# The BKL Conjectures for Spatially Homogeneous Spacetimes

**ABSTRACT** We rigorously construct and control a generic class of spatially homogeneous (Bianchi VIII and Bianchi IX) vacuum spacetimes that exhibit the oscillatory BKL phenomenology. We investigate the causal structure of these spacetimes and show that there is a "particle horizon". Comment: 49 pages; Section 7 (Causal structure and particle horizons) added, small changes in other sections to support Section 7

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**ABSTRACT:**We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation. Comment: Some updates. A shortened version, to appear in the Bulletin of the AMS, is available online at http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2010-01304-504/2010; - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**We consider cosmological models of Bianchi type. In particular, we are interested in the α-limit dynamics near the Kasner circle of equilibria for Bianchi classes VIII and IX. They correspond to cosmological models close to the big-bang singularity. We prove the existence of a codimension-one family of solutions that limit, for t → −∞, onto a heteroclinic 3-cycle to the Kasner circle of equilibria. The theory extends to arbitrary heteroclinic chains that are uniformly bounded away from the three critical Taub points on the Kasner circle, in particular to all closed heteroclinic cycles of the Kasner map.Communications in Mathematical Physics 01/2011; 305(1):59-83. · 1.97 Impact Factor

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arXiv:1005.4908v2 [gr-qc] 10 Jun 2010

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The BKL Conjectures for

Spatially Homogeneous Spacetimes

Michael Reiterer, Eugene Trubowitz

Department of Mathematics, ETH Zurich, Switzerland

Abstract:We rigorouslyconstructandcontrola genericclass ofspatiallyhomogeneous

(Bianchi VIII and Bianchi IX) vacuum spacetimes that exhibit the oscillatory BKL

phenomenology. We investigate the causal structure of these spacetimes and show that

there is a “particle horizon”.

1. Introduction

The goal of this paper is to rigorously construct and explicitly control a generic class of

solutions Φ = α ⊕ β : [0,∞) → R3⊕ R3, with independent variable τ ∈ [0,∞) and

with1(α1+ α2+ α3)|τ=0< 0, to the autonomous system of six ordinary differential

equations

0 = −d

0 = −d

def

= {(1,2,3),(2,3,1),(3,1,2)},subject to the quadratic constraint2

0 = α2α3+α3α1+α1α2−(β1)2−(β2)2−(β3)2+2β2β3+2β3β1+2β1β2 (1.1c)

Here, α = (α1,α2,α3), β = (β1,β2,β3). The system (1.1) are the vacuum Einstein

equations for spatially homogeneous (Bianchi) spacetimes, see Proposition 2.1.

ThepioneeringcalculationsandheuristicpictureofBelinskii, Khalatnikov,Lifshitz3

[BKL1] and Misner [Mis] suggest that a generic class of solutions to (1.1) are oscilla-

tory as τ → +∞ and that the dynamics of one degree of freedom is closely related

to the discrete dynamics of the Gauss map G(x) =

dταi− (βi)2+ (βj)2+ (βk)2− 2βjβk

dτβi+ βiαi

(1.1a)

(1.1b)

for all (i,j,k) ∈ C

1

x− ⌊1

x⌋, a non-invertible map

1If τ ?→ Φ(τ) is a solution to (1.1), so is τ ?→ −Φ(−τ). The condition (α1+α2+α3)|τ=0< 0 breaks

this symmetry. Solutions to (1.1) with (α1+ α2+ α3)|τ=0< 0 do not break down in finite positive time,

that is, they extend to [0,∞). A proof of this fact is given later in this introduction.

2As a quadratic form on R3⊕ R3, the right hand side of (1.1c) has signature (+,+,−,−,−,−).

3The work of Belinskii, Khalatnikov, Lifshitz concerns general (inhomogeneous) spacetimes, but relies

on intuition about the homogeneous case.

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from (0,1)\Q to itself. Every element of (0,1)\ Q admits a unique infinite continued

fraction expansion

?k1,k2,k3,...? =

1

k1+

1

k2+

1

k3+...

(1.2)

where (kn)n≥1are strictly positive integers. The Gauss map is the left-shift,

G??k1,k2,k3,...??= ?k2,k3,k4,...?

(1.3)

Rigorous results about spatially homogeneous spacetimes have been obtained by

Rendall [Ren] and Ringstr¨ om [Ri1], [Ri2]. See also Heinzle and Uggla [HU2]. We

refer to the very readable paper [HU1] for a detailed discussion.

The first rigorous proofs that there exist spatially homogeneous vacuum spacetimes

whose asymptotic behavior is related, in a precise sense, to iterates of the Gauss map,

have been obtained recently by B´ eguin [Be] and by Liebscher, H¨ arterich, Webster and

Georgi [LHWG]. These theorems apply to a dense subset of (0,1)\Q. A basic restric-

tion of both these works is that the sequence (kn)n≥1has to be bounded, a condition

fulfilled only by a Lebesgue measurezero subset of (0,1)\Q. The results of the present

paperapplyto anysequence(kn)n≥1that growsat most polynomially.Thecorrespond-

ing subset of (0,1) \ Q has full Lebesgue measure one.

We point out some properties of the system (1.1a), (1.1b), not assuming (1.1c):

(i) The right hand side of (1.1c) is a conserved quantity.

(ii) If τ ?→ Φ(τ) is a solution, so is τ ?→ pΦ(pτ + q), for all p,q ∈ R.

(iii) The signatures (sgnβ1,sgnβ2,sgnβ3) are constant.

(iv)

(v) We have4 d

d

dτ|β1β2β3|2= 2(α1+ α2+ α3)|β1β2β3|2.

dτ(α1+ α2+ α3) ≥ −3|β1β2β3|2/3.

If in addition we assume (1.1c), then:

(vi)

d

dτ(α1+ α2+ α3) = α2α3+ α3α1+ α1α2≤1

Let Φ = α⊕β be any solution to (1.1), that is (1.1a), (1.1b), (1.1c), on the half-open

interval [0,τ1) with 0 < τ1< ∞. Set α / = α1+ α2+ α3and suppose α /(0) < 0. Then

α /(τ) ≤ −|α /(0)|/?1 +1

by (vi). Consequently,|β1β2β3| is bounded,by (iv), and α / is boundedbelow, by (v), on

[0,τ1). The constraint(1.1c) implies that5(α1)2+(α2)2+(α3)2≤ 6|β1β2β3|2/3+α /2

is bounded. Now (1.1b) implies that (β1)2+ (β2)2+ (β3)2is bounded. Therefore,

solutions to (1.1) with α /(0) < 0 can be extended to [0,∞). The solutions considered

in this paper belong to this general class. We are especially interested in their τ → +∞

asymptotics.

3(α1+ α2+ α3)2.

3|α /(0)|τ?< 0

for all τ ∈ [0,τ1)

(1.4)

4(β1)2+(β2)2+(β3)2−2β2β3−2β3β1−2β1β2+3|β1β2β3|2/3≥ 0 holds for all β1,β2,β3∈ R,

see [HU1]. The only nontrivial cases are β1,β2,β3> 0 or β1,β2,β3< 0. In these cases, the inequality is

a direct consequence of the polynomial identity

x6+ y6+ z6− 2y3z3− 2z3x3− 2x3y3+ 3x2y2z2=

1

2

?x2+y2+z2+yz+zx+xy??

(y−z)2(y+z−x)2+(z−x)2(z+x−y)2+(x−y)2(x+y−z)2?

5Use 2(α2α3+ α3α1+ α1α2) = α /2− (α1)2− (α2)2− (α3)2.

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For every solution to (1.1) with α /(0) < 0, as in the last paragraph, the half-infinite

interval [0,∞) actually corresponds to a finite physical duration of the associated spa-

tially homogeneousvacuum spacetime (given in Proposition 2.1). In fact, an increasing

affine parameter along the timelike geodesics orthogonal to the level sets of τ is given

by τ ?→?τ

called Bianchi VIII or IX models). We now give an informal description of the solu-

tions that we construct, the phenomenologicalpicture of [BKL1]. The structure of each

of these solutions is described by three sequences of compact subintervals (Ij)j≥1,

(Bj)j≥1, (Sj)j≥1of [0,∞), for which:

(a.1) The left endpoint of I1is the origin, and the right endpoint of Ij, henceforth

denoted τj, coincides with the left endpoint of Ij+1, for all j ≥ 1. Set τ0= 0.

(a.2)?

(a.4) Sjis the closed interval of all points between Bjand Bj+1, for all j ≥ 1.

Here is a picture:

0exp(1

2

?s

0α /)ds, with uniform upper bound 6|α /(0)|−1, by (1.4).

In this paper, we consider only solutions to (1.1) for which β1,β2,β3 ?= 0 (also

j≥1Ij= [0,∞), that is, limj→+∞τj= +∞.

(a.3) Bjis contained in the interior of Ij, and 0 < |Bj| ≪ |Ij|, for all j ≥ 1.

Ij

Bj+1

Sj+1

Bj

Ij+1

Sj

τj

τj+1

Let S3be the set of all permutations (a,b,c) of the triple (1,2,3). The solution is

further described by a sequence (πj)j≥1in S3, with πj= (a(j),b(j),c(j)), so that:

(b.1) On Ij, the components βb(j), βc(j)are so small in absolute value that the local

dynamics of Φ = α ⊕ β is essentially unaffected if βb(j), βc(j)are set equal to

zero in the four equations (1.1a) and (1.1c).

(b.2) On Ij\ Bj, the component βa(j)is so small in absolute value that the local dy-

namics of Φ = α ⊕ β is essentially unaffected if βa(j)is set equal to zero in the

four equations (1.1a) and (1.1c). The component βa(j)is not small on Bj, but the

mixed products βa(j)βb(j)and βa(j)βc(j)are still small.

(b.3) Items (b.1) and (b.2) imply that mixed products of components of β are small on

all of [0,∞), and that all three components of β are small on?

(b.5) None of the properties listed so far distinguishes b(j) from c(j). By (b.4), this

ambiguity can be consistently eliminated by stipulating b(j) = a(j + 1).

j≥1Sj.

(b.4) a(j) ?= a(j + 1) for all j ≥ 1.

We can draw the following heuristic consequences from the eight heuristic properties

above. Separately on each interval Sj, j ≥ 1:

(c.1) The components of α are essentially constant, by (1.1a) and (b.3), and log|β1|,

log|β2|, log|β3| are essentially linear functions with slopes α1, α2, α3, by (1.1b).

(c.2) The constraint (1.1c) essentially reduces to α2α3+ α3α1+ α1α2 = 0. As be-

fore, we require α / = α1+ α2+ α3 < 0. Furthermore, we make the generic

assumption that all components of α are nonzero. These conditions imply that

two components of α are negative, one component of α is positive, and the sum

of any two is negative.

(c.3) The single positive component of α has to be αb(j)= αa(j+1). In fact, we know

that |βa(j+1)| is very small on Sjbut is not small on Bj+1. Therefore, the slope

of log|βa(j+1)|, which is αa(j+1)by (c.1), has to be positive on Sj.

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(c.4) By the last three items and (b.4), there is at most one point in Sjwhere |βa(j)| =

|βa(j+1)|. By (b.1), (b.2), there is such a point, because |βa(j)| is going from not

small to small on Sj, and |βa(j+1)| is going from small to not small on Sj. By

convention, this point is τj.

Separately on each interval Ij, j ≥ 1 (in particular on Bj⊂ Ij):

(d.1) αa(j)+ αb(j)and αa(j)+ αc(j)are essentially constant, by (1.1a), (b.1), and

they are both negative, by (c.1), (c.2). Also, log|βa(j)βb(j)|, log|βa(j)βc(j)| are

essentially linear functionswith slopes αa(j)+αb(j)and αa(j)+αc(j), by (1.1b).

(d.2) Essentially (αa(j)+αb(j))(αa(j)+αc(j)) = (αa(j))2+(βa(j))2, by (1.1c). Since

the left hand side is essentially constant by (d.1), so is the right hand side.

(d.3) By (d.1),it only remains to understandthe behaviorof αa(j), βa(j). By (1.1a), we

essentially have

d

dταa(j)= −(βa(j))2

d

dτβa(j)= αa(j)βa(j)

(1.5)

A special solution is αa(j)= −tanhτ and βa(j)= ±sechτ = ±(coshτ)−1.

The general solution is obtained from the special solution by applying the affine

symmetry transformation (ii) above, with p > 0. Since Bjis essentially the in-

terval on which |βa(j)| is not small, see (b.2), we must have p ∼ |Bj|−1(here ∼

means “same order of magnitude”). See [BKL1], Section 3, in particular pages

534 and 535.

(d.4) Recall (c.1). By (d.3), we have αa(j)|Sj−1= −αa(j)|Sj, since the hyperbolic

tangent just flips the sign. Therefore, by (d.1), the net change across Bjof the

components of α, from right to left, is given by

αa(j)|Sj−1= αa(j)|Sj− 2αa(j)|Sj

αb(j)|Sj−1= αb(j)|Sj+ 2αa(j)|Sj

αc(j)|Sj−1= αc(j)|Sj+ 2αa(j)|Sj

These equations make sense only for j ≥ 2, since S0has not been defined.

In this paper, we turn the heuristic picture of [BKL1], sketched above,into a mathemat-

ically rigorous one, globally on [0,∞), for a generic class of solutions. The first step is

to construct a discrete dynamical system, that maps the state Φ(τj) to the state Φ(τj−1)

at the earlier time τj−1 < τj, for all j ≥ 1. That is, the construction proceeds from

right-to-left, beginning at τ = +∞. We refer to the discrete dynamical system maps as

transfer maps.

For each j ≥ 0, two components of β(τj) have the same absolute value, see (c.4),

and Φ(τj) satisfies the constraint (1.1c). Therefore, the states of the discrete dynamical

system have 4 continuous degrees of freedom. By the symmetry (ii), the transfer maps

commute with rescalings. Taking the quotient, one obtains a 3-dimensional discrete

dynamical system. The three “dimensionless” quantities that we use to parametrize the

discrete states are denoted fj= (hj,wj,qj). Morally, they are interpreted as follows:

• hj∼ |Bj|/|Ij| > 0. In the billiard picture of [Mis], it is the dimensionless ratio of

the collision and free-motion times. By (a.3), one has 0 < hj≪ 1. In fact, hjis the

all-important small parameter in our construction. It goes to zero rapidly as j → ∞.

This is necessary for us to make a global construction on [0,∞). The precise rate

depends on the sequence (kn)n≥1. The rate is the same as in Proposition 4.4, up to

even smaller corrections.

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• The components of α are essentially constant on Sjand subject to the reduced con-

straint equation in (c.2). Thus, modulo the scaling symmetry (ii), only one degree of

freedomis requiredto parametrizeα|Sj. We use wj≈ −(αb(j)/(αa(j)+αb(j)))|Sj.

By (c.2)and(c.3),we havewj> 0. Theleft-to-rightdiscretedynamicsof wj(which

is opposite to the right-to-left direction of our transfer maps) is closely related to a

variant of the Gauss map, sometimes referred to as the BKL map or Kasner map.

• The meaning of qjwill be explained in a more indirect way. As pointed out above,

theleft-to-rightdynamicsofwjisrelatedtotheGauss map,whichis anon-invertible

left-shift, see (1.3). The non-invertibilityof the Gauss map seems to be at odds with

the invertible dynamics of the system of ordinary differential equations (1.1). The

parameterqjis introducedso that the joint left-to-rightdiscretedynamicsof(wj,qj)

is closely related to the left-shift on two-sided sequences (kn)n∈Zof strictly positive

integers, which is invertible. Accordingly, the right-to-left transfer maps are related

to the right shift on two-sided sequences (kn)n∈Z.

This concludes the informal discussion. We emphasize that the notation used above is

specific to the introduction. In particular, (Ij)j≥1, (Bj)j≥1, (Sj)j≥1do not appear in

the main text. Starting from Section 2, all the notation is introduced from scratch.

We now state simplified, self-contained versions of our results. References to their

stronger counterparts are given. Here is a short guide:

Definition 1.1 (equivalent to Definition 3.12). Introduces the state vectors Φ⋆(π,f,σ∗)

of the 3-dimensionaldiscrete dynamical system. The dynamics of the signature vec-

tor σ∗is trivial, by (iii), but it affects the dynamics of (π,f) in a non-trivial way.

Definition 1.2 (this is Definition 3.16). Introduces explicit maps PL, QL, λLthat turn

out to be verygoodapproximationsto the transfermaps.It is shown in Section4 that

iteratesofQLcanbeunderstoodintermsoftheGaussmap/continuedfractionsand,

by a change of variables, in terms of solutions to certain linear equations.

Definition 3.19 (only in the main text). The essential smallness condition on h > 0 is

quantitatively encoded in an open subset F ⊂ (0,1) × (0,∞) × ((0,∞) \ {1}). It

determines the domain of definition of the transfer maps.

Proposition 1.1 (slimmed-down version of Proposition 3.3). It asserts the existence of

transfer maps. The pair (PL,Π) and the triple (PL,Π,Λ) constitute the transfer

maps for the 3-dimensional and 4-dimensional systems, respectively, and they are

very close to (PL,QL) and (PL,QL,λL). Explicit error bounds and precise esti-

mates for the transfer solution appear only in the full version, Proposition 3.3.

Theorem 1.1 (simplified version of Theorems 6.2, 6.3). Gives a generic class of iterates

to (PL,Π) that are super-exponentially close to iterates of (PL,QL). That is, it

asserts the existence of solutions to the 3-dimensional discrete dynamical system.

The overview is as follows. Every solution to the 3-dimensional discrete dynamical

system as in Theorem 1.1 can be lifted to a unique solution to the 4-dimensional dis-

crete dynamical system, up to an overall scale, through the map Λ in Proposition 1.1.

This solution corresponds to the sequence of states (Φ(τj))j≥0in the informal discus-

sion. Proposition 1.1 gives solutions to (1.1) on compact intervals that connect next-

neighbor states. Symmetry (ii) is used to translate these compact intervals and place

them next to each other, beginning at τ = 0, just like the (Ij)j≥1in the informal dis-

cussion. As in (a.2) of the informal discussion, the union of these intervals is indeed

[0,∞), and a semi-global solution to (1.1) is obtained. To see this, denote the states

by λjΦ⋆(πj,gj,σ∗) with λj > 0 and πj ∈ S3and gj = (h′

j ≥ 0. One has λj = Λ[πj+1,σ∗](gj+1)λj+1 ≥ λj+1 by the definition of Λ and

j,w′

j,q′

j) ∈ F, where