Claus Gerhardt- Transition from big crunch to big bang in brane cosmology

8/3/2019 Claus Gerhardt- Transition from big crunch to big bang in brane cosmology

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c 2004 International PressAdv. Theor. Math. Phys. 8 (2004) 319343

Transition from big crunch to big

bang in brane cosmology

Claus Gerhardt

Institut fur Angewandte MathematikRuprecht-Karls-Universitat69120 Heidelberg, Germany

[email protected]

Abstract

We consider branes N = I S0, where S0 is an ndimensionalspace form, not necessarily compact, in a Schwarzschild-AdS(n+2) bulkN. The branes have a big crunch singularity. If a brane is an ARWspace, then, under certain conditions, there exists a smooth naturaltransition flow through the singularity to a reflected brane N, whichhas a big bang singularity and which can be viewed as a brane in areflected Schwarzschild-AdS(n+2) bulk N. The joint branes N N canthus be naturally embedded in R2 S0, hence there exists a secondpossibility of defining a smooth transition from big crunch to big bangby requiring that N N forms a C-hypersurface in R2 S0. Thislast notion of a smooth transition also applies to branes that are notARW spaces, allowing a wide range of possible equations of state.

0 Introduction

The problem of finding a smooth transition from a spacetime N with a bigcrunch to a spacetime N with a big bang singularity has been the focus ofsome recent works in general relativity, see e.g., [6, 8] and the referencestherein. For abstract spacetimes, i.e., for spacetimes that are not embeddedin a bulk space, it is even a non-trivial question how to define a smoothtransition.

e-print archive: http://lanl.arXiv.org/abs/gr-qc/0404061


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320 TRANSITION FROM BIG CRUNCH TO BIG BANG.. .

In two recent papers [2, 4] we studied this problem for a special class ofspacetimes, so-called ARW spaces, and used the inverse mean curvature flowto prove that, by reflecting the spaces, a smooth transition from big crunchto big bang is possible.

In this paper we look at branes in a Schwarzschild-AdS(n+2) bulk N,where the branes are assumed to lie in the black hole region, i.e., the radialcoordinate is the time function. For those branes that are ARW spaces thetransition results from [4] can be applied to conclude that a smooth transitionflow from a brane N to a properly reflected brane N exists. However, theassumption that the branes are ARW spaces reduces the number of possiblebranes drastically, cf. Theorem 0.4 and Theorem 0.5. Fortunately, in thecase of embedded spacetimes, it is possible to define a transition throughthe singularity without using the inverse mean curvature flow.

Let N be a Schwarzschild-AdS(n+2) bulk space with a black hole singu-larity in r = 0. We assume that the radial coordinate r is negative, r < 0.Then, by switching the light cone and changing r to r we obtain a reflected

Schwarzschild-AdS(n+2) bulk space N with a white hole singularity in r = 0.These two bulk spacetimes can be pasted together such that N N is asmooth manifold, namely, R2 S0, which has a metric singularity in r = 0.In the black (white) hole region r is the time function and it is smooth acrossthe singularity.

Now, let us consider branes N in the black hole region of N. Thesebranes need not to be ARW spaces, they are only supposed to satisfy thefirst of the five assumptions imposed for ARW spaces. We call those branesto be of class (B), cf. Definition 0.1.

The relation between the geometry of the branes and physics is governedby the Israel junction condition.

We shall prove existence and transition through the singularity onlyfor single branes, but this does include a two branes or multiple branesconfigurationwhere then each brane has to be treated separately.

Moreover, in the equation of state

p = n

(0.1)

variable are allowed

= 0 + , 0 = const, (0.2)where = (log(r)) is defined in the bulk.


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CLAUS GERHARDT 321

Branes of class (B) exist in the whole black hole region ofN, they stretchfrom r = 0, the black hole singularity, to r = r0, the event horizon.

The branes whose existence is proved in Theorem 3.1 automatically havea big crunch singularity in r = 0, since this is the initial condition for theordinary differential equation that has to be solved. The branes are givenby an embedding

y() = (r(), t(), xi), a < < 0, (0.3)

with a big crunch singularity in = 0 such that r(0) = 0.

From the modified Friedmann equation we shall deduce that the limit

lim0

t() = t0 (0.4)

exists and without loss of generality we shall suppose t0 = 0.

We then shall define a brane N N by reflection

y() = (r(), t(), xi), 0 < < a. (0.5)

The two branes N, N can be pasted together to yield at least a Lipschitzhypersurface in R2 S0. If this hypersurface is of class C, then we shallspeak of a smooth transition from big crunch to big bang.

To prove the smoothness we reparametrize N N by using r as a newparameter instead of . The old brane N is then expressed as

y(r) = (r, t(r), xi), r < 0, (0.6)

and the reflected

N as

y(r) = (r, t(r), xi), r > 0. (0.7)

Hence, N N is a smooth hypersurface, if dtdr

is a smooth and evenfunction in r in a small neighbourhood of r = 0, < r < .

If dtdr is not an even function in r, then NN will not be C hypersurface.

Examples can easily be constructed.

For branes, that are ARW spaces, the transition flow provides an alter-nate criterion for a smooth transition through the singularity. The reflected

brane N that is used in this process is the same as in the above descriptionof pasting together the two embeddings.


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A version of the modified Friedmann equation, equation (1.18), plays acentral role in determining if the transition is smooth, namely, the quantityfef has to be a smooth and even function in the variable ef in order thatthe transition flow is smooth, and a smooth and even function in the variabler = ef, if the joint embedding is to represent a smooth hypersurface inR2 S0.

The metric in the bulk space N is given by

ds2 = h1dr2 + hdt2 + r2ijdxidxj, (0.8)

where (ij) is the metric of an n- dimensional space form S0, the radialcoordinate r is assumed to be negative, r < 0, and h(r) is defined by

h = m(r)(n1) + 2n(n+1)r

2 , (0.9)

where m > 0 and 0 are constants, and = 1, 0, 1 is the curvature ofS0. We note that we assume that there is a black hole region, i.e., if = 0,then we have to suppose = 1.

We consider branes N contained in the black hole region {r0 < r < 0}.N will be a globally hyperbolic spacetime N = I S0 with metric

ds2 = e2f((dx0)2 + ijdxidxj) (0.10)

such thatf = f(x0) = log(r(x0)). (0.11)

We may assume that the time variable x0 = maps N on the interval(a, 0). In = 0 we have a big crunch singularity induced by the black hole.

The relation between geometry and physics is governed by the Israel junction conditions

h Hg = (T g), (0.12)

where h is the second fundamental form of N, H = gh the mean

curvature, = 0 a constant, T the stress energy tensor of a perfect fluidwith an equation of state

p = n, (0.13)

and the tension of the brane.

One of the parameters used in the definition of (n +1)-dimensional ARW

spaces is a positive constant , which is best expressed as = 12(n + 2). (0.14)


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If N would satisfy the Einstein equation of a perfect fluid with an equationof state

p = n

, (0.15)

then would be defined with the help of in (0.15), cf. [2, Section 9].

ARW spaces with compact S0 also have a future mass m > 0, which isdefined by

m = lim

M

Gef, (0.16)

where G is the Einstein tensor and where the limit is defined with respectto a sequence of spacelike hypersurfaces running into the future singularity,cf. [5].

For ARW spaces with non-compact S0 we simply call the limit

lim|f|2ef = m, (0.17)

which exists by definition and is positive, mass.

The most general branes that we consider are branes of class (B), theyare supposed to satisfy only the first of the five conditions that are imposedon ARW spaces, cf. Definition 0.8.

Let us formulate this condition as

0.1 Definition. A globally hyperbolic spacetime N, N = I S0, I =(a, b), the metric of which satisfies (0.10), with f = f(x0), is said to be ofclass (B), if there exist positive constants and m such that

limb

|f|2e2f = m > 0. (0.18)

We also say that N is of class (B) with constant and call m the mass ofN,though, even in the case of compact S0, the relation (0.16) is defined onlyunder special circumstances.

The time function in spacetimes of class (B) has finite future range, cf. [2,Lemma 3.1], thus we mayand shallassume that b = 0 and I = (a, 0).

By considering branes of class (B) instead of ARW spaces a larger rangeof equation of states is possible

p =

n. (0.19)We shall also consider variable .


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0.2 Lemma. Let T be the divergence free stress energy tensor of aperfect fluid in N with an equation of state

p = n, (0.20)

where = 0 + (f) and 0 = const. Assume that is smooth satisfying

limt(t) = 0 (0.21)

and let = (t) be a primitive of such that

limt

(t) = 0. (0.22)

Then satisfies the conservation law

= 0e(n+0)f, (0.23)

where 0 is a constant.

A proof will be given in Section 1.

0.3 Remark. Since the branes, we shall consider, always satisfy theassumptions

lim0

f = , f > 0, (0.24)

it is possible to define = (r), r = ef, by

(r) = (log(r)). (0.25)

We also call a primitive of .

The main results of this paper can now be summarized in the followingfour theorems.

0.4 Theorem. Let N be a brane contained in the black hole region ofN. Let n 3 and assume that = (t) satisfies

|Dm(t)| cm m N. (0.26)

(i) If 0, then = 12 (n 1) is the only possible value such that N isan ARW space.


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(ii) On the other hand, if we set = 12(n 1), then N is an ARW spaceif and only if the following conditions are satisfied

0 = n12 and

= 0, if 3 < n N,

R, if n = 3,(0.27)

|D

m

t | cme

(n1)t

m N

(0.28)and

limt

(t)e(n1)t (0.29)

exists, or

0 (n 1) and R, 3 n N. (0.30)

If the condition (0.27) holds, then the mass m of N is larger than m

m = m + 2

n220, (0.31)

where 0 is an integration constant of e(n+0)f

e

. In the other cases wehave m = m.

(iii) There exists a smooth transition flow from N to a reflected brane N,if the primitive can be viewed as a smooth and even function in ef, andprovided the following conditions are valid

n = 3, 0 = n12 , R, (0.32)

or

n > 3, 0 = n12 , = = 0, = 1, (0.33)

orn = 3, 0 = m(n 1) + 1, 2 m N, R, (0.34)

or

n > 3, 0 = m(n 1) + 1, 2 m N,2

n(n+1) = 2, (0.35)

where in case = = 0, we have to assume = 1.

(iv) Since an ARW brane is also a brane of class (B), the smooth transi-tion results from Theorem 0.7, (i), are valid, if the primitive can be viewedas a smooth and even function in the variable r = ef.

(v) For the specified values of 0 and the branes do actually exist.


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0.5 Theorem. Assume that vanishes in a neighbourhood of. Thena brane N N is an ARW space with = 12(n 1) if and only if = n,0 = 1, and

= m20 . (0.36)

The mass m of N is then equal to

m =

2

n2

2

0. (0.37)

(i) There exists a smooth transition flow, if

2n(n+1) =

2. (0.38)

(ii) Since N is also a brane of class (B), the smooth transition result from Theorem 0.7, (ii), is valid, i.e., the joint branes N N form a C-hypersurface inR2 S0, if n 3 is odd, since can be viewed as a smoothand even function in r.

A brane with the specified values does actually exist.

0.6 Theorem. A brane N N satisfying an equation of state with = 0 + , where satisfies the conditions of Lemma 0.2, is of class (B)with constant > 0 if and only if

= n12 and 0 n12 , (0.39)

or = n + 0 1 and 0 >

n12 . (0.40)

In both cases the tension R can be arbitrary. Branes with the specifiedvalues do actually exist.

0.7 Theorem. Let N be a brane of class (B) as described in the pre-ceding theorem and assume that the corresponding function satisfies theconditions stated in (2.7) resp. (2.8), which more or less is tantamount torequiring that is smooth and even as a function of r. Then N can bereflected to yield a brane N N. The joint branes N N form a C-hypersurface inR2 S0 provided the following conditions are valid

(i) If = n12 and 0 n12 , then the relations

n12 odd, 0 odd, and R, (0.41)

orn odd, 0 Z, and = 0 (0.42)


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CLAUS GERHARDT 327

should hold.

(ii) If = n + 0 1 and 0 > n12 , then the relations

n odd, 0 odd, and R, (0.43)

or

n odd, 0 Z, and = 0 (0.44)

should be valid.

For the convenience of the reader we repeat the definition of ARW spaces,slightly modified to include the case of non-compact S0.

0.8 Definition. A globally hyperbolic spacetime N, dim N = n + 1,is said to be asymptotically Robertson-Walker (ARW) with respect to thefuture, if a future end ofN, N+, can be written as a product N+ = [a, b)S0,where S0 is a Riemannian space, and there exists a future directed time

function = x0 such that the metric in N+ can be written as

ds2 = e2{(dx0)2 + ij(x0, x)dxidxj}, (0.45)

where S0 corresponds to x0 = a, is of the form

(x0, x) = f(x0) + (x0, x), (0.46)

and we assume that there exists a positive constant c0 and a smooth Rie-mannian metric ij on S0 such that

limb

e = c0 limb

ij(, x) = ij(x), (0.47)

andlimb

f() = . (0.48)

Without loss of generality we shall assume c0 = 1. Then N is ARW withrespect to the future, if the metric is close to the Robertson-Walker metric

ds2 = e2f{dx02

+ ij(x)dxidxj} (0.49)

near the singularity = b. By close we mean that the derivatives of arbitraryorder with respect to space and time of the conformal metric e2fg in

(0.45) should converge to the corresponding derivatives of the conformal limitmetric in (0.49), when x0 tends to b. We emphasize that in our terminology


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Robertson-Walker metric does not necessarily imply that (ij) is a metric ofconstant curvature, it is only the spatial metric of a warped product.

We assume, furthermore, that f satisfies the following five conditions

f > 0, (0.50)

there exists R such that

n + 2 > 0 limb

|f|2e(n+2)f = m > 0. (0.51)

Set = 12(n + 2), then there exists the limit

limb

(f + |f|2) (0.52)

and|Dm (f

+ |f|2)| cm|f|m m 1, (0.53)

as well as|Dm f| cm|f

|m m 1. (0.54)

If S0 is compact, then we call N a normalized ARW spacetime, ifS0

det ij = |S

n|. (0.55)

0.9 Remark. The special branes we consider are always Robertson-Walker spaces, i.e., in order to prove that they are also ARW spaces we onlyhave to show that f satisfies the five conditions stated above.

1 The modified Friedmann equation

The Israel junction condition (0.12) is equivalent to

h = (T 1nTg +

n g), (1.1)

where T = T .

Assuming the stress energy tensor to be that of a perfect fluid

T00 = , Ti = p

i , (1.2)

satisfying an equation of statep =

n, (1.3)


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we finally obtainhij =

n

( + )gij (1.4)

for the spatial components of the second fundamental form.

Let us label the coordinates in N as (ya) = (yr, yt, yi) (r,t,xi). Thenwe consider embeddings of the form

y = y(, xi) = (r(), t(), xi) (1.5)

x0 = should be the time function on the brane which is chosen such thatthe induced metric can be written as

ds2 = r2((dx0)2 + ijdxidxj). (1.6)

We also assume that r > 0. Notice that r < 0, so that (x) is a futureoriented coordinate system on the brane. If we set f = log(r), then theinduced metric has the form as indicated in (0.10).

Let us point out that this choice of implies the relation

r2 = h1r2 h|t|2, (1.7)

sinceg00 = y, y, (1.8)

where we use a dot or a prime to indicate differentiation with respect to unless otherwise specified.

Since the time function in ARW spaces or in spaces of class (B) has afinite future range, cf. [2, Lemma 3.1], we assume without loss of generality

that the embedding is defined in I S0 with I = (a, 0).

The only non-trivial tangent vector of N is

y = (r, t, 0 . . . , 0), (1.9)

and hence a covariant normal (a) of N is given by

(t, r, 0, . . . , 0), (1.10)

where is a normalization factor, and the contravariant normal vector isgiven by

(a

) = r1

(hdt

d, h1

r, 0, . . . , 0), (1.11)in view of (1.7).


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CLAUS GERHARDT 331

Proof. From (1.16) and (1.17) we derive

|f|2e2f = me(2(n1))f + 2n(n+1)e

2(+1)f e2f

+ 2

n2(20e

2(+1(n+0))fe2

+ 20e(2+2(n+0))fe + 2e2(+1)f).

(1.18)

Differentiating both sides and dividing by 2f yields

(f + |f|2)e2f =

m( (n1)2 )e(2(n1))f + 2(+1)

n(n+1)e2(+1)f e2f

+ 2

n2(20(+ 1 (n + 0))e

2(+1(n+0))fe2

20e2(+1(n+0))fe2

+ 0(2+ 2 (n + 0))e(2+2(n+0))fe

0e(2+2(n+0))fe

+ 2(+ 1)e2(+1)f).

(1.19)

If N is an ARW space, then the left-hand side of (1.18) has to convergeto a positive constant, if goes to zero, and f + |f|2 has to converge toa constant.

Thus we deduce that all exponents ofef on the right-hand side of (1.18)have to be non-negative. Dividing now equation (1.19) by e2f, and using the

fact that the terms involving can be neglected, since vanishes sufficientlyfast near , we see that the coefficients of all powers of ef, which have thecommon factor

2

n2, are non-negative, hence we must have = 12(n 1), for

otherwise we get a contradiction.

1.2 Lemma. Let = 12(n 1), R and assume that (0.26) is valid.ThenN is an ARW space if and only if the following conditions are satisfied

0 = n12 and

= 0, if 3 < n N,

R, if n = 3,(1.20)

and the relations (0.28) and (0.29) hold, or

0 (n 1) and R, 3 n N. (1.21)


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Proof. Let n 3 and R be arbitrary. If the left-hand side of (1.18)

converges, then the exponents of the terms with the common factor 2

n2have

to be non-negative, since the exponents of the terms with the factor 0 cantbe both negative and equal, so that they might cancel each other. Hencethere must hold

0 n12 . (1.22)

Moreover, after dividing (1.19) by e2f we see that either 0 = n12 or

n + 0 1, (1.23)

i.e.,0 (n 1). (1.24)

In case 0 = n12 , we deduce from (1.19) that either = 0 or

0 2 n + n12 = n32 , (1.25)

i.e., n = 3 must be valid, if = 0.

If these necessary conditions are satisfied, then we can express f+ |f|2

in the formf + |f|2 = 1

ne2f

+ 2

n2(20e

(n1)fe2 + 20e

0e + 2(+ 1)e2f)

(1.26)

in case 0 = n12 , where we note that = 0, ifn > 3, and = 1, if n = 3,

and in the form

f + |f|2 = 1n

e2f

+ 2

n2(c1e

21fe2 c2e21fe2 + c3e

(1+1)fe

c4e(1+1)fe + c5e

2f)

(1.27)

with constants ci, 1, such that 1 0, if0 (n 1).

Thus the remaining conditions for f in the definition of ARW spaces areautomatically satisfied, in view of the conditions (0.26), (0.28), and (0.29).

On the other hand, it is immediately clear that the conditions in thelemma are also sufficient provided we have a solution of equation (1.16) suchthat

lim0

f = and f < 0. (1.28)


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The existence of such a solution will be shown in Theorem 3.1.

Next let us examine the possibility that N is an ARW space with =n12 .

1.3 Lemma. Let = n12 , and suppose that vanishes in a neighbour-

hood of . ThenN is an ARW space with constant if and only if = n, = 1, and < 0 is fine tuned to

= m20 , (1.29)

where 0 is the integration constant in (1.17).

Proof. Let N be an ARW space with = n12 , then we conclude from (1.18)that all exponents of ef had to be non-negative or

2 (n + 0) = (n 1), (1.30)

i.e., 0 = 1, and had to be fine tuned as indicated in (1.29). If all exponentswere non-negative, then we would use (1.19) to deduce the same result as in(1.30) with the corresponding value for .

Hence, in any case we must have 0 = 1 and as in (1.29). Insertingthese values in (1.18) we conclude

0 = 2+ 2 2(n + 1), (1.31)

i.e., = n.

The conditions in the lemma are therefore necessary. Suppose they aresatisfied, then we deduce

lim|f|2e2f = 2

n220 (1.32)

and

f + |f|2 = 2n

e2f + 2

n22(n + 1)e2f, (1.33)

if f is close to , from which we immediately infer that N is an ARWspace.

For the existence result we refer to Theorem 3.1.


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1.4 Remark. (i) If in Lemma 1.2 resp. Lemma 1.3 0 is equal to theisolated values 0 =

n12 resp. 0 = 1, then the future mass ofN is different

from the mass m of N, namely, in case 0 = n12 and =

n12 we get

m = m + 2

n220 > m, (1.34)

and in case 0 = 1 and = n

m = 2

n220. (1.35)

(ii) In the case = n12 , the value of is equal to the value that onewould get assuming the Einstein equations were valid in N, where the stressenergy tensor would be that of a perfect fluid with an equation of state

p = n

(1.36)

such that = 1, since then N would be an ARW space satisfying (0.14), cf.

[2, Section 9].

Furthermore, the classical Friedmann equation has the form

|f|2 = + ne2f = + n0e

(2(n+))f. (1.37)

Thus, by identifying the leading powers of ef on the right-hand side of(1.16) and (1.37), we see that in case = 1 there should hold

(n + 1) = 2(n + 0), (1.38)

i.e.,0 =

n12 . (1.39)

Therefore, if we want to interpret the modified Friedmann equation inanalogy to the classical Friedmann equation, then we have to choose 0 =n12 , if the relation (0.14) serves as a guiding principle.

Let us conclude these considerations on branes that are ARW spaces withthe following lemma

1.5 Lemma. The branes N which are ARW spaces satisfy the timelikeconvergence condition near the singularity.


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CLAUS GERHARDT 335

Proof. We apply the relation connecting the Ricci tensors of two conformalmetrics. Let

() = (1, ui) and v = 1 ijuiuj 1 |Du|2 > 0, (1.40)

then

R

= Rijui

uj

(n 1)(f

|f

|2

) + v2

(f

(n 1)|f

|2

)

(n 1)|Du|2 (n 2)f + (n 1)|f|2|Du|2,(1.41)

hence the result.

Branes of class (B)

Next we shall examine branes that are of class (B). Let us start with aproof of Lemma 0.2

Proof of Lemma 0.2. The equation T0; = 0 implies

0 = (n + )f. (1.42)

If would vanish in a point, then it would vanish identically, in view ofGronwalls lemma. Hence, we may assume that > 0, so that

(log ) = (n + 0)f f, (1.43)

from which the conservation law (0.16) follows immediately.

In the following two lemmata we shall assume that T satisfies an equa-tion of state with = 0 + , such that , and satisfy the relations(0.21), (0.22) and (0.25).

1.6 Lemma. Let = n12 and = 0 + . Then the brane N is of class(B) with constant if and only if

0 n12 . (1.44)

Proof. The left hand-side of equation (1.18) converges to a positive constantif and only if 0 satisfies (1.44).


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1.7 Lemma. Let = n12 , > 0, and = 0 + . Then the brane Nis of class (B) with constant if and only if

= n + 0 1 (1.45)

and

0 > n1

2 . (1.46)

Proof. (i) Let N be of class (B) with constant = n12 , then

n + 0 > 0 (1.47)

must be valid, for otherwise we get a contradiction. Therefore

2(+ 1 (n + 0)) (1.48)

is the smallest exponent on the right-hand side of (1.18) and has thus tovanish, i.e.,

= n + 0 1. (1.49)

The other exponents have to be non-negative, i.e.,

2 > n 1 (1.50)

and

2 n + 0 or = 0. (1.51)

The inequality (1.50) is equivalent to (1.46) which in turn implies (1.51).

(ii) The conditions (1.45) and (1.46) are sufficient, since then

2 > n 1 (1.52)

and

m = 2

n220. (1.53)

Similarly as before branes of class (B) do actually exist, cf. Theorem 3.1.


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CLAUS GERHARDT 337

2 Transition from big crunch to big bang

Branes that are ARW spaces

For a brane N that is an ARW space, the results of [4] can be applied.Define a reflected spacetime N by switching the lightcone and changing the

time function to x0 = x0, then N and N can be pasted together at thesingularity {0} S0 yielding a smooth manifold with a metric singularitywhich is a big crunch, when viewed from N, and a big bang, when viewedfrom N. Moreover, there exists a natural transition flow of class C3 acrossthe singularity which is defined by rescaling an appropriate inverse meancurvature flow.

This transition flow is of class C, if the quantity fef, or equivalently,|f|2e2f can be viewed as a smooth and even function in the variable ef,cf. [4, the remarks before Theorem 2.1].

For the branes considered in Theorem 0.4 and Theorem 0.5 |f|2e2f is

a smooth and even function in ef.

In the following we shall see that the reflected spacetime can be viewedas a brane in the Schwarzschild-AdS(n+2) space N which is a reflection ofN.N is obtained by switching the lightcone in N and by changing the radial

coordinate r to r. The singularity in r = 0 is then a white hole singularity.

Moreover, ARW branes are also branes of class (B), i.e., the transitionresults for those branes, which we shall prove next, also apply.

Branes of class (B)

The brane N is given by

y() = (r(), t(), xi). (2.1)

Since the coordinates (xi) do not change, let us write y() = (r(), t()).

We shall see in a moment that t() converges, if tends to zero, hence,let us set this limit equal to zero.

Now we define the reflection N of N by

y() = (r(), t()) (2.2)


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for > 0. The result is a brane in N which can be pasted continuously toN.

The embeddings of N resp. N in N resp. N are also embeddings inR2 S0, endowed with the Riemannian product metric, and now it makes

sense to ask, if the joint hypersurfaces N N form a smooth hypersurfacein R2 S0.

The smoothness ofN N in R2 S0 we interpret as a smooth transitionfrom big crunch to big bang.

To prove the smoothness we have to parameterize y = y() with respectto r. This is possible, since

drd = f

ef > 0. (2.3)

Then we have y(r) = (r, t(r)) and we need to examine

dt

dr =

dt

d

d

dr =

dt

d

1

fef . (2.4)

Define(r) = h(r)e(n1)f

= m + 2n(n+1)e

(n+1)f e(n1)f,(2.5)

then we obtain from (1.15) and (1.17)

dtdr

= n

( + 0e(n+0)fe)e

nf

f

= n

(e(n+)f + 0e(0)fe) 1

fef.

(2.6)

This immediately shows that in all cases | dtdr

| is uniformly bounded, hencelimr0 t(r) exists.

The joint embedding N N in R2 S0 will be of class C, ifdtdr

can beviewed as a smooth and even function in r = ef.

We shall assume that the function = (r) in (0.25) is either even andsmooth in (r0, r0)

C((r0, r0)) is even (2.7)

or that C((r0

, 0]) such that

|Dm(0)| = 0 m N. (2.8)


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CLAUS GERHARDT 339

In the latter case we can extend as an even and smooth function to r > 0by setting

(r) = (r) for r > 0. (2.9)

Then we can prove

2.1 Theorem. Let N be a brane of class (B) satisfying an equation of

state with = 0 + , so that vanishes at the singularity. The correspond-ing function should satisfy the conditions (2.7) or (2.8). ThenN can bereflected as described above to yield a brane N N. The joint branes N N form a C- hypersurface in R2 S0 provided the following conditions arevalid

(i) If = n12 and 0 n12 , then the relations

n12 odd, 0 odd, and R, (2.10)

orn odd, 0 Z, and = 0 (2.11)

should hold.

(ii) If = n + 0 1 and 0 > n12 , then the relations

n odd, 0 odd, and R, (2.12)

orn odd, 0 Z. and = 0 (2.13)

should be valid.

Proof. (i) Let =

n1

2 and 0

n1

2 . We have to show that

dt

dr is asmooth and even function in r. From equation (2.6) we deduce

dtdr

= n

((r)(n+)f + 0e(0)fe)11, (2.14)

where

2 = |f|2e2f = me(2(n1))f + 2n(n+1)e

2(+1)f e2f

+ 2

n2(20e

2(+1(n+0))fe2

+ 20e(2+2(n+0))fe + 2e2(+1)f).

(2.15)

The right-hand side of (2.15) has to be an even and smooth function inr = ef. By assumption is already an even and smooth function or can be


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extended to such a function, hence we have to guarantee that the exponentsof ef are all even.

A thorough examination of the exponents reveals that this is the case, ifthe conditions (2.10) or (2.11) are fulfilled. Notice that

n1

2odd = n odd 3n1

2even. (2.16)

(ii) Identical proof.

3 Existence of the branes

We want to prove that for the specified values of 0 and corresponding embedded branes N in the black hole region of N satisfying the Israeljunction condition do exist.

Evidently it will be sufficient to solve the modified Friedmann equation(1.16) on an interval I = (a, 0] such that, if we set r = ef, then

lim0

r() = 0, lima

r() = r0, (3.1)

where r0 is the (negative) black hole radius, i.e., {r = r0} equals the horizon.

We shall assume the most general equation of state that we consideredin the previous sections, i.e., should be of the form = 0 + (f), cf.Lemma 0.2, and the corresponding primitive = (r) should be smooth in

r0 < r < 0.

Let us look at equation (1.18). Setting = ef we deduce that it can berewritten as

22 = F(), (3.2)

where F = F(t) is defined on an interval J = [0 t < t0) and t0 is given by

t0 = (r0). (3.3)

F is continuous in J and smooth in

J. Furthermore, it satisfies

F > 0 and F(0) = m > 0. (3.4)


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CLAUS GERHARDT 341

If we succeed in solving the equation

1 =

F() (3.5)

with initial value (0) = 0 on an interval I = (a, 0] such that C0(I)

C(

I) and such that the relations (3.1) are valid for r = 1

, then themodified Friedmann equation is solved by f = 1 log and f satisfies

lim0

f = , f > 0. (3.6)

To solve (3.5) let us make a variable transformation , so that wehave to solve

1 =

F() (3.7)

with initial value (0) = 0 on an interval I = [0, a).

3.1 Theorem. The equation (3.7) with initial value (0) = 0 has a

solution C1(I) C(

I), where I = [0, a) is an interval such that therelations (3.1) are satisfied byr =

1

with obvious modifications resulting from the variable transformation .

Proof. Let 0 < t < t0 be arbitrary and define J = [0, t]. Then there arepositive constants c1, c2 such that

c21 F(t) c22 t J

. (3.8)

Let > 0 be small, then the differential equation (3.7) with initial value(0) = has a smooth, positive solution in 0 < , where isdetermined by the requirement

() t

0 < . (3.9)

In view of the estimates (3.8) we immediately conclude that

c1 () + c2 0 < , (3.10)

hence, if we choose the maximal possible, then there exists 0 > 0 suchthat

0 > 0 . (3.11)

Now, using well-known a priori estimates, we deduce that for any interval

I

(0, 0) we have||m,I cm(I

) m N (3.12)


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independently of . Moreover, since is uniformly Lipschitz continuous in[0, 0], we infer that

lim0

= (3.13)

exists such that C1([0, 0]) C((0, 0)) and solves (3.7) with initialvalue (0) = 0.

can be defined on a maximal interval interval [0, a), where a is de-termined by

lima

() = or lima

F() = 0. (3.14)

Obviously, there exists 0 < a a such that

lima

() = (r0). (3.15)

Hence the proof of the theorem is completed.

3.2 Remark. If a < a, then we cannot conclude that we have defined

branes extending beyond the horizon, since our embedding deteriorates whenthe horizon is approached, as a look at the equations (1.15) and (1.16) re-veals. Since h vanishes on the horizon, either dt

dbecomes unbounded or

( + ) would tend to zero, in which case F() would vanish, i.e., if a < a,then necessarily

lima

| dtd

| = (3.16)

has to b e valid. An extension of the brane past the horizon will require adifferent embedding or at least a different coordinate system in the bulk.Notice that

g00 = y, y = r2 = 0, (3.17)

so that it is more or less a coordinate singularity.

Acknowledgement. This work has been supported by the DeutscheForschungsgemeinschaft.

References

[1] Claus Gerhardt, Hypersurfaces of prescribed curvature in Lorentzianmanifolds, Indiana Univ. Math. J. 49 (2000), 11251153, pdf file.

[2] , The inverse mean curvature flow in ARW spaces - transitionfrom big crunch to big bang, 2004, 39 pages, math.DG/0403485.


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[3] , The inverse mean curvature flow in cosmological spacetimes,2004, 24 pages, math.DG/0403097.

[4] , The inverse mean curvature flow in Robertson-Walker spacesand its application to cosmology, 2004, 9 pages, gr-qc/0404112.

[5] , The mass of a Lorentzian manifold, 2004, 14 pages,

math.DG/0403002.

[6] Justin Khoury, A briefing on the ekpyrotic/cyclic universe, 2004, 8 pages,astro-ph/0401579.

[7] David Langlois, Brane cosmology: an introduction, 2002, 32 pages, hep-th/0209261.

[8] Neil Turok and Paul J. Steinhardt, Beyond inflation: A cyclic universescenario, 2004, 27 pages, hep-th/0403020.

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Claus Gerhardt- Transition from big crunch to big bang in brane cosmology