# Stability Within $T^{2}$-Symmetric Expanding Spacetimes

Beverly K. Berger  Edward L. Ginzton Laboratory
Stanford University
Stanford, CA 94305-4088
USA<EMAIL_ADDRESS>,  James Isenberg  Department of Mathematics
University of Oregon
Eugene, OR 97403-1222
USA<EMAIL_ADDRESS>and  Adam Layne  Department of Mathematics
KTH Royal Institute of Technology
100 44 Stockholm
Sweden<EMAIL_ADDRESS>
###### Abstract.

We prove a nonpolarised analogue of the asymptotic characterization of
$T^{2}$-symmetric Einstein Flow solutions completed recently by LeFloch and
Smulevici. In this work, we impose a condition weaker than polarisation and so
our result applies to a larger class. We obtain similar rates of decay for the
normalized energy and associated quantities for this class. We describe
numerical simulations which indicate that there is a locally attractive set
for $T^{2}$-symmetric solutions not covered by our main theorem. This local
attractor is distinct from the local attractor in our main theorem, thereby
indicating that the polarised asymptotics are unstable.

##  1\. introduction

There exist broad conjectures about the expanding direction behavior of vacuum
spacetimes with closed 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 locally
stable behavior in the expanding direction, but that well-known subclasses are
unstable. These results should be compared to models with matter [16, 23, 22,
21] where spatially homogeneous solutions are known to be stable. It is also
important to note that the behavior of these models in the direction of the
singularity is not sensitive to the presence of most types of matter [3].

In the special case that the spacetime has spatial topology $T^{3}$, admits
two spacelike Killing vector fields (such spacetimes are called _$T^{2}$
-symmetric_), and satisfies a further technical condition (that the spacetime
is _polarised_) results of [13] show that there is a local attractor of the
Einstein Flow in the expanding direction. It is natural to ask whether the
condition that the spacetime be polarised is necessary. Do spacetimes on
$T^{3}$ with two spacelike Killing vector fields necessarily become
effectively polarised? Do they then flow to the polarised attractor?

We partially resolve these questions by analytic and numerical means. Our main
theorem states that solutions which are not polarised have the expanding
direction asymptotics of polarised solutions if they satisfy a certain weaker
condition: that one of the two conserved quantities of the flow vanishes. We
call such solutions _$B_{0}$_ or _$B=0$ solutions_. The conserved quantity $B$
vanishes for all polarised solutions; the set of $B=0$ solutions is of
codimension one in the space of all solutions in these coordinates while the
set of polarised solutions is of infinite codimension.

It was shown in [6] that $T^{2}$-symmetric vacuum spacetimes posess a global
foliation; all such Einstein Flows have a metric of the form

(1.1) $\displaystyle g=$ $\displaystyle
e^{\widehat{l}-V+4\tau}\left(-d\tau^{2}+e^{2(\rho-\tau)}d\theta^{2}\right)+e^{V}\left[dx+Qdy+(G+QH)d\theta\right]^{2}+e^{-V+2\tau}\left[dy+Hd\theta\right]^{2}$

where $\partial_{x}$ and $\partial_{y}$ are the Killing vector fields. The
area of the $\\{\partial_{x},\partial_{y}\\}$ orbit is $e^{2\tau}$, so the
singularity occurs as $\tau\to-\infty$ and the spacetime expands as
$\tau\to\infty$. Relative to the coordinates $t,P,\alpha,\lambda$ used in
[18], our quantities are given by

(1.4) $\displaystyle\begin{array}[]{rlrl}\tau:=&\log
t,&\rho:=&-\frac{1}{2}\log\alpha\\\ V:=&P+\log
t,&\widehat{l}:=&P+\frac{1}{2}\lambda-\frac{3}{2}\log t.\end{array}$

See the Appendix for a complete concordance of notations between the cited
papers and the present work. In the coordinates (1.1), the Einstein Flow is

(1.5) $\displaystyle\partial_{\tau}\left(e^{\rho}V_{\tau}\right)=$
$\displaystyle\partial_{\theta}\left(e^{2\tau-\rho}V_{\theta}\right)+e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)$
(1.6) $\displaystyle\partial_{\tau}\left(e^{\rho+2(V-\tau)}Q_{\tau}\right)=$
$\displaystyle\partial_{\theta}\left(e^{-\rho+2V}Q_{\theta}\right)$ (1.7)
$\displaystyle\widehat{l}_{\tau}+\rho_{\tau}+2=$
$\displaystyle\frac{1}{2}\left[V_{\tau}^{2}+e^{2(\tau-\rho)}V_{\theta}^{2}+e^{2(V-\tau)}\left(Q_{\tau}^{2}+e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\right]$
(1.8) $\displaystyle\rho_{\tau}=$ $\displaystyle K^{2}e^{\widehat{l}}$ (1.9)
$\displaystyle\widehat{l}_{\theta}=$ $\displaystyle
V_{\theta}V_{\tau}+e^{2(V-\tau)}Q_{\theta}Q_{\tau}.$

The last equation is the momentum constraint, and it is preserved by the
evolution equations. Equation (1.8) is a consequence of the constraints;
$\rho$ satisfies a wave equation similar to (1.5) which can be derived as a
consequence of (1.7) and (1.8), so we take equations (1.5) through (1.8) to be
the evolution equations instead. There are, in addition, evolution equations
for $G,H$, but these may be integrated once $V,Q,\rho,\widehat{l}$ have been
found, so these latter four functions are the ones of interest. As a
consequence of (1.5) and (1.6), there are two conserved quantities along the
flow:

(1.10) $\displaystyle A:=$
$\displaystyle\int_{S^{1}}e^{\rho}\left(V_{\tau}-e^{2(V-\tau)}Q_{\tau}Q\right)\,d\theta$
(1.11) $\displaystyle B:=$
$\displaystyle\int_{S^{1}}e^{\rho+2(V-\tau)}Q_{\tau}\,d\theta.$

The condition $Q\equiv 0$ is often imposed when studying these solutions in
the collapsing direction. Such solutions are called _polarised_. (Note that
all polarised solutions have $B=0$, but not all $B=0$ solutions are
polarised.) The constant $K$ is, without loss of generality, that “twist
constant” which cannot in general be made to vanish by a coordinate
transformation. The $T^{3}$ _Gowdy models_ [10] are those for which $K=0$.
$T^{2}$-symmetric spacetimes which are polarised, half polarised [11, 8], or
Gowdy have been studied extensively in the contracting direction (e.g. [1]).
We are here concerned only with the expanding direction.

KasnerPH$T^{2}$-symmetric$B_{0}$ Kasner$B_{0}$ PH$B_{0}$
$T^{2}$-symmetricpolarised Kasnerpolarised PHpolarised $T^{2}$-symmetric (a)
Figure 1. The classes of Einstein Flow solutions discussed in this paper, and
their inclusions. We have omitted the Gowdy models, which are not the focus of
this work.

The _Kasner_ models are those which are spatially homogeneous
($\widehat{l},V,Q$ are independent of $\theta$) and satisfy $K=0$. Let us note
that, in our coordinates, polarised Kasner solutions [12] take the form

(1.12) $\displaystyle
V=a\tau+b,\quad\widehat{l}=\left(\frac{1}{2}a^{2}-2\right)\tau+c$

for some constants $a,b,c\in\mathbb{R}$. The Gowdy models contain all Kasners,
and in the expanding direction the dynamics of Gowdy solutions are known [17],
[19] and appear to be very different from those of non-Gowdy solutions. Non-
Gowdy solutions such that $\widehat{l},V,Q$ are independent of $\theta$ are
called _pseudo-homogeneous_ or _PH_. This definition appears in [18], where it
is shown that the future asymptotics are of the form

(1.13) $\displaystyle\left|V-(a\tau+b)\right|\to
0,\quad\left|\widehat{l}-\left(\left[\frac{1}{2}a^{2}-2\right]\tau+c\right)\right|\to
0,\quad a\in(-2,2).$

That is, PH solutions have asymptotics of the same form as a Kasner solution,
but the value of $V_{\tau}$ at $\tau=\infty$ is restricted.

In contrast to these examples, in [13] the authors find a set of non-Gowdy,
polarised solutions such that

(1.14) $\displaystyle\left|V-b\right|\to 0,\quad\left|\widehat{l}-c\right|\to
0.$

The results in [18] and [13] are much more detailed than the above statements;
we give this simple description only to demonstrate that an instability
arises; no polarised Kasner or PH solutions can have future behavior of the
form (1.14). The relationships between these sets of solutions are given in
Figure 1.

Previous to this work, numerical simulations conducted by Berger [4, 5]
indicated that all $T^{2}$-symmetric solutions, without regard to the
polarisation or smallness conditions imposed in [13], flowed toward the
polarised attractor (1.14). In addition, in [18] it is shown that within the
neighborhood of each polarised PH solution is a polarised non-PH solution with
future asymptotics of the form (1.14).

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 (1.7) by $J$. The idea of the proof is to treat the
asymptotic regime of the solution as a wave equation for $V,Q$ coupled to an
ordinary differential equation (up to some error terms) for the means in the
$\theta$-direction of $e^{\rho},e^{\widehat{l}},J$. The smallness assumptions
are then used to guarantee that the errors decay, and so the behavior of the
means of $e^{\rho},e^{\widehat{l}},J$ approaches the behavior of the solution
of the ODE. When we use the word “attractor” here, we refer to the dynamics of
the $e^{\rho},e^{\widehat{l}},J$ system; a solution
$V^{\prime},Q^{\prime},\rho^{\prime},\widehat{l}^{\prime}$ is not generally a
proper attractor of the flow in the sense that

(1.15)
$\displaystyle\|V-V^{\prime}\|+\|Q-Q^{\prime}\|+\|\rho-\rho^{\prime}\|+\|\widehat{l}-\widehat{l}^{\prime}\|\longrightarrow
0$

in $C^{k}$ norm.

Our main theorem states roughly that the condition $B=0$ suffices to ensure
that a solution has polarised asymptotics if it begins sufficiently close to
the asymptotic regime. In the latter portion of the paper, we present
numerical evidence that the condition $B=0$ is necessary for the solution to
have polarised asymptotics and flow toward the polarised attractor. There
appears to be an attractor for solutions satisfying $B\neq 0$, which shares
some formal properties with the $B=0$ attractor. However, such solutions 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 with non-Gowdy, non-PH solutions; that is, solutions with $K\neq 0$
and $\int_{S^{1}}e^{\rho}\,d\theta$ unbounded as $\tau\to\infty$. In this
case, we shift $\widehat{l}$ by a constant

(1.16) $\displaystyle l:=\widehat{l}+\log(K^{2}/2)$

so that

(1.17)
$\displaystyle\widehat{l}_{\theta}=l_{\theta},\quad\widehat{l}_{\tau}=l_{\tau}\quad\text{and}\quad\rho_{\tau}=e^{l}.$

In the rest of the paper, we assume solutions are non-Gowdy and so change
variables to $l$ to avoid writing factors of $K$.

Before proceeding with the proof, it is important to note that there is some
very interesting work on the rescaling limits of certain expanding spacetimes
[14, 15]. The earlier of these works uses techniques from the study of Ricci
Flow to analyze the rescaling limits of CMC-foliated expanding spacetimes. The
latter 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 solutions considered in
this paper.

Acknowledgements. We are grateful to David Maxwell, Peng Lu, Paul T Allen,
Florian Beyer, Piotr Chruściel, Anna Sakovich, and Hans Ringström for
providing useful comments on various parts of this project.

The second and third authors were supported by NSF grants DMS-1263431 and
PHY-1306441. This article was in part written during a stay of the third
author at the Erwin Schrödinger Institute in Vienna during the thematic
program ‘Geometry and Relativity’. This paper incorporates material that
previously appeared in the third author’s dissertation which was submitted to
the Department of Mathematics in partial fulfillment of the requirements for
the degree of Doctor of Philosophy at the University of Oregon.

##  2\. Preliminary Computations

Before proceeding with the proof of the main theorem, we define the energy
under consideration and calculate its evolution. It is useful to have notation
for the mean of a function in the $\theta$-direction.

######  Definition 2.1 ($S^{1}$-mean).

For $f\colon S^{1}\to\mathbb{R}$, let

(2.1) $\displaystyle\langle f\rangle:=\int_{S^{1}}f(\theta)\,d\theta.$

Note that in [13], the authors choose to use the volume form
$e^{\rho}\,d\theta$ for their mean. Our choice is almost identical to that
used in [18], but we normalize so that $\int_{S^{1}}d\theta=1$. Either choice
would suffice.

Define the following energy

(2.2) $\displaystyle J:=$
$\displaystyle\frac{1}{2}\left[V_{\tau}^{2}+e^{2(\tau-\rho)}V_{\theta}^{2}+e^{2(V-\tau)}\left(Q_{\tau}^{2}+e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\right],$
(2.3) $\displaystyle E:=$
$\displaystyle\int_{S^{1}}e^{\rho-2\tau}J\,d\theta=\frac{1}{2}\int_{S^{1}}e^{\rho-2\tau}V_{\tau}^{2}+e^{-\rho}V_{\theta}^{2}+e^{2V-4\tau+\rho}Q_{\tau}^{2}+e^{2(V-\tau)-\rho}Q_{\theta}^{2}\,d\theta,$

and the $S^{1}$-volume

(2.4) $\displaystyle\Pi:=$ $\displaystyle\langle
e^{\rho}\rangle=\int_{S^{1}}e^{\rho}\,d\theta.$

Note that equation (1.7) now reads $l_{\tau}+\rho_{\tau}+2=J$. We use the
terms $V$-energy and $Q$-energy loosely to refer to
$V_{\tau}^{2}+e^{2(\tau-\rho)}V_{\theta}^{2}$ and
$e^{2(V-\tau)}\left(Q_{\tau}^{2}+e^{2(\tau-\rho)}Q_{\theta}^{2}\right)$,
respectively. One may compute using the evolution equations for $V$ and $Q$
that

(2.5) $\displaystyle\partial_{\tau}\left(e^{\rho}J\right)=$ $\displaystyle
2e^{\rho}J-\rho_{\tau}e^{\rho}J-e^{\rho}V_{\tau}^{2}-e^{2V-\rho}Q_{\theta}^{2}+\partial_{\theta}\left(e^{2\tau-\rho}V_{\theta}V_{\tau}+e^{2V-\rho}Q_{\theta}Q_{\tau}\right)$

so the energy $E$ evolves by

(2.6) $\displaystyle
E_{\tau}=\int_{S^{1}}-\rho_{\tau}e^{\rho-2\tau}J-e^{\rho-2\tau}V_{\tau}^{2}-e^{2(V-\tau)-\rho}Q_{\theta}^{2}\,d\theta.$

The terms $-e^{\rho-2\tau}V_{\tau}^{2}-e^{2(V-\tau)-\rho}Q_{\theta}^{2}$
appearing here are undesirable for proving energy inequalities. This
necessitates the modification of $E$ by a term which trades $V_{\tau}^{2}$ for
$V_{\theta}^{2}$. This is the main topic of Section 3.

##  3\. corrections and their bounds

Define the correction

(3.1) $\displaystyle\Lambda:=$
$\displaystyle\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}\left(V-\langle
V\rangle-1\right)e^{\rho}\,d\theta.$

Corrections to the energy of essentially this form were used previously in the
Gowdy case [17] and in the existing results on $T^{2}$-symmetric spacetimes
[18, 13]. Our definition differs only slightly from those previously 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, which may be bounded. At the same time, the
correction has better decay properties than the energy, and so we are able to
draw conclusions about the energy in the expanding direction.

To trade $V_{\tau}^{2}$ for $V_{\theta}^{2}$ and $Q_{\tau}^{2}$ for
$Q_{\theta}^{2}$, it would be more natural to consider the corrections

(3.2) $\displaystyle\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}\left(V-\langle
V\rangle\right)e^{\rho}\,d\theta,\quad\text{and}\quad\frac{1}{2}e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)}Q_{\tau}\left(Q-\langle
Q\rangle\right)e^{\rho}\,d\theta$

separately as other authors have done. Then, by differentiating the
$Q$-correction one would hope to obtain terms of the form
$Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}$, perhaps with a leading factor.
Our definition exploits the fact that (1.5) contains exactly the expression
that we would like to obtain from the $Q$-correction.

######  Lemma 3.1.

Consider a non-Gowdy $T^{2}$-symmetric Einstein Flow. The correction defined
in (3.1) evolves by

(3.3) $\displaystyle\partial_{\tau}\Lambda\ =$
$\displaystyle-2\Lambda-\frac{1}{2}\left\langle
e^{-\rho}V_{\theta}^{2}\right\rangle+\frac{1}{2}\left\langle
e^{\rho-2\tau}V_{\tau}^{2}\right\rangle-\frac{1}{2}\left\langle
V_{\tau}\right\rangle\left\langle e^{\rho-2\tau}V_{\tau}\right\rangle$ (3.4)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle-1\right)\,d\theta.$

###### Proof.

We compute straightforwardly using equations (1.5), (1.6) and integration by
parts. From the definition of $\Lambda$ we have

(3.5) $\displaystyle\partial_{\tau}\Lambda=$
$\displaystyle-2\Lambda+\frac{1}{2}e^{-2\tau}\int_{S^{1}}(e^{\rho}V_{\tau})_{\tau}\left(V-\left\langle
V\right\rangle-1\right)\,d\theta+\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}\partial_{\tau}\left(V-\left\langle
V\right\rangle-1\right)e^{\rho}\,d\theta$ (3.6) $\displaystyle=$
$\displaystyle-2\Lambda$ (3.7)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}\left[e^{2\tau}\left(e^{-\rho}V_{\theta}\right)_{\theta}+e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\right]\left(V-\left\langle
V\right\rangle-1\right)\,d\theta$ (3.8)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}\partial_{\tau}\left(V-\left\langle
V\right\rangle-1\right)e^{\rho}\,d\theta$ (3.9) $\displaystyle=$
$\displaystyle-2\Lambda+\frac{1}{2}e^{-2\tau}\int_{S^{1}}-e^{2\tau}e^{-\rho}V_{\theta}^{2}\,d\theta+\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}^{2}e^{\rho}\,d\theta$
(3.10)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle-1\right)\,d\theta$ (3.11) $\displaystyle-\left\langle
V_{\tau}\right\rangle\left\langle\frac{1}{2}e^{\rho-2\tau}V_{\tau}\right\rangle$

which completes the proof. ∎

We modify the energy $E$ by $\Lambda$. It is then desirable to know that
$\Lambda$ has better decay than $E$. To that end, note that

(3.12) $\displaystyle\|V-\langle
V\rangle\|_{C^{0}}\lesssim\int_{S^{1}}|V_{\theta}|\,d\theta\leq\left(\int_{S^{1}}V_{\theta}^{2}e^{-\rho}\,d\theta\right)^{1/2}\Pi^{1/2}\leq(\Pi
E)^{1/2}.$

As is standard (cf. [20]), we use the notation $f\lesssim h$ to mean that
there is a universal constant $C>0$ such that $f\leq Ch$.

One finds the following bound using Hölder’s Inequality.

######  Lemma 3.2 ([18], Lemma 72).

Consider a non-Gowdy $T^{2}$-symmetric Einstein Flow. Then

(3.13)
$\displaystyle\left|\Lambda+\left\langle\frac{1}{2}e^{\rho-2\tau}V_{\tau}\right\rangle\right|=\left|\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}\left(V-\langle
V\rangle\right)e^{\rho}\,d\theta\right|\lesssim e^{-\tau}\Pi E$

For the following bound on the $Q$ correction, cf. [18] Lemma 73, where the
author assumes a uniform bound on $\Pi$ which we don’t assume here. The proof
is essentially the same.

######  Lemma 3.3.

For any a non-Gowdy $T^{2}$-symmetric Einstein Flow,

(3.14)
$\displaystyle\left|e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)}Q_{\tau}\left(Q-\langle
Q\rangle\right)e^{\rho}\,d\theta\right|\lesssim e^{-\tau}e^{2(\Pi E)^{1/2}}\Pi
E$

###### Proof.

Note that we have already bounded $\|V-\langle V\rangle\|_{C_{0}}$ in equation
(3.12), and so we may commute out factors of $e^{V}$ to obtain

(3.15) $\displaystyle\left\|e^{V}\left(Q-\langle
Q\rangle\right)\right\|_{C^{0}}=$ $\displaystyle\left\|e^{V-\langle
V\rangle+\langle V\rangle}\left(Q-\langle Q\rangle\right)\right\|_{C^{0}}$
(3.16) $\displaystyle=$ $\displaystyle e^{\|V-\langle
V\rangle\|_{C^{0}}}e^{\langle V\rangle}\left\|Q-\langle
Q\rangle\right\|_{C^{0}}$ (3.17) $\displaystyle\leq$ $\displaystyle
e^{2\|V-\langle
V\rangle\|_{C^{0}}}\left(\int_{S^{1}}e^{2V}Q_{\theta}^{2}e^{-\rho}\,d\theta\right)^{1/2}\Pi^{1/2}$
(3.18) $\displaystyle\leq$ $\displaystyle e^{2\|V-\langle
V\rangle\|_{C^{0}}}e^{\tau}\left(E\Pi\right)^{1/2}$

via Hölder’s inequality. So we may compute, using the bound on $\|V-\langle
V\rangle\|_{C_{0}}$, Hölder’s inequality, and the definition of $E$

(3.19)
$\displaystyle\left|e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)}Q_{\tau}\left(Q-\langle
Q\rangle\right)e^{\rho}\,d\theta\right|\lesssim$ $\displaystyle
e^{-4\tau}\left\|e^{V}\left(Q-\langle
Q\rangle\right)\right\|_{C^{0}}\left|\int_{S^{1}}e^{V}Q_{\tau}e^{\rho}\,d\theta\right|$
(3.20) $\displaystyle\lesssim$ $\displaystyle e^{2\|V-\langle
V\rangle\|_{C^{0}}}e^{-3\tau}E^{1/2}\Pi^{1/2}\left|\int_{S^{1}}e^{V}Q_{\tau}e^{\rho}\,d\theta\right|$
(3.21) $\displaystyle\leq$ $\displaystyle e^{2\|V-\langle
V\rangle\|_{C^{0}}}e^{-\tau}E\Pi$ (3.22) $\displaystyle\lesssim$
$\displaystyle e^{-\tau}e^{2(\Pi E)^{1/2}}\Pi E.\qed$

We only need the $Q$ correction for the following identity, which follows
directly from the definitions of the conserved quantities $A,B$:

(3.23) $\displaystyle\left\langle e^{\rho}V_{\tau}\right\rangle=A+B\langle
Q\rangle+\int_{S^{1}}e^{2(V-\tau)}Q_{\tau}\left(Q-\langle
Q\rangle\right)e^{\rho}\,d\theta.$

For $B_{0}$ solutions, however, we use the bound on the $Q$ correction to
obtain the following bound

(3.24) $\displaystyle\left|\left\langle
e^{\rho-2\tau}V_{\tau}\right\rangle\right|-e^{-2\tau}|A|\lesssim
e^{-\tau}e^{2(\Pi E)^{1/2}}\Pi E$

which together with (3.13) yields the desired estimate on the correction.

######  Proposition 3.4.

For any a non-Gowdy, $B_{0}$ $T^{2}$-symmetric Einstein Flow,

(3.25) $\displaystyle\left|\Lambda\right|-\frac{e^{-2\tau}}{2}|A|\lesssim
e^{-\tau}\left(1+e^{2(\Pi E)^{1/2}}\right)\Pi E.$

The correction $\Lambda$ introduces significant new error terms after
differentiation. However, these terms have good bounds, and the modified
energy $E+\Lambda$ has significantly better properties upon comparison to $E$
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, $\Pi$ and $E$ satisfy an ODE.
While this is true asymptotically, it is more useful to compute with an energy
which has been modified by the correction.

One computes that

(4.1) $\displaystyle\left(E+\Lambda\right)_{\tau}=$
$\displaystyle\int_{S^{1}}-e^{\rho-2\tau}\rho_{\tau}J-e^{\rho-2\tau}V_{\tau}^{2}-e^{2(V-\tau)-\rho}Q_{\theta}^{2}\,d\theta-2\Lambda$
(4.2)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}-e^{2\tau}e^{-\rho}V_{\theta}^{2}\,d\theta+\frac{1}{2}e^{-2\tau}\int_{S^{1}}V_{\tau}^{2}e^{\rho}\,d\theta$
(4.3)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle-1\right)\,d\theta$ (4.4) $\displaystyle-\left\langle
V_{\tau}\right\rangle\left\langle\frac{1}{2}e^{\rho-2\tau}V_{\tau}\right\rangle$
(4.5) $\displaystyle=$
$\displaystyle-\left(1+\frac{\Pi_{\tau}}{\Pi}\right)\left(E+\Lambda\right)+\left(\frac{\Pi_{\tau}}{\Pi}E-\int_{S^{1}}e^{\rho-2\tau}\rho_{\tau}J\,d\theta\right)-\left(1-\frac{\Pi_{\tau}}{\Pi}\right)\Lambda$
(4.6)
$\displaystyle+\frac{1}{2}e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle\right)\,d\theta-\left\langle
V_{\tau}\right\rangle\left\langle\frac{1}{2}e^{\rho-2\tau}V_{\tau}\right\rangle.$

The leading term on the right leads us to the ansatz that
$\Pi\left(E+\Lambda\right)$ (and so $\Pi E$) should decay like $e^{-\tau}$.
Accordingly, define the corrected, normalized energy
$H:=\Pi\left(E+\Lambda\right)$. One computes that

(4.7) $\displaystyle\partial_{\tau}\left(e^{\tau}H\right)=$ $\displaystyle
e^{\tau}H+e^{\tau}\Pi_{\tau}\left(E+\Lambda\right)+e^{\tau}\Pi\left(E+\Lambda\right)_{\tau}$
(4.8) $\displaystyle=$ $\displaystyle
e^{\tau}\Pi\left(\left(E+\Lambda\right)\left(1+\frac{\Pi_{\tau}}{\Pi}\right)+\left(E+\Lambda\right)_{\tau}\right)$
(4.9) $\displaystyle=$ $\displaystyle
e^{\tau}\Pi\left[\left(\frac{\Pi_{\tau}}{\Pi}E-\int_{S^{1}}e^{\rho-2\tau}\rho_{\tau}J\,d\theta\right)-\left(1-\frac{\Pi_{\tau}}{\Pi}\right)\Lambda\right.$
(4.10)
$\displaystyle\left.+\frac{1}{2}e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle\right)\,d\theta-\left\langle
V_{\tau}\right\rangle\left\langle\frac{1}{2}e^{\rho-2\tau}V_{\tau}\right\rangle\right]$

The ansatz in the local stability proof is that $e^{\tau}H$ is of constant
order. The proof is via a bootstrap argument, where we bound all of the terms
of $\partial_{\tau}(e^{\tau}H)$ in terms of $\Pi,E,H$ and $\tau$. The
following Proposition deals with each of these error terms.

######  Proposition 4.1.

Consider the evolution of a $B_{0}$ solution with initial data given at time
$\tau=s_{0}$. Let $\rho_{0}:=\min_{\theta\in S^{1}}\rho(\theta,s_{0})$. The
following estimates hold.

(4.11)
$\displaystyle\left|\frac{\Pi_{\tau}}{\Pi}E-\int_{S^{1}}e^{\rho-2\tau}\rho_{\tau}J\,d\theta\right|\lesssim$
$\displaystyle E\int_{S^{1}}e^{\rho-\tau}\rho_{\tau}J\,d\theta,$ (4.12)
$\displaystyle\left|\left(1-\frac{\Pi_{\tau}}{\Pi}\right)\Lambda\right|\lesssim$
$\displaystyle|A|e^{-2\tau}\left(1+\frac{\Pi_{\tau}}{\Pi}\right)+e^{-\tau}\left(1+e^{2(\Pi
E)^{1/2}}\right)\left(\Pi+\Pi_{\tau}\right)E,$ (4.13)
$\displaystyle\left|\left\langle
V_{\tau}\right\rangle\left\langle\frac{1}{2}e^{\rho-2\tau}V_{\tau}\right\rangle\right|\lesssim$
$\displaystyle e^{-\rho_{0}/2}e^{-\tau}\left(|A|+e^{\tau}e^{2(\Pi E)^{1/2}}\Pi
E\right)E^{1/2},$

and

(4.14)
$\displaystyle\left|e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle\right)\,d\theta\right|\lesssim$
$\displaystyle\Pi^{1/2}E^{3/2}.$

###### Proof.

For (4.11), using Young’s inequality, we note that

(4.15) $\displaystyle|l_{\theta}|\leq$
$\displaystyle|V_{\tau}V_{\theta}|+|e^{V-\tau}Q_{\tau}e^{V-\tau}Q_{\theta}|$
(4.16) $\displaystyle=$
$\displaystyle|e^{(\rho-\tau)/2}V_{\tau}e^{-(\rho-\tau)/2}V_{\theta}|+|e^{V-\tau}e^{(\rho-\tau)/2}Q_{\tau}e^{V-\tau}e^{-(\rho-\tau)/2}Q_{\theta}|$
(4.17) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\left[e^{\rho-\tau}V_{\tau}^{2}+e^{\tau-\rho}V_{\theta}^{2}+e^{2(V-\tau)}e^{\rho-\tau}Q_{\tau}^{2}+e^{2(V-\tau)}e^{\tau-\rho}Q_{\theta}^{2}\right]$
(4.18) $\displaystyle=$ $\displaystyle e^{\rho-\tau}J.$

Thus we may use the Poincaré inequality to compute that

(4.19)
$\displaystyle\left|\frac{\Pi_{\tau}}{\Pi}E-\int_{S^{1}}e^{\rho-2\tau}\rho_{\tau}J\,d\theta\right|=$
$\displaystyle\Pi^{-1}\left|\Pi_{\tau}E-\Pi\int_{S^{1}}e^{\rho-2\tau}\rho_{\tau}J\,d\theta\right|$
(4.20) $\displaystyle=$
$\displaystyle\Pi^{-1}\left|\int_{S^{1}}\int_{S^{1}}e^{\rho(\phi)}e^{\rho(\theta)-2\tau}J(\theta)\left(\rho_{\tau}(\phi)-\rho_{\tau}(\theta)\right)\,d\phi
d\theta\right|$ (4.21) $\displaystyle\leq$
$\displaystyle\Pi^{-1}\left|\int_{S^{1}}\int_{S^{1}}e^{\rho(\phi)}e^{\rho(\theta)-2\tau}J(\theta)\sup_{a,b\in
S^{1}}|\rho_{\tau}(a)-\rho_{\tau}(b)|\,d\phi d\theta\right|$ (4.22)
$\displaystyle=$
$\displaystyle\Pi^{-1}\Pi\int_{S^{1}}e^{\rho(\theta)-2\tau}J(\theta)\,d\theta\sup_{a,b\in
S^{1}}|\rho_{\tau}(a)-\rho_{\tau}(b)|$ (4.23) $\displaystyle\lesssim$
$\displaystyle E\int_{S^{1}}\rho_{\tau}|l_{\theta}|\,d\theta$ (4.24)
$\displaystyle\leq$ $\displaystyle
E\int_{S^{1}}\rho_{\tau}e^{\rho-\tau}J\,d\theta.$

Inequality (4.12) follows directly from inequality (3.25). To prove (4.14), we
first commute out the $V$-mean.

(4.25)
$\displaystyle\left|e^{-2\tau}\int_{S^{1}}e^{2(V-\tau)+\rho}\left(Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\left(V-\left\langle
V\right\rangle\right)\,d\theta\right|$ (4.26) $\displaystyle\leq$
$\displaystyle e^{-2\tau}\|V-\langle
V\rangle\|_{C^{0}}\int_{S^{1}}e^{2(V-\tau)+\rho}\left|Q_{\tau}^{2}-e^{2(\tau-\rho)}Q_{\theta}^{2}\right|\,d\theta$
(4.27) $\displaystyle\lesssim$ $\displaystyle e^{-2\tau}(\Pi
E)^{1/2}\int_{S^{1}}e^{\rho}e^{2(V-\tau)}\left(Q_{\tau}^{2}+e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\,d\theta$
(4.28) $\displaystyle\lesssim$ $\displaystyle(\Pi
E)^{1/2}\int_{S^{1}}e^{\rho-2\tau}J\,d\theta$ (4.29) $\displaystyle=$
$\displaystyle\Pi^{1/2}E^{3/2}.$

Lastly, for (4.13) recall that $\rho$ is increasing and compute that

(4.30) $\displaystyle\left|\left\langle
V_{\tau}\right\rangle\right|\leq\left(\int_{S^{1}}V_{\tau}^{2}e^{\rho}\,d\theta\right)^{1/2}\left(\int_{S^{1}}e^{-\rho}\,d\theta\right)^{1/2}\leq
e^{-\rho_{0}/2}e^{\tau}\left(\int_{S^{1}}V_{\tau}^{2}e^{\rho-2\tau}\,d\theta\right)^{1/2}\lesssim
e^{-\rho_{0}/2}e^{\tau}E^{1/2}$

and use (3.24). This completes the proof. ∎

Now that we have an energy satisfying a good differential equation with good
bounds on the error, we proceed to the linearization.

##  5\. Linearization

In [13], the authors present an argument that certain asymptotic rates of
$\Pi,E$ should be preferred, based on the assumption that $e^{\tau}H$ should
be of constant order. In this section we briefly summarize that argument as it
appears in our context.

######  Definition 5.1.

Let $Y:=\left\langle e^{l+\rho+2\tau}\right\rangle$.

This quantity has been previously considered; see [7] where (modulo factors of
$e^{\tau}$) it is called the “twist potential.”

Note that we have defined $Y$ so that $Y_{\tau}=\langle
e^{l+\rho+2\tau}(l_{\tau}+\rho_{\tau}+2)\rangle=\langle
e^{l+\rho+2\tau}J\rangle$. We want to form a system of ordinary differential
equations from the means, however. So we distribute the integral over the
product, introducing the error term $\Omega$. One computes

(5.1) $\displaystyle\Pi_{\tau}=$ $\displaystyle e^{-2\tau}Y$ (5.2)
$\displaystyle Y_{\tau}=$ $\displaystyle e^{2\tau}EY\Pi^{-1}+\Omega$

where

(5.3) $\displaystyle\Omega:=\left\langle
e^{l+\rho+2\tau}J\right\rangle-e^{2\tau}EY\Pi^{-1}$

is an error term satisfying

(5.4) $\displaystyle\left|\Omega\right|\leq e^{4\tau}E\left\langle
e^{l+\rho-\tau}J\right\rangle=e^{\tau}EY_{\tau}.$

Note that our quantity $E$ contains the terms $Q_{\theta}$ and $Q_{\tau}$, and
so is not identical to the energy in [13]. Nonetheless, the quantities
$\Pi,Y,E$ satisfy similar relations to the relations that LeFloch and
Smulevici’s quantities do. Normalizing, we compute that

(5.5) $\displaystyle\partial_{\tau}\left(e^{-\tau}H^{-1/2}\Pi\right)=$
$\displaystyle
e^{-\tau}H^{-1/2}\Pi_{\tau}-e^{-\tau}H^{-1/2}\Pi-\frac{1}{2}e^{-\tau}H^{-1/2}\Pi\frac{H_{\tau}}{H}$
(5.6) $\displaystyle=$
$\displaystyle\left(e^{-3\tau}H^{-1/2}Y\right)+\left(e^{-\tau}H^{-1/2}\Pi\right)\left(-1-\frac{1}{2}\frac{H_{\tau}}{H}\right)$
(5.7) $\displaystyle\partial_{\tau}\left(e^{-3\tau}H^{-1/2}Y\right)=$
$\displaystyle
e^{-3\tau}H^{-1/2}Y_{\tau}-3e^{-3\tau}H^{-1/2}Y-\frac{1}{2}e^{-3\tau}H^{-1/2}Y\frac{H_{\tau}}{H}$
(5.8) $\displaystyle=$ $\displaystyle
e^{-3\tau}H^{-1/2}\left(e^{2\tau}EY\Pi^{-1}+\Omega\right)+\left(e^{-3\tau}H^{-1/2}Y\right)\left(-3-\frac{1}{2}\frac{H_{\tau}}{H}\right)$
(5.9) $\displaystyle=$
$\displaystyle\frac{\left(e^{-3\tau}H^{-1/2}Y\right)}{\Pi^{2}}e^{2\tau}\Pi
E+\left(e^{-3\tau}H^{-1/2}Y\right)\left(-3-\frac{1}{2}\frac{H_{\tau}}{H}\right)+e^{-3\tau}H^{-1/2}\Omega$
(5.10) $\displaystyle=$
$\displaystyle\frac{\left(e^{-3\tau}H^{-1/2}Y\right)}{\left(e^{-\tau}H^{-1/2}\Pi\right)^{2}}\frac{\Pi
E}{H}+\left(e^{-3\tau}H^{-1/2}Y\right)\left(-3-\frac{1}{2}\frac{H_{\tau}}{H}\right)+e^{-3\tau}H^{-1/2}\Omega$
(5.11) $\displaystyle=$
$\displaystyle\frac{\left(e^{-3\tau}H^{-1/2}Y\right)}{\left(e^{-\tau}H^{-1/2}\Pi\right)^{2}}+\left(e^{-3\tau}H^{-1/2}Y\right)\left(-3-\frac{1}{2}\frac{H_{\tau}}{H}\right)+e^{-3\tau}H^{-1/2}\Omega$
(5.12)
$\displaystyle+\frac{\left(e^{-3\tau}H^{-1/2}Y\right)}{\left(e^{-\tau}H^{-1/2}\Pi\right)^{2}}\left(\frac{\Pi
E}{H}-1\right).$

We insert our ansätze that $\frac{H_{\tau}}{H}\to-1$,
$e^{-3\tau}H^{-1/2}\Omega\to 0$, and $\left(\frac{\Pi E}{H}-1\right)\to 0$, to
obtain the ODE

(5.13) $\displaystyle\partial_{\tau}{\overline{c}}=$
$\displaystyle{\overline{d}}+{\overline{c}}\left(-\frac{1}{2}\right)$ (5.14)
$\displaystyle\partial_{\tau}{\overline{d}}=$
$\displaystyle\frac{{\overline{d}}}{{\overline{c}}^{2}}+{\overline{d}}\left(-\frac{5}{2}\right)$

which has a fixed point at

(5.15)
$\displaystyle{\overline{c}}=\frac{2}{\sqrt{10}},\quad{\overline{d}}=\frac{1}{\sqrt{10}}.$

So we conjecture that the quantities

(5.16) $\displaystyle
c:=\frac{\Pi}{e^{\tau}\sqrt{H}}-\frac{2}{\sqrt{10}},\quad
d:=\frac{Y}{e^{3\tau}\sqrt{H}}-\frac{1}{\sqrt{10}}$

decay and compute the evolution of these quantities using (5.1) and (5.2). We
find that

(5.25) $\displaystyle\partial_{\tau}\left(\begin{array}[]{l}c\\\
d\end{array}\right)=$ $\displaystyle\left(\begin{array}[]{ll}-1/2&1\\\
-5/2&0\end{array}\right)\left(\begin{array}[]{l}c\\\
d\end{array}\right)-\frac{1}{2}\partial_{\tau}\log\left(e^{\tau}H\right)\left(\begin{array}[]{l}c\\\
d\end{array}\right)$ (5.30)
$\displaystyle-\frac{1}{2}\partial_{\tau}\log\left(e^{\tau}H\right)\left(\begin{array}[]{l}\frac{2}{\sqrt{10}}\\\
\frac{1}{\sqrt{10}}\end{array}\right)+\left(\begin{array}[]{l}0\\\
\frac{f(d,c)}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}+\left(\displaystyle\frac{\Pi
E}{H}-1\right)\frac{d+\frac{1}{\sqrt{10}}}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}+\displaystyle\frac{\Omega}{e^{3\tau}H^{1/2}}\end{array}\right)$

where $f(c,d)=\frac{10c-10d+3\sqrt{10}}{4}c^{2}-\sqrt{10}cd$ has vanishing
linear part. Let

(5.35)
$\displaystyle\widetilde{\Omega}:=-\frac{1}{2}\partial_{\tau}\log\left(e^{\tau}H\right)\left(\begin{array}[]{l}\frac{2}{\sqrt{10}}\\\
\frac{1}{\sqrt{10}}\end{array}\right)+\left(\begin{array}[]{l}0\\\
\frac{f(d,c)}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}+\left(\displaystyle\frac{\Pi
E}{H}-1\right)\frac{d+\frac{1}{\sqrt{10}}}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}+\displaystyle\frac{\Omega}{e^{3\tau}H^{1/2}}\end{array}\right)$

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 $B_{0}$ $T^{2}$-symmetric solution. Provided the
corrected energy $H$ is positive, one has for $s\geq s_{0}$

(5.40) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{(s_{0}-s)/4}\left(\frac{e^{s_{0}}H(s_{0})}{e^{s}H(s)}\right)^{1/2}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s_{0})+\int^{s}_{s_{0}}e^{(\tau-s)/4}\left(\frac{e^{\tau}H(\tau)}{e^{s}H(s)}\right)^{1/2}|\omega(\tau)|\,d\tau,$

where

(5.41) $\displaystyle|\omega|\lesssim\left|\widetilde{\Omega}\right|.$

Quickly note a bound on one of the terms appearing in $\widetilde{\Omega}$.

######  Lemma 5.2.

Consider the evolution of a $B_{0}$ $T^{2}$-symmetric solution. The following
estimate holds.

(5.42) $\displaystyle\left|e^{-3\tau}H^{-1/2}\Omega\right|\lesssim$
$\displaystyle e^{-2\tau}|H|^{-1/2}EY_{\tau}.$

The proof of this lemma proceeds in the same manner as the proof of inequality
(4.11). The remaining three terms in $\widetilde{\Omega}$ are estimated
directly. In the next section, we perform a bootstrap argument to bound these
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 means of the energy, the $S^{1}$ volume, and their
derivatives. We then use a bootstrap argument to show that these assumptions
are improved. The reason for obtaining the estimates of Lemma 4.1 is to bound
the evolution of the corrected energy $H$. Let us discuss how that proof goes.
We have computed $\partial_{\tau}\left(e^{\tau}H\right)$ in equation (4.9).
Note that we may bound the right side of that equation by an expression of the
form

(6.1)
$\displaystyle\left|\partial_{\tau}\left(e^{\tau}H\right)\right|\lesssim$
$\displaystyle e^{\tau}\Pi EF+\widetilde{F}=e^{\tau}H\frac{\Pi
E}{H}F+\widetilde{F}$

where, using the results of Lemma 4.1 we can write

(6.2) $\displaystyle F:=$
$\displaystyle\int_{S^{1}}e^{\rho-\tau}\rho_{\tau}J\,d\theta+e^{-\tau}\left(1+e^{2(\Pi
E)^{1/2}}\right)\left(\Pi+\Pi_{\tau}\right)+(\Pi
E)^{1/2}+e^{-\rho(s_{0})/2}e^{2(\Pi E)^{1/2}}\Pi E^{1/2}$

and

(6.3)
$\displaystyle\widetilde{F}:=|A|\Pi\left(e^{-\tau}\left(1+\frac{\Pi_{\tau}}{\Pi}\right)+e^{-\rho_{0}/2}E^{1/2}\right).$

Note that $F$ and $\widetilde{F}$ are nonnegative. We are then concerned with
the quantities

(6.4)
$\displaystyle\int_{s_{0}}^{\infty}F(\tau)\,d\tau,\quad\text{and}\quad\int_{s_{0}}^{\infty}\widetilde{F}(\tau)\,d\tau$

which bound the evolution of $e^{\tau}H$ in the bootstrap proof.

First, however, we need the following version of Grönwall’s Lemma, the proof
of which is straightforward.

######  Lemma 6.1 (Grönwall’s Inequality).

Let $\alpha,\beta,f$ be nonnegative smooth functions on the interval
$[s_{0},s]$. Suppose $f$ satisfies the differential inequality

(6.5) $\displaystyle|f^{\prime}|\leq\alpha+\beta f.$

Then

(6.6)
$\displaystyle\left|f(s)-f(s_{0})\right|\leq-f(s_{0})+\left(f(s_{0})+\int_{s_{0}}^{s}\alpha(t)\,dt\right)\exp\left(\int_{s_{0}}^{s}\beta(t)\,dt\right).$

######  Lemma 6.2.

There exist constants $\epsilon,C_{1}>0$, $M>1$, a time $s_{0}>0$ depending on
$\epsilon$, and an open set of $B_{0}$ Einstein Flows satisfying the following
conditions at time $\tau=s_{0}$:

(6.7) $\displaystyle|A|<$ $\displaystyle 1$ (6.8)
$\displaystyle\rho_{0}:=\inf_{S^{1}}\rho>$ $\displaystyle 0$ (6.9)
$\displaystyle|c|<$ $\displaystyle\epsilon$ (6.10) $\displaystyle|d|<$
$\displaystyle\epsilon$ (6.11) $\displaystyle\left|\frac{\Pi E}{H}-1\right|<$
$\displaystyle 1,$ (6.12) $\displaystyle\frac{1}{2}\epsilon^{-1}<e^{s_{0}}<$
$\displaystyle 2\epsilon^{-1}$ (6.13) $\displaystyle
e^{s_{0}}H(s_{0})+C_{1}\epsilon^{1/2}<M\epsilon e^{s_{0}}$ (6.14)
$\displaystyle 0<\frac{1}{M}<$ $\displaystyle
e^{s_{0}}H(s_{0})-C_{1}\epsilon^{1/2}.$

Furthermore, for all $\tau\in[s_{0},\infty)$, the following weaker estimates
hold:

(6.15) $\displaystyle|c|<$ $\displaystyle\epsilon^{1/4}$ (6.16)
$\displaystyle|d|<$ $\displaystyle\epsilon^{1/4}$ (6.17)
$\displaystyle\left|\frac{\Pi E}{H}-1\right|<$ $\displaystyle 3$ (6.18)
$\displaystyle\frac{1}{2}\left(e^{s_{0}}H(s_{0})-C_{1}\epsilon^{1/2}\right)<e^{\tau}H(\tau)<$
$\displaystyle 2\left(e^{s_{0}}H(s_{0})+C_{1}\epsilon^{1/2}\right)$

######  Remark 1.

Assumptions (6.7) and (6.8) are not strictly needed. One could omit these
assumptions and instead gain terms involving $A,\rho_{0}$ in inequalities
(6.13), (6.14), and (6.18). We have added these assumptions just to simplify
the notation.

The technique of proof is a straightforward “open closed” argument:

  1. (1)

Suppose estimates (6.15) to (6.18) are satisfied for $\tau\in[s_{0},s)$.

  2. (2)

We improve each of the five estimates (6.15) to (6.18) at $\tau=s$ by choosing
$\epsilon$ small.

###### Proof.

Initial Estimates:

From assumptions (6.15) to (6.18), we have that

(6.19) $\displaystyle e^{-\tau}\lesssim H\lesssim\epsilon e^{s_{0}-\tau},$

and

(6.20)
$\displaystyle\left|\frac{\Pi}{e^{\tau}\sqrt{H}}-\frac{2}{\sqrt{10}}\right|=\left|c\right|<\epsilon^{1/4},\quad\left|\frac{Y}{e^{3\tau}\sqrt{H}}-\frac{1}{\sqrt{10}}\right|=\left|d\right|<\epsilon^{1/4}$

so

(6.21) $\displaystyle\Pi\lesssim$
$\displaystyle\left(\frac{2}{\sqrt{10}}+\epsilon^{1/4}\right)e^{\tau}H^{1/2}\leq\left(\frac{2}{\sqrt{10}}+\epsilon^{1/4}\right)\epsilon^{1/2}e^{(s_{0}+\tau)/2}\lesssim\epsilon^{1/2}e^{s_{0}/2}e^{\tau/2},$
(6.22) $\displaystyle e^{2\tau}\Pi_{\tau}=Y\lesssim$
$\displaystyle\left(\frac{1}{\sqrt{10}}+\epsilon^{1/4}\right)e^{3\tau}H^{1/2}\lesssim\left(\frac{1}{\sqrt{10}}+\epsilon^{1/4}\right)\epsilon^{1/2}e^{(5\tau+s_{0})/2}\lesssim\epsilon^{1/2}e^{s_{0}/2}e^{5\tau/2}.$

Note that (6.17) implies that $\Pi E\lesssim H$ on this interval, which
implies that

(6.23) $\displaystyle\Pi E\lesssim\epsilon
e^{s_{0}-\tau}<\epsilon,\quad\text{and}\quad 1+e^{2(\Pi E)^{1/2}}\lesssim
1+e^{\epsilon^{1/2}}\lesssim 1$

for sufficiently small $\epsilon$. The bound on $\Pi$ and the fact that
$\Pi,Y>0$ together imply that, for $a<0$,

(6.24) $\displaystyle\int_{s_{0}}^{s}e^{(a-1/2)\tau}\Pi_{\tau}\,d\tau\lesssim$
$\displaystyle\epsilon^{1/2}e^{s_{0}/2}\int_{s_{0}}^{s}e^{a\tau}\,d\tau\leq
C(a)\epsilon^{1/2}e^{(a+1/2)s_{0}}$

and similarly

(6.25) $\displaystyle\int_{s_{0}}^{s}e^{(a-5/2)\tau}Y_{\tau}\,d\tau\lesssim$
$\displaystyle
e^{(a-5/2)s}Y(s)-e^{(a-5/2)s_{0}}Y(s_{0})-(a-5/2)\int_{s_{0}}^{s}e^{(a-5/2)\tau}Y\,d\tau$
(6.26) $\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}\left[e^{as+s_{0}/2}-(a-5/2)e^{s_{0}/2}\int_{s_{0}}^{s}e^{a\tau}\,d\tau\right]$
(6.27) $\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}\left[e^{as+s_{0}/2}-\frac{a-5/2}{a}\left(e^{as+s_{0}/2}-e^{(a+1/2)s_{0}}\right)\right]$
(6.28) $\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}\left[e^{as+s_{0}/2}+C(a)e^{(a+1/2)s_{0}}\right]$
(6.29) $\displaystyle\lesssim$ $\displaystyle
C(a)\epsilon^{1/2}e^{\left(a+\frac{1}{2}\right)s_{0}}.$

Bound on $\Lambda$:

To improve inequality (6.17), first note that $\left|\frac{\Pi
E}{H}-1\right|=\frac{\Pi}{H}|\Lambda|$. Then we may use the estimate of the
correction in inequality (3.25) to obtain

(6.30) $\displaystyle\frac{\Pi}{H}|\Lambda|\lesssim$
$\displaystyle\frac{\Pi}{H}\left[e^{-2\tau}|A|+e^{-\tau}\left(1+e^{2(\Pi
E)^{1/2}}\right)\Pi E\right]$ (6.31) $\displaystyle\lesssim$
$\displaystyle\frac{\Pi}{H}\left[e^{-2\tau}+e^{-\tau}\Pi E\right]$ (6.32)
$\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}e^{s_{0}/2}e^{3\tau/2}\left[e^{-2\tau}+e^{s_{0}-2\tau}\epsilon\right]$
(6.33) $\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}e^{s_{0}/2}e^{-\tau/2}+\epsilon^{3/2}e^{3s_{0}/2}e^{-\tau/2}$
(6.34) $\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}+\epsilon^{3/2}e^{s_{0}}$ (6.35)
$\displaystyle\lesssim$ $\displaystyle\epsilon^{1/2}$

since $H^{-1}\lesssim e^{\tau}$. Thus we may ensure

(6.36) $\displaystyle\left|\frac{\Pi E}{H}-1\right|<2$

for $\epsilon$ small.

An Upper and Lower Bound on $H$:

For the energy $H$ we have the following estimate:

(6.37)
$\displaystyle\left|\partial_{\tau}\left(e^{\tau}H\right)\right|\lesssim
e^{\tau}H\frac{\Pi E}{H}F+\widetilde{F}\lesssim$ $\displaystyle
e^{\tau}HF+\widetilde{F}$

That is, there is a constant $C>0$ such that

(6.38) $\displaystyle\left|\partial_{\tau}\left(e^{\tau}H\right)\right|\leq$
$\displaystyle C\left(e^{\tau}HF+\widetilde{F}\right).$

The quantities $F$ and $\widetilde{F}$ are the nonnegative quantities defined
in equations (6.2) and (6.3). We then apply Lemma 6.1 to obtain the upper
bound

(6.39) $\displaystyle e^{s}H(s)\leq$
$\displaystyle\left(e^{s_{0}}H(s_{0})+C\int_{s_{0}}^{s}\widetilde{F}\,d\tau\right)\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right)$

and the lower bound

(6.40) $\displaystyle e^{s}H(s)\geq$ $\displaystyle
2e^{s_{0}}H(s_{0})-\left(e^{s_{0}}H(s_{0})+C\int_{s_{0}}^{s}\widetilde{F}\,d\tau\right)\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right).$

What we want, then, is for $\int_{s_{0}}^{\infty}\widetilde{F}\,d\tau$ to be
bounded and for $\int_{s_{0}}^{\infty}F\,d\tau\to 0$ as $\epsilon\to 0$.

Recall that we have assumed $e^{-\rho_{0}/2}<1$ and $|A|<1$. We compute

(6.41) $\displaystyle\widetilde{F}=$
$\displaystyle|A|\Pi\left(e^{-\tau}\left(1+\frac{\Pi_{\tau}}{\Pi}\right)+e^{-\rho_{0}/2}E^{1/2}\right)$
(6.42) $\displaystyle<$ $\displaystyle
e^{-\tau}\left(\Pi+\Pi_{\tau}\right)+\Pi^{1/2}(\Pi E)^{1/2}$ (6.43)
$\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}e^{s_{0}/2}e^{-\tau/2}+\epsilon^{3/4}e^{3s_{0}/4}e^{-\tau/4}$

so

(6.44) $\displaystyle\int_{s_{0}}^{\infty}\widetilde{F}\,d\tau\lesssim$
$\displaystyle\epsilon^{1/2}+\epsilon^{3/4}e^{s_{0}/2}\lesssim\epsilon^{1/4}.$

Let $C_{1}$ be the product of $C$ and the constant associated to the
$\lesssim$ in inequality (6.44).

Inequality (6.39) becomes

(6.45) $\displaystyle e^{s}H(s)\leq$
$\displaystyle\left(e^{s_{0}}H(s_{0})+C_{1}\epsilon^{1/4}\right)\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right)$

and the lower bound (6.40) becomes

(6.46) $\displaystyle e^{s}H(s)\geq$ $\displaystyle
2e^{s_{0}}H(s_{0})-\left(e^{s_{0}}H(s_{0})+C_{1}\epsilon^{1/4}\right)\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right)$

Now we turn to the bound on $F$.

(6.47) $\displaystyle F=$
$\displaystyle\int_{S^{1}}e^{\rho-\tau}\rho_{\tau}J\,d\theta+e^{-\tau}\left(1+e^{2(\Pi
E)^{1/2}}\right)\left(\Pi+\Pi_{\tau}\right)+(\Pi
E)^{1/2}+e^{-\rho_{0}/2}e^{2(\Pi E)^{1/2}}\Pi E^{1/2}$ (6.48)
$\displaystyle\lesssim$ $\displaystyle
e^{-3\tau}\int_{S^{1}}e^{\rho+l+2\tau}J\,d\theta+e^{-\tau}\left(\Pi+\Pi_{\tau}\right)+(\Pi
E)^{1/2}+\Pi E^{1/2}$ (6.49) $\displaystyle\lesssim$ $\displaystyle
e^{-3\tau}Y_{\tau}+\epsilon^{1/2}e^{s_{0}/2}e^{-\tau/2}+\epsilon^{3/4}e^{3s_{0}/4}e^{-\tau/4}$

We have previously bounded the integral of the latter terms in time by
$\epsilon^{1/4}$, so it remains to compute

(6.50)
$\displaystyle\int_{s_{0}}^{\infty}e^{-3\tau}Y_{\tau}\,d\tau\lesssim\epsilon^{1/2}.$

So

(6.51) $\displaystyle\int_{s_{0}}^{\infty}F\,d\tau\lesssim\epsilon^{1/4}.$

Thus, in total for $H$, we have

(6.52) $\displaystyle
e^{s}H(s)<\frac{3}{2}\left(e^{s_{0}}H(s_{0})+C_{1}\epsilon^{1/4}\right)$

when we choose $\epsilon$ small enough that
$\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right)<\frac{3}{2}$.

Turning to the lower bound, it is useful to define $N:=e^{s_{0}}H(s_{0})$ and
$L:=C_{1}\epsilon^{1/4}$. Note assumption (6.14) implies that

(6.53)
$\displaystyle\frac{1}{4(N+L)}\left(5N+3L\right)>1+\frac{1}{4M(N+L)}>1+\frac{1}{4M^{2}}$

so we take $\epsilon$ small enough that

(6.54) $\displaystyle
1+\frac{1}{4M^{2}}>\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right).$

The lower bound from Grönwall’s inequality takes the form

(6.55) $\displaystyle e^{s}H(s)\geq$ $\displaystyle
2N-\left(N+L\right)\exp\left(C\int_{s_{0}}^{s}F\,d\tau\right)>2N-\frac{1}{4}\left(5N+3L\right)=\frac{3}{4}(N-L)$

which improves the lower bound on $e^{\tau}H(\tau)$.

Bounds on $\Pi,Y$:

Let us determine what the smallness assumptions of Lemma 6.2 imply for the
error term of the ODE system of Section 5. Recall the conclusion of
Proposition 5.1: if $H>0$, then

(6.60) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{(s_{0}-s)/4}\left(\frac{e^{s_{0}}H(s_{0})}{e^{s}H(s)}\right)^{1/2}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s_{0})+\int^{s}_{s_{0}}e^{(\tau-s)/4}\left(\frac{e^{\tau}H(\tau)}{e^{s}H(s)}\right)^{1/2}|\omega(\tau)|\,d\tau,$

where

(6.65) $\displaystyle|\omega|\lesssim\left|\widetilde{\Omega}\right|=$
$\displaystyle\left|-\frac{1}{2}\partial_{\tau}\log\left(e^{\tau}H\right)\left(\begin{array}[]{l}\frac{2}{\sqrt{10}}\\\
\frac{1}{\sqrt{10}}\end{array}\right)+\left(\begin{array}[]{l}0\\\
\frac{f(d,c)}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}+\left(\displaystyle\frac{\Pi
E}{H}-1\right)\frac{d+\frac{1}{\sqrt{10}}}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}+\displaystyle\frac{\Omega}{e^{3\tau}H^{1/2}}\end{array}\right)\right|$

and

(6.66) $\displaystyle\left|e^{-3\tau}H^{-1/2}\Omega\right|\lesssim$
$\displaystyle e^{-2\tau}|H|^{-1/2}EY_{\tau}.$

To begin with, note that $e^{\tau}H(\tau)$ has both upper and lower bounds,
and so both terms of the form
$\left(\frac{e^{\tau}H(\tau)}{e^{s}H(s)}\right)^{1/2}$ can be bounded above by
a constant:

(6.71) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{(s_{0}-s)/4}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s_{0})+\int^{s}_{s_{0}}e^{(\tau-s)/4}|\omega(\tau)|\,d\tau$
(6.72) $\displaystyle\lesssim$
$\displaystyle\epsilon+\int^{s}_{s_{0}}e^{(\tau-s)/4}|\omega(\tau)|\,d\tau.$

To finish the bootstrap, we must bound the right side of this inequality
strictly below $\epsilon^{1/4}$. We deal with each of the 4 summands in
$\omega$ in the remainder of the proof.

The contribution to the right side of (6.60) from the error term
$e^{-3\tau}H^{-1/2}\Omega$ is

(6.73)
$\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|e^{-3\tau}H^{-1/2}\Omega\right|\,d\tau\lesssim$
$\displaystyle
e^{-s/4}\int^{s}_{s_{0}}e^{-7\tau/4}\left|H^{-1/2}\Pi^{-1}\right|\Pi
EY_{\tau}\,d\tau$ (6.74) $\displaystyle=$ $\displaystyle
e^{-s/4}\int^{s}_{s_{0}}e^{-11\tau/4}\left|\frac{\Pi
E}{H}\frac{1}{c+\frac{2}{\sqrt{10}}}\right|Y_{\tau}\,d\tau$ (6.75)
$\displaystyle\lesssim$ $\displaystyle
e^{-s/4}\int^{s}_{s_{0}}e^{-11\tau/4}\left|\frac{\Pi
E}{H}Y_{\tau}\right|\,d\tau$ (6.76) $\displaystyle\lesssim$ $\displaystyle
e^{-s/4}\int^{s}_{s_{0}}e^{(-1/4-5/2)\tau}Y_{\tau}\,d\tau$ (6.77)
$\displaystyle\lesssim$ $\displaystyle\epsilon^{1/2}e^{(s_{0}-s)/4}$ (6.78)
$\displaystyle<$ $\displaystyle\epsilon^{1/2}$

where we have used the fact that
$\frac{e^{\tau}H^{-1/2}}{c+\frac{2}{\sqrt{10}}}=\Pi^{-1}$ and the bootstrap
assumptions.

The contribution from $\left(\frac{\Pi
E}{H}-1\right)\frac{d+\frac{1}{\sqrt{10}}}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}$
is

(6.79) $\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|\left(\frac{\Pi
E}{H}-1\right)\frac{d+\frac{1}{\sqrt{10}}}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}\right|\,d\tau\lesssim$
$\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\epsilon^{1/2}\,d\tau$ (6.80)
$\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}\left(1-e^{(s_{0}-s)/4}\right)$ (6.81)
$\displaystyle<$ $\displaystyle\epsilon^{1/2}.$

Turning to $\frac{f(d,c)}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}$, we recall
that $f$ has vanishing linear part, so

(6.82)
$\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|\frac{f(d,c)}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}\right|\,d\tau\lesssim$
$\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\epsilon^{1/2}\,d\tau\lesssim\epsilon^{1/2}\left(1-e^{(s_{0}-s)/4}\right)<\epsilon^{1/2}$

To bound $\partial_{\tau}\log\left(e^{\tau}H\right)$, note that $e^{\tau}H$
has a lower bound, and use the estimates on $F$ and $\widetilde{F}$ obtained
above to compute

(6.83) $\displaystyle\left|\partial_{\tau}\log\left(e^{\tau}H\right)\right|=$
$\displaystyle\frac{1}{e^{\tau}H}\left|\partial_{\tau}(e^{\tau}H)\right|$
(6.84) $\displaystyle\lesssim$
$\displaystyle\frac{1}{e^{\tau}H}\left(e^{\tau}HF+\widetilde{F}\right)$ (6.85)
$\displaystyle\lesssim$ $\displaystyle F+\widetilde{F}$ (6.86)
$\displaystyle\lesssim$ $\displaystyle
e^{-3\tau}Y_{\tau}+\epsilon^{1/2}e^{s_{0}/2}e^{-\tau/2}+\epsilon^{3/4}e^{3s_{0}/4}e^{-\tau/4}$

So the contribution to (6.60) is

(6.87)
$\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\left(e^{-3\tau}Y_{\tau}+\epsilon^{1/2}e^{s_{0}/2}e^{-\tau/2}+\epsilon^{3/4}e^{3s_{0}/4}e^{-\tau/4}\right)\,d\tau\lesssim$
$\displaystyle\epsilon^{1/2}+e^{-s/4}\int^{s}_{s_{0}}e^{-\tau/4-5/2\tau}Y_{\tau}\,d\tau$
(6.88) $\displaystyle\lesssim$
$\displaystyle\epsilon^{1/2}+e^{(s_{0}-s)/4}\epsilon^{1/2}$ (6.89)
$\displaystyle\lesssim$ $\displaystyle\epsilon^{1/2}.$

Combining these estimates, we have from inequality (6.71) that

(6.94) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{(s_{0}-s)/4}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s_{0})+\int^{s}_{s_{0}}e^{(\tau-s)/4}|\omega(\tau)|\,d\tau$
(6.95) $\displaystyle\lesssim$ $\displaystyle\epsilon+\epsilon^{1/2}$ (6.96)
$\displaystyle\lesssim$ $\displaystyle\epsilon^{1/2}.$

This improves the bootstrap inequality on $c,d$.

Thus we have improved all of the bootstrap inequalities, and the proof is
complete. ∎

##  7\. Asymptotic Behavior

We are now in a position to present the $B_{0}$ version of the main result of
[13]. In particular, for $T^{2}$-symmetric vacuum spacetimes satisfying $B=0$,
we find rates of growth/decay in the expanding direction for the
$\theta$-direction volume, the normalized energy, and their derivatives. In
going from the polarised to $B_{0}$ case, we appear to lose some of the fine
grained asymptotics of $V$ and its mean. Forthcoming work will 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 $\epsilon_{0}$ such that if $0\leq\epsilon\leq\epsilon_{0}$,
for any $B_{0}$ initial data set satisfying the smallness conditions of Lemma
6.2, the associated solution satisfies for $\tau\in[s_{0},\infty)$.

(7.3) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|\lesssim$ $\displaystyle e^{-\tau/4}$ (7.4)
$\displaystyle\left|e^{\tau}H-C_{\infty}^{2}\right|\lesssim$ $\displaystyle
e^{-\tau/4}$ (7.5)
$\displaystyle\left|e^{-\tau/2}\Pi-\frac{2}{\sqrt{10}}C_{\infty}\right|\lesssim$
$\displaystyle e^{-\tau/4}$ (7.6)
$\displaystyle\left|e^{-5\tau/2}Y-\frac{1}{\sqrt{10}}C_{\infty}\right|\lesssim$
$\displaystyle e^{-\tau/4}$ (7.7)
$\displaystyle\left|e^{3\tau/2}E-\frac{\sqrt{10}}{2}C_{\infty}\right|\lesssim$
$\displaystyle e^{-\tau/4}$ (7.8) $\displaystyle\left|\langle
l\rangle-l\right|\lesssim$ $\displaystyle e^{-\tau/2}$ (7.9)
$\displaystyle\left|e^{l}-\frac{1}{2}\right|\lesssim$ $\displaystyle
e^{-\tau/4}$ (7.10) $\displaystyle\left|\langle
V\rangle-V\right|+\left|e^{V-\tau}\left(\langle
Q\rangle-Q\right)\right|\lesssim$ $\displaystyle e^{-\tau/2}$ (7.11)
$\displaystyle\left|\Pi^{-1}e^{\rho}-e^{\rho_{\infty}}\right|\lesssim$
$\displaystyle e^{-\tau/2}$

for some $C_{\infty}>0$ and $\rho_{\infty}\colon S^{1}\to\mathbb{R}$.

###### Proof.

The proof proceeds as in [13]. First, observe that inequalities (6.21) and
(6.22) imply that

(7.12) $\displaystyle e^{-\tau}\Pi+e^{-\tau}\Pi_{\tau}+e^{-3\tau}Y\lesssim
e^{-\tau/2}.$

Furthermore, $\Pi E\lesssim e^{-\tau}$ and $e^{\tau}H$ is bounded above and
below by positive constants. On the other hand

(7.13)
$\displaystyle\left|\partial_{\tau}\left(e^{\tau}H\right)\right|\lesssim
e^{\tau}HF+\widetilde{F}\lesssim F+\widetilde{F}\lesssim
e^{-\tau/4}+e^{-3\tau}Y_{\tau}.$

The right side is integrable in $\tau$, so let
$C_{\infty}:=\lim_{\tau\to\infty}\sqrt{e^{\tau}H}$. Then

(7.14)
$\displaystyle\left|C_{\infty}^{2}-e^{\tau}H\right|\lesssim\int_{\tau}^{\infty}\left|\partial_{\tau}\left(e^{s}H\right)\right|\,ds\lesssim
e^{-\tau/4}$

giving (7.4).

Note that (6.60) now reads

(7.17) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{-s/4}+\int^{s}_{s_{0}}e^{(\tau-s)/4}|\omega(\tau)|\,d\tau,$

and that all of the terms of
$\int^{s}_{s_{0}}e^{(\tau-s)/4}|\omega(\tau)|\,d\tau$ are now bounded by
$e^{-s/4}$ with the exception of

(7.22)
$\displaystyle\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|\frac{f(d,c)}{\left(c+\frac{2}{\sqrt{10}}\right)^{2}}\right|\,d\tau\lesssim\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|^{2}\,d\tau\lesssim\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|\,d\tau$

since $\left|\left(c,d\right)\right|\lesssim 1$. So

(7.25) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{-s/4}+\int^{s}_{s_{0}}e^{(\tau-s)/4}|\omega(\tau)|\,d\tau$ (7.28)
$\displaystyle\lesssim$ $\displaystyle
e^{-s/4}+\int^{s}_{s_{0}}e^{(\tau-s)/4}\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|\,d\tau.$

Applying the integral version of Grönwall’s inequality gives

(7.31) $\displaystyle\left|\left(\begin{array}[]{l}c\\\
d\end{array}\right)\right|(s)\lesssim$ $\displaystyle
e^{-s/4}+\int^{s}_{s_{0}}e^{-\tau/4}e^{(\tau-s)/4}\exp\left(\int^{s}_{\tau}e^{(r-s)/4}\,dr\right)\,d\tau$
(7.32) $\displaystyle\lesssim$ $\displaystyle
e^{-s/4}+e^{-s/4}\int^{s}_{s_{0}}\exp\left(\int^{s}_{\tau}e^{(r-s)/4}\,dr\right)\,d\tau$
(7.33) $\displaystyle\lesssim$ $\displaystyle e^{-s/4}+se^{-s/4}$ (7.34)
$\displaystyle\lesssim$ $\displaystyle e^{\left(\delta-\frac{1}{4}\right)s}$

for any $\delta>0$. Inserting this improved estimate into (7.22) and applying
Grönwall’s inequality again gives (7.3). Combining this with (7.4) yields
(7.5) and (7.6).

Recall that $H=\Pi(E+\Lambda)$ and

(7.35) $\displaystyle\left|\Lambda\right|\lesssim
e^{-2\tau}|A|+e^{-\tau}\left(1+e^{2(\Pi E)^{1/2}}\right)\Pi E\lesssim
e^{-2\tau}.$

Then combine (7.4) and (7.5) to obtain (7.7). The estimate (7.8) follows from
(4.18) and (7.7). Estimate (7.10) follows directly from the Poincaré
inequality and the bound on $E$.

To estimate $e^{l}$ let us note that

(7.36) $\displaystyle\left|\Pi e^{l}-e^{-2\tau}Y\right|=$
$\displaystyle\left|\int_{S^{1}}e^{\rho(\varphi)+l(\varphi)}\left(e^{l(\varphi)-l(\theta)}-1\right)\,d\varphi\right|\lesssim
e^{-2\tau}Y.$

One can then combine (7.36), (7.5), and (7.6) to find that $\sup_{S^{1}}e^{l}$
is bounded by a constant. Then, we estimate again

(7.37) $\displaystyle\left|\Pi e^{l}-e^{-2\tau}Y\right|=$
$\displaystyle\left|\int_{S^{1}}e^{\rho(\varphi)}\left(e^{l(\varphi)}-e^{l(\theta)}\right)\,d\varphi\right|\lesssim\Pi\left(\sup_{S^{1}}e^{l}\right)e^{\tau}E\lesssim
1$

and combine (7.37), (7.5), and (7.6) to obtain (7.9). The proof of (7.11) is
identical to the one in [13], since we have the same bounds on
$\rho_{\theta\tau}$. ∎

##  8\. Numerical Evidence

The full Einstein Flow is a large, quasilinear system of partial differential
equations about which it is difficult even to make conjectures. This remains
true even in the simplified $T^{2}$-symmetric case considered in this work. It
has been crucial to this work to base our conjectures on evidence garnered
from numerical simulations of $T^{2}$-symmetric Einstein Flows. We summarize
this numerical work in this section. A more detailed 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 $T^{2}$-symmetric spacetimes in the contracting direction [7], and
then later in the expanding direction. We reimplemented this code in
OCaml111OCaml is a general purpose programming language developed primarily at
INRIA. See https://ocaml.org/., and made a number of modifications to improve
the accuracy and speed. Most importantly, we developed code to produce
solutions of the $T^{2}$-symmetric constraint equation via a random process,
which allowed us to probe the behavior of generic $T^{2}$-symmetric Einstein
Flows.

We have developed code which samples the constraint submanifold for the
$T^{2}$-symmetric Einstein Field Equations in a fairly generic manner. We have
then evolved these initial data using a finite difference method. This generic
sampling has been a crucial element allowing us to determine that the
assumption $B=0$ was necessary for our main theorem, and otherwise develop our
intuition about the solutions. The simulations have the expected convergence
properties upon refining the spatial resolution so we are confident that they
are accurate approximations of solutions. To obtain confidence that our
simulations depict behavior which is generic for the class under
consideration, we simulated on the order of 20 randomly chosen initial
constraints solutions in each of the following classes: polarised, $B_{0}$,
and $B\neq 0$ $T^{2}$-symmetric. The qualitative behavior depicted in Figures
2 through 4 is observed to be the same for all simulations in that class.

It has been useful to plot the evolution of the following quantities along
each of the numerical solutions.

(8.1) $\displaystyle
S:=\partial_{\tau}\int_{S^{1}}l\,e^{\rho-\tau/2}d\theta,\quad$ $\displaystyle
T:=\partial_{\tau}\int_{S^{1}}\rho\,e^{\rho-\tau/2}d\theta,$ (8.2)
$\displaystyle
E_{V}:=\int_{S^{1}}\left[V_{\tau}^{2}+e^{2(\tau-\rho)}V_{\theta}^{2}\right]\,e^{\rho-\tau/2}d\theta,\quad$
$\displaystyle
E_{Q}:=\int_{S^{1}}e^{2(V-\tau)}\left(Q_{\tau}^{2}+e^{2(\tau-\rho)}Q_{\theta}^{2}\right)\,e^{\rho-\tau/2}d\theta,$
(8.3) $\displaystyle
W:=\log\left|\int_{S^{1}}V_{\tau}\,e^{\rho-\tau/2}d\theta\right|$

These are not the quantities that were used in the proof of our main theorem,
but they capture the dynamics of the system. The volume form
$e^{\rho-\tau/2}d\theta$ is used to smooth out the graphs (the integrals
generally oscillate without this normalization).

(a) polarised (b) $B=0$ (c) $B\neq 0$

Figure 2. $S$ and $T$ flow toward a spiral sink, regardless of polarisation or
the value of $B$. Although $l_{\tau},\rho_{\tau}$ converge to the same values
in all cases, the volume form $e^{\rho-\tau/2}\,d\theta$ causes the variables
used in the plots to flow toward different values.

(a) polarised (b) $B=0$ (c) $B\neq 0$

Figure 3. For polarised solutions, $E=E_{V}$ which converges to a constant.
For $B=0$ solutions, $E$ and the $V$ and $Q$ energies all converge to
constants. For $B\neq 0$ solutions, however, although the total energy
converges, $E_{V}$ and $E_{Q}$ do not; they oscillate with amplitude which
does not decay and period matching the period of the sink in Figure 2.

(a) polarised (b) $B=0$ (c) $B\neq 0$

Figure 4. For $B=0$ solutions (including polarised), $V_{\tau}\to 0$
exponentially. For $B\neq 0$ solutions, however, $V_{\tau}$ appears to
converge to a nonzero constant.

In [13], the authors are able to determine the first order behavior of the
energy and $\Pi$, but also the first order behavior of $V$ and the rate of its
decay to the mean value. We have generalized their results on the asymptotic
values of the energy, $\Pi$ as well as the decay of $V$ and $Q$ to their means
to the $B_{0}$ case, but so far have been unable to derive other estimates for
$V$ and $Q$. However, the numerical solutions that we have found have the
property that there are constants $a,C_{V}$ such that

(8.4) $\displaystyle\left|\langle V\rangle-C_{V}\tau-a\right|=O(e^{-\tau/2})$

and that

(8.7) $\displaystyle C_{V}=\left\\{\begin{array}[]{ll}0&\text{if }B=0\\\
\frac{1}{2}&\text{if }B\neq 0\end{array}\right..$

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 many important special subsets of solutions.
Unfortunately, authors have used many different coordinates for exactly the
same set of spacetimes, and this document adds yet another set of coordinates.
As an aid to the reader who wishes to read the cited works together, we
provide in this appendix a concordance of notations 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 derived in [6]. This paper, [6], [7] and [18] use
coordinates for $T^{2}$-symmetric Einstein Flows which are completely general.
The analysis in [13] applies only to polarised $T^{2}$-symmetric Einstein
Flows, and so relies on the assumption that some metric components vanish
identically. In [17], future asymptotics of Gowdy solutions are derived. The
notation used there is exactly the notation of [18] if one imposes the
conditions $\alpha\equiv 1,K=0$ 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. All of the expressions in a given row are equal. For
example, the function called $P$ in [18] has the expression $2U-\log R$ in
[13]. Since [13] only deals with polarised flows, the expressions in this
column are only equal to those in other documents if the polarization
condition is imposed.

A

[6] [7] [18] [13] this document $\log t$ $-\tau$ $\log t$ $\log R$ $\tau$
[0.5pt/5pt] $2U$ $P-\tau$ $P+\log t$ $2U$ $V$ [0.5pt/5pt] $\alpha^{-1/2}$
$2\pi_{\lambda}$ $\alpha^{-1/2}$ $a^{-1}$ $e^{\rho}$ [0.5pt/5pt]
$\displaystyle\frac{K^{2}}{2}t^{-2}\alpha e^{2\nu}$
$\displaystyle\frac{K^{2}}{2}e^{P+\frac{1}{2}\lambda+3\tau/2}$
$\displaystyle\frac{K^{2}}{2}t^{-3/2}e^{P+\frac{1}{2}\lambda}$
$\displaystyle\frac{K^{2}}{2}R^{-2}e^{2\eta}$ $e^{l}$ [0.5pt/5pt] $2tU_{t}$
$1-\displaystyle\frac{\pi_{P}}{2\pi_{\lambda}}$ $tP_{t}+1$ $2RU_{R}$
$V_{\tau}$ [0.5pt/5pt] $\displaystyle
t^{-1}\int_{S^{1}}\alpha^{-1/2}e^{4U}Q_{t}\,d\theta$
$\displaystyle-\int_{S^{1}}\pi_{Q}\,d\theta$
$\displaystyle\int_{S^{1}}\alpha^{-1/2}e^{2P}tQ_{t}\,d\theta$ $0$
$\displaystyle\int_{S^{1}}e^{\rho+2(V-\tau)}Q_{\tau}\,d\theta=:B$ [0.5pt/5pt]

A

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