Stability Within T 2 -Symmetric Expanding Spacetimes
SSTABILITY WITHIN T -SYMMETRIC EXPANDING SPACETIMES BEVERLY K. BERGER, JAMES ISENBERG, AND ADAM LAYNE
Abstract.
We prove a nonpolarised analogue of the asymptotic characterization of T -symmetricEinstein Flow solutions completed recently by LeFloch and Smulevici. In this work, we impose acondition weaker than polarisation and so our result applies to a larger class. We obtain similar ratesof decay for the normalized energy and associated quantities for this class. We describe numericalsimulations which indicate that there is a locally attractive set for T -symmetric solutions not coveredby our main theorem. This local attractor is distinct from the local attractor in our main theorem,thereby indicating that the polarised asymptotics are unstable. introduction There exist broad conjectures about the expanding direction behavior of vacuum spacetimes withclosed Cauchy surfaces [2, 9], but currently little is known about some of the most elementary examples.Recent results [13, 18] have demonstrated that certain vacuum cosmological models demonstrate locallystable behavior in the expanding direction, but that well-known subclasses are unstable. These resultsshould be compared to models with matter [16, 23, 22, 21] where spatially homogeneous solutions areknown to be stable. It is also important to note that the behavior of these models in the direction of thesingularity is not sensitive to the presence of most types of matter [3].In the special case that the spacetime has spatial topology T , admits two spacelike Killing vector fields(such spacetimes are called T -symmetric ), and satisfies a further technical condition (that the spacetimeis polarised ) results of [13] show that there is a local attractor of the Einstein Flow in the expandingdirection. It is natural to ask whether the condition that the spacetime be polarised is necessary. Dospacetimes on T with two spacelike Killing vector fields necessarily become effectively polarised? Dothey then flow to the polarised attractor?We partially resolve these questions by analytic and numerical means. Our main theorem states thatsolutions which are not polarised have the expanding direction asymptotics of polarised solutions if theysatisfy a certain weaker condition: that one of the two conserved quantities of the flow vanishes. We callsuch solutions B or B = 0 solutions . The conserved quantity B vanishes for all polarised solutions; theset of B = 0 solutions is of codimension one in the space of all solutions in these coordinates while theset of polarised solutions is of infinite codimension.It was shown in [6] that T -symmetric vacuum spacetimes posess a global foliation; all such EinsteinFlows have a metric of the form g = e (cid:98) l − V +4 τ (cid:16) − dτ + e ρ − τ ) dθ (cid:17) + e V [ dx + Qdy + ( G + QH ) dθ ] + e − V +2 τ [ dy + Hdθ ] (1.1)where ∂ x and ∂ y are the Killing vector fields. The area of the { ∂ x , ∂ y } orbit is e τ , so the singularityoccurs as τ → −∞ and the spacetime expands as τ → ∞ . Relative to the coordinates t, P, α, λ used in[18], our quantities are given by τ := log t, ρ := − log αV := P + log t, (cid:98) l := P + λ − log t. Date : December 20, 2018. a r X i v : . [ m a t h . A P ] D ec asner PH T -symmetric B Kasner B PH B T -symmetricpolarised Kasner polarised PH polarised T -symmetric Figure 1.
The classes of Einstein Flow solutions discussed in this paper, and theirinclusions. We have omitted the Gowdy models, which are not the focus of this work.See the Appendix for a complete concordance of notations between the cited papers and the presentwork. In the coordinates (1.1), the Einstein Flow is ∂ τ ( e ρ V τ ) = ∂ θ (cid:0) e τ − ρ V θ (cid:1) + e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) (1.2) ∂ τ (cid:16) e ρ +2( V − τ ) Q τ (cid:17) = ∂ θ (cid:0) e − ρ +2 V Q θ (cid:1) (1.3) (cid:98) l τ + ρ τ + 2 = 12 (cid:104) V τ + e τ − ρ ) V θ + e V − τ ) (cid:16) Q τ + e τ − ρ ) Q θ (cid:17)(cid:105) (1.4) ρ τ = K e (cid:98) l (1.5) (cid:98) l θ = V θ V τ + e V − τ ) Q θ Q τ . The last equation is the momentum constraint, and it is preserved by the evolution equations. Equation(1.5) is a consequence of the constraints; ρ satisfies a wave equation similar to (1.2) which can be derivedas a consequence of (1.4) and (1.5), so we take equations (1.2) through (1.5) to be the evolution equationsinstead. There are, in addition, evolution equations for G, H , but these may be integrated once
V, Q, ρ, (cid:98) l have been found, so these latter four functions are the ones of interest. As a consequence of (1.2) and(1.3), there are two conserved quantities along the flow: A := (cid:90) S e ρ (cid:16) V τ − e V − τ ) Q τ Q (cid:17) dθB := (cid:90) S e ρ +2( V − τ ) Q τ dθ. The condition Q ≡ polarised . (Note that all polarised solutions have B = 0, but not all B = 0 solutionsare polarised.) The constant K is, without loss of generality, that “twist constant” which cannot ingeneral be made to vanish by a coordinate transformation. The T Gowdy models [10] are those forwhich K = 0. T -symmetric spacetimes which are polarised, half polarised [11, 8], or Gowdy have beenstudied extensively in the contracting direction (e.g. [1]). We are here concerned only with the expandingdirection.The Kasner models are those which are spatially homogeneous ( (cid:98) l, V, Q are independent of θ ) andsatisfy K = 0. Let us note that, in our coordinates, polarised Kasner solutions [12] take the form V = aτ + b, (cid:98) l = (cid:18) a − (cid:19) τ + c for some constants a, b, c ∈ R . The Gowdy models contain all Kasners, and in the expanding directionthe dynamics of Gowdy solutions are known [17], [19] and appear to be very different from those ofnon-Gowdy solutions. Non-Gowdy solutions such that (cid:98) l, V, Q are independent of θ are called pseudo-homogeneous or PH . This definition appears in [18], where it is shown that the future asymptotics areof the form | V − ( aτ + b ) | → , (cid:12)(cid:12)(cid:12)(cid:12)(cid:98) l − (cid:18)(cid:20) a − (cid:21) τ + c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) → , a ∈ ( − , . That is, PH solutions have asymptotics of the same form as a Kasner solution, but the value of V τ at τ = ∞ is restricted. n contrast to these examples, in [13] the authors find a set of non-Gowdy, polarised solutions suchthat | V − b | → , (cid:12)(cid:12)(cid:12)(cid:98) l − c (cid:12)(cid:12)(cid:12) → . (1.6)The results in [18] and [13] are much more detailed than the above statements; we give this simpledescription only to demonstrate that an instability arises; no polarised Kasner or PH solutions can havefuture behavior of the form (1.6). The relationships between these sets of solutions are given in Figure1. Previous to this work, numerical simulations conducted by Berger [4, 5] indicated that all T -symmetric solutions, without regard to the polarisation or smallness conditions imposed in [13], flowedtoward the polarised attractor (1.6). In addition, in [18] it is shown that within the neighborhood ofeach polarised PH solution is a polarised non-PH solution with future asymptotics of the form (1.6).Before giving a description of our main theorem, let us note the sense in which we use the word“attractor.” Our technique of proof follows [13]. Let us denote the right side of (1.4) by J . The ideaof the proof is to treat the asymptotic regime of the solution as a wave equation for V, Q coupled toan ordinary differential equation (up to some error terms) for the means in the θ -direction of e ρ , e (cid:98) l , J .The smallness assumptions are then used to guarantee that the errors decay, and so the behavior of themeans of e ρ , e (cid:98) l , J approaches the behavior of the solution of the ODE. When we use the word “attractor”here, we refer to the dynamics of the e ρ , e (cid:98) l , J system; a solution V (cid:48) , Q (cid:48) , ρ (cid:48) , (cid:98) l (cid:48) is not generally a properattractor of the flow in the sense that (cid:107) V − V (cid:48) (cid:107) + (cid:107) Q − Q (cid:48) (cid:107) + (cid:107) ρ − ρ (cid:48) (cid:107) + (cid:107) (cid:98) l − (cid:98) l (cid:48) (cid:107) −→ C k norm.Our main theorem states roughly that the condition B = 0 suffices to ensure that a solution haspolarised asymptotics if it begins sufficiently close to the asymptotic regime. In the latter portion ofthe paper, we present numerical evidence that the condition B = 0 is necessary for the solution to havepolarised asymptotics and flow toward the polarised attractor. There appears to be an attractor forsolutions satisfying B (cid:54) = 0, which shares some formal properties with the B = 0 attractor. However, suchsolutions flow away from the B = 0 attractor, and so the B = 0 asymptotics appear to be unstable.Since the future behavior of Gowdy and PH solutions is understood, we are only concerned withnon-Gowdy, non-PH solutions; that is, solutions with K (cid:54) = 0 and (cid:82) S e ρ dθ unbounded as τ → ∞ . In thiscase, we shift (cid:98) l by a constant l := (cid:98) l + log( K / (cid:98) l θ = l θ , (cid:98) l τ = l τ and ρ τ = e l . In the rest of the paper, we assume solutions are non-Gowdy and so change variables to l to avoid writingfactors of K .Before proceeding with the proof, it is important to note that there is some very interesting work onthe rescaling limits of certain expanding spacetimes [14, 15]. The earlier of these works uses techniquesfrom the study of Ricci Flow to analyze the rescaling limits of CMC-foliated expanding spacetimes. Thelatter work is concerned with the extent to which rescaling limits of the spacetimes considered in [13]have a nonzero Einstein tensor. It is likely that this result can be generalized to the class of solutionsconsidered in this paper. Acknowledgements.
We are grateful to David Maxwell, Peng Lu, Paul T Allen, Florian Beyer, PiotrChru´sciel, Anna Sakovich, and Hans Ringstr¨om for providing useful comments on various parts of thisproject.The second and third authors were supported by NSF grants DMS-1263431 and PHY-1306441. Thisarticle was in part written during a stay of the third author at the Erwin Schr¨odinger Institute in Vi-enna during the thematic program ‘Geometry and Relativity’. This paper incorporates material thatpreviously appeared in the third author’s dissertation which was submitted to the Department of Math-ematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Universityof Oregon. . Preliminary Computations
Before proceeding with the proof of the main theorem, we define the energy under consideration andcalculate its evolution. It is useful to have notation for the mean of a function in the θ -direction. Definition 2.1 ( S -mean) . For f : S → R , let (cid:104) f (cid:105) := (cid:90) S f ( θ ) dθ. Note that in [13], the authors choose to use the volume form e ρ dθ for their mean. Our choice is almostidentical to that used in [18], but we normalize so that (cid:82) S dθ = 1. Either choice would suffice.Define the following energy J := 12 (cid:104) V τ + e τ − ρ ) V θ + e V − τ ) (cid:16) Q τ + e τ − ρ ) Q θ (cid:17)(cid:105) ,E := (cid:90) S e ρ − τ J dθ = 12 (cid:90) S e ρ − τ V τ + e − ρ V θ + e V − τ + ρ Q τ + e V − τ ) − ρ Q θ dθ, and the S -volume Π := (cid:104) e ρ (cid:105) = (cid:90) S e ρ dθ. Note that equation (1.4) now reads l τ + ρ τ + 2 = J . We use the terms V -energy and Q -energy looselyto refer to V τ + e τ − ρ ) V θ and e V − τ ) (cid:0) Q τ + e τ − ρ ) Q θ (cid:1) , respectively. One may compute using theevolution equations for V and Q that ∂ τ ( e ρ J ) =2 e ρ J − ρ τ e ρ J − e ρ V τ − e V − ρ Q θ + ∂ θ (cid:0) e τ − ρ V θ V τ + e V − ρ Q θ Q τ (cid:1) so the energy E evolves by E τ = (cid:90) S − ρ τ e ρ − τ J − e ρ − τ V τ − e V − τ ) − ρ Q θ dθ. The terms − e ρ − τ V τ − e V − τ ) − ρ Q θ appearing here are undesirable for proving energy inequalities. Thisnecessitates the modification of E by a term which trades V τ for V θ . This is the main topic of Section3. 3. corrections and their bounds Define the correction Λ := 12 e − τ (cid:90) S V τ ( V − (cid:104) V (cid:105) − e ρ dθ. (3.1)Corrections to the energy of essentially this form were used previously in the Gowdy case [17] and inthe existing results on T -symmetric spacetimes [18, 13]. Our definition differs only slightly from thosepreviously used. Differentiating (3.1) and using integration by parts yields the two components of the V -energy, but with opposite sign. This allows us to replace time derivatives by space derivatives, whichmay be bounded. At the same time, the correction has better decay properties than the energy, and sowe are able to draw conclusions about the energy in the expanding direction.To trade V τ for V θ and Q τ for Q θ , it would be more natural to consider the corrections12 e − τ (cid:90) S V τ ( V − (cid:104) V (cid:105) ) e ρ dθ, and 12 e − τ (cid:90) S e V − τ ) Q τ ( Q − (cid:104) Q (cid:105) ) e ρ dθ separately as other authors have done. Then, by differentiating the Q -correction one would hope toobtain terms of the form Q τ − e τ − ρ ) Q θ , perhaps with a leading factor. Our definition exploits the factthat (1.2) contains exactly the expression that we would like to obtain from the Q -correction. Lemma 3.1.
Consider a non-Gowdy T -symmetric Einstein Flow. The correction defined in (3.1) evolves by ∂ τ Λ = − − (cid:10) e − ρ V θ (cid:11) + 12 (cid:10) e ρ − τ V τ (cid:11) − (cid:104) V τ (cid:105) (cid:10) e ρ − τ V τ (cid:11) + 12 e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) − dθ. roof. We compute straightforwardly using equations (1.2), (1.3) and integration by parts. From thedefinition of Λ we have ∂ τ Λ = −
2Λ + 12 e − τ (cid:90) S ( e ρ V τ ) τ ( V − (cid:104) V (cid:105) − dθ + 12 e − τ (cid:90) S V τ ∂ τ ( V − (cid:104) V (cid:105) − e ρ dθ = − e − τ (cid:90) S (cid:104) e τ (cid:0) e − ρ V θ (cid:1) θ + e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17)(cid:105) ( V − (cid:104) V (cid:105) − dθ + 12 e − τ (cid:90) S V τ ∂ τ ( V − (cid:104) V (cid:105) − e ρ dθ = −
2Λ + 12 e − τ (cid:90) S − e τ e − ρ V θ dθ + 12 e − τ (cid:90) S V τ e ρ dθ + 12 e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) − dθ − (cid:104) V τ (cid:105) (cid:28) e ρ − τ V τ (cid:29) which completes the proof. (cid:3) We modify the energy E by Λ. It is then desirable to know that Λ has better decay than E . To thatend, note that (cid:107) V − (cid:104) V (cid:105)(cid:107) C (cid:46) (cid:90) S | V θ | dθ ≤ (cid:18)(cid:90) S V θ e − ρ dθ (cid:19) / Π / ≤ (Π E ) / . (3.2)As is standard (cf. [20]), we use the notation f (cid:46) h to mean that there is a universal constant C > f ≤ Ch .One finds the following bound using H¨older’s Inequality. Lemma 3.2 ([18], Lemma 72) . Consider a non-Gowdy T -symmetric Einstein Flow. Then (cid:12)(cid:12)(cid:12)(cid:12) Λ + (cid:28) e ρ − τ V τ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) e − τ (cid:90) S V τ ( V − (cid:104) V (cid:105) ) e ρ dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ Π E (3.3)For the following bound on the Q correction, cf. [18] Lemma 73, where the author assumes a uniformbound on Π which we don’t assume here. The proof is essentially the same. Lemma 3.3.
For any a non-Gowdy T -symmetric Einstein Flow, (cid:12)(cid:12)(cid:12)(cid:12) e − τ (cid:90) S e V − τ ) Q τ ( Q − (cid:104) Q (cid:105) ) e ρ dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ e E ) / Π E Proof.
Note that we have already bounded (cid:107) V − (cid:104) V (cid:105)(cid:107) C in equation (3.2), and so we may commute outfactors of e V to obtain (cid:13)(cid:13) e V ( Q − (cid:104) Q (cid:105) ) (cid:13)(cid:13) C = (cid:13)(cid:13)(cid:13) e V −(cid:104) V (cid:105) + (cid:104) V (cid:105) ( Q − (cid:104) Q (cid:105) ) (cid:13)(cid:13)(cid:13) C = e (cid:107) V −(cid:104) V (cid:105)(cid:107) C e (cid:104) V (cid:105) (cid:107) Q − (cid:104) Q (cid:105)(cid:107) C ≤ e (cid:107) V −(cid:104) V (cid:105)(cid:107) C (cid:18)(cid:90) S e V Q θ e − ρ dθ (cid:19) / Π / ≤ e (cid:107) V −(cid:104) V (cid:105)(cid:107) C e τ ( E Π) / via H¨older’s inequality. So we may compute, using the bound on (cid:107) V − (cid:104) V (cid:105)(cid:107) C , H¨older’s inequality, andthe definition of E (cid:12)(cid:12)(cid:12)(cid:12) e − τ (cid:90) S e V − τ ) Q τ ( Q − (cid:104) Q (cid:105) ) e ρ dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ (cid:13)(cid:13) e V ( Q − (cid:104) Q (cid:105) ) (cid:13)(cid:13) C (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e V Q τ e ρ dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e (cid:107) V −(cid:104) V (cid:105)(cid:107) C e − τ E / Π / (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e V Q τ e ρ dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ e (cid:107) V −(cid:104) V (cid:105)(cid:107) C e − τ E Π (cid:46) e − τ e E ) / Π E. (cid:3) e only need the Q correction for the following identity, which follows directly from the definitionsof the conserved quantities A, B : (cid:104) e ρ V τ (cid:105) = A + B (cid:104) Q (cid:105) + (cid:90) S e V − τ ) Q τ ( Q − (cid:104) Q (cid:105) ) e ρ dθ. For B solutions, however, we use the bound on the Q correction to obtain the following bound (cid:12)(cid:12)(cid:10) e ρ − τ V τ (cid:11)(cid:12)(cid:12) − e − τ | A | (cid:46) e − τ e E ) / Π E (3.4)which together with (3.3) yields the desired estimate on the correction. Proposition 3.4.
For any a non-Gowdy, B T -symmetric Einstein Flow, | Λ | − e − τ | A | (cid:46) e − τ (cid:16) e E ) / (cid:17) Π E. (3.5)The correction Λ introduces significant new error terms after differentiation. However, these termshave good bounds, and the modified energy E + Λ has significantly better properties upon comparisonto E alone. The evolution of this modified energy is the focus of the next chapter.4. The corrected energy
One would like to show that, up to error terms, Π and E satisfy an ODE. While this is true asymp-totically, it is more useful to compute with an energy which has been modified by the correction.One computes that( E + Λ) τ = (cid:90) S − e ρ − τ ρ τ J − e ρ − τ V τ − e V − τ ) − ρ Q θ dθ − e − τ (cid:90) S − e τ e − ρ V θ dθ + 12 e − τ (cid:90) S V τ e ρ dθ + 12 e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) − dθ − (cid:104) V τ (cid:105) (cid:28) e ρ − τ V τ (cid:29) = − (cid:18) τ Π (cid:19) ( E + Λ) + (cid:18) Π τ Π E − (cid:90) S e ρ − τ ρ τ J dθ (cid:19) − (cid:18) − Π τ Π (cid:19) Λ+ 12 e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) ) dθ − (cid:104) V τ (cid:105) (cid:28) e ρ − τ V τ (cid:29) . The leading term on the right leads us to the ansatz that Π ( E + Λ) (and so Π E ) should decay like e − τ .Accordingly, define the corrected, normalized energy H := Π ( E + Λ). One computes that ∂ τ ( e τ H ) = e τ H + e τ Π τ ( E + Λ) + e τ Π ( E + Λ) τ = e τ Π (cid:18) ( E + Λ) (cid:18) τ Π (cid:19) + ( E + Λ) τ (cid:19) = e τ Π (cid:20)(cid:18) Π τ Π E − (cid:90) S e ρ − τ ρ τ J dθ (cid:19) − (cid:18) − Π τ Π (cid:19) Λ(4.1) + 12 e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) ) dθ − (cid:104) V τ (cid:105) (cid:28) e ρ − τ V τ (cid:29)(cid:21) The ansatz in the local stability proof is that e τ H is of constant order. The proof is via a bootstrapargument, where we bound all of the terms of ∂ τ ( e τ H ) in terms of Π , E, H and τ . The followingProposition deals with each of these error terms. roposition 4.1. Consider the evolution of a B solution with initial data given at time τ = s . Let ρ := min θ ∈ S ρ ( θ, s ) . The following estimates hold. (cid:12)(cid:12)(cid:12)(cid:12) Π τ Π E − (cid:90) S e ρ − τ ρ τ J dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) E (cid:90) S e ρ − τ ρ τ J dθ, (4.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Π τ Π (cid:19) Λ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) | A | e − τ (cid:18) τ Π (cid:19) + e − τ (cid:16) e E ) / (cid:17) (Π + Π τ ) E, (4.3) (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) V τ (cid:105) (cid:28) e ρ − τ V τ (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − ρ / e − τ (cid:16) | A | + e τ e E ) / Π E (cid:17) E / , (4.4) and (cid:12)(cid:12)(cid:12)(cid:12) e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) ) dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) Π / E / . (4.5) Proof.
For (4.2), using Young’s inequality, we note that | l θ | ≤| V τ V θ | + | e V − τ Q τ e V − τ Q θ | = | e ( ρ − τ ) / V τ e − ( ρ − τ ) / V θ | + | e V − τ e ( ρ − τ ) / Q τ e V − τ e − ( ρ − τ ) / Q θ |≤ (cid:104) e ρ − τ V τ + e τ − ρ V θ + e V − τ ) e ρ − τ Q τ + e V − τ ) e τ − ρ Q θ (cid:105) = e ρ − τ J. (4.6)Thus we may use the Poincar´e inequality to compute that (cid:12)(cid:12)(cid:12)(cid:12) Π τ Π E − (cid:90) S e ρ − τ ρ τ J dθ (cid:12)(cid:12)(cid:12)(cid:12) =Π − (cid:12)(cid:12)(cid:12)(cid:12) Π τ E − Π (cid:90) S e ρ − τ ρ τ J dθ (cid:12)(cid:12)(cid:12)(cid:12) =Π − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S (cid:90) S e ρ ( φ ) e ρ ( θ ) − τ J ( θ ) ( ρ τ ( φ ) − ρ τ ( θ )) dφdθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Π − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S (cid:90) S e ρ ( φ ) e ρ ( θ ) − τ J ( θ ) sup a,b ∈ S | ρ τ ( a ) − ρ τ ( b ) | dφdθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =Π − Π (cid:90) S e ρ ( θ ) − τ J ( θ ) dθ sup a,b ∈ S | ρ τ ( a ) − ρ τ ( b ) | (cid:46) E (cid:90) S ρ τ | l θ | dθ ≤ E (cid:90) S ρ τ e ρ − τ J dθ.
Inequality (4.3) follows directly from inequality (3.5). To prove (4.5), we first commute out the V -mean. (cid:12)(cid:12)(cid:12)(cid:12) e − τ (cid:90) S e V − τ )+ ρ (cid:16) Q τ − e τ − ρ ) Q θ (cid:17) ( V − (cid:104) V (cid:105) ) dθ (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − τ (cid:107) V − (cid:104) V (cid:105)(cid:107) C (cid:90) S e V − τ )+ ρ (cid:12)(cid:12)(cid:12) Q τ − e τ − ρ ) Q θ (cid:12)(cid:12)(cid:12) dθ (cid:46) e − τ (Π E ) / (cid:90) S e ρ e V − τ ) (cid:16) Q τ + e τ − ρ ) Q θ (cid:17) dθ (cid:46) (Π E ) / (cid:90) S e ρ − τ J dθ =Π / E / . Lastly, for (4.4) recall that ρ is increasing and compute that |(cid:104) V τ (cid:105)| ≤ (cid:18)(cid:90) S V τ e ρ dθ (cid:19) / (cid:18)(cid:90) S e − ρ dθ (cid:19) / ≤ e − ρ / e τ (cid:18)(cid:90) S V τ e ρ − τ dθ (cid:19) / (cid:46) e − ρ / e τ E / and use (3.4). This completes the proof. (cid:3) Now that we have an energy satisfying a good differential equation with good bounds on the error,we proceed to the linearization. . Linearization
In [13], the authors present an argument that certain asymptotic rates of Π , E should be preferred,based on the assumption that e τ H should be of constant order. In this section we briefly summarizethat argument as it appears in our context. Definition 5.1.
Let Y := (cid:10) e l + ρ +2 τ (cid:11) . This quantity has been previously considered; see [7] where (modulo factors of e τ ) it is called the “twistpotential.”Note that we have defined Y so that Y τ = (cid:104) e l + ρ +2 τ ( l τ + ρ τ + 2) (cid:105) = (cid:104) e l + ρ +2 τ J (cid:105) . We want to form asystem of ordinary differential equations from the means, however. So we distribute the integral overthe product, introducing the error term Ω. One computesΠ τ = e − τ Y (5.1) Y τ = e τ EY Π − + Ω(5.2)where Ω := (cid:10) e l + ρ +2 τ J (cid:11) − e τ EY Π − is an error term satisfying | Ω | ≤ e τ E (cid:10) e l + ρ − τ J (cid:11) = e τ EY τ . Note that our quantity E contains the terms Q θ and Q τ , and so is not identical to the energy in [13].Nonetheless, the quantities Π , Y, E satisfy similar relations to the relations that LeFloch and Smulevici’squantities do. Normalizing, we compute that ∂ τ (cid:16) e − τ H − / Π (cid:17) = e − τ H − / Π τ − e − τ H − / Π − e − τ H − / Π H τ H = (cid:16) e − τ H − / Y (cid:17) + (cid:16) e − τ H − / Π (cid:17) (cid:18) − − H τ H (cid:19) ∂ τ (cid:16) e − τ H − / Y (cid:17) = e − τ H − / Y τ − e − τ H − / Y − e − τ H − / Y H τ H = e − τ H − / (cid:0) e τ EY Π − + Ω (cid:1) + (cid:16) e − τ H − / Y (cid:17) (cid:18) − − H τ H (cid:19) = (cid:0) e − τ H − / Y (cid:1) Π e τ Π E + (cid:16) e − τ H − / Y (cid:17) (cid:18) − − H τ H (cid:19) + e − τ H − / Ω= (cid:0) e − τ H − / Y (cid:1)(cid:0) e − τ H − / Π (cid:1) Π EH + (cid:16) e − τ H − / Y (cid:17) (cid:18) − − H τ H (cid:19) + e − τ H − / Ω= (cid:0) e − τ H − / Y (cid:1)(cid:0) e − τ H − / Π (cid:1) + (cid:16) e − τ H − / Y (cid:17) (cid:18) − − H τ H (cid:19) + e − τ H − / Ω+ (cid:0) e − τ H − / Y (cid:1)(cid:0) e − τ H − / Π (cid:1) (cid:18) Π EH − (cid:19) . We insert our ans¨atze that H τ H → − e − τ H − / Ω →
0, and (cid:0) Π EH − (cid:1) →
0, to obtain the ODE ∂ τ c = d + c (cid:18) − (cid:19) ∂ τ d = dc + d (cid:18) − (cid:19) which has a fixed point at c = 2 √ , d = 1 √ . So we conjecture that the quantities c := Π e τ √ H − √ , d := Ye τ √ H − √ ecay and compute the evolution of these quantities using (5.1) and (5.2). We find that ∂ τ (cid:18) cd (cid:19) = (cid:18) − / − / (cid:19) (cid:18) cd (cid:19) − ∂ τ log ( e τ H ) (cid:18) cd (cid:19) − ∂ τ log ( e τ H ) (cid:32) √ √ (cid:33) + f ( d,c ) (cid:16) c + √ (cid:17) + (cid:18) Π EH − (cid:19) d + √ (cid:16) c + √ (cid:17) + Ω e τ H / where f ( c, d ) = c − d +3 √ c − √ cd has vanishing linear part. Let (cid:101) Ω := − ∂ τ log ( e τ H ) (cid:32) √ √ (cid:33) + f ( d,c ) (cid:16) c + √ (cid:17) + (cid:18) Π EH − (cid:19) d + √ (cid:16) c + √ (cid:17) + Ω e τ H / denote the error term of this approximation. In the end, the following estimate is obtained (cf. [13],Proposition 5.1). Proposition 5.1.
Consider the evolution of a B T -symmetric solution. Provided the corrected energy H is positive, one has for s ≥ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e ( s − s ) / (cid:18) e s H ( s ) e s H ( s ) (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) + (cid:90) ss e ( τ − s ) / (cid:18) e τ H ( τ ) e s H ( s ) (cid:19) / | ω ( τ ) | dτ, where | ω | (cid:46) (cid:12)(cid:12)(cid:12)(cid:101) Ω (cid:12)(cid:12)(cid:12) . Quickly note a bound on one of the terms appearing in (cid:101) Ω. Lemma 5.2.
Consider the evolution of a B T -symmetric solution. The following estimate holds. (cid:12)(cid:12)(cid:12) e − τ H − / Ω (cid:12)(cid:12)(cid:12) (cid:46) e − τ | H | − / EY τ . The proof of this lemma proceeds in the same manner as the proof of inequality (4.2). The remainingthree terms in (cid:101)
Ω are estimated directly. In the next section, we perform a bootstrap argument to boundthese errors, provided the initial data is sufficiently close to the asymptotic behavior.6.
The Bootstrap
The technique of proof follows [13]. The idea is to impose some smallness assumptions on the meansof the energy, the S volume, and their derivatives. We then use a bootstrap argument to show thatthese assumptions are improved. The reason for obtaining the estimates of Lemma 4.1 is to bound theevolution of the corrected energy H . Let us discuss how that proof goes. We have computed ∂ τ ( e τ H )in equation (4.1). Note that we may bound the right side of that equation by an expression of the form | ∂ τ ( e τ H ) | (cid:46) e τ Π EF + (cid:101) F = e τ H Π EH F + (cid:101) F where, using the results of Lemma 4.1 we can write F := (cid:90) S e ρ − τ ρ τ J dθ + e − τ (cid:16) e E ) / (cid:17) (Π + Π τ ) + (Π E ) / + e − ρ ( s ) / e E ) / Π E / (6.1)and (cid:101) F := | A | Π (cid:18) e − τ (cid:18) τ Π (cid:19) + e − ρ / E / (cid:19) . (6.2)Note that F and (cid:101) F are nonnegative. We are then concerned with the quantities (cid:90) ∞ s F ( τ ) dτ, and (cid:90) ∞ s (cid:101) F ( τ ) dτ which bound the evolution of e τ H in the bootstrap proof.First, however, we need the following version of Gr¨onwall’s Lemma, the proof of which is straightfor-ward. emma 6.1 (Gr¨onwall’s Inequality) . Let α, β, f be nonnegative smooth functions on the interval [ s , s ] .Suppose f satisfies the differential inequality | f (cid:48) | ≤ α + βf. Then | f ( s ) − f ( s ) | ≤ − f ( s ) + (cid:18) f ( s ) + (cid:90) ss α ( t ) dt (cid:19) exp (cid:18)(cid:90) ss β ( t ) dt (cid:19) . Lemma 6.2.
There exist constants (cid:15), C > , M > , a time s > depending on (cid:15) , and an open set of B Einstein Flows satisfying the following conditions at time τ = s : | A | < ρ := inf S ρ > | c | <(cid:15) | d | <(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) Π EH − (cid:12)(cid:12)(cid:12)(cid:12) < , (cid:15) − < e s < (cid:15) − e s H ( s ) + C (cid:15) / < M (cid:15)e s (6.5) 0 < M A, ρ in inequalities (6.5), (6.6), and (6.9). We have added these assumptionsjust to simplify the notation.The technique of proof is a straightforward “open closed” argument:(1) Suppose estimates (6.7) to (6.9) are satisfied for τ ∈ [ s , s ).(2) We improve each of the five estimates (6.7) to (6.9) at τ = s by choosing (cid:15) small. Proof. Initial Estimates: From assumptions (6.7) to (6.9), we have that e − τ (cid:46) H (cid:46) (cid:15)e s − τ , and (cid:12)(cid:12)(cid:12)(cid:12) Π e τ √ H − √ (cid:12)(cid:12)(cid:12)(cid:12) = | c | < (cid:15) / , (cid:12)(cid:12)(cid:12)(cid:12) Ye τ √ H − √ (cid:12)(cid:12)(cid:12)(cid:12) = | d | < (cid:15) / so Π (cid:46) (cid:18) √ 10 + (cid:15) / (cid:19) e τ H / ≤ (cid:18) √ 10 + (cid:15) / (cid:19) (cid:15) / e ( s + τ ) / (cid:46) (cid:15) / e s / e τ/ , (6.10) e τ Π τ = Y (cid:46) (cid:18) √ 10 + (cid:15) / (cid:19) e τ H / (cid:46) (cid:18) √ 10 + (cid:15) / (cid:19) (cid:15) / e (5 τ + s ) / (cid:46) (cid:15) / e s / e τ/ . (6.11) Note that (6.8) implies that Π E (cid:46) H on this interval, which implies thatΠ E (cid:46) (cid:15)e s − τ < (cid:15), and 1 + e E ) / (cid:46) e (cid:15) / (cid:46) or sufficiently small (cid:15) . The bound on Π and the fact that Π , Y > a < (cid:90) ss e ( a − / τ Π τ dτ (cid:46) (cid:15) / e s / (cid:90) ss e aτ dτ ≤ C ( a ) (cid:15) / e ( a +1 / s and similarly (cid:90) ss e ( a − / τ Y τ dτ (cid:46) e ( a − / s Y ( s ) − e ( a − / s Y ( s ) − ( a − / (cid:90) ss e ( a − / τ Y dτ (cid:46) (cid:15) / (cid:20) e as + s / − ( a − / e s / (cid:90) ss e aτ dτ (cid:21) (cid:46) (cid:15) / (cid:20) e as + s / − a − / a (cid:16) e as + s / − e ( a +1 / s (cid:17)(cid:21) (cid:46) (cid:15) / (cid:104) e as + s / + C ( a ) e ( a +1 / s (cid:105) (cid:46) C ( a ) (cid:15) / e ( a + ) s . Bound on Λ : To improve inequality (6.8), first note that (cid:12)(cid:12) Π EH − (cid:12)(cid:12) = Π H | Λ | . Then we may use theestimate of the correction in inequality (3.5) to obtainΠ H | Λ | (cid:46) Π H (cid:104) e − τ | A | + e − τ (cid:16) e E ) / (cid:17) Π E (cid:105) (cid:46) Π H (cid:2) e − τ + e − τ Π E (cid:3) (cid:46) (cid:15) / e s / e τ/ (cid:2) e − τ + e s − τ (cid:15) (cid:3) (cid:46) (cid:15) / e s / e − τ/ + (cid:15) / e s / e − τ/ (cid:46) (cid:15) / + (cid:15) / e s (cid:46) (cid:15) / since H − (cid:46) e τ . Thus we may ensure (cid:12)(cid:12)(cid:12)(cid:12) Π EH − (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) small. An Upper and Lower Bound on H : For the energy H we have the following estimate: | ∂ τ ( e τ H ) | (cid:46) e τ H Π EH F + (cid:101) F (cid:46) e τ HF + (cid:101) F That is, there is a constant C > | ∂ τ ( e τ H ) | ≤ C (cid:16) e τ HF + (cid:101) F (cid:17) . The quantities F and (cid:101) F are the nonnegative quantities defined in equations (6.1) and (6.2). Wethen apply Lemma 6.1 to obtain the upper bound e s H ( s ) ≤ (cid:18) e s H ( s ) + C (cid:90) ss (cid:101) F dτ (cid:19) exp (cid:18) C (cid:90) ss F dτ (cid:19) (6.12) and the lower bound e s H ( s ) ≥ e s H ( s ) − (cid:18) e s H ( s ) + C (cid:90) ss (cid:101) F dτ (cid:19) exp (cid:18) C (cid:90) ss F dτ (cid:19) . (6.13) What we want, then, is for (cid:82) ∞ s (cid:101) F dτ to be bounded and for (cid:82) ∞ s F dτ → (cid:15) → e − ρ / < | A | < 1. We compute (cid:101) F = | A | Π (cid:18) e − τ (cid:18) τ Π (cid:19) + e − ρ / E / (cid:19) Now we turn to the bound on F . F = (cid:90) S e ρ − τ ρ τ J dθ + e − τ (cid:16) e E ) / (cid:17) (Π + Π τ ) + (Π E ) / + e − ρ / e E ) / Π E / (cid:46) e − τ (cid:90) S e ρ + l +2 τ J dθ + e − τ (Π + Π τ ) + (Π E ) / + Π E / (cid:46) e − τ Y τ + (cid:15) / e s / e − τ/ + (cid:15) / e s / e − τ/ We have previously bounded the integral of the latter terms in time by (cid:15) / , so it remains tocompute (cid:90) ∞ s e − τ Y τ dτ (cid:46) (cid:15) / . So (cid:90) ∞ s F dτ (cid:46) (cid:15) / . Thus, in total for H , we have e s H ( s ) < (cid:16) e s H ( s ) + C (cid:15) / (cid:17) when we choose (cid:15) small enough that exp (cid:16) C (cid:82) ss F dτ (cid:17) < .Turning to the lower bound, it is useful to define N := e s H ( s ) and L := C (cid:15) / . Noteassumption (6.6) implies that14( N + L ) (5 N + 3 L ) > M ( N + L ) > M so we take (cid:15) small enough that1 + 14 M > exp (cid:18) C (cid:90) ss F dτ (cid:19) . The lower bound from Gr¨onwall’s inequality takes the form e s H ( s ) ≥ N − ( N + L ) exp (cid:18) C (cid:90) ss F dτ (cid:19) > N − 14 (5 N + 3 L ) = 34 ( N − L )which improves the lower bound on e τ H ( τ ). Bounds on Π , Y : Let us determine what the smallness assumptions of Lemma 6.2 imply for theerror term of the ODE system of Section 5. Recall the conclusion of Proposition 5.1: if H > (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e ( s − s ) / (cid:18) e s H ( s ) e s H ( s ) (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) + (cid:90) ss e ( τ − s ) / (cid:18) e τ H ( τ ) e s H ( s ) (cid:19) / | ω ( τ ) | dτ, (6.15) where | ω | (cid:46) (cid:12)(cid:12)(cid:12)(cid:101) Ω (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∂ τ log ( e τ H ) (cid:32) √ √ (cid:33) + f ( d,c ) (cid:16) c + √ (cid:17) + (cid:18) Π EH − (cid:19) d + √ (cid:16) c + √ (cid:17) + Ω e τ H / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nd (cid:12)(cid:12)(cid:12) e − τ H − / Ω (cid:12)(cid:12)(cid:12) (cid:46) e − τ | H | − / EY τ . To begin with, note that e τ H ( τ ) has both upper and lower bounds, and so both terms of theform (cid:16) e τ H ( τ ) e s H ( s ) (cid:17) / can be bounded above by a constant: (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e ( s − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) + (cid:90) ss e ( τ − s ) / | ω ( τ ) | dτ (6.16) (cid:46) (cid:15) + (cid:90) ss e ( τ − s ) / | ω ( τ ) | dτ. To finish the bootstrap, we must bound the right side of this inequality strictly below (cid:15) / . Wedeal with each of the 4 summands in ω in the remainder of the proof.The contribution to the right side of (6.15) from the error term e − τ H − / Ω is (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12) e − τ H − / Ω (cid:12)(cid:12)(cid:12) dτ (cid:46) e − s/ (cid:90) ss e − τ/ (cid:12)(cid:12)(cid:12) H − / Π − (cid:12)(cid:12)(cid:12) Π EY τ dτ = e − s/ (cid:90) ss e − τ/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Π EH c + √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y τ dτ (cid:46) e − s/ (cid:90) ss e − τ/ (cid:12)(cid:12)(cid:12)(cid:12) Π EH Y τ (cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:46) e − s/ (cid:90) ss e ( − / − / τ Y τ dτ (cid:46) (cid:15) / e ( s − s ) / <(cid:15) / where we have used the fact that e τ H − / c + √ = Π − and the bootstrap assumptions.The contribution from (cid:0) Π EH − (cid:1) d + √ (cid:16) c + √ (cid:17) is (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Π EH − (cid:19) d + √ (cid:16) c + √ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:46) (cid:90) ss e ( τ − s ) / (cid:15) / dτ (cid:46) (cid:15) / (cid:16) − e ( s − s ) / (cid:17) <(cid:15) / . Turning to f ( d,c ) (cid:16) c + √ (cid:17) , we recall that f has vanishing linear part, so (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( d, c ) (cid:16) c + √ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:46) (cid:90) ss e ( τ − s ) / (cid:15) / dτ (cid:46) (cid:15) / (cid:16) − e ( s − s ) / (cid:17) < (cid:15) / To bound ∂ τ log ( e τ H ), note that e τ H has a lower bound, and use the estimates on F and (cid:101) F obtained above to compute | ∂ τ log ( e τ H ) | = 1 e τ H | ∂ τ ( e τ H ) | (cid:46) e τ H (cid:16) e τ HF + (cid:101) F (cid:17) (cid:46) F + (cid:101) F (cid:46) e − τ Y τ + (cid:15) / e s / e − τ/ + (cid:15) / e s / e − τ/ o the contribution to (6.15) is (cid:90) ss e ( τ − s ) / (cid:16) e − τ Y τ + (cid:15) / e s / e − τ/ + (cid:15) / e s / e − τ/ (cid:17) dτ (cid:46) (cid:15) / + e − s/ (cid:90) ss e − τ/ − / τ Y τ dτ (cid:46) (cid:15) / + e ( s − s ) / (cid:15) / (cid:46) (cid:15) / . Combining these estimates, we have from inequality (6.16) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e ( s − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) + (cid:90) ss e ( τ − s ) / | ω ( τ ) | dτ (cid:46) (cid:15) + (cid:15) / (cid:46) (cid:15) / . This improves the bootstrap inequality on c, d .Thus we have improved all of the bootstrap inequalities, and the proof is complete. (cid:3) Asymptotic Behavior We are now in a position to present the B version of the main result of [13]. In particular, for T -symmetric vacuum spacetimes satisfying B = 0, we find rates of growth/decay in the expanding directionfor the θ -direction volume, the normalized energy, and their derivatives. In going from the polarised to B case, we appear to lose some of the fine grained asymptotics of V and its mean. Forthcoming workwill describe the behavior of V and Q , and the dependence of that behavior on the conserved quantity B . Given our estimates above, the proof of the theorem is nearly identical to the polarised case. Theorem 7.1. There exists an (cid:15) such that if ≤ (cid:15) ≤ (cid:15) , for any B initial data set satisfying thesmallness conditions of Lemma 6.2, the associated solution satisfies for τ ∈ [ s , ∞ ) . (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ/ (7.1) (cid:12)(cid:12) e τ H − C ∞ (cid:12)(cid:12) (cid:46) e − τ/ (7.2) (cid:12)(cid:12)(cid:12)(cid:12) e − τ/ Π − √ C ∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ/ (7.3) (cid:12)(cid:12)(cid:12)(cid:12) e − τ/ Y − √ C ∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ/ (7.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e τ/ E − √ C ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ/ (7.5) |(cid:104) l (cid:105) − l | (cid:46) e − τ/ (7.6) (cid:12)(cid:12)(cid:12)(cid:12) e l − (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ/ (7.7) |(cid:104) V (cid:105) − V | + (cid:12)(cid:12) e V − τ ( (cid:104) Q (cid:105) − Q ) (cid:12)(cid:12) (cid:46) e − τ/ (7.8) (cid:12)(cid:12) Π − e ρ − e ρ ∞ (cid:12)(cid:12) (cid:46) e − τ/ (7.9) for some C ∞ > and ρ ∞ : S → R .Proof. The proof proceeds as in [13]. First, observe that inequalities (6.10) and (6.11) imply that e − τ Π + e − τ Π τ + e − τ Y (cid:46) e − τ/ . Furthermore, Π E (cid:46) e − τ and e τ H is bounded above and below by positive constants. On the other hand | ∂ τ ( e τ H ) | (cid:46) e τ HF + (cid:101) F (cid:46) F + (cid:101) F (cid:46) e − τ/ + e − τ Y τ . The right side is integrable in τ , so let C ∞ := lim τ →∞ √ e τ H . Then (cid:12)(cid:12) C ∞ − e τ H (cid:12)(cid:12) (cid:46) (cid:90) ∞ τ | ∂ τ ( e s H ) | ds (cid:46) e − τ/ giving (7.2). ote that (6.15) now reads (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e − s/ + (cid:90) ss e ( τ − s ) / | ω ( τ ) | dτ, and that all of the terms of (cid:82) ss e ( τ − s ) / | ω ( τ ) | dτ are now bounded by e − s/ with the exception of (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( d, c ) (cid:16) c + √ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:46) (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:46) (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dτ (7.10)since | ( c, d ) | (cid:46) 1. So (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e − s/ + (cid:90) ss e ( τ − s ) / | ω ( τ ) | dτ (cid:46) e − s/ + (cid:90) ss e ( τ − s ) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dτ. Applying the integral version of Gr¨onwall’s inequality gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) cd (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ( s ) (cid:46) e − s/ + (cid:90) ss e − τ/ e ( τ − s ) / exp (cid:18)(cid:90) sτ e ( r − s ) / dr (cid:19) dτ (cid:46) e − s/ + e − s/ (cid:90) ss exp (cid:18)(cid:90) sτ e ( r − s ) / dr (cid:19) dτ (cid:46) e − s/ + se − s/ (cid:46) e ( δ − ) s for any δ > 0. Inserting this improved estimate into (7.10) and applying Gr¨onwall’s inequality againgives (7.1). Combining this with (7.2) yields (7.3) and (7.4).Recall that H = Π( E + Λ) and | Λ | (cid:46) e − τ | A | + e − τ (cid:16) e E ) / (cid:17) Π E (cid:46) e − τ . Then combine (7.2) and (7.3) to obtain (7.5). The estimate (7.6) follows from (4.6) and (7.5). Estimate(7.8) follows directly from the Poincar´e inequality and the bound on E .To estimate e l let us note that (cid:12)(cid:12) Π e l − e − τ Y (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e ρ ( ϕ )+ l ( ϕ ) (cid:16) e l ( ϕ ) − l ( θ ) − (cid:17) dϕ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) e − τ Y. (7.11)One can then combine (7.11), (7.3), and (7.4) to find that sup S e l is bounded by a constant. Then, weestimate again (cid:12)(cid:12) Π e l − e − τ Y (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S e ρ ( ϕ ) (cid:16) e l ( ϕ ) − e l ( θ ) (cid:17) dϕ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) Π (cid:18) sup S e l (cid:19) e τ E (cid:46) ρ θτ . (cid:3) Numerical Evidence The full Einstein Flow is a large, quasilinear system of partial differential equations about which it isdifficult even to make conjectures. This remains true even in the simplified T -symmetric case consideredin this work. It has been crucial to this work to base our conjectures on evidence garnered from numericalsimulations of T -symmetric Einstein Flows. We summarize this numerical work in this section. A moredetailed discussion of the numerical methods and results is the subject of a forthcoming paper.Our code is a reimplementation of one previously developed by Berger to simulate T -symmetricspacetimes in the contracting direction [7], and then later in the expanding direction. We reimplementedthis code in OCaml , and made a number of modifications to improve the accuracy and speed. Mostimportantly, we developed code to produce solutions of the T -symmetric constraint equation via arandom process, which allowed us to probe the behavior of generic T -symmetric Einstein Flows. OCaml is a general purpose programming language developed primarily at INRIA. See https://ocaml.org/ . e have developed code which samples the constraint submanifold for the T -symmetric EinsteinField Equations in a fairly generic manner. We have then evolved these initial data using a finitedifference method. This generic sampling has been a crucial element allowing us to determine that theassumption B = 0 was necessary for our main theorem, and otherwise develop our intuition about thesolutions. The simulations have the expected convergence properties upon refining the spatial resolutionso we are confident that they are accurate approximations of solutions. To obtain confidence that oursimulations depict behavior which is generic for the class under consideration, we simulated on the orderof 20 randomly chosen initial constraints solutions in each of the following classes: polarised, B , and B (cid:54) = 0 T -symmetric. The qualitative behavior depicted in Figures 2 through 4 is observed to be thesame for all simulations in that class.It has been useful to plot the evolution of the following quantities along each of the numerical solutions. S := ∂ τ (cid:90) S l e ρ − τ/ dθ, T := ∂ τ (cid:90) S ρ e ρ − τ/ dθ,E V := (cid:90) S (cid:104) V τ + e τ − ρ ) V θ (cid:105) e ρ − τ/ dθ, E Q := (cid:90) S e V − τ ) (cid:16) Q τ + e τ − ρ ) Q θ (cid:17) e ρ − τ/ dθ,W := log (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S V τ e ρ − τ/ dθ (cid:12)(cid:12)(cid:12)(cid:12) These are not the quantities that were used in the proof of our main theorem, but they capture thedynamics of the system. The volume form e ρ − τ/ dθ is used to smooth out the graphs (the integralsgenerally oscillate without this normalization). - (a) polarised - (b) B = 0 - (c) B (cid:54) = 0 Figure 2. S and T flow toward a spiral sink, regardless of polarisation or the value of B . Although l τ , ρ τ converge to the same values in all cases, the volume form e ρ − τ/ dθ causes the variables used in the plots to flow toward different values. τ (a) polarised τ (b) B = 0 τ (c) B (cid:54) = 0 Figure 3. For polarised solutions, E = E V which converges to a constant. For B = 0solutions, E and the V and Q energies all converge to constants. For B (cid:54) = 0 solutions,however, although the total energy converges, E V and E Q do not; they oscillate withamplitude which does not decay and period matching the period of the sink in Figure2. τ - - - - (a) polarised 10 20 τ - - - - (b) B = 0 10 20 τ - - - - (c) B (cid:54) = 0 Figure 4. For B = 0 solutions (including polarised), V τ → B (cid:54) = 0solutions, however, V τ appears to converge to a nonzero constant.In [13], the authors are able to determine the first order behavior of the energy and Π, but also thefirst order behavior of V and the rate of its decay to the mean value. We have generalized their resultson the asymptotic values of the energy, Π as well as the decay of V and Q to their means to the B case,but so far have been unable to derive other estimates for V and Q . However, the numerical solutionsthat we have found have the property that there are constants a, C V such that |(cid:104) V (cid:105) − C V τ − a | = O ( e − τ/ )and that C V = (cid:26) B = 0 if B (cid:54) = 0 . More detailed descriptions of the numerical results will be given in future work. Appendix A. Concordance of notations between [6] , [7] , [18] , and [13]The Einstein Flows under consideration in the this work have been studied extensively, including manyimportant special subsets of solutions. Unfortunately, authors have used many different coordinates forexactly the same set of spacetimes, and this document adds yet another set of coordinates. As an aid tothe reader who wishes to read the cited works together, we provide in this appendix a concordance ofnotations used in the most frequently cited of these works.To the best of our knowledge, all of the works in the table rely on the foliation and equations derivedin [6]. This paper, [6], [7] and [18] use coordinates for T -symmetric Einstein Flows which are completelygeneral. The analysis in [13] applies only to polarised T -symmetric Einstein Flows, and so relies onthe assumption that some metric components vanish identically. In [17], future asymptotics of Gowdysolutions are derived. The notation used there is exactly the notation of [18] if one imposes the conditions α ≡ , K = 0 so we omit it from the table.In the table below, each column uses the notation internal to the document named in the first row. Allof the expressions in a given row are equal. For example, the function called P in [18] has the expression2 U − log R in [13]. Since [13] only deals with polarised flows, the expressions in this column are onlyequal to those in other documents if the polarization condition is imposed. ][ ][ ][ ] t h i s d o c u m e n t l og t − τ l og t l og R τ U P − τ P + l og t U V α − / π λ α − / a − e ρ K t − α e ν K e P + λ + τ / K t − / e P + λ K R − e η e l t U t − π P π λ t P t + R U R V τ t − (cid:90) S α − / e U Q t d θ − (cid:90) S π Q d θ (cid:90) S α − / e P t Q t d θ (cid:90) S e ρ + ( V − τ ) Q τ d θ = : B eferences [1] Ellery Ames, Florian Beyer, James Isenberg, and Philippe G. LeFloch. Quasilinear hyperbolic Fuchsian systems andAVTD behavior in T -symmetric vacuum spacetimes. Ann. Henri Poincar´e , 14(6):1445–1523, 2013.[2] Michael T. Anderson. On long-time evolution in general relativity and geometrization of 3-manifolds. Comm. Math.Phys. , 222(3):533–567, 2001.[3] V. A. Belinski˘ı and I. M. Khalatnikov. Effect of scalar and vector fields on the nature of the cosmological singularity. ˇZ. `Eksper. Teoret. 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Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hyper-surfaces: topologies and boundary conditions. Ann. Physics , 83:203–241, 1974.[11] James Isenberg and Vincent Moncrief. Asymptotic behaviour in polarized and half-polarized U(1) symmetric vacuumspacetimes. Classical Quantum Gravity , 19(21):5361–5386, 2002.[12] Edward Kasner. Solutions of the Einstein equations involving functions of only one variable. Trans. Amer. Math. Soc. ,27(2):155–162, 1925.[13] Philippe LeFloch and Jacques Smulevici. Future asymptotics and geodesic completeness of polarized T2-symmetricspacetimes. Anal. PDE , 9(2):363–395, 2016.[14] J. Lott. Collapsing in the Einstein flow. ArXiv e-prints , January 2017.[15] J. Lott. Backreaction in the future behavior of an expanding vacuum spacetime. Classical Quantum Gravity ,35(3):035010, 10, 2018.[16] Katharina Radermacher. On the Cosmic No-Hair Conjecture in T2-symmetric non-linear scalar field spacetimes. 2017.[17] Hans Ringstr¨om. On a wave map equation arising in general relativity. Comm. Pure Appl. Math. , 57(5):657–703, 2004.[18] Hans Ringstr¨om. Instability of Spatially Homogeneous Solutions in the Class of T -Symmetric Solutions to Einstein’sVacuum Equations. Comm. Math. Phys. , 334(3):1299–1375, 2015.[19] Hans Ringstr¨om. Linear systems of wave equations on cosmological backgrounds with convergent asymptotics. ArXive-prints , July 2017.[20] Igor Rodnianski and Jared Speck. Stable Big Bang Formation in Near-FLRW Solutions to the Einstein-Scalar Fieldand Einstein-Stiff Fluid Systems. ArXiv e-prints , July 2014.[21] Igor Rodnianski and Jared Speck. A regime of linear stability for the Einstein-scalar field system with applications tononlinear big bang formation. Ann. of Math. (2) , 187(1):65–156, 2018.[22] Igor Rodnianski and Jared Speck. Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field andEinstein-stiff fluid systems. Selecta Math. (N.S.) , 24(5):4293–4459, 2018.[23] Jared Speck. The nonlinear future stability of the FLRW family of solutions to the Euler-Einstein system with apositive cosmological constant. Selecta Math. (N.S.) , 18(3):633–715, 2012. Edward L. Ginzton Laboratory, Stanford University, Stanford, CA 94305-4088, USA E-mail address : [email protected] Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA E-mail address : [email protected] Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden E-mail address : [email protected]@kth.se