# Enhanced Dissipation and Transition Threshold for the Poiseuille Flow in a
Periodic Strip

Augusto Del Zotto111Department of Mathematics, Imperial College London, SW7
2AZ, UK. E-mail<EMAIL_ADDRESS>
###### Abstract

We consider the solution to the 2D Navier-Stokes equations around the
Poiseuille flow $(y^{2},0)$ on $\mathbb{T}\times\mathbb{R}$ with small
viscosity $\nu>0$. Via a hypocoercivity argument, we prove that the
$x-$dependent modes of the solution to the linear problem undergo the enhanced
dissipation effect with a rate proportional to $\nu^{\frac{1}{2}}$. Moreover,
we study the nonlinear enhanced dissipation effect and we establish a
transition threshold of $\nu^{\frac{2}{3}+}$. Namely, when the perturbation of
the Poiseuille flow is size at most $\nu^{\frac{2}{3}+}$, its size remains so
for all times and the enhanced dissipation persists with a rate proportional
to $\nu^{\frac{1}{2}}$.

##  1 Introduction

We study the 2D Navier-Stokes equations

$\left\\{\begin{array}[]{l}\partial_{t}U+(U\cdot\nabla)U+\nabla P-\nu\Delta
U=0,\\\ \nabla\cdot U=0,\\\ \end{array}\right.$ (1)

defined on the domain $\mathbb{T}\times\mathbb{R}$. $U=(U_{1},U_{2})$ is the
velocity vector field, $P$ is the scalar pressure and $\nu$ is the viscosity
coefficient of the fluid, proportional to the inverse of the Reynolds number.
Defining the vorticity of $U$ as $\Omega=\nabla^{\perp}\cdot U$, where
$\nabla^{\perp}=(-\partial_{y},\partial_{x})$, it is possible to remove the
pressure term and to rewrite the above system as

$\left\\{\begin{array}[]{l}\partial_{t}\Omega+U\cdot\nabla\Omega-\nu\Delta\Omega=0,\\\
\Omega=\Delta\Psi,\\\ U=\nabla^{\perp}\Psi.\end{array}\right.$ (2)

Here $\Psi$ is the corresponding stream function for the vector field $U$. It
is easy to see that the so called Poiseuille Flow

$U_{P}=(y^{2},0),\qquad\Omega_{P}=-2y,$

is a stationary solution of (2). To study the dynamic near the Poiseuille flow
we consider a small perturbation of it. We set $U=U_{P}+u$, so
$\Omega=\Omega_{P}+\omega$, where $u,\omega$ are the perturbations for the
velocity field and the vorticity, respectively. The vorticity formulation for
$\omega$ reads as follow

$\left\\{\begin{array}[]{l}\partial_{t}\omega+y^{2}\partial_{x}\omega-2\partial_{x}\psi-\nu\Delta\omega=-u\cdot\nabla\omega,\\\
\omega=\Delta\psi,\\\ u=\nabla^{\perp}\psi.\end{array}\right.$ (3)

The boundary conditions here are set to be periodic for the $x$ variable,
while for the $y$ direction $\omega$ is assumed to have sufficient decay at
infinity. The stability of the Poiseuille flow can then be seen as the decay
of the solution to (3). An overview of the paper is presented in the following
subsections of the introduction. The first part is devoted to the analysis of
the linear problem while the second part focuses on the transition threshold
problem for the fully non linear equation.

###  1.1 Linear enhanced dissipation and estimates for $\mathcal{L}_{\nu}$

We define the linear operator

$\mathcal{L}_{\nu}=-y^{2}\partial_{x}+2\partial_{x}\Delta^{-1}+\nu\Delta$ (4)

associated to the linearized counterpart of (3). In the first part, we
establish some decay estimates for the semigroup generated by
$\mathcal{L}_{\nu}$ in the usual $L^{2}$ norm. Our first result is the
following, here $\mathbb{P}_{\neq}$ is the projection on the nonzero
$x$-Fourier modes.

######  Theorem 1.1.

Let $\nu<1$ and $g\in L^{2}(\mathbb{T}\times\mathbb{R})$. There exist
constants $C_{0},c_{0}>0$ independent of $\nu$ such that

$\left\|e^{\mathcal{L}_{\nu}t}\mathbb{P}_{\neq}(g)\right\|_{L^{2}}\leq
C_{0}e^{-c_{0}\nu^{\frac{1}{2}}t}\left\|\mathbb{P}_{\neq}(g)\right\|_{L^{2}},$
(5)

for all $t\geq 0.$

This result gives a quantitative estimate of the linear _enhanced dissipation_
effect. Indeed, the timescale obtained for the nonzero modes of the initial
datum is proportional to $\nu^{-1/2}$, which is much faster than the heat
equation timescale, proportional to $\nu^{-1}$.

We refer to enhanced dissipation as the phenomenon where the mixing properties
of the fluid allow to improve the natural heat dissipation timescale
$\mathcal{O}(\nu^{-1})$ to a faster timescale $\mathcal{O}(d(\nu)^{-1})$ that
satisfies

$\lim_{\nu\to 0}\frac{\nu}{d(\nu)}=0.$

This phenomenon has been widely studied in the physics literature, see for
example [11, 15, 16] and mathematics literature [7, 8]. In the context of the
Navier-Stokes equations near shear flows, we cite results for the well known
Couette flow [4, 19], with a dissipation timescale $\mathcal{O}(\nu^{-1/3})$,
and the Kolmogorov flow [12, 20, 21], where the rate is known to be
$\mathcal{O}(\nu^{-1/2})$. Regarding the Poiseuille flow, the first linear
enhanced dissipation result was given by Coti Zelati, Elgindi and Widmayer [9]
for the unbounded 2D domain $(x,y)\in\mathbb{T}\times\mathbb{R}$. In their
paper the linear enhanced dissipation effect is established around the
Poiseuille flow. The rate obtained is proportional to
$\nu^{1/2}(1+|\log\nu|)^{-1}$ in the weighted $L^{2}$ space with norm

$\left\|f\right\|^{2}_{X}=\left\|f\right\|^{2}_{L^{2}}+\left\|yf\right\|_{L^{2}}^{2}.$

Ding and Lin [10] proved the same decay rate, without the logarithmic
correction, for the Poiseuille flow in a bounded 2D channel
$\mathbb{T}\times[-1,1].$ We cite also the paper by Chen, Wei and Zhang [6],
where a $\mathcal{O}(\nu^{-1/2})$ rate is obtained for the 3D pipe Poiseuille
flow. The approach used in the last two papers is completely different from
the hypocoercivity method. It relies on resolvent estimates and a Gearhart-
Prüss type Theorem introduced by Wei in [18].

Our first result is a sharpening of [9], indeed we are able to remove the
logarithmic correction and to get a decay rate of $\mathcal{O}(\nu^{-1/2})$ in
$L^{2}(\mathbb{T}\times\mathbb{R})$. The proof of Theorem 1.1 relies on a
hypocoercivity argument [17], similar to the one in [9]. Here, we construct an
energy functional following Wei and Zhang idea [20], namely, each term of the
functional has a time dependent weight. Theorem 1.1 follows then from an
iteration argument. Furthermore, thanks to the time dependent weights of the
energy functional, we are able to prove additional estimates on the semigroup
$e^{\mathcal{L}_{\nu}t}$ generated by the linearized operator
$\mathcal{L}_{\nu}$. These estimates play a crucial role in establishing the
transition threshold.

###  1.2 Nonlinear enhanced dissipation and transition threshold

Our second result concerns the transition threshold for the 2D Poiseuille
flow. The asymptotic stability of fluid motion between parallel plates was
firstly analyzed by Kelvin [13], who introduced the following concept: the
stability may depend on the viscosity coefficient $\nu$ in such a way that the
stability threshold decreases whenever $\nu$ decreases. The mathematical
formulation of this problem can be given as follows. Given a norm
$\left\|\cdot\right\|_{X}$, find a $\gamma=\gamma(X)$ such that

$\begin{array}[]{ccc}\left\|u\right\|_{X}\leq\nu^{\gamma}&\Rightarrow&stability,\\\
\left\|u\right\|_{X}\gg\nu^{\gamma}&\Rightarrow&instability,\end{array}$

here $u$ is a perturbation of the flow.

The transition threshold problem for the Couette flow has been deeply studied
recently [1, 2, 3, 5], in both the frameworks of Sobolev spaces and Gevrey
classes. In the 2D case the transition threshold is known to be
$\gamma\leq\frac{1}{3}$, [14]. For the 2D Kolmogorov flow in the periodic box,
it holds that $\gamma\leq\frac{2}{3}+\varepsilon$ for any $\varepsilon>0$, see
[21]. For the Poiseuille flow, it has been proved by Coti Zelati, Elgindi and
Widmayer [9] that $\gamma\leq\frac{3}{4}+\varepsilon$ in
$\mathbb{T}\times\mathbb{R}$ and by Ding and Lin [10] that
$\gamma\leq\frac{3}{4}$ in $\mathbb{T}\times[-1,1]$ with Navier-slip boundary
conditions, i.e. $\omega(\pm 1)=\psi(\pm 1)=0$.

In the second part of this paper we are going to show a transition threshold
for the Poiseuille Flow on $\mathbb{T}\times\mathbb{R}$ with
$\gamma\leq\frac{2}{3}+\varepsilon.$ We are able to prove the following
Theorem using the linear enhanced dissipation and the estimates on the
semigroup generated by the linearized operator.

######  Theorem 1.2.

There exists constants $\varepsilon_{0}\in(0,1),C_{1}>0,c_{1}>0$ such that for
all $0<\nu<1$ and for every $\omega_{in}\in L^{2}$ with

$\left\|\omega_{in}\right\|_{L^{2}}\leq\varepsilon_{0}(1+|\log\nu|^{\frac{1}{2}})^{-1}\nu^{2/3},$

the solution $\omega$ of (3) is global in time with the bound

$\left\|\mathbb{P}_{\neq}(\omega)(t)\right\|_{L^{2}}\leq
C_{1}e^{-c_{1}\nu^{1/2}t}\left\|\mathbb{P}_{\neq}(\omega_{in})\right\|_{L^{2}}.$

The proof is based on careful estimates of the non-zero modes of the nonlinear
term in (3). We remark that our theorem gives a better transition threshold
for the planar Poiseuille flow in $\mathbb{T}\times\mathbb{R}$, bringing it
from $\nu^{\frac{3}{4}+}$ to $\nu^{\frac{2}{3}+}$, more precisely
$(1+|\log\nu|^{\frac{1}{2}})^{-1}\nu^{\frac{2}{3}}$.

#### Structure of the paper

In Section 2 we define the modified energy functional (11) and we prove
Theorem 2.1. We then deduce the enhanced dissipation for the linear problem
(10). Section 3 is devoted to prove Lemma 3.1, which establishes additional
estimates on the semigroup generated by the linearized operator. Section 4
concludes the paper and contains the proof of Theorem 4.1.

#### Notation

Throughout this paper we use

$\nabla_{k}=\mathbb{P}_{k}\nabla=(ik,\partial_{y})$

for the projected gradient onto the $\pm k$-th $x$-Fourier modes and

$\Delta_{k}=\mathbb{P}_{k}\Delta=-k^{2}+\partial_{y}^{2}$

for the projected Laplacian. Moreover, we use $C>0$ to indicate a constant
independent of $\nu,k$ and $t$ that may vary line by line. We also denote
$\left\|\cdot\right\|=\left\|\cdot\right\|_{L^{2}}$ the usual $L^{2}$ norm.

##  2 Hypocoercivity Estimates

For a $x$-periodic function $f$ we write its Fourier expansion as

$f(t,x,y)=\sum_{j\in\mathbb{Z}}a_{j}(t,y)e^{ijx},\qquad
a_{j}(t,y)=\frac{1}{2\pi}\int_{\mathbb{T}}f(t,x,y)e^{-ijx}\text{d}x.$ (6)

For $k\in\mathbb{N}_{0}$ we set

$f_{k}(t,x,y)=\sum_{|j|=k}a_{j}(t,y)e^{ijx},$ (7)

so that

$f(t,x,y)=\sum_{k\in\mathbb{N}_{0}}f_{k}(t,x,y)$ (8)

can be expressed as a sum of real valued functions localized in $x$-frequency
on a single band $\pm k$, $k\in\mathbb{N}_{0}$. We also introduce the
following operators: given a function $f$ we define

$\mathbb{P}_{0}(f)=\frac{1}{2\pi}\int_{\mathbb{T}}f(x,y)\text{d}x,\qquad\mathbb{P}_{\neq}(f)=f-\mathbb{P}_{0}(f),$
(9)

and for any $k\in\mathbb{N}_{0}$, we denote $\mathbb{P}_{k}$ the projection to
the sum of the $\pm k$-th Fourier modes in $x$. We start by considering the
linearized system associated to (3) with initial datum $g\in L^{2}$, i.e.

$\left\\{\begin{array}[]{l}\partial_{t}\omega+y^{2}\partial_{x}\omega-2\partial_{x}\psi-\nu\Delta\omega=0,\\\
\Delta\psi=\omega,\\\ \omega|_{t=0}=g.\end{array}\right.$ (10)

The solution is given by $\omega(t)=e^{\mathcal{L}_{\nu}t}g$ for all $t\geq
0,$ where $\mathcal{L}_{\nu}$ is the linear operator (4). To establish decay
estimates for the semigroup we proceed by defining the following energy
functional, where $\alpha,\beta,\gamma$ have to be determined,

$\Phi(t)=\frac{1}{2}\left\|\omega\right\|^{2}+\frac{1}{2}\alpha\nu
t\left\|\nabla\omega\right\|^{2}+2\beta\nu
t^{2}\langle\partial_{y}\omega,y\partial_{x}\omega\rangle+\frac{1}{2}\gamma\nu
t^{3}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right].$
(11)

This energy functional resembles the one used in [9] and differs from it by
the time dependent weights. This modification is the key to achieve enhanced
dissipation in the $L^{2}$ norm and to remove the $1+|\log\nu|$ correction
obtained in [9].

######  Theorem 2.1.

Fix $0<\varepsilon<\frac{1}{36}$ and define
$\alpha=\varepsilon^{2},\beta=\varepsilon^{3},\gamma=16\varepsilon^{4}$. Then

$\alpha<\frac{1}{2},\quad 16\beta^{2}\leq\alpha\gamma,\quad
2\alpha^{2}<\beta<\frac{1}{4}\alpha,\quad 9\gamma<4\beta,$ (12)

and it follows that $\Phi(t)$ satisfies

$\Phi(t)\geq\frac{1}{2}\left\|\omega\right\|^{2}+\frac{1}{4}\alpha\nu
t\left\|\nabla\omega\right\|^{2}+\frac{1}{4}\gamma\nu
t^{3}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right]$
(13)

and

$\Phi^{\prime}(t)\leq-\gamma\nu t^{3}\left\|\partial_{x}\omega\right\|^{2}$
(14)

for every $t\geq 0.$

Once this statement is proved, linear enhanced dissipation follows directly.

######  Theorem 2.2.

Let $\nu<1$ and $g\in L^{2}$. There exist constants $C_{0},c_{0}>0$
independent of $\nu$ such that

$\left\|e^{\mathcal{L}_{\nu}t}\mathbb{P}_{k}(g)\right\|\leq
C_{0}e^{-c_{0}\nu^{\frac{1}{2}}|k|^{\frac{1}{2}}t}\left\|\mathbb{P}_{k}(g)\right\|,$
(15)

for all $t\geq 0.$ Combining all non-zero modes together we get

$\left\|e^{\mathcal{L}_{\nu}t}\mathbb{P}_{\neq}(g)\right\|\leq
C_{0}e^{-c_{0}\nu^{\frac{1}{2}}t}\left\|\mathbb{P}_{\neq}(g)\right\|.$ (16)

In addiction, for the non-zero modes of the velocity $u_{k}$, we have the
following decay

$\left\|\mathbb{P}_{k}(u)(t)\right\|^{2}\leq\frac{2}{\gamma\nu
k^{2}t^{3}}\left\|\mathbb{P}_{k}(g)\right\|^{2},$ (17)

for all $t\geq 0.$

We briefly give the proof. To simplify the notation we will write $f_{k}$
instead of $\mathbb{P}_{k}(f)$.

###### Proof of Theorem 2.2.

Since the equations decouple in $k$, we can apply separately Theorem 2.1 to
the $\pm k$-th $x$-frequency and get

$\Phi_{k}(t)\geq\frac{1}{2}\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|^{2}\quad\text{and}\quad\Phi^{\prime}_{k}(t)\leq-\gamma\nu^{2}k^{2}t^{3}\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|^{2}.$
(18)

Then,

$\displaystyle\frac{1}{2}\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|^{2}$
$\displaystyle\leq\Phi_{k}(t)$
$\displaystyle=\Phi_{k}(0)+\int_{0}^{t}\Phi_{k}^{\prime}(s)\text{d}s$
$\displaystyle\leq\frac{1}{2}\left\|g_{k}\right\|^{2}-\int_{0}^{t}\gamma\nu^{2}k^{2}s^{3}\left\|e^{\mathcal{L}_{\nu}s}g_{k}\right\|^{2}\text{d}s$
$\displaystyle\leq\frac{1}{2}\left\|g_{k}\right\|^{2}-\frac{\gamma}{4}\nu^{2}k^{2}t^{4}\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|^{2}.$

Rearranging this inequality we obtain

$\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|^{2}\leq\frac{1}{1+\frac{\gamma}{2}\nu^{2}k^{2}t^{4}}\left\|g_{k}\right\|^{2}.$
(19)

To prove the enhanced dissipation we proceed by iteration. Fix a time $t_{0}$
for which

$\frac{1}{1+\frac{\gamma}{2}\nu^{2}k^{2}t_{0}^{4}}=\frac{1}{2}.$

Now, for every time $t>t_{0}$, write it as $t=\lfloor t_{0}^{-1}t\rfloor
t_{0}+t^{*}$, with $t^{*}\in[0,t_{0})$. Using the fact that (10) is
autonomous, we deduce from (19) that for every $s\in[0,t_{0}]$

$\left\|e^{\mathcal{L}_{\nu}(t_{0}+s)}g_{k}\right\|^{2}\leq\frac{1}{1+\frac{\gamma}{2}\nu^{2}k^{2}t_{0}^{4}}\left\|e^{\mathcal{L}_{\nu}s}g_{k}\right\|^{2}\leq\frac{1}{2}\left\|e^{\mathcal{L}_{\nu}s}g_{k}\right\|^{2}.$
(20)

Then, using the fact that $t\mapsto\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|$
is decreasing, by iteration we have

$\displaystyle\left\|e^{\mathcal{L}_{\nu}t}g_{k}\right\|^{2}$
$\displaystyle\leq\left\|e^{\mathcal{L}_{\nu}\lfloor t_{0}^{-1}t\rfloor
t_{0}}g_{k}\right\|^{2}$ $\displaystyle\leq\left(\frac{1}{2}\right)^{\lfloor
t_{0}^{-1}t\rfloor}\left\|g_{k}\right\|^{2}$ $\displaystyle\leq
C_{0}e^{-c_{0}\nu^{\frac{1}{2}}|k|^{\frac{1}{2}}t}\left\|g_{k}\right\|^{2}.$

Finally, from the monotonicity of $\Phi(t)$ we can deduce (17). Indeed,

$\frac{1}{2}\gamma\nu
t^{3}\left\|\partial_{x}\nabla\psi\right\|^{2}\leq\left\|g\right\|^{2},$ (21)

and recalling that

$\left\|u_{k}\right\|^{2}=\left\|\nabla_{k}^{\perp}\psi_{k}\right\|^{2}=\left\|\nabla_{k}\psi_{k}\right\|^{2},$

we have the last inequality.

∎

In order to prove Theorem 2.1 we state some preliminary identities that will
be used to compute the derivative of the functional $\Phi$.

######  Proposition 2.3.

Let $\omega$ be a solution to (10). Then the following holds:

$\frac{1}{2}\frac{\text{d}}{\text{d}t}\left\|\omega\right\|^{2}=-\nu\left\|\nabla\omega\right\|^{2};$
(22)
$\frac{1}{2}\frac{\text{d}}{\text{d}t}\left\|\nabla\omega\right\|^{2}=-\nu\left\|\Delta\omega\right\|^{2}-2\langle
y\partial_{x}\omega,\partial_{y}\omega\rangle;$ (23)
$\frac{\text{d}}{\text{d}t}\langle\partial_{y}\omega,y\partial_{x}\omega\rangle=-2\left\|y\partial_{x}\omega\right\|^{2}-4\left\|\partial_{xy}\psi\right\|^{2}-2\nu\langle\Delta\omega,y\partial_{xy}\omega\rangle;$
(24)
$\frac{1}{2}\frac{\text{d}}{\text{d}t}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right]=-\nu\left\|\partial_{x}\omega\right\|^{2}-\nu\left\|y\partial_{x}\nabla\omega\right\|^{2}.$
(25)

The proof of these identities can be found in [9, Lemma 2.4]. We proceed with
the proof of Theorem 2.1.

###### Proof of Theorem 2.1.

For sake of clarity we recall here that $\Phi(t)$ is defined as

$\Phi(t)=\frac{1}{2}\left\|\omega\right\|^{2}+\frac{1}{2}\alpha\nu
t\left\|\nabla\omega\right\|^{2}+2\beta\nu
t^{2}\langle\partial_{y}\omega,y\partial_{x}\omega\rangle+\frac{1}{2}\gamma\nu
t^{3}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right].$
(26)

Using the Cauchy-Schwarz inequality and the Young inequality we get

$2\beta t^{2}|\langle\partial_{y}\omega,y\partial_{x}\omega\rangle|\leq 2\beta
t^{2}\left\|\partial_{y}\omega\right\|\left\|y\partial_{x}\omega\right\|\leq\frac{\alpha
t}{4}\left\|\partial_{y}\omega\right\|^{2}+\frac{4\beta^{2}t^{3}}{\alpha}\left\|y\partial_{x}\omega\right\|^{2}.$
(27)

Plugging this into $\Phi(t)$ we get

$\Phi(t)\geq\frac{1}{2}\left\|\omega\right\|^{2}+\frac{1}{4}\alpha\nu
t\left\|\nabla\omega\right\|^{2}+\frac{1}{2}\left(\gamma-\frac{8\beta^{2}}{\alpha}\right)\nu
t^{3}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right],$
(28)

hence the lower bound. We now proceed by computing the derivative of the
functional. By Proposition 2.3 we have

$\displaystyle\frac{\text{d}}{\text{d}t}\Phi(t)$
$\displaystyle=-\nu\left\|\nabla
w\right\|^{2}+\frac{\alpha\nu}{2}\left\|\nabla w\right\|^{2}+\alpha\nu
t\left(-\nu\left\|\Delta w\right\|^{2}-2\langle
y\partial_{x},\partial_{y}w\rangle\right)+4\beta
t\nu\langle\partial_{y}w,y\partial_{x}w\rangle$ $\displaystyle\qquad+2\beta\nu
t^{2}\left(-2\nu\langle\Delta
w,y\partial_{x}\partial_{y}w\rangle-2\left\|y\partial_{x}w\right\|^{2}-4\left\|\partial_{x}\partial_{y}\psi\right\|^{2}\right)$
$\displaystyle\qquad+\frac{3}{2}\gamma
t^{2}\nu\left(\left\|y\partial_{x}w\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right)+\gamma\nu
t^{3}\left(-\nu\left\|y\partial_{x}\nabla
w\right\|^{2}-\nu\left\|\partial_{x}w\right\|^{2}\right).$

After rearranging all the terms we obtain

$\frac{\text{d}}{\text{d}t}\Phi(t)=I_{1}+I_{2}+I_{3}+I_{4}+I_{5},$ (29)

where

$\displaystyle I_{1}$ $\displaystyle=-\frac{\nu}{2}\left\|\nabla
w\right\|^{2}+\alpha\nu\left\|\nabla
w\right\|^{2}-\gamma\nu^{2}t^{3}\left\|\partial_{x}\omega\right\|^{2}$
$\displaystyle I_{2}$
$\displaystyle=-\alpha\nu^{2}t\left\|\Delta\omega\right\|^{2}-4\beta\nu^{2}t^{2}\langle\Delta\omega,y\partial_{xy}\omega\rangle-\gamma\nu^{2}t^{3}\left\|y\partial_{x}\nabla\omega\right\|^{2}$
$\displaystyle I_{3}$
$\displaystyle=-\frac{\nu}{2}\left\|\nabla\omega\right\|^{2}-2\alpha\nu
t\langle y\partial_{x},\partial_{y}w\rangle-\beta\nu
t^{2}\left\|y\partial_{x}\omega\right\|^{2}-2\beta\nu
t^{2}\left\|\partial_{xy}\psi\right\|^{2}$ $\displaystyle I_{4}$
$\displaystyle=-\frac{\nu\alpha}{2}\left\|\nabla\omega\right\|^{2}+4\beta\nu
t\langle y\partial_{x},\partial_{y}w\rangle-\beta\nu
t^{2}\left\|y\partial_{x}\omega\right\|^{2}-2\beta\nu
t^{2}\left\|\partial_{xy}\psi\right\|^{2}$ $\displaystyle I_{5}$
$\displaystyle=-2\nu\beta
t^{2}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\partial_{xy}\psi\right\|^{2}\right]+\frac{3}{2}\gamma\nu
t^{2}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right]$

Recalling now the conditions (12) on the constants $\alpha,\beta,\gamma$,
namely

$\alpha<\frac{1}{2},\quad 16\beta^{2}\leq\alpha\gamma,\quad
2\alpha^{2}<\beta<\frac{1}{4}\alpha,\quad 9\gamma<4\beta,$

we use again the Cauchy-Schwarz inequality and the Young inequality to get

$I_{1}\leq-\gamma\nu^{2}t^{3}\left\|\partial_{x}\omega\right\|^{2};$ (30)
$I_{2}\leq-\frac{1}{2}\alpha\nu^{2}t\left\|\Delta\omega\right\|^{2}-\left(\gamma-\frac{8\beta^{2}}{\alpha}\right)\nu^{2}t^{3}\left\|y\nabla\partial_{x}\omega\right\|^{2}<0;$
(31)
$I_{3}\leq-\frac{\nu}{4}\left\|\nabla\omega\right\|^{2}-(\beta-2\alpha^{2})\nu
t^{2}\left\|y\partial_{x}\omega\right\|^{2}-2\beta\nu
t^{2}\left\|\partial_{xy}\psi\right\|^{2}<0;$ (32)
$I_{4}\leq-\frac{\nu}{2}(\alpha-4\beta)\left\|\nabla\omega\right\|^{2}-\frac{\beta}{2}\nu
t^{2}\left\|y\partial_{x}\omega\right\|^{2}-2\beta\nu
t^{2}\left\|\partial_{xy}\psi\right\|^{2}<0.$ (33)

Moreover, from

$\langle\partial_{xy}\psi,y\partial_{x}\omega\rangle=-\frac{1}{2}\left\|\partial_{xy}\psi\right\|^{2}+\frac{1}{2}\left\|\partial_{xx}\psi\right\|^{2}$
(34)

we can deduce that

$\left\|\nabla\partial_{x}\psi\right\|^{2}\leq\left\|y\partial_{x}\omega\right\|^{2}+3\left\|\partial_{xy}\psi\right\|^{2},$
(35)

hence

$I_{5}\leq-\left(2\beta-\frac{9}{2}\gamma\right)\nu
t^{2}\left[\left\|y\partial_{x}\omega\right\|^{2}+2\left\|\nabla\partial_{x}\psi\right\|^{2}\right]\leq
0.$ (36)

Combining all together we get the upper bound on $\Phi^{\prime}(t)$. ∎

##  3 Additional Linear Estimates

In this section we give some estimates on the semigroup generated by the
linearized operator $\mathcal{L}_{\nu}.$ These estimates play a crucial role
in the estimates for the nonlinear term in the full perturbed system.

######  Lemma 3.1.

Let $g\in L^{2}$ such that $\mathbb{P}_{0}(g)=0$. The following estimates hold
for every $T>0$:

$\int_{0}^{T}\left\|\nabla(e^{\mathcal{L}_{\nu}t}g)\right\|^{2}\leq\frac{1}{2}\nu^{-1}\left\|g\right\|^{2};$
(37) $\int_{0}^{T}\left\|\partial_{x}(e^{\mathcal{L}_{\nu}t}g)\right\|\leq
C\nu^{-2/3}\left\|g\right\|;$ (38)
$\int_{0}^{T}\left\|\nabla\Delta^{-1}(e^{\mathcal{L}_{\nu}t}\mathbb{P}_{\neq}(g))\right\|^{2}_{L^{\infty}}\leq
C(|\log\nu|+1)\nu^{-1/3}\left\|\mathbb{P}_{\neq}(g)\right\|^{2}.$ (39)

###### Proof.

Throughout the proof, $T$ will be any positive time. The first estimate
follows directly from the energy inequality for the linearized problem

$\left\|e^{\mathcal{L}_{\nu}t}g\right\|^{2}+2\nu\int_{0}^{t}\left\|\nabla(e^{\mathcal{L}_{\nu}t}g)\right\|^{2}\leq\left\|g\right\|^{2}$
(40)

and it hold for any $g\in L^{2}$. For the second estimate we note that, for
any $k\neq 0$,

$\left\\{\begin{array}[]{l}\left\|\partial_{x}(e^{\mathcal{L}_{\nu}t}g)\right\|^{2}=k^{2}\left\|e^{\mathcal{L}_{\nu}t}g\right\|^{2}\leq
k^{2}e^{-2\nu k^{2}t}\left\|g\right\|^{2},\\\
\left\|\partial_{x}(e^{\mathcal{L}_{\nu}t}g)\right\|^{2}=k^{2}\left\|e^{\mathcal{L}_{\nu}t}g\right\|^{2}\leq
k^{2}e^{-\gamma|k|^{\frac{1}{2}}\nu^{\frac{1}{2}}t}\left\|g\right\|^{2}.\\\
\end{array}\right.$ (41)

Using the fact that the function $x^{n}e^{-x}$ is bounded for $x\geq 0$ for
every $n\in\mathbb{N}$, we have

$\left\\{\begin{array}[]{l}k^{2}\left\|e^{\mathcal{L}_{\nu}t}g\right\|^{2}\leq\frac{C}{\nu
t}\left\|g\right\|^{2},\\\
k^{2}\left\|e^{\mathcal{L}_{\nu}t}g\right\|^{2}\leq\frac{C}{\nu^{2}t^{4}}\left\|g\right\|^{2}\end{array}\right.$
(42)

and hence

$\left\|\partial_{x}(e^{\mathcal{L}_{\nu}t}g)\right\|=|k|\left\|e^{\mathcal{L}_{\nu}t}g\right\|\leq
C\min\left\\{\frac{1}{\sqrt{\nu t}},\frac{1}{\nu
t^{2}}\right\\}\left\|g\right\|.$ (43)

We have that

$\displaystyle\min\left\\{\frac{1}{\sqrt{\nu t}},\frac{1}{\nu
t^{2}}\right\\}=\left\\{\begin{aligned} (\nu t)^{-\frac{1}{2}}\quad\text{for
}t\leq\nu^{-\frac{1}{3}},\\\ (\nu t^{2})^{-1}\quad\text{for
}t\geq\nu^{-\frac{1}{3}},\\\ \end{aligned}\right.$ (44)

and so

$\displaystyle\int_{0}^{T}\left\|\partial_{x}(e^{\mathcal{L}_{\nu}t}g)\right\|$
$\displaystyle\leq C\left(\int_{0}^{\nu^{-\frac{1}{3}}}\frac{1}{\sqrt{\nu
t}}\text{d}t+\int_{\nu^{-\frac{1}{3}}}^{T}\frac{1}{\nu
t^{2}}\text{d}t\right)\left\|g\right\|$ $\displaystyle\leq
C\left(\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}}+\nu^{-1}\nu^{\frac{1}{3}}\right)\left\|g\right\|$
$\displaystyle\leq C\nu^{-\frac{2}{3}}\left\|g\right\|.$

For the third estimate, we apply the Minkowski inequality and we reduce to

$\int_{0}^{T}\left\|\nabla\Delta^{-1}(e^{\mathcal{L}_{\nu}t}g)\right\|^{2}_{L^{\infty}}\leq\left(\sum_{k>0}\left(\int_{0}^{T}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\text{d}t\right)^{\frac{1}{2}}\right)^{2}.$
(45)

Consider
$\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}$
for $k\neq 0$. Using the one dimensional Gagliardo-Niremberg-Sobolev
inequality we have

$\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\leq
C\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|\left\|(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|.$
(46)

Moreover, interpolating the $L^{2}$ norm we get

$\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\leq
C\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{\frac{3}{2}}\left\|\nabla_{k}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{\frac{1}{2}}.$
(47)

Combining (46) with (21) we deduce

$\left\\{\begin{array}[]{l}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\leq(\nu
k^{2}t^{3})^{-\frac{1}{2}}\left\|g_{k}\right\|^{2},\\\
\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\leq|k|^{-1}\left\|g_{k}\right\|^{2},\\\
\end{array}\right.$ (48)

where the second inequality follows from
$|k|\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|\leq\left\|(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|\leq\left\|g_{k}\right\|$.
Hence, we deduce that

$\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\leq
C|k|^{-1}\min\\{1,(\nu t^{3})^{-\frac{1}{2}}\\}\left\|g_{k}\right\|^{2}.$ (49)

As before, using (49) we have

$\displaystyle\int_{0}^{T}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{2}_{L^{\infty}}$
$\displaystyle\leq C|k|^{-1}\left(\int_{0}^{T}\min\\{1,(\nu
t^{3})^{-\frac{1}{2}}\\}\text{d}t\right)\left\|g_{k}\right\|^{2}$
$\displaystyle\leq
C|k|^{-1}\left(\nu^{-\frac{1}{3}}+\int_{\nu^{-\frac{1}{3}}}^{T}(\nu
t^{3})^{-\frac{1}{2}}\text{d}t\right)\left\|g_{k}\right\|^{2}$
$\displaystyle\leq
C|k|^{-1}\left(\nu^{-\frac{1}{3}}+\nu^{-\frac{1}{2}}\nu^{\frac{1}{6}}\right)\left\|g_{k}\right\|^{2}$
$\displaystyle\leq C|k|^{-1}\nu^{-\frac{1}{3}}\left\|g_{k}\right\|^{2}.$

Using (47) and the Hölder inequality we obtain

$\displaystyle\int_{0}^{T}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{2}_{L^{\infty}}$
$\displaystyle\leq
C\int_{0}^{T}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{\frac{3}{2}}\left\|\nabla_{k}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\int_{0}^{T}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{2}\right)^{\frac{3}{4}}\left(\int_{0}^{T}\left\|\nabla_{k}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|^{2}\right)^{\frac{1}{4}}$
$\displaystyle\leq C\left(\int_{0}^{T}|k|^{-2}\min\\{1,(\nu
t^{3})^{-1}\\}\left\|g_{k}\right\|^{2}\text{d}t\right)^{\frac{3}{4}}\nu^{-\frac{1}{4}}\left\|g_{k}\right\|^{\frac{1}{2}}$
$\displaystyle\leq
C|k|^{-\frac{3}{2}}\nu^{-\frac{1}{2}}\left\|g_{k}\right\|^{2}.$

So, from (45) and using the Cauchy-Schwarz inequality we deduce

$\displaystyle\int_{0}^{T}\left\|\nabla\Delta^{-1}(e^{\mathcal{L}_{\nu}t}g)\right\|_{L^{\infty}}^{2}$
$\displaystyle\leq
C\left(\sum_{k>0}\left(\int_{0}^{T}\left\|\nabla_{k}\Delta_{k}^{-1}(e^{\mathcal{L}_{\nu}t}g)_{k}\right\|_{L^{\infty}}^{2}\right)^{\frac{1}{2}}\right)^{2}$
$\displaystyle\leq
C\left(\sum_{0<k<\nu^{-\frac{1}{3}}}\left(k^{-1}\nu^{-\frac{1}{3}}\left\|g_{k}\right\|^{2}\right)^{\frac{1}{2}}+\sum_{k>\nu^{-\frac{1}{3}}}\left(k^{-\frac{3}{2}}\nu^{-\frac{1}{2}}\left\|g_{k}\right\|^{2}\right)^{\frac{1}{2}}\right)^{2}$
$\displaystyle\leq
C\left(\nu^{-\frac{1}{3}}\sum_{0<k<\nu^{-\frac{1}{3}}}k^{-1}+\nu^{-\frac{1}{2}}\sum_{k>\nu^{-\frac{1}{3}}}k^{-\frac{3}{2}}\right)\sum_{k\neq
0}\left\|g_{k}\right\|^{2}$ $\displaystyle\leq
C\left(\nu^{-\frac{1}{3}}|\log\nu|+\nu^{-\frac{1}{2}}\nu^{\frac{1}{6}}\right)\left\|g\right\|^{2}$
$\displaystyle\leq C\nu^{-\frac{1}{3}}(1+|\log\nu|)\left\|g\right\|^{2}.$

∎

######  Remark 3.2.

Regarding the third estimate, note that using only (46) leads to the series
$\sum_{k>0}\frac{1}{k}$, which is clearly not convergent. The logarithmic
correction term arises when we use (47) to overcome this problem.

##  4 Nonlinear Transition Threshold

Before proving the main result we start by considering the nonlinear problem
(3). We decompose $\omega=\omega_{s}+\tilde{\omega}$, as well as
$\psi=\psi_{s}+\tilde{\psi}$ and $u=u_{s}+\tilde{u}$, where

$\omega_{s}=\mathbb{P}_{0}\omega=\frac{1}{2\pi}\int_{\mathbb{T}}\omega\text{d}x$

is the shear part of the flow while

$\quad\tilde{\omega}=\mathbb{P}_{\neq}\omega=(1-\mathbb{P}_{0})\omega$

is the non shear part. It follows that $\omega_{s}$ and $\tilde{\omega}$
satisfy

$\displaystyle\left\\{\begin{aligned}
&\partial_{t}\omega_{s}-\nu\partial_{y}^{2}\omega_{s}=-\mathbb{P}_{0}(\tilde{u}\cdot\nabla\tilde{\omega}),\\\
&\partial_{t}\tilde{\omega}+y^{2}\partial_{x}\tilde{\omega}-2\partial_{x}\tilde{\psi}-\nu\Delta\tilde{\omega}=-\mathbb{P}_{\neq}(u_{s}\partial_{x}\tilde{\omega}+\tilde{u}\cdot\nabla\omega).\end{aligned}\right.$
(50)

Before proving Theorem 4.1, we consider the following equation

$\partial_{t}\tilde{\omega}-\mathcal{L}_{\nu}\tilde{\omega}=f,$ (51)

where $\mathcal{L}_{\nu}$ is the linearized operator defined in (4) and $f$ is
a time-dependent function. Using the Duhamel formula we can write the solution
for every $t\geq 0$ as

$\tilde{\omega}(t)=e^{t\mathcal{L}_{\nu}}\tilde{\omega}(0)+e^{t\mathcal{L}_{\nu}}\int_{0}^{t}e^{-s\mathcal{L}_{\nu}}f(s)\text{d}s.$
(52)

Using Lemma 3.1 we obtain similar estimates for the solution
$\tilde{\omega}(t)$ for any arbitrary $T>0$. We consider firstly

$\int_{0}^{T}\left\|\partial_{x}\tilde{\omega}\right\|\text{d}t\leq\int_{0}^{T}\left\|\partial_{x}e^{t\mathcal{L}_{\nu}}\tilde{\omega}(0)\right\|\text{d}t+\int_{0}^{T}\left\|\partial_{x}e^{t\mathcal{L}_{\nu}}\int_{0}^{t}e^{-s\mathcal{L}_{\nu}}f(s)\text{d}s\right\|\text{d}t.$
(53)

The first term on the right hand side can be bounded using (38) as

$\int_{0}^{T}\left\|\partial_{x}e^{t\mathcal{L}_{\nu}}\tilde{\omega}(0)\right\|\text{d}t\leq
C\nu^{-2/3}\left\|\tilde{\omega}(0)\right\|$ (54)

while for the second term we use the Minkowski integral inequality and
estimate (38) again as follows

$\displaystyle\begin{aligned}
\int_{0}^{T}\left\|\partial_{x}e^{t\mathcal{L}_{\nu}}\int_{0}^{t}e^{-s\mathcal{L}_{\nu}}f(s)\text{d}s\right\|\text{d}t&=\int_{0}^{T}\left\|\int_{0}^{t}\partial_{x}e^{(t-s)\mathcal{L}_{\nu}}f(s)\text{d}s\right\|\text{d}t\\\
&\leq\int_{0}^{T}\int_{0}^{T}\left\|\partial_{x}e^{(t-s)\mathcal{L}_{\nu}}f(s)\right\|\text{d}t\text{d}s\\\
&\leq C\nu^{-2/3}\int_{0}^{T}\left\|f(s)\right\|\text{d}s.\end{aligned}$ (55)

Combining the two estimates we have

$\nu^{2/3}\int_{0}^{T}\left\|\partial_{x}\tilde{\omega}\right\|\leq
C\left(\left\|\tilde{\omega}(0)\right\|+\int_{0}^{T}\left\|f(s)\right\|\text{d}s\right).$
(56)

Analogously, using estimate (39) we have

$\displaystyle\begin{aligned}
\left(\int_{0}^{T}\left\|\nabla\Delta^{-1}\tilde{\omega}(t)\right\|_{L^{\infty}}^{2}\text{d}t\right)^{1/2}&\leq\left(\int_{0}^{T}\left\|\nabla\Delta^{-1}e^{t\mathcal{L}_{\nu}}\tilde{\omega}(0)\right\|^{2}\text{d}t\right)^{1/2}\\\
&\quad+\left(\int_{0}^{T}\left\|\nabla\Delta^{-1}e^{t\mathcal{L}_{\nu}}\int_{0}^{t}e^{-s\mathcal{L}_{\nu}}f(s)\text{d}s\right\|\text{d}t\right)^{1/2}\\\
&\leq
C(|\log\nu|+1)^{1/2}\nu^{-1/6}\left(\left\|\tilde{\omega}(0)\right\|+\int_{0}^{T}\left\|f(s)\right\|\text{d}s\right)\end{aligned}$
(57)

and we deduce

$\displaystyle\begin{aligned}
\nu^{2/3}\int_{0}^{T}\left\|\partial_{x}\tilde{\omega}\right\|\text{d}t+(|\log\nu|+1)^{-1/2}\nu^{1/6}\left(\int_{0}^{T}\left\|\nabla\Delta^{-1}\tilde{\omega}(t)\right\|_{L^{\infty}}^{2}\text{d}t\right)^{1/2}&\\\
\leq
C\left\|\tilde{\omega}(0)\right\|+&C\int_{0}^{T}\left\|f(s)\right\|\text{d}s.\end{aligned}$
(58)

We are ready to prove the nonlinear threshold theorem.

######  Theorem 4.1.

There exists constants $\varepsilon_{0}\in(0,1),C_{1}>0,c_{1}>0$ such that for
all $\nu<1$ for every $\omega_{in}\in L^{2}$ with

$\left\|\omega_{in}\right\|\leq\varepsilon_{0}(1+|\log\nu|^{\frac{1}{2}})^{-1}\nu^{2/3},$

the solution of (3) is global in time with the bound

$\left\|\tilde{\omega}\right\|\leq
C_{1}e^{-c_{1}\nu^{1/2}t}\left\|\tilde{\omega}_{in}\right\|.$

###### Proof.

Denote by $\mathcal{L}_{\nu}$ the linearized operator, we have

$\partial_{t}\tilde{\omega}-\mathcal{L}_{\nu}\tilde{\omega}=-\mathbb{P}_{\neq}(u_{s}\partial_{x}\tilde{\omega}+\tilde{u}\cdot\nabla\omega),$
(59)

thus, applying Duhamel formula we obtain

$\tilde{\omega}(t)=e^{t\mathcal{L}_{\nu}}\tilde{\omega}_{in}-e^{t\mathcal{L}_{\nu}}\int_{0}^{t}e^{-s\mathcal{L}_{\nu}}\mathbb{P}_{\neq}(u_{s}\partial_{x}\tilde{\omega}+\tilde{u}\cdot\nabla\omega)\text{d}s.$
(60)

Taking the $L^{2}$ norm of the solution and using the linear enhanced
dissipation estimates (16) of the solution operator $e^{\mathcal{L}_{\nu}t}$
leads to

$\displaystyle\begin{aligned} \left\|\tilde{\omega}(t)\right\|&\leq
C_{0}e^{-c_{0}\nu^{\frac{1}{2}}t}\left\|\tilde{\omega}_{in}\right\|+\int_{0}^{t}\left\|u_{s}\partial_{x}\tilde{\omega}+\tilde{u}\cdot\nabla\omega\right\|\text{d}s\\\
&\leq
C_{0}e^{-c\nu^{\frac{1}{2}}t}\left\|\tilde{\omega}_{in}\right\|+\int_{0}^{t}\left\|u_{s}\partial_{x}\tilde{\omega}\right\|\text{d}s+\int_{0}^{t}\left\|\tilde{u}\cdot\nabla\omega\right\|\text{d}s,\end{aligned}$
(61)

where $c_{0},C_{0}$ are the constants defined in Theorem 1.1. Defining

$A(t)=\nu^{\frac{2}{3}}\int_{0}^{t}\left\|\partial_{x}\tilde{\omega}\right\|+\nu^{\frac{1}{6}}(|\log\nu|+1)^{-\frac{1}{2}}\left(\int_{0}^{t}\left\|\tilde{u}\right\|_{L^{\infty}}^{2}\text{d}s\right)^{\frac{1}{2}},$
(62)

it follows from (58) that

$A(t)\leq
K\left(\left\|\tilde{\omega}_{in}\right\|+\int_{0}^{t}\left\|u_{s}\partial_{x}\tilde{\omega}\right\|\text{d}s+\int_{0}^{t}\left\|\tilde{u}\cdot\nabla\omega\right\|\text{d}s\right),$
(63)

for a fixed positive constant $K.$ To achieve nonlinear enhanced dissipation
we proceed as follows. For the second term on the right hand side in (61), we
have by using the definition of $A(t)$ and the hypothesis

$\displaystyle\begin{aligned}
\int_{0}^{t}\left\|u_{s}\partial_{x}\tilde{\omega}\right\|\text{d}s&\leq\int_{0}^{t}\left\|u_{s}\right\|_{L^{\infty}}\left\|\partial_{x}\tilde{\omega}\right\|\text{d}s\\\
&\leq
C\left\|\omega_{in}\right\|\int_{0}^{t}\left\|\partial_{x}\tilde{\omega}\right\|\text{d}s\\\
&\leq C\nu^{-\frac{2}{3}}\left\|\omega_{in}\right\|A(t)\\\
&\leq\frac{1}{8K}A(t),\end{aligned}$ (64)

where in the last inequality we use that $\varepsilon_{0}\ll 1$. Analogously,
for the third term on the right hand side in (61), we have

$\displaystyle\begin{aligned}
\int_{0}^{t}\left\|\tilde{u}\cdot\nabla\omega\right\|&\leq\int_{0}^{t}\left\|\tilde{u}\right\|_{L^{\infty}}\left\|\nabla\omega\right\|\\\
&\leq\left(\int_{0}^{t}\left\|\tilde{u}\right\|_{L^{\infty}}^{2}\right)^{\frac{1}{2}}\left(\int_{0}^{t}\left\|\nabla\omega\right\|^{2}\right)^{\frac{1}{2}}\\\
&\leq\left(\nu^{-\frac{1}{6}}(|\log\nu|+1)^{\frac{1}{2}}A(t)\right)\left(\nu^{-\frac{1}{2}}\left\|\omega_{in}\right\|\right)\\\
&\leq\nu^{-\frac{2}{3}}(|\log\nu|+1)^{\frac{1}{2}}\left\|\omega_{in}\right\|A(t)\\\
&\leq\frac{1}{8K}A(t).\end{aligned}$ (65)

From (63), (64) and (65), we can deduce that

$A(t)\leq K\left\|\tilde{\omega}_{in}\right\|+\frac{1}{4}A(t),$ (66)

hence

$A(t)\leq\frac{4}{3}K\left\|\tilde{\omega}_{in}\right\|.$ (67)

Moreover we have

$\int_{0}^{t}\left\|u_{s}\partial_{x}\tilde{\omega}+\tilde{u}\cdot\nabla\omega\right\|\text{d}s\leq\int_{0}^{t}\left\|u_{s}\partial_{x}\tilde{\omega}\right\|\text{d}s+\int_{0}^{t}\left\|\tilde{u}\cdot\nabla\omega\right\|\text{d}s\leq\frac{1}{4K}A(t)\leq\frac{1}{3}\left\|\tilde{\omega}_{in}\right\|.$
(68)

Fix now $t_{0}$ such that
$C_{0}e^{-c_{0}\nu^{\frac{1}{2}}t_{0}}\leq\frac{1}{6}$, we have that

$\left\|\tilde{\omega}(t_{0})\right\|\leq\frac{1}{6}\left\|\tilde{\omega}_{in}\right\|+\frac{1}{3}\left\|\tilde{\omega}_{in}\right\|=\frac{1}{2}\left\|\tilde{\omega}_{in}\right\|$
(69)

and

$\left\|\tilde{\omega}(t)\right\|\leq C\left\|\tilde{\omega}_{in}\right\|,$
(70)

for all $0<t<t_{0}$. To conclude the proof we use the same iteration argument
used for the linear enhanced dissipation. To do so, we prove that for every
$s\geq 0$ it holds

$\left\|\tilde{\omega}(s+t_{0})\right\|\leq\frac{1}{2}\left\|\tilde{\omega}(s)\right\|$
(71)

and

$\left\|\tilde{\omega}(s+t^{*})\right\|\leq
C\left\|\tilde{\omega}(s)\right\|,$ (72)

for any $t^{*}\in(0,t_{0}].$ Hence, by iteration, it follows that

$\left\|\tilde{\omega}(t)\right\|\leq C\left\|\tilde{\omega}(\lfloor
t_{0}^{-1}t\rfloor t_{0})\right\|\leq C\left(\frac{1}{2}\right)^{\lfloor
t_{0}^{-1}t\rfloor}\left\|\tilde{\omega}_{in}\right\|\leq
C_{1}e^{-c_{1}\nu^{\frac{1}{2}}t}\left\|\tilde{\omega}_{in}\right\|$ (73)

for all $t>0$, and the theorem is proved. Estimates (71) and (72) follow from
the previous computations done for $s=0$ and the fact that
$\left\|\omega_{s}(t)\right\|\leq\left\|\omega_{in}\right\|$. Indeed, we have

$\tilde{\omega}(s+t^{*})=e^{t^{*}\mathcal{L}_{\nu}}\tilde{\omega}(s)-e^{t^{*}\mathcal{L}_{\nu}}\int_{0}^{t^{*}}e^{-\tau\mathcal{L}_{\nu}}\mathbb{P}_{\neq}(u_{s}\partial_{x}\tilde{\omega}+\tilde{u}\cdot\nabla\omega)(\tau+s)\text{d}\tau$
(74)

and hence, analogously to (61)

$\left\|\tilde{\omega}(s+t^{*})\right\|\leq
e^{-c\nu^{\frac{1}{2}}t^{*}}\left\|\tilde{\omega}(s)\right\|+\int_{0}^{t^{*}}\left\|u_{s}\partial_{x}\tilde{\omega}(\tau+s)\right\|\text{d}\tau+\int_{0}^{t^{*}}\left\|\tilde{u}\cdot\nabla\omega(\tau+s)\right\|\text{d}\tau.$
(75)

In order to proceed with the same argument, we notice that

$\left\|u_{s}(\tau+s)\right\|_{L^{\infty}}^{2}\leq
C\left\|u_{s}(\tau+s)\right\|\left\|\omega_{s}(\tau+s)\right\|\leq
C\left\|\omega_{s}(\tau+s)\right\|^{2}\leq C\left\|\omega_{in}\right\|^{2}$
(76)

where the last inequality follow from the fact that
$t\mapsto\left\|\omega(t)\right\|$ is decreasing. Moreover we have

$\left\|\omega(s+t^{*})\right\|^{2}+\frac{\nu}{2}\int_{0}^{t^{*}}\left\|\nabla\omega(\tau+s)\right\|^{2}\text{d}\tau\leq\left\|\omega(s)\right\|^{2}\leq\left\|\omega_{in}\right\|^{2}.$
(77)

Hence, defining, as done in (62),

$B(t^{*};s)=\nu^{\frac{2}{3}}\int_{0}^{t^{*}}\left\|\partial_{x}\tilde{\omega}(\tau+s)\right\|\text{d}\tau+\nu^{\frac{1}{6}}(|\log\nu|+1)^{-\frac{1}{2}}\left(\int_{0}^{t^{*}}\left\|\tilde{u}(\tau+s)\right\|_{L^{\infty}}^{2}\text{d}\tau\right)^{\frac{1}{2}},$
(78)

we can deduce the analogous estimate (67),

$B(t^{*};s)\leq\frac{4}{3}K\left\|\tilde{\omega}(s)\right\|$ (79)

and finally

$\left\|\tilde{\omega}(s+t^{*})\right\|\leq
C_{0}e^{-c_{0}\nu^{1/2}t^{*}}\left\|\tilde{\omega}(s)\right\|+\frac{1}{3}\left\|\tilde{\omega}(s)\right\|$
(80)

from which (71), (72) follow. ∎

#### Acknowledgments

The author wants to thank Michele Coti Zelati for his support and all the
useful and stimulating discussions about this problem.

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# Diffeomorphism invariance and general covariance: a pedagogical introduction

Mateo Casariego<EMAIL_ADDRESS>Instituto Superior
Técnico, Universidade de Lisboa, Portugal  Physics of Information and Quantum
Technologies Group, Centro de Física e Engenharia de Materiais Avançados
(CeFEMA), Portugal  PQI – Portuguese Quantum Institute, Portugal

###### Abstract

Diffeomorphism invariance is a feature that gets sometimes highlighted as
something with profound implications in the physics of spacetime. Moreover, it
is often wrongly associated exclusively with General Relativity. The fact that
diffeomorphism invariance and general covariance are used interchangeably does
not help. Here, we attempt at clarifying these concepts.

##  I Introduction

Back in 2015, when what follows was written, it seemed to me that, in the
context of General Relativity (GR), the notions of (general) covariance and
diffeomorphism invariance were confusing to me. In the literature, an partly,
perhaps, because of Einstein himself [2], the terms were used as synonyms.
Here, we revisit our personal notes: an attempt at a self-contained discussion
of the differences between the two, stressing the importance of active
diffeomorphism invariance as one of the most unique features of GR. It will be
assumed that the reader has a fair level of familiarity with GR and
differential geometry. Still, the aimed audience for this article is not the
specialist. Rather, we hope that our exposition will help an undergraduate
student understand – and consequently admire – the notion of diffeomorphism
invariance and, more importantly, why it is something unique of GR. We
apologize for the short and certainly not up to date references111After all,
this is almost ten years old work!. Good complementary references can be found
in [3, 4, 1, 5, 6, 7, 8].

##  II Tensors and notation

Tensors are defined with respect to some group of transformations. A relation
between tensors still holds if we transform them with respect to the group
used to define them. In Minkowski spacetime this group is the Poincaré
group222Boosts (in the Lorentz way), rotations, and translations.: if we apply
a Poincaré transformation to a Poincaré tensor, we will still have a Poincaré
tensor. Contrary to elementary intuition, partial derivatives are not
necessarily invariant operations, group-tensor-wise. In Minkowski spacetime we
have a `Poincaré covariant derivative', which happens to be the same as the
partial derivative333In coordinates such that
$\mathrm{d}s^{2}=\eta_{\mu\nu}\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}$.; as
long as we use it in our tensorial equations, their `covariance' is
guaranteed. In GR, tensors are general: they are defined to behave as tensor
fields under any (local) invertible coordinate transformation. The `general
covariant derivative' (we will drop the `general' part) is the right
derivative to use to preserve the tensorial character of our equations.

In a coordinate chart, the covariant derivative reads

${\bm{\nabla}}_{\bm{X}}\bm{t}=X^{a}\nabla_{a}\bm{t},$ (1)

where the subscript $a$ in $\nabla_{a}$ is a covariant index rather than a
vector field (it really means ${{\nabla}}_{\mathop{}\\!{\bm{\partial}}_{a}}$).
Defining the connection coefficients $\Gamma^{c}_{\,ab}(x)$ as

$\Gamma^{c}_{\;ab}(x)\mathop{}\\!{\bm{\partial}}_{c}\equiv\nabla_{a}(\mathop{}\\!{\bm{\partial}}_{b}),$
(2)

we get the well-known expression for the covariant derivative along the
coordinate axis $\mathop{}\\!{\bm{\partial}}_{a}$ of the vector field
$\bm{V}$:

${{\nabla}}_{a}\bm{V}=\left((\partial_{a}V^{c})+\Gamma^{c}_{\;ab}V^{b}\right)\mathop{}\\!{\bm{\partial}}_{c}.$
(3)

We will assume a Levi-Civita connection, and denote covariant differentiation
of tensor $\bm{t}$ along the basis vector field
$\mathop{}\\!{\bm{\partial}}_{a}$ as $\bm{t}_{;a}$. For example, for a scalar
field $\phi$ we have $\nabla_{a}\nabla^{a}\phi=\phi_{;a}^{\,\,\,;a}.$

###### Definition 1

A spacetime consists on the pair $\mathop{}\\!{(\mathcal{M},\bm{g})}$, where
$\mathcal{M}$ is a four-dimensional real smooth (_i.e._ ,
$\mathcal{C}^{\infty}$) manifold equipped with a Lorentzian metric444See below
for a definition of a metric as a mapping. $\bm{g}$.

##  III General covariance and diffeomorphism invariance

Our (simplified) principle of general covariance can be stated as follows:

> _A theory is_ generally covariant iff every equation describing its `laws'
> looks the same in all frames of reference, i.e., _equations are relations
> between (general) tensors_.

Let's assume we are interested in writing equations for some scalar field
$\phi$. Then, our theory would be generally covariant if given a spacetime
$\mathop{}\\!{(\mathcal{M},\bm{g})}$ – which we assume to be a solution to
Einstein's field equations –– the equations of motion (EOM) of the scalar take
the same form in any coordinate system. For example, the equation
$\phi_{;a}^{\,\,\,\,;a}=0$ is generally covariant, while
$\phi_{,a}^{\,\,\,\,,a}=0$ is not.

Now, since a change of coordinates is a diffeomorphism from $\mathcal{M}$ to
itself, we could say that the EOM of the field are _diffeomorphism invariant_.
This is true if we regard the diffeomorphism as a _passive_ one, _i.e._ , such
that all it does is `permute' the points of the manifold. In other words:
giving different names to points should not affect the physics.
There is nothing deep in this general covariance feature of equations. One can
`covariantize' everything in Minkowski spacetime to make equations look `GR-
like' without any effort whatsoever. As stated in [5], Kretschmann even
claimed that _any_ theory can be made generally covariant, although this might
require quite a lot of mathematical ingenuity. We have not found a proof of
this, though it seems to be generally accepted [4] that this is the case.

_Active_ diffeomorphisms are quite different, however, and not every theory is
invariant under their action. When people say that GR is a _diffeomorphism
invariant_ theory, what they really mean is that GR is invariant under
_active_ diffeomorphisms. This happens to have profound implications in the
interpretation of spacetime, as opposed to the `triviality' of general
covariance, which is physically meaningless.
Perhaps the source of confusion stems from the fact that active and passive
diffeomorphisms are merely different interpretations of a single mathematical
transformation (a diffeomorphism). For instance, a diffeomorphism locally
given by smooth maps

$f:x^{a}\longrightarrow f^{a}(x)$ (4)

produces a new metric $\tilde{g}_{ab}$ from an old metric $g_{ab}$ by means of
the usual chage of coordinates formula:

$\tilde{g}_{ab}(x):=\frac{\partial f^{c}(x)}{\partial x^{a}}\frac{\partial
f^{d}(x)}{\partial x^{b}}g_{cd}(f(x)).$ (5)

However, there is nothing in this formula telling us whether we should
interpret the diffeomorphism of Eq. (4) as a passive or as an active one. A
mathematical formula _per se_ does not `mean' anything: it is pure structure
555Although in Tegmark’s ‘mathematical universe’ [10] a formula can actually
mean ‘everything’, including the interpretation (‘baggage’) that we humans
give to the formula itself. In this hypothesis, “ _…our successful [physical]
theories are not mathematics approximating physics, but mathematics
approximating mathematics_ ”. We shall not worry about this, even when in
principle we endow the _External Reality Hypothesis_ , that Tegmark claims to
imply by necessity the _Mathematical Universe Hypothesis_..
At this point, to see the difference between passive and active
diffeomorphisms, it is convenient to switch to a coordinate-free formulation.
In order to do so, let us first define a metric $\bm{g}$ in such a coordinate-
free way 666There are more rigorous and abstract definitions in terms of fiber
products and tangent bundles. For us, this version will suffice.:

###### Definition 2

A _metric_ $\bm{g}$ is a map from points $p\in\mathcal{M}$ to the tensor
product of the cotangent space at those points, _i.e._ ,

$\displaystyle\bm{g}:\,$
$\displaystyle\mathcal{M}\rightarrow{TM}^{*}\otimes{TM}^{*}$ (6)
$\displaystyle p\longmapsto\bm{g}(p).$ (7)

By Sylvester's law of inertia 777Essentially, a theorem stating that the
number of pluses and minuses in a quadratic metric depends only on whether you
come from high energy physics, or GR. we can write:

$\bm{g}=-\bm{\Theta}_{0}\otimes\bm{\Theta}_{0}+\sum_{i=1}^{3}\bm{\Theta}_{i}\otimes\bm{\Theta}_{i},$
(8)

where $\bm{\Theta}_{k}$ ($k=0,\ldots,3$) form a basis of one-forms (the
subindex is merely a label, _not_ a covariant index). Then, if
$\mathbf{E}_{l}$ ($l=0,\ldots,3$) are the dual basis of vector fields, we can
define a contraction of the metric in such a way that it gives a _function_
$s(p)$ as an output 888We omit the details. The idea is that the contraction
should be defined in such a way that each one-form gets paired with one dual
vector field. Each pairing produces a function. The tensor product of
functions becomes trivially equivalent to simple multiplication of functions
and, since a sum of smooth functions is a smooth function, we get the desired
result.. This function is smooth and it can be used to define a notion of
distance $d_{\bm{g}}(p,p^{\prime})$ between any two points $p$ and
$p^{\prime}$ on the manifold by integration of the one-form $\mathrm{d}s$:

$d_{\bm{g}}(p,p^{\prime})\equiv\int_{p}^{p^{\prime}}\mathrm{d}s.$ (9)

It is in this sense that we can say that the metric defines a map from the
Cartesian product of $\mathcal{M}$ with itself to the real set:

$\displaystyle d_{\bm{g}}:\,$
$\displaystyle\mathcal{M}\times\mathcal{M}\rightarrow\mathbb{R}$ (10)
$\displaystyle(p,p^{\prime})\longmapsto d_{\bm{g}}(p,p^{\prime}).$ (11)

Take $\bm{g}$ to be a solution to Einstein's field equations (EFE)
$\mathbf{G}=8\pi\mathbf{T}$. Then $\mathop{}\\!{(\mathcal{M},\bm{g})}$ is a
valid spacetime and every classical equation of motion of test particles will
have a unique solution in some region, provided that this region can be
foliated into spacelike hypersurfaces. It is clear that a passive
diffeomorphism gives the same physical situation, because the way we label the
points of $\mathcal{M}$ is not associated to any physical observable
999Although some labellings are more sensible than others and can make our
lives much easier. Also, two different coordinate systems related by a
diffeomorphism may have ‘problems’ around certain points. In those cases, we
have to resort to _local_ diffeomorphisms. (_e.g._ , the transition from
Cartesian to polar coordinates is not diffeomorphic when we are close to the
origin.).
Now, consider a diffeomorphism $\Phi$ from $\mathcal{M}$ to itself. In
general, we have

$d_{\bm{g}}(p,p^{\prime})\neq d_{\bm{g}}(\Phi^{-1}(p),\Phi^{-1}(p^{\prime})).$
(12)

However, since $\Phi^{-1}$ is a smooth map, it ought to be possible to define
a _new_ metric $\tilde{\bm{g}}$ on $\mathcal{M}$ such that its associated
distance function is given by

$d_{\tilde{\bm{g}}}(p,p^{\prime})\equiv
d_{\bm{g}}(\Phi^{-1}(p),\Phi^{-1}(p^{\prime})).$ (13)

The pair $(\mathcal{M},\tilde{\bm{g}})$ is still a valid spacetime and the
claim is that it is physically indistinguishable from
$\mathop{}\\!{(\mathcal{M},\bm{g})}$. This is equivalent to saying that
$\tilde{\bm{g}}$ solves the _same_ equations (Einstein's) as $\bm{g}$ does.
This is what the sentence ``GR is a diffeomorphism invariant theory'' really
means and it is clearly not a trivial fact (as passive diffeomorphism
invariance is). Invariance under passive diffeomorphisms talks about the
_form_ of equations. Invariance under active diffeomorphisms tells us
something about the _mathematical structure_ of the theory and inevitably
hints towards a _relational_ 101010In [7, Sec. 6.2.] it is argued that the
correct statement is that GR suggests a _structuralist_ view rather than a
_relational_ one: “ _(…) that the observables are indifferent to matters of
spacetime point role does not imply there are no spacetime points_ ”. For us,
the word ‘relation’ is much more suggestive than ‘structure’ in the sense that
we aim to picture the physical world as a network of information being
exchanged between observers. Yet, this is not the place to discuss ‘observers’
and ‘information’: that would inevitably lead us to adopt an interpretation of
the density operator in rigged Hilbert spaces. interpretation: as long as the
relations between entities remain the same, it does not matter `where' or
`how' they are distributed.
To put it in an intuitive, physical way: imagine a two-dimensional grid with
objects on it. For simplicity, assume that time is `frozen' and that the
picture is static (see Fig. 1). We are standing somewhere _inside_ the grid
and looking at things. When we change our position with respect to the grid,
the objects will appear to move with respect to each other but they will
remain fixed to their position on the grid, because if we move with respect to
the grid we can always say that it's the grid that moved with respect to us
(if there were such an absolute frame of reference). This is a passive
diffeomorphism: we haven't really changed anything, just our point of view.
Thus, we shouldn't expect to get a different set of equations of motion.

Figure 1: A passive diffeomorphism, acting on a static space for simplicity,
just changes our point of view: Observer $O_{1}$ uses an artificial background
to label events $A$ and $B$ –simply positions of objects– as $a_{1}$ anb
$b_{1}$. Observer $O_{2}$ does the same, obtaining $a_{2}\neq a_{1}$ and
$b_{2}\neq b_{1}$, respectively. The passive diffeomorphism is an invertible
map that relates two descriptions of the same situation, and its effect is
pictured as a set of black arrows. The white grid in which `things happen' is
taken to represent a `two-dimensional ruler' for the spacetime $\mathcal{M}$,
_i.e._ , a metric. The artificial background can be thought as the `language'
and the passive diffeomorphism as the `dictionary'.

Now imagine that we stay fixed to our position in the grid but the objects
move in a smooth but otherwise arbitrary way with respect to it, and
–consequently– with respect to us. Then it is not so trivial to `see' that the
physical phenomena predicted by GR remains unchanged, _i.e._ , to see that it
is possible to find a new grid (a new metric) which, used as a measuring tool
for the new distribution of the objects (the `energy-matter'), will give us
the same exact physics as we had before.
Let us illustrate this ideas with two complementary figures of an active
diffeomorphism. In Fig. 2 we emphasized the active part of the diffeomorphism,
in the sense that if does move the matter `around'. But even if some sort of
diabolical $n$-dimensional creature (with $n\geq 5$) performed (instantaneous)
rearrangements of the matter in the universe we wouldn't be able to tell from
the inside 111111As far as GR is concerned, of course.. In a sense, that evil
creature would be mapping our entire 4-dimensional universe to a different,
but physically indistinguishable one, so she would see something like what is
shown in Fig. 2.

Figure 2: An active diffeomorphism acting on a solution to EFE produces
another solution: Observer at point $p$ describes the relations between points
$p_{A}$ and $p_{B}$ by means of a metric $\bm{g}$ pictured as a red grid. An
active diffeomorphism _moves_ the points to $p^{\prime}_{A}$ and
$p^{\prime}_{B}$. Then, there exist a _different_ metric $\tilde{\bm{g}}$
(green lattice) such that the relations between the new points are physically
indistinguishable from the previous case. Observer $p$ has then the freedom to
choose which description to use. Figure 3: The action of the diffeomorphism
$\Phi:\mathcal{M}\rightarrow\mathcal{M}$ on the spacetime
$\mathop{}\\!{(\mathcal{M},\bm{g})}$ produces a new spacetime
$(\mathcal{M},\tilde{\bm{g}})$ that is physically equivalent to the former. A
point $p\in\mathcal{M}$ gets moved to $\Phi(p)=p^{\prime}\in\mathcal{M}$ and
the metric, represented as a grid, gets `moved' to a new one, $\tilde{\bm{g}}$
under the action of $\Phi^{*}$, a map induced from $\Phi$ that maps metrics to
metrics. This map is the pull-back and it is an isomorphism between products
of cotangent spaces at points $p$ and $\Phi(p)$.

In the language of gauge theories, it is often stated that
$G\equiv\text{Diff}(\mathcal{M})$ is the _gauge group_ of GR. This means that
there is some sort of `redundancy' in the theory (or extra degrees of
freedom), since two mathematically different solutions
$\mathop{}\\!{(\mathcal{M},\bm{g})}$ and $(\mathcal{M},\tilde{\bm{g}})$ are
considered physically indistinguishable if they are related to each other by
an (active) element of $G$. Gauging this group, _i.e._ , making it _local_ ,
leads to GR. Then, once is `forced' to write down equations that are gauge
invariant, because the choice of a gauge is completely arbitrary. The same
thing happens in electrodynamics, as we learn when we study quantum field
theory: when one writes down the Lagrangian for a _free_ electron (in flat
spacetime) and imposes –based on common sense– that the physics (that is, the
Lagrangian) remains invariant under _local_ phase shifts (represented by
elements of the group $U(1)$), then one naturally picks up a new vector field
$\mathbf{A}$, that also has to transform in a nice way and that we call the
`gauge field'. The free field then couples to the gauge field, producing an
_interaction_ term in the Lagrangian that is gauge-invariant. Neither the
original electron field, nor the gauge field are gauge invariant, but the way
they appear in the Lagrangian is through a gauge invariant term. One then
proceeds to define gauge invariant things like the tensor field
$\mathbf{F}:=\mathrm{d}\mathbf{A}$ and claims that the physics of the theory
is contained _only_ in those objects.
Likewise, diffeomorphisms in GR are regarded as extra, unphysical degrees of
freedom: the physics must be contained only in gauge-invariant quantities.
This is in flagrant contrast with what experience tells us: in `real life'
things are constrained to fixed frames of reference, and one can measure
`gauge-variant' [8] quantities, such as the energy, proper time, the electric
field, and so on.
Systems _couple_ to each other through certain fields, which are gauge-
variant. By means of these couplings, `relative observables' appear in the
equations. In [8], the author claims that this leads to a _relational_
interpretation:

> ``The fact that the world is well described by gauge theories expresses the
> fact that the quantities we deal with in the world are generally quantities
> that pertain to relations between different parts of the world, that is,
> which are defined across subsystems.''

##  IV An example

Take for instance two isolated observers, Felice and Dob, moving along world
lines $x_{F}^{a}(\tau_{F})$ and $x_{D}^{a}(\tau_{D})$, parametrised by their
proper times. Between them there is a region in which there is a gravitational
field described by some metric $\bm{g}$. _We_ 121212We italize for obvious
reasons: we have more information than the total information of the two
_isolated_ observers: we know the metric! In Ref. [9] – where the author
discusses the ‘problem of time’, the dichotomy between the notion of time as
seen by us, ‘god-like’ observers and that based on the information available
to the true observers is achieved by the consistent use of _theorists’ time_
and _participant-observers’ time_. Likewise, in Ref. [10], the author uses the
terms ‘bird’ and ‘frog’. In the philosophy of physics literature, this vantage
point of view is often referred to as ‘Archimedean point’, of _punctus
Archimedis_.  `know' this, but they don't: they are isolated from the rest of
the world in small elevators without windows. Each observer uses a clock to
measure their proper time. We know how to write this: it is the line integral
of the gravitational field along a single world line. For Felice we have that
the time elapsed between events $P,Q\in x_{F}^{a}(\tau_{F})$ is

$\Delta\tau_{F}(P,Q)=\int_{P}^{Q}\operatorname{d}\tau_{F}\sqrt{-g_{ab}(x_{F}(\tau_{F}))\frac{\operatorname{d}x_{F}^{a}(\tau_{F})}{\operatorname{d}\tau_{F}}\frac{\operatorname{d}x_{F}^{b}(\tau_{F})}{\operatorname{d}\tau_{F}}}.$
(14)

The last expression is obviously gauge-invariant, but it doesn't correspond to
what Felice would write. How would she know what values to assign to the
events $P,Q$ if she only has access to her small, sad elevator? Instead, she
would simply get a number by looking at a clock: something that she would
write as

$\Delta\tau_{F}=\int_{0}^{1}\operatorname{d}\tau_{F}\sqrt{-g_{00}(\tau_{F},0,0,0)},$
(15)

where the integration limits denote the same events $P,Q$ but as described by
Felice: when the clock hand is here and when it is there. We set these to be 0
and 1 with no loss of generality. For her, there is no way of knowing the
function $g_{00}$ for arbitrary values of the arguments. Through local
experiments, all she can do is erect a local coordinate system (a tetrad) and
assume that there might be a metric $g_{ab}$ out there. Then, that
hypothesised metric should enter her description of time in the way of
equation Eq. (15) above. This number $\Delta\tau_{F}$ is clearly gauge-
dependent as it involves a component of the metric tensor in a particular
frame of reference. The same thing would happen to Dob. Let us assume that he
is also measuring the time lapsed between events $P$ and $Q$. In other words,
the world lines of Felice and Dob meet at these points. They don't have to
collide: passing each other by a few meters will do.
For Dob, things are completely similar: he measures some gauge-variant number
$\Delta\tau_{D}$.

It is only when the two observers exchange information _through_ the causal
structure given by the gravitational field – _i.e._ , by _coupling_ to the
field– that they can describe the situation with the gauge-invariant quantity
of equation Eq. (14). In other words, based on the knowledge of $g_{ab}$, they
can find and predict the gauge-invariant function $\tau_{F}(\tau_{D})$ (or its
inverse $\tau_{D}(\tau_{F})$).
Gauge-variant quantities can be used to construct gauge-invariant ones by
promoting the original system of two isolated _local_ observers to a larger
system given by them _and_ the field. In practice, this promotion is
effectively achieved by the exchange of information between the two observers.
In many situations, this exchange of information –signalling– involves
_classical_ physics, as in the following example (borrowed from [8]): an
observer on Earth with two similar clocks throws one of them up in the air
(proper time $\tau^{\prime}$) and, when it comes back, he compares the value
of $\tau^{\prime}$ with $\tau$ (proper time on Earth). GR correctly predicts
the function $\tau(\tau^{\prime})$ through the knowledge of $g_{ab}$. GR
cannot predict $\tau$ or $\tau^{\prime}$ alone, although both of them are
measurable. Hence the _relational_ nature of gravity.
In Fig. 2 above we already saw this from the mathematical point of view:
diffeomorphism invariance of EFE means that only relations between events can
be predicted.
Let us quote some examples that Rovelli [Rovelli:1999hz] gives as physical
gauge-invariant quantities that can be predicted by GR and confirmed
experimentally:

> ``Examples of diff-invariant quantities (…) are the Earth-Venus distance
> during the last solar eclipses, the number of pulses of a pulsar in a binary
> system that reach the Earth during one revolution of the system (that is,
> between two Doppler maxima), the energy deposited on a gravitational antenna
> by a gravitational wave pulse and, in fact, any significative physical
> quantity measured in general relativistic experimental or observational
> physics.''

When the information exchanged behaves quantum mechanically, things become far
more subtle, as we should expect. This is because in most of the
interpretations of quantum mechanics (QM), the world splits into two: the
system under observation (SUO) and the observer (O), which is linked to an
apparatus. We shall now adopt the terminology of [9]: By `theorists' we will
mean _us_ , _i.e._ , beings with access to the whole theory (what we have
called `god-like observers'); and by `participant-observers' we will mean the
real observers, that describe the world around them collecting information by
means of experiments.
From the theorist's point of view, in GR is possible to enlarge systems so
that, through the coupling of $g_{ab}$ to the original system, gauge-invariant
quantities involving gauge-variant objects can be calculated. The calculation
of gauge-invariant quantities means that _predictions_ can be made. There is,
in principle, nothing wrong with this, for if the assumptions made by the
theorist were wrong, experiment will tell. Thus, a theorist can couple distant
objects in the universe by means of the theory of GR, make predictions, and
then ask their experimental colleagues to test them. In a way, the role of the
theorist is completely irrelevant insofar as the outcomes are concerned: there
is only one way things can happen in a `block universe'. This is a clear
consequence of the classicality of GR: the observer does not affect the SUO,
which evolves in a predetermined way regardless of what `questions' (_i.e._ ,
measurements) are asked.

We shall disregard the problem of the `conscious observer'. In principle, an
observer could be `anything'. What makes humans different from, say, turtles,
is that the former can perform _experiments_ , recording data from
measurements in notebooks; whereas the latter, being an `unconscious' system,
may not have a memory and no notebook available. One could say that any
interaction represents a measurement. But it is clear that a collection of
measurements is not an experiment. Let us close this rather philosophical
paper with some words by Anderson [9]:

> ``There is a tendency to anthropomorphize observers which associates with
> them a connotation of awareness. This connotation is wholly undeserved, and
> unnecessarily complicates the language of measurement and observation. I
> shall speak of observers even in the absence of human beings. An observer is
> simply a subsystem whose state we choose to focus on as it interacts with
> other subsystems.''

## Acknowledgments

I thank George Jaroszkiewicz, my advisor while at Nottingham.

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