and transition through the singularity only
for single branes, but this does include a tw o branes or m ultiple branes
configuration–w here then each brane has to be treated separately.
M oreover, in the equation of state
p = ?
n ?
(0.1 )
v a ria b le ? are allow ed
? = ?0 + ?, ?0 =const,
(0.2)
w here ? = ?(lo g (-r)) is defined in the bulk.
Page 3
CL A U S G E R H A R D T
321
B ran es of class (B) exist in the w hole black hole region of N, th ey stretch
fro m r = 0, th e b lack h ole sin gu larity, to r = r0,the event horizon.
T he branes w hose existence is proved in T heorem 3.1 autom atically have
a big crunch singularity in r = 0, sin ce th is is th e initia l co nd itio n for the
ordinary differential equation that has to be solved. T he branes are given
by an em bedding
y(t)=(r(t),t(t),x i), -a< t< 0,
(0.3)
w ith a big crunch singularity in t =0such that r(0) = 0.
From the m odified Friedm ann equation w e shalldeduce that the lim it
lim
t?0
t(t) = t0
(0.4 )
exists and w ithout loss of generality w e shall supp ose t0 = 0.
W e then shalldefine a brane ˆN ?ˆN by reflection
y(t)=(-r(-t),-t(-t),x i), 0 < t< a.
(0.5 )
T he tw o branes N,ˆN can be pasted together to yield at least a Lipschitz
h y p ersu rface in R2 × S 0. If this hyp ersurface is of class C8, then w e shall
speak ofa sm ooth transition from big crunch to big bang.
T o prove the sm oothness w e reparam etrize N ? ˆN b y u sin g r as a new
param eter instead of t. T he old brane N is then expressed as
y(r)=(r, t(r),x i), r < 0,
(0.6 )
and the reflected ˆN as
y(r)=(r,-t(-r),x i), r > 0.
(0.7 )
H ence, N ?ˆN is a sm ooth hypersurface, if d t
d r is a sm ooth and even
fu n ctio n in r in a sm allneighbourhood of r = 0, -?< r< ?.
If d t
d r is not an even function in r,then N?ˆN w ill n o t b e C8 h y p ersu rfa ce.
E xam ples can easily be constructed.
For branes,that are A RW spaces, the transition flow provides an alter-
nate criterion for a sm ooth transition through the singularity. T he reflected
brane ˆN that is used in this process is the sam e as in the above description
of pasting together the tw o em beddings.
Page 4
322
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
A version of the m odified Friedm ann equation, equation (1.18), plays a
central role in determ ining if the transition is sm ooth, nam ely, the quantity
f e˜?f has to be a sm ooth and even function in the variable e˜?f in order that
the transition flow is sm ooth,and a sm ooth and even function in the variable
r = -ef , if the joint em bedding is to represent a sm ooth hypersurface in
R2 × S0.
T he m etric in the bulk space N is given by
d˜s
2 = -˜h-1dr
2 + ˜hdt2 + r2
sijdxidxj,
(0.8 )
w here (sij) is the m etric of an n- dim ensional space form S0, the radial
coordinate r is assum ed to be negative, r < 0,and ˜h(r) is defined by
˜h = m(-r)-(n-1) +
2
n(n+ 1)
?r2 - ˜?,
(0.9 )
w here m > 0 and ? = 0 are constants, and ˜? = -1,0,1 is the curvature of
S0. W e note that w e assum e that there is a black hole region, i.e., if ? = 0,
then w e have to suppose ˜? = 1.
W e consider branes N contained in the black hole region {r0 < r< 0}.
N w ill b e a globally hyp erb olic spacetim e N = I × S0 w ith m etric
d¯s
2 = e2f (-(dx0)2 + sijdxidxj)
(0.10)
su ch th a t
f = f(x0)=log(-r(x0)).
(0.11)
W e m ay assum e that the tim e variable x0 = t m aps N on the interval
(-a,0). In t = 0 w e have a big crunch singularity induced by the black hole.
T he relation between geom etry and physics is governed by the Israel
ju n ctio n co n d itio n s
ha ß - H¯ga ß = ?(Ta ß - s¯ga ß ),
(0.12)
w here ha ß is the second fundam ental form of N, H = ¯ga ß ha ß the m ean
curvature, ? = 0 a constant, Ta ß the stress energy tensor of a perfect fluid
w ith an equation ofstate
p = ?
n
?,
(0.13)
and s the tension of the brane.
O ne ofthe param eters used in the definition of(n+1)-d im en sion al A R W
spaces is a positive constant ˜?, w hich is best expressed as
˜? = 1
2
(n + ˜? - 2).
(0.14 )
Page 5
CL A U S G E R H A R D T
323
If N w ould satisfy the E instein equation of a perfect fluid w ith an equation
of state
p = ˜?
n
?,
(0.1 5 )
th en ˜? w ould be defined w ith the help of ˜? in (0.15), cf. [2, Section 9].
A RW spaces w ith com pact S0 also have a future m ass ˜m > 0, w hich is
defined by
˜m = lim ?M
Gaß?a?ße˜?f ,
(0.1 6 )
w here Gaß is the E instein tensor and w here the lim it is defined w ith respect
to a sequence of spacelike hypersurfaces running into the future singularity,
cf. [5].
For A R W spaces w ith non-com pact S0 w e sim ply call the lim it
lim |f |2e˜? f = ˜m,
(0.1 7 )
w hich exists by definition and is positive, m ass.
T he m ost general branes that w e consider are branes of class (B), th ey
are supposed to satisfy only the first of the five conditions that are im posed
on A R W spaces, cf. D efinition 0.8.
L et us form ulate this condition as
0.1 Definition. A globally hyperbolic spacetim e N, N = I × S0, I =
(a, b), the m etric of w hich satisfies (0.10), w ith f = f(x0), is said to be of
class (B), if th ere exist p ositive con stants ˜? and ˜m su ch th a t
lim
t?b
|f |2e2˜? f = ˜m > 0.
(0.1 8 )
W e also say that N is of class (B) w ith constant ˜? and call ˜m the m ass of N,
though, even in the case of com pact S0, the relation (0.16) is defined only
u n d er sp ecial circu m stan ces.
T he tim e function in spacetim es ofclass (B) h a s fi n ite fu tu re ra n g e, cf. [2,
Lem m a 3.1], thus w e m ay— and shall— assum e that b =0and I = (-a,0).
B y con sid erin g b ran es of class (B) instead of A R W spaces a larger range
of equ ation of states is p ossib le
p = ?
n
?.
(0.1 9 )
W e shallalso consider variable ?.
Page 6
324
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
0.2 Lemma. Let Taß be the divergence free stress energy tensor of a
perfect fluid in N w ith an equation of state
p = ?
n
?,
(0.20)
w here ? = ?0 + ?(f) and ?0 =const. A ssum e that ? is sm ooth satisfying
lim
t?-8
?(t) = 0
(0.21 )
and let ˜µ = ˜µ(t) be a prim itive of ? su c h th at
lim
t?-8
˜µ(t)=0.
(0.22)
Th e n ? satisfies the conservation law
? = ?0e-(n+?0)f-˜µ
,
(0.23)
w here ?0 is a constant.
A proofw illbe given in Section 1.
0.3 R emark. Since the branes, w e shall consider, alw ays satisfy the
a ssu m p tio n s
lim
t?0
f = -o, -f > 0,
(0.24)
it is p ossib le to d efi n e µ = µ(r), r = -ef ,by
µ(r)=˜µ(lo g (-r)).
(0.25 )
W e also call µ a prim itive of ?.
T he m ain results of this paper can now be sum m arized in the follow ing
four theorem s.
0.4 T heorem. Let N be a brane contained in the black hole region of
N. Let n ^ 3 and assum e that ? = ?(t) satisfi e s
\Dm?(t)\ = cm
V m G N.
(0.26 )
(i) If s ^ 0, then ˜? = 1
2
(n - 1) is the only possible value such that N is
an ARW space.
Page 7
CL A U S G E R H A R D T
325
(ii) O n the other hand, if w e set ˜? = 1
2
(n - 1), then N is an ARW space
if and only if the follow ing conditions are satisfied
?0 = -n-1
2
and
{s = 0, if 3 < n ? N,
s ? R, if n = 3,
(0 .27 )
|Dm
t ?| = cme
(n-1)t
? m ? N
(0 .28 )
a nd
lim
t? -8
?(t)e-(n-1)t
(0 .29 )
exists, or
?0 = -(n - 1) and s ? R, ? 3 = n ? N.
(0 .30 )
If the condition (0 .27 ) holds, then the m ass ˜m of N is larger than m
˜m = m + ?2
n2 ?
2
0,
(0 .31 )
w here ?0 is an integration constant of ?e(n+?0)f e˜µ. In the other cases w e
have ˜m = m.
(iii) There exists a sm ooth transition flow from N to a reflected braneˆN,
if the prim itive µ can be view ed as a sm ooth and even function in e˜? f , and
provided the follow ing conditions are valid
n = 3, ?0 = -n-1
2
, s ? R,
(0 .32)
or
n > 3, ?0 = -n-1
2
, ? = s = 0, ˜? = 1,
(0 .33)
or
n = 3, ?0 = -m(n - 1)+1, 2 = m ? N, s ? R,
(0 .34 )
or
n > 3, ?0 = -m(n - 1)+1, 2 = m ? N,
2
n(n+1)
? = -s
2
,
(0 .35)
w here in case ? = s = 0, w e have to assum e ˜? = 1.
(iv ) Since an A R W brane is also a brane of class (B), the sm ooth transi-
tio n resu lts fro m Th eo rem 0 .7, (i), are valid, if the prim itive µ can be view ed
as a sm ooth and even function in the variable r = -ef .
(v ) For the specified values of ?0 a nd s the branes do actually exist.
Page 8
326
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
0.5 Theorem. A ssum e that ? vanishes in a neighbourhood of -8. Then
a brane N ? ˆN is an ARW space w ith ˜? = 1
2
(n - 1) if and only if ˜? = n,
?0 = 1, and
s = - m
2?0
.
(0.36)
The m ass ˜m of N is then equal to
˜m = ?
2
n
2 ?2
0.
(0.37 )
(i) There exists a sm ooth transition flow , if
2
n(n+ 1)
? = -s
2
.
(0.38 )
(ii) S ince N is also a brane of class (B), the sm ooth transition result
fro m Th eo re m 0.7, (ii), is valid, i.e., the joint branes N ? ˆN fo rm a C8-
hypersurface in R2 × S 0, if n = 3 is odd, since µ can be view ed as a sm ooth
and even function in r.
A brane w ith the specified values does actually exist.
0.6 Theorem. A brane N ? N satisfying an equation of state w ith
? = ?0 + ?, w here ? satisfies the conditions of Lem m a 0.2, is of class (B)
w ith constant ˜? > 0 if and only if
˜? = n-1
2
and ?0 = -n-1
2
,
(0.39 )
or
˜? = n + ?0 - 1 and ?0 > -n-1
2
.
(0.4 0)
In both cases the tension s ? R can be arbitrary. B ranes w ith the specified
values do actually exist.
0.7 Theorem. Let N be a brane of class (B) as described in the pre-
ceding theorem and assum e that the corresponding function µ sa tisfi e s th e
conditions stated in (2.7 ) re sp . (2.8 ), w hich m ore or less is tantam ount to
requiring that µ is sm ooth and even as a function of r. Then N can be
reflected to yield a brane ˆN ? ˆN. The joint branes N ? ˆN fo rm a C8-
hypersurface in R2 × S 0 provided the follow ing conditions are valid
(i) If ˜? = n-1
2
a nd ?0 = -n-1
2
, then the relations
n-1
2
odd, ?0 odd, and s ? R,
(0.4 1 )
or
nodd, ?0 ? Z, and s = 0
(0.4 2)
Page 9
CL A U S G E R H A R D T
327
sh o u ld h o ld .
(ii) If ˜? = n + ?0 - 1 and ?0 > -n-1
2 , then the relations
n o dd, ?0 o dd, and s ? R,
(0 .4 3)
or
n o dd, ?0 ? Z, and s = 0
(0 .4 4 )
should be valid.
F or th e con ven ien ce of th e read er w e rep eat th e d efi n ition of A R W sp aces,
slightly m odified to include the case of non-com pact S0.
0.8 Definition. A globally hyperbolic spacetim e N, dim N = n + 1,
is said to be asym ptotically R obertson-W alker (A RW ) w ith respect to the
fu tu re, if a fu tu re en d of N, N+,can be w ritten as a product N+ = [a, b)× S0,
w here S0 is a R iem annian space, and there exists a future directed tim e
fu n ctio n t = x0 such that the m etric in N+ can be w ritten as
d?s2 = e2 ˜?{-(dx0)2 + sij(x0,x)dxidxj},
(0 .4 5 )
w here S0 corresp on d s to x0 = a, ˜? is of th e form
˜
?(x0,x) = f(x0) + ?(x0,x),
(0 .4 6 )
and w e assum e that there exists a positive constant c0 and a sm ooth R ie-
m annian m etric ¯sij on S0 su ch th a t
lim
t^b
e? = c0
?
lim
t^b
sij(t,x)=¯sij(x),
(0 .4 7)
and
lim
t^b
f(t) = -8 .
(0 .4 8 )
W ithout loss ofgenerality w e shallassum e c0 =1. T hen N is A R W w ith
respect to the future, if the m etric is close to the R obertson-W alker m etric
d¯s2 = e2f {-dx02 + ¯sij(x)dxidxj}
(0 .4 9 )
n ea r th e sin g u la rity t = b.By close w e m ean th at th e d erivatives of arb itrary
order w ith respect to space and tim e of the conform al m etric e-2f ?ga ß in
(0.45) sh ou ld converge to th e corresp on d in g d erivatives of th e con form allim it
m etric in (0.49), w hen x0 ten d s to b. W e em phasize that in our term inology
Page 10
328
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
R obertson-W alker m etric does not necessarily im ply that (¯sij) is a m etric of
con stan t cu rvatu re, it is on ly th e sp atial m etric of a w arp ed p ro d u ct.
W e assum e,furtherm ore,that f satisfi es th e follow in g fi ve con d ition s
-f > 0,
(0.5 0)
there exists ˜? G R su ch th at
n + ˜? - 2 > 0 ?
lim
t?b
\f \2
e
(n+ ˜?-2)f = ˜m > 0.
(0.5 1)
Set ˜? = 1
2
(n + ˜? - 2), then there exists the lim it
lim
t?b
(f + ˜?\f \2)
(0.5 2)
an d
\Dm
t (f + ˜?\f \2)\ = cm\f \m
Vm ^ 1,
(0.5 3)
as w ell as
\Dm
t f\ = cm\f \m
Vm ^ 1.
(0.5 4 )
If S0 is com pact, then w e call N a n o rm a lized A R W sp acetim e, if
?S0 vd et ¯sij = \Sn\.
(0.5 5 )
0.9 Remark. T he special branes w e consider are alw ays R obertson-
W alker sp aces, i.e., in ord er to p rove th at th ey are also A R W sp aces w e on ly
have to show that f satisfies the five conditions stated above.
1 The modified Friedman n eq u ation
T h e Israel ju n ction con d ition (0.12) is eq u ivalen t to
haß = ?(Taß - 1
nT ¯gaß + s
n ¯gaß ),
(1.1)
w here T = Ta
a .
A ssum ing the stress energy tensor to be that of a perfect fluid
T0
0 = -?, Ta
i = pda
i ,
(1.2)
satisfying an equation of state
p = ?
n
?,
(1.3)
Page 11
CL A U S G E R H A R D T
329
w e finally obtain
hij = ?
n
(? + s)¯gij
(1 .4 )
for the spatial com ponents of the second fundam entalform .
L et u s lab el th e coord in ates in N as (ya)=(yr,yt,yi) = (r, t, xi). T hen
w e consider em beddings of the form
y = y(t,xi)=(r(t),t(t),xi)
(1 .5 )
x0 = t should be the tim e function on the brane w hich is chosen such that
the induced m etric can be w ritten as
d¯s2 = r2(-(dx0)2 + sijdxidxj).
(1 .6 )
W e also assum e that ?r > 0. N otice that r < 0, so that (xa) is a future
oriented coordinate system on the brane. If w e set f = lo g (-r), then the
induced m etric has the form as indicated in (0.10).
L et us p oint out that this choice of t im plies the relation
r2 = ˜h-1 ?r2 - ˜h|t |2,
(1 .7 )
sin ce
¯g00 = < ?y, ?y>,
(1 .8 )
w here w e use a dot or a prim e to indicate differentiation w ith respect to t
u n less oth erw ise sp ecifi ed .
Since the tim e function in A RW spaces or in spaces of class (B) has a
fi n ite fu tu re ran ge, cf. [2, L em m a 3.1], w e assu m e w ith ou t loss of gen erality
that the em bedding is defined in I × S 0 w ith I = (-a,0).
T he only non-trivial tangent vector of N is
y = (r ,t ,0...,0),
(1 .9)
and hence a covariant norm al(?a) of N is given by
?(-t ,r ,0,...,0),
(1 .1 0)
w here ? is a norm alization factor, and the contravariant norm al vector is
given by
(?a) = -r-1(˜h d t
d t
, ˜h-1 ?r,0,...,0),
(1 .1 1 )
in view of (1.7).
Page 12
330
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
T he norm al vector ? of the brane is spacelike, i.e., the G auß form ula
reads
yaß = -haß?,
(1 .1 2 )
w e refer to [1, Section 2] for our conventions.
W e also em phasize that w e have neither specified the sign nor the actual
value of ?, i.e., it is irrelevant w hich norm al w e use in the G auß form ula.
To determ ine hij w e use
yt
ij = yt
,ij + ˜Gt
bcyb
i yc
j -¯G?
ijyt
? = -r-1 ?r d t
d t
sij,
(1 .1 3)
and w e conclude
hij = -˜h d t
d t
sij,
(1 .1 4 )
in view of (1.12),(1.13)and the assum ption ?r =0,and from (1.4)w e further
deduce
-˜hr-2 d t
d t
= ?
n
(? + s),
(1 .1 5 )
or,by taking (1.7) into account,
|f |2 -˜h = ?2
n2 (? + s)2
e
2f
.
(1 .1 6 )
T his is the m odified Friedm ann equation.
B ranes that are A R W spaces
L et us first consider branes that are A R W spaces.
Since the bulk is an E instein space, the left-hand side of (0.12) is diver-
gence free,as can be easily deduced w ith the help of the C odazziequation,
i.e., Taß is also divergence free,and hence ? satisfies the conservation law
?e
(n+?0)f
e
˜µ = con st = ?0,
(1 .1 7 )
cf. L em m a 0.2; a proof w ill b e given later.
In order to find out under w hich conditions the brane is an A R W space,
w e d istin gu ish b etw een th e cases s = 0and s < 0. T he latter choice violates
the approxim ation of the classical Friedm ann equation for sm all ?.
1.1 Lemma. Let s = 0, then ˜? =
1
2
(n - 1) is the only possible value
such that N can be an ARW space.
Page 13
CL A U S G E R H A R D T
331
P roof. From (1.16) and (1.17) w e derive
|f |2
e
2˜?f = me(2˜?-(n-1))f +
2
n(n+1)
? e
2(˜?+1)f - ˜? e2˜?f
+ ?2
n2 (?2
0e2(˜?+1-(n+?0))f e-2˜µ
+ 2s?0e(2˜?+2-(n+?0))f e-˜µ + s2e2(˜?+1)f ).
(1.18 )
D ifferentiating both sides and dividing by 2f y ie ld s
(f + ˜?|f |2)e2˜?f =
m(˜? -
(n-1)
2
)e(2˜?-(n-1))f +
2(˜?+1)
n(n+1)
? e2(˜?+1)f - ˜?˜? e2˜?f
+ ?2
n2 (?2
0(˜? + 1 - (n + ?0))e2(˜?+1-(n+?0))f
e-2˜µ
- ??2
0e2(˜?+1-(n+?0))f e-2˜µ
+ s?0(2 ˜? + 2 - (n + ?0))e(2˜?+2-(n+?0))f e-˜µ
- ?s?0e
(2˜?+2-(n+?0))f
e-˜µ
+ s2(˜? + 1)e2(˜?+1)f ).
(1.19 )
If N is an A R W space, then the left-hand side of (1.18) has to converge
to a positive constant, if t goes to zero, and f + ˜?|f |2 has to converge to
a constant.
T hus w e deduce that allexponents of ef on the right-hand side of (1.18)
have to be non-negative. D ividing now equation (1.19)by e2˜?f ,and using the
fact th at th e term s involvin g ? can b e n eglected , sin ce ? van ish es su ffi cien tly
fast n ear -8 , w e see th at th e coeffi cients of all p ow ers of ef ,w hich have the
com m on factor ?2
n2 , are non-negative, hence w e m ust have ˜? = 1
2
(n - 1), fo r
otherw ise w e get a contradiction.
1.2 Lemma. Let ˜? =
1
2
(n - 1), s ? R and assum e that (0 .2 6 ) is valid.
Th e n N is an ARW space if and only if the follow ing conditions are satisfied
?0 = -n-1
2
and
{s = 0, if 3 < n ? N,
s ? R, if n = 3,
(1.2 0 )
and the relations (0 .2 8 ) and (0 .2 9 ) hold, or
?0 = -(n - 1) a n d s ? R, ? 3 = n ? N.
(1.2 1)
Page 14
332
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
P roof. L et n = 3 and s ? R be arbitrary. If the left-hand side of (1.18)
converges,then the exponents ofthe term s w ith the com m on factor ?2
n2 have
to b e n on -n egative, sin ce th e exp on ents of th e term s w ith th e factor ?0 can’t
be both negative and equal, so that they m ight cancel each other. H ence
there m ust hold
?0 = -n-1
2
.
(1.22)
M oreover, after dividing (1.19) by e2˜? f w e see that either ?0 = -n-1
2
or
n + ?0 = 1,
(1.23)
i.e.,
?0 = -(n - 1).
(1.24 )
In case ?0 = -n-1
2
, w e deduce from (1.19) that either s = 0 or
0 = 2 - n + n-1
2
= -n-3
2
,
(1.25 )
i.e., n = 3 m u st b e valid , if s = 0.
If th ese n ecessary con d ition s are satisfi ed , th en w e can exp ress f +˜?|f |2
in the form
f + ˜?|f |2 = 1
n
? e2f - ˜?˜?
+ ?2
n2 (-? ?2
0e-(n-1)f e-2˜µ + 2s?0e-˜µ
- ? s?0e-˜µ + s2(˜? + 1)e2f )
(1.26 )
in case ?0 = -n-1
2
,w here w e note that s = 0, if n > 3,and ˜? = 1, if n = 3,
and in the form
f + ˜?|f |2 = 1
n
? e2f - ˜?˜?
+ ?2
n2 (c1e2?1f e-2˜µ - ? c2e2?1f e-2˜µ + c3e(?1+ 1)f e-˜µ
- ? c4e
(?1+ 1)f
e-˜µ + c5e
2f )
(1.27 )
w ith constants ci,?1,such that ?1 = 0, if ?0 = -(n - 1).
T hus the rem aining conditions for f in the definition of A R W spaces are
autom atically satisfied,in view ofthe conditions (0.26),(0.28),and (0.29).
O n the other hand, it is im m ediately clear that the conditions in the
lem m a are also suffi cient provided w e have a solution ofequation (1.16) such
th a t
lim
t?0
f = -8
and f < 0.
(1.28 )
Page 15
CL A U S G E R H A R D T
333
T he existence of such a solution w illbe show n in T heorem 3.1.
N ext let us exam ine the possibility that N is an A RW space w ith ˜? \=
n-1
2 .
1.3 Lemma. Let ˜? \= n-1
2 , and suppose that ? vanishes in a neighbour-
hood of -o . Then N is an ARW space w ith constant ˜? if and only if ˜? = n,
? = 1, and s < 0 is fine tuned to
s = - m
2?0 ,
(1 .29 )
w here ?0 is the integration constant in (1 .1 7 ).
P roof. L et N be an A R W space w ith ˜? \= n-1
2 ,then w e conclude from (1.18)
that all exp onents of ef had to be non-negative or
2 - (n + ?0) = -(n - 1),
(1 .30)
i.e., ?0 =1,and s h ad to b e fi n e tu n ed as in d icated in (1.29). If all ex p on en ts
w ere non-negative, then w e w ould use (1.19) to deduce the sam e result as in
(1.30) w ith the corresponding value for s.
H ence, in any case w e m ust have ?0 =1 and s as in (1.29). Inserting
these values in (1.18) w e conclude
0 = 2˜? + 2 - 2(n +1),
(1 .31 )
i.e., ˜? = n.
T he conditions in the lem m a are therefore necessary. Suppose they are
satisfied,then w e deduce
lim \f \2e2˜? f = ?2
n2 ?2
0
(1 .32)
and
f + ˜?\f \2 = 2
n
? e2f - ˜?˜? + ?2
n2 s2(n +1)e2f ,
(1 .33)
if f is close to -o , from w hich w e im m ediately infer that N is an A RW
space.
For the existence result w e refer to T heorem 3.1.
Page 16
334
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
1.4 Remark. (i) If in L em m a 1.2 resp. L em m a 1.3 ?0 is equal to the
iso la te d v a lu e s ?0 = -n-1
2
resp . ?0 = 1,then the future m ass of N is d iff eren t
from the m ass m of N,nam ely,in case ?0 = -n-1
2
and ˜? = n-1
2
w e get
˜m = m + ?2
n2 ?
2
0 > m,
(1 .34)
and in case ?0 =1 and ˜? = n
˜m = ?2
n2 ?
2
0.
(1 .35 )
(ii) In the case ˜? = n-1
2
, the value of ˜? is equal to the value that one
w ould get assum ing the E instein equations w ere valid in N, w h ere th e stress
energy tensor w ould be that of a perfect fluid w ith an equation of state
p = ˜?
n
?
(1 .36 )
such that ˜? = 1, sin ce th en N w ould b e an A R W space satisfying (0.14), cf.
[2 , S ectio n 9 ].
Furtherm ore,the classical Friedm ann equation has the form
|f |2 = -˜? + ?
n
?e
2f = -˜? + ?
n
?0e
(2-(n+ ˜?))f
.
(1 .37 )
T hus, by identifying the leading pow ers of ef on the right-hand side of
(1.16) and (1.37),w e see that in case ˜? =1 there should hold
-(n +1)= -2(n + ?0),
(1 .38 )
i.e.,
?0 = -n-1
2
.
(1 .39 )
T herefore, if w e w ant to interpret the m odified Friedm ann equation in
analogy to the classical Friedm ann equation, then w e have to choose ?0 =
-n-1
2
, if the relation (0.14) serves as a guiding principle.
L et u s con clu d e th ese con sid eration s on b ran es th at are A R W sp aces w ith
the follow ing lem m a
1.5 L emma. Th e b ra n e s N w hich are ARW spaces satisfy the tim elike
convergence condition near the singularity.
Page 17
CL A U S G E R H A R D T
335
P roof. W e apply the relation connecting the R icci tensors of tw o conform al
m etrics. L et
(?a) = (1,ui) and v = 1 - sijuiuj = 1 - |D u|2 > 0,
(1.4 0)
th en
¯Raß?a?ß = Rijuiuj - (n - 1)(f - |f |2) + v2(-f - (n - 1)|f |2)
= (n - 1)˜?|D u|2 - (n - 2)f + (n - 1)|f |2|D u|2,
(1.4 1)
hence the result.
B ranes of class (B)
N ext w e shall exam ine branes that are of class (B). L et us start w ith a
proof of L em m a 0.2
P roof of Lem m a 0.2. T he equation T
?
0;? = 0 im p lies
0 = - ?? - (n + ?)f ?.
(1.4 2 )
If ? w ould vanish in a point,then it w ould vanish identically, in view of
G ronw all’s lem m a. H ence, w e m ay assum e that ? > 0,so that
(lo g ?) = -(n + ?0)f - ?f ,
(1.4 3)
from w hich the conservation law (0.16) follow s im m ediately.
In the follow ing tw o lem m ata w e shallassum e that Taß satisfies an equa-
tion of state w ith ? = ?0 + ?, such that ?, ˜µ and µ satisfy the relations
(0.2 1), (0.2 2 ) a n d (0.2 5).
1.6 Lemma. Let ˜? = n-1
2
and ? = ?0 +?. Then the brane N is of class
(B) w ith constant ˜? if and only if
?0 = -n-1
2
.
(1.4 4 )
P roof. T h e left h an d -sid e of equ ation (1.18) converges to a p ositive con stant
ifand only if ?0 sa tisfi es (1.4 4 ).
Page 18
336
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
1.7 Lemma. Let ˜? = n-1
2 , ? > 0, and ? = ?0 + ?. Then the brane N
is of class (B) w ith constant ˜? if and only if
˜? = n + ?0 - 1
(1.4 5 )
and
?0 > -n-1
2 .
(1.4 6)
P roof. (i) L et N b e of class (B) w ith constant ˜? = n-1
2 , th en
n + ?0 > 0
(1.4 7 )
m ust b e valid, for otherw ise w e get a contradiction. T herefore
2 (˜? + 1 - (n + ?0))
(1.4 8 )
is the sm allest exponent on the right-hand side of (1.18) and has thus to
v a n ish , i.e .,
˜? = n + ?0 - 1.
(1.4 9 )
T he other exponents have to be non-negative, i.e.,
2 ˜?>n - 1
(1.5 0)
and
2 ˜? = n + ?0
or s = 0.
(1.5 1)
T h e in equ ality (1.50) is equ ivalent to (1.46) w h ich in tu rn im p lies (1.51).
(ii) T h e con d ition s (1.45) an d (1.46) are su ffi cient, sin ce th en
2 ˜?>n - 1
(1.5 2 )
and
˜m = ?2
n2 ?2
0.
(1.5 3)
S im ilarly as b efore b ran es of class (B)do actually exist,cf.T heorem 3.1.
Page 19
CL A U S G E R H A R D T
337
2 Transition from big crunch to big bang
B ranes that are A R W spaces
For a brane N that is an A R W space, the results of [4] can be applied.
D efine a reflected spacetim e ˆN by sw itching the lightcone and changing the
tim e function to x0 = -x0, then N and ˆN can be pasted together at the
sin g u la rity {0} × S 0 yielding a sm ooth m anifold w ith a m etric singularity
w hich is a big crunch, w hen view ed from N, and a big bang, w hen view ed
fro m ˆN. M oreover, there exists a naturaltransition flow of class C3 across
the singularity w hich is defined by rescaling an appropriate inverse m ean
curvature flow .
T his transition flow is of class C8, if th e q u a n tity f e˜?f , o r eq u iva len tly,
|f |2e2˜?f can be view ed as a sm ooth and even function in the variable e˜?f ,
cf. [4, the rem arks before T heorem 2.1].
For the branes considered in T heorem 0.4 and T heorem 0.5 |f |2e2˜?f is
a sm ooth and even function in e˜?f .
In the follow ing w e shall see that the reflected spacetim e can be view ed
as a b ran e in th e S ch w arzsch ild -A d S (n+ 2) space ˆN w hich is a reflection of N.
ˆN is obtained by sw itching the lightcone in N and by changing the radial
coordinate r to -r. T he singularity in r = 0 is th en a w h ite h ole sin gu larity.
M oreover, A R W branes are also branes of class (B), i.e., th e tran sition
results for those branes, w hich w e shall prove next, also apply.
B ranes of class (B )
T he brane N is given by
y(t)=(r(t),t(t),xi).
(2 .1 )
Since the coordinates (xi) do not change, let us w rite y(t)=(r(t),t(t)).
W e shallsee in a m om ent that t(t) converges, if t tends to zero, hence,
let us set this lim it equal to zero.
N ow w e define the reflection ˆN of N b y
y(t)=(-r(-t),-t(-t))
(2 .2 )
Page 20
338
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
fo r t > 0. T he result is a brane in ˆN w hich can be pasted continuously to
N.
T he em beddings of N resp . ˆN in N resp . ˆN are also em beddings in
R2 × S 0, endow ed w ith the R iem annian product m etric, and now it m akes
sense to ask, if the joint hypersurfaces N ?ˆN form a sm ooth hypersurface
in R2 × S 0.
T he sm oothness of N ?ˆN in R2 × S 0 w e interpret as a sm ooth transition
from big crunch to big bang.
T o prove the sm oothness w e have to param eterize y = y(t) w ith resp ect
to r. T h is is p o ssib le, sin ce
dr
dt
= -f ef > 0.
(2 .3)
T hen w e have y(r)=(r, t(r)) and w e need to exam ine
dt
dr
= dt
dt
dt
dr
= - dt
dt
1
f ef .
(2 .4 )
D efine
?(r)=˜h(r)e(n-1)f
= m +
2
n(n+1)
? e
(n+1)f - ˜? e(n-1)f
,
(2 .5 )
then w e obtain from (1.15) and (1.17)
?dt
dr
= ?
n
(s + ?0e-(n+?0)f
e-˜µ)en f
f
= ?
n
(se(n+˜?)f + ?0e
(˜?-?0)f
e-˜µ) 1
f e˜? f .
(2 .6 )
T hisim m ediately show sthat in allcases |dt
dr
| is u n ifo rm ly b o u n d ed , h en ce
lim r?0 t(r) ex ists.
T he joint em bedding N ?ˆN in R2 × S 0 w ill b e of class C8, if dt
dr
can b e
view ed as a sm ooth and even function in r = -ef .
W e shall assum e that the function µ = µ(r) in (0.25) is either even and
sm ooth in (-r0,r0)
µ ? C8((-r0,r0)) is e v e n
(2 .7 )
or that µ ? C8((-r0,0]) such that
|Dmµ(0)| = 0 ? m ? N.
(2 .8)
Page 21
CL A U S G E R H A R D T
339
In the latter case w e can extend µ as an even and sm ooth function to r > 0
b y settin g
µ(r) = µ(-r) fo r r > 0.
(2 .9)
T hen w e can prove
2.1 Theorem. Let N be a brane of class (B) satisfying an equation of
state w ith ? = ?0 +?, so that ? vanishes at the singularity. The correspond-
in g fun c tio n µ should satisfy the conditions (2 .7 ) or (2 .8 ). Then N can be
reflected as described above to yield a brane ˆN ?ˆN. The joint branes N ? ˆN
form a C8- hypersurface in R2 × S 0 provided the follow ing conditions are
v a lid
(i) If ˜? = n-1
2
and ?0 = -n-1
2
, then the relations
n-1
2
odd, ?0 odd, and s ? R,
(2 .10)
or
nodd, ?0 ? Z, and s = 0
(2 .11)
should hold.
(ii) If ˜? = n + ?0 - 1 and ?0 > -n-1
2
, then the relations
nodd, ?0 odd, and s ? R,
(2 .12 )
or
nodd, ?0 ? Z. and s = 0
(2 .13)
should be valid.
P roof.
”
(i)“ L et ˜? = n-1
2
and ?0 = -n-1
2
. W e have to show that d t
d r
is a
sm ooth and even function in r. From equation (2.6) w e deduce
d t
d r
= -?
n
(s(-r)(n+˜?)f + ?0e
(˜?-?0)f
e-µ)?-1?-1,
(2 .14 )
w here
?2 = |f |2e2˜?f = m e(2˜?-(n-1))f +
2
n(n+1)
? e2(˜?+1)f - ˜? e2˜?f
+ ?2
n2 (?2
0e
2(˜?+1-(n+?0))f
e-2µ
+ 2s?0e
(2˜?+2-(n+?0))f
e-µ + s2
e
2(˜?+1)f ).
(2 .15 )
T he right-hand side of (2.15) has to be an even and sm ooth function in
r = -ef . B y assum ption µ is already an even and sm ooth function or can be
Page 22
340
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
extended to such a function,hence w e have to guarantee that the exponents
of ef are all even.
A thorough exam ination of the exponents reveals that this is the case,if
th e con d ition s (2.10) or (2.11) are fu lfi lled . N otice th at
n-1
2
odd =? n odd
?
3n-1
2
ev en .
(2 .1 6 )
”
(ii)“ Id en tica l p ro o f.
3 Existen ce of th e b ran es
W e want to prove that for the specified values of ?0 a n d co rresp o n d in g
˜? em bedded branes N in the black hole region of N satisfying the Israel
junction condition do exist.
E vidently it w ill be suffi cient to solve the m odified Friedm ann equation
(1.16) on an interval I = (-a,0] su ch th at, if w e set r = -ef , th en
lim
t?0
r(t)=0,
lim
t?-a
r(t) = r0,
(3.1 )
w here r0 is th e (n eg a tiv e) b la ck h o le ra d iu s, i.e., {r = r0} eq u a ls th e h o rizo n .
W e shall assum e the m ost general equation of state that w e considered
in the previous sections, i.e., ? should be of the form ? = ?0 + ?(f), cf.
Lem m a 0.2,and the corresponding prim itive µ = µ(r) should be sm ooth in
r0 < r< 0.
L et us look at equation (1.18). Setting ? = e˜?f w e deduce that it can be
rew ritten as
˜?-2 ??2 = F(?),
(3.2 )
w here F = F(t) is defined on an interval J = [0 = t< t0) and t0 is given by
t0 = (-r0)˜?.
(3.3)
F is co n tin u o u s in J and sm ooth in
?
J. F u rth erm ore, it satisfi es
F > 0 and F(0) = ˜m > 0.
(3.4)
Page 23
CL A U S G E R H A R D T
341
If w e succeed in solving the equation
˜?-1 ?? = -vF(?)
(3.5 )
w ith in itial valu e ?(0) = 0 on an interval I = (-a,0]such that ? ? C0(I) n
C8(
?
I ) and such that the relations (3.1) are valid for r = -?˜?-1 ,then the
m odified Friedm ann equation is solved by f = -˜?-1 lo g ? and f satisfies
lim
t?0
f = -8 , -f > 0.
(3.6 )
T o solve (3.5) let us m ake a variable transform ation t ? -t,so that w e
have to solve
˜?-1 ?? = vF(?)
(3.7 )
w ith in itial valu e ?(0) = 0 on an interval I = [0,a).
3.1 Theorem. T he equation (3.7 ) w ith initial value ?(0) = 0 has a
so lu tio n ? ? C1(I) n C8(
?
I ), w here I = [0,a) is an interval such that the
re la tio ns (3.1) are satisfied by r = -?˜?-1 w ith o bvio u s mod ifi ca tio ns resu lting
from the variable transformation t ? -t.
P roof. L et 0 < t < t0 be arbitrary and define J = [0,t ]. T hen there are
p o sitiv e co n sta n ts c1,c2 su ch th a t
c2
1 = F(t) = c2
2
? t ? J .
(3.8 )
L et ? > 0 b e sm all, th en th e d iff erential equ ation (3.7) w ith in itial valu e
??(0) = ? has a sm ooth, positive solution ?? in 0 = t < t?, w here t? is
determ ined by the requirem ent
??(t) = t
? 0 = t< t?.
(3.9 )
In view of the estim ates (3.8) w e im m ediately conclude that
˜?c1t = ??(t) = ? + ˜?c2t
? 0 = t< t?,
(3.10)
hence, if w e choose the m axim al t? possible, then there exists t0 > 0 such
th a t
t? = t0 > 0
? ?.
(3.11)
N ow ,using w ell-know n a prioriestim ates,w e deduce that for any interval
I ? (0,t0) w e have
|??|m,I = cm(I )
? m ? N
(3.12 )
Page 24
342
TR A N SITION F R OM B IG CR U N CH TO B IG B A N G ...
in d ep en d en tly o f ?. M oreover, since ?? is uniform ly L ipschitz continuous in
[0,t0], w e in fer th at
lim
??0
?? = ?
(3.1 3)
exists such that ? ? C1([0,t0]) n C8((0,t0)) and ? solves (3.7) w ith initial
value ?(0) = 0.
? can be defined on a m axim al interval interval [0,a*), w here a* is de-
term ined by
lim
t?a*
?(t) = 8 o r lim
t?a*
F(?)=0.
(3.1 4)
O bviously, there exists 0 < a = a* su ch th a t
lim
t?a
?(t)=(-r0)˜?.
(3.1 5 )
H ence the proofofthe theorem is com pleted.
3.2 Remark. If a< a*, then w e cannot conclude that w e have defined
b ran es ex ten d in g b eyon d th e h orizon , sin ce ou r em b ed d in g d eteriorates w h en
the horizon is approached, as a look at the equations (1.15) and (1.16) re-
veals. Since ˜h vanishes on the horizon, either d t
d t
becom es unbounded or
(? + s) w ould tend to zero, in w hich case F(?) w ou ld van ish , i.e., if a< a*,
th en n ecessa rily
lim
t?- a
| d t
d t
| = 8
(3.1 6 )
has to be valid. A n extension of the brane past the horizon w ill require a
different em bedding or at least a different coordinate system in the bulk.
N otice that
¯g00 = < ?y, ?y> = -r2 = 0,
(3.1 7 )
so that it is m ore or less a coordinate singularity.
A ckn ow led gemen t. T his w ork has been supported by the D eutsche
F o rsch u n g sg em ein sch a ft.
References
[1] C laus G erhardt, H ypersurfaces of prescribed curvature in Lorentzian
m a n ifo ld s, Indiana U niv.M ath.J. 4 9 (20 0 0 ), 1 1 25 – 1 1 5 3, p d f fi le.
[2]
, The inverse m ean curvature flow in A RW spaces - transition
from big crunch to big bang, 2004, 39 pages, m ath.D G /0403485.
Page 25
CL A U S G E R H A R D T
343
[3]
, T he inverse m ean curvature flow in cosm ological spacetim es,
2004, 24 pages, m ath.D G /0403097.
[4]
, T he inverse m ean curvature flow in R obertson-W alker spaces
and its application to cosm ology, 2004, 9 pages, gr-qc/0404112.
[5 ]
, T he m ass of a Lorentzian m anifold, 2004, 14 pages,
m ath.D G /0403002.
[6] Justin K houry, A briefing on the ekpyrotic/cyclic universe, 2004, 8 p ages,
a stro -p h / 0 40 1 5 7 9 .
[7] D avid L anglois, B rane cosm ology: an introduction, 2002, 32 pages, hep-
th/0209261.
[8] N eil T urok and P aul J. Steinhardt, B eyond inflation: A cyclic universe
scena rio, 2004, 27 pages, hep-th/0403020. |
