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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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