# Gravity entanglement, quantum reference systems, degrees of freedom

Marios Christodoulouab, Andrea Di Biagioa, Richard Howlcd, Carlo Rovelliefg
aInstitute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian
Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria  bVienna Center
for Quantum Science and Technology (VCQ), Faculty of Physics, University of
Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria  cQuantum Group, Department
of Computer Science, University of Oxford, Wolfson Building, Parks Road,
Oxford, OX1 3QD, United Kingdom  dQICI Quantum Information and Computation
Initiative, Department of Computer Science, The University of Hong Kong,
Pokfulam Road, Hong Kong  eAix-Marseille University, Université de Toulon,
CPT-CNRS, F-13288 Marseille, France.  fDepartment of Philosophy and the Rotman
Institute of Philosophy, 1151 Richmond St. N London N6A5B7, Canada  gPerimeter
Institute, 31 Caroline Street N, Waterloo ON, N2L2Y5, Canada

(August 27, 2024)

###### Abstract

We discuss the physical interpretation of the gravity mediated entanglement
effect. We show how to read it in terms of quantum reference systems. We
pinpoint the single gravitational degree of freedom mediating the
entanglement. We clarify why the distinction between longitudinal and
transverse degrees of freedom is irrelevant for the interpretation of the
results. We discuss the relation between the LOCC theorem and the
interpretation of the effect, its different relevance for, respectively, the
quantum gravity and quantum information communities, and the reason for the
excitement raised by the prospect of detection.

##  I Introduction

The suggestion that a detection of Gravity Mediated Entanglement (GME) might
be achieved in the lab Bose  _et al._ (2017); Marletto and Vedral (2017) has
raised much interest and a lively discussion on the precise significance of
this detection (see Huggett  _et al._ (2022) and full references there).
Suggested interpretations range from evidence for the quantum nature of
geometry Christodoulou and Rovelli (2019); Christodoulou  _et al._ (2022), to
agnosticism about the quantum or classical nature of the ‘true’ gravitational
degrees of freedom Anastopoulos and Hu (2018); Fragkos  _et al._ (2022). Here
we contribute to the discussion with several observations.

We consider the simplest version of GME: two particles of mass $m$ and
positions $X$ and $Y$ are each set into a quantum superposition of two
positions, say $X_{\pm}$ and $Y_{\pm}$, and kept in this state for a time $t$.
Then the wave packets of both are recombined. In one of the four branches the
particles are kept at a short distance $d$ from each other. The gravitational
potential energy $\delta E=-Gm^{2}/d$ in this branch, where $G$ is the Newton
constant, gives rise to a difference in the total energy with respect to the
other branches, that causes a relative shift in the phase of the branch

$\delta\phi=-\frac{\delta E}{\hbar}t=\frac{Gm^{2}t}{\hbar
d}=\left(\frac{m}{m_{\mathrm{Pl}}}\right)^{2}\frac{ct}{d},$ (1)

where $c$ is the speed of light and $m_{\mathrm{Pl}}$ the Planck mass. The
effect of the phase in one branch is that the state of the two particles can
get entangled. (For complete quantum field theoretical accounts of the effect,
see for instance Christodoulou  _et al._ (2022); Chen  _et al._ (2022a).) The
entanglement can be detected by measuring degrees of freedom entangled with
the position (for instance a spin) and checking the violation of the Bell
inequalities. This detection is evidence that the gravitational field is not
classical, since two systems cannot be entangled by interacting via a third
classical system. Since we have learned from general relativity that the
spacetime geometry measured by rods and clocks is determined by the
gravitational field, a measurement of the GME effect is a detection of quantum
properties of the physical geometry. This perspective has raised the lively
interest of the quantum gravity community.

A number of cautionary remarks regarding the interpretation of the effect have
been raised:

  1. (a)

The experiment is in the non-relativistic regime: it cannot probe relativistic
aspects of quantum gravity.

  2. (b)

In this regime the notion of field can be dropped altogether and interactions
can be described as instantaneous at a distance. Hence the experiment cannot
probe aspects of the field nature of gravity.

  3. (c)

The experiment can be described using the field’s longitudinal modes only.
These are ‘enslaved’ to matter, they are not part of the radiative modes
generally considered in field quantization. Hence the effect does not test
‘genuine’ quantum aspects of gravity Anastopoulos and Hu (2018); Anastopoulos
_et al._ (2021).

  4. (d)

The impossibility of entangling degrees of freedom with a classical mediator
can be shown in the context of the LOCC protocol Horodecki  _et al._ (2009),
but the hypotheses of this protocol are not clearly satisfied in the
experiment.

In the following, we offer some considerations on the GME that shed light on
these issues. We discuss in particular the following questions

  1. i.

What exactly is the degree of freedom mediating entanglement, if any?

  2. ii.

What is the physical significance of the split between longitudinal and
transverse modes?

  3. iii.

What is the exact difference between this and past experiments involving
gravity and quantum matter?

  4. iv.

What does the experiment exactly provide evidence for, by itself or with
additional assumptions?

Some light on these questions is shed by viewing the experiment from the
‘quantum reference frames’ perspective Rovelli (1991); Giacomini  _et al._
(2019), as we do in the next section.

##  II Quantum reference systems account of GME

A fundamental tenet of general relativity is that the laws of physics are the
same in all reference systems, where by ‘reference system’ is meant here an
arbitrary (smooth) choice of spacetime coordinates. The relevance of reference
systems for quantum mechanics and quantum gravity has long been noticed and
has been recently exploited in a number of interesting situations Aharonov, Y.
, Susskind (1967); Aharonov, Y., Susskind (1967); Aharonov, Y. Kaufherr
(1984); Rovelli (1991); Bartlett, S. D., Rudolph, T., Spekkens (2007); Palmer,
M. C., Girelli, F., Bartlett (2014); Castro-Ruiz  _et al._ (2020); Pereira, S.
T. Angelo (2015); Giacomini  _et al._ (2019). Since coordinates are arbitrary
labels, nothing forbids us from using coordinates determined by the particles,
rather than Cartesian coordinates determined by distances. Let us apply this
logic to the GME setting.

We introduce general coordinates $x(X)$ such that $x(X)=0$ and $x(Y)=1$. In
words, we can choose a coordinate system in which one particle has coordinate
$x=0$ at all times and the other has coordinate $x=1$ at all times. We can
change the spatial coordinates only, without touching the time coordinate
(defined a physical clock).

This can be done in all four branches where the particles are at respective
Cartesian coordinate positions $(X_{i},Y_{j})$, where $i,j\in\\{+,-\\}$. Since
the relation between the Cartesian and the new coordinates is different in
each branch, the change of coordinates is different in the different branches,
and the metric will then be different in the different branches. Let
$g^{(i,j)}_{ab}(x)$ be the metric in the $(i,j)$ branch, in these coordinates.

The distance between the two particles is then given by the standard general
relativistic formula for the distance, which is the length of the shortest
curve $\gamma$ on a fixed time surface. This definition is diffeomorphism
invariant in this context. Adapting the coordinates so that the curve $\gamma$
is along the $x$ direction, the distance is

$d_{(i,j)}(t)=\int_{0}^{1}\sqrt{g^{(i,j)}_{xx}(x,t)}\ \differential x.$ (2)

This formulation shows that the relevant degree of freedom for the experiment
is a specific function of the gravitational field $g_{ab}(x,t)$: the line
integral $d$, which has a different (time dependent) value $d_{(i,j)}(t)$ in
each branch. In this coordinate system, the state of the system during the
experiment can be written as

$|\psi(t)\rangle\ =\ \
|X\\!=\\!0\rangle\otimes|Y\\!=\\!1\rangle\otimes\sum_{{(i,j)}}|d_{(i,j)}(t)\rangle$
(3)

where $|X=0\rangle$ and $|Y=1\rangle$ are the states of the particles and
$|d\rangle$ is a (non normalised) quantum state of the variable $d$ of the
gravitational field.111Notice that both superposition and entanglement are
frame-dependent, as nicely illustrated for example in Giacomini  _et al._
(2019).

This drastic effect of the choice of reference frame brings forth the fact
that in a gravitational context the physical degrees of freedom of the
particles cannot be disentangled from those of the geometry. The relevant
(relative) variable is the distance between the two masses, which is a non
local function of the gravitational field. Since the energy in each branch
depends on this distance, the phase difference between the branches follows.

Formally, the Hilbert space of the particles’ positions plus the gravitational
field is restricted by the diffeomorphism constraint, which forces the state
to be independent from changes of coordinates. This allows us to consider only
diffeomorphism invariant quantities, and $d$ is such a quantity: a relational
quantity, that depends on both field and particles.

If the particles have a degree of freedom entangled with their distance, for
instance a spin variable –or any other path dependent observable– that takes
the values $\pm$ in the respective $\pm$ branches, then this degree of freedom
gets entangled with $d$:

$|\psi(t)\rangle\ =\
\sum_{i,j\in\\{+,-\\}}|X\\!=\\!0,i\rangle\otimes|Y\\!=\\!1,j\rangle\otimes|d_{(i,j)}(t)\rangle$
(4)

Notice that in Cartesian coordinates the information about the distance is
coded into the coordinate positions of the particle, and the spins get
entangled with this position. In the reference frame we have chosen here,
instead, the spins are directly entangled with the relevant variable, $d(t)$,
whose dependence on the gravitational field is manifest. The coordinate
position of the particles, which is pure gauge, is not affected by the
dynamics and does not get entangled with anything.

Since $d(t)$ is time dependent during the experiment, the non-commuting
observable that allows the mediating system to transfer entanglement are $d$
and its conjugate variable $p_{d}$.222In the Hamiltonian formulation of
general relativity, the conjugate momentum to the spatial metric, hence also
$d_{p}$, is related to the extrinsic curvature of spacetime Misner  _et al._
(1973). When the particle is split into a superposition of distances, say with
a Stern-Gerlach apparatus, the variable $d$ changes with time. This can be
modelled as an interaction between the particle’s spin and $d$, which has the
result of modifying the value of $d$. (The particles’s position is gauge fixed
in this reference frame.) If the initial spin of one particle is in the state
$\ket{s_{o}}=c_{+}\ket{+}+c_{-}\ket{-}$ and the initial distance is $d_{o}$,
the interaction’s evolution $U$ leads to

$U|s_{o},d_{o}\rangle=\sum_{i}c_{i}\,|i,d_{i}\rangle.$ (5)

Similarly, the evolution entangles the second particle’s spin with $d$. Thus,
the spins of the two particles will also get entangled _through_ $d$, giving
rise to the four branches.

The identification of $d$, namely the _distance_ between the particles, as the
relevant entangling variable, emphasizes the fact that the GME effects can be
interpreted as probing quantum properties of the spacetime _geometry_. Of
course in physics geometry is operationally _defined_ by measurements
performed with material instruments, and the astounding empirical success of
general relativity supports the fertility of Einstein’s idea that this
geometry is a manifestation of the gravitational field. Here we see all this
in place. The GME apparatus can be seen as a devise indirectly measuring a
distance, and the effect of its being in superposition.

##  III The non-relativistic limit and the implications of the experiment

In the previous section, we have employed general relativistic concepts and
assumed that these can be used in this context. In particular, we have assumed
that the gravitational field and its function $d$ are dynamical variables. The
experiment, however, is in the non-relativistic limit. As such, it can be
described in non-relativistic terms. In this language, each particle acts
instantaneously on the other with a force. Such action-at-a distance describes
well the world in this approximation. In this picture, there is no mediator of
the interaction and no degree of freedom associated to a mediator.

This fact has raised a certain confusion in the discussion about the GME
experiments. More precisely, a positive outcome of the experiment is
compatible with a world where gravity has no special quantum properties and is
an instantaneous interaction at a distance.

Why then these experiments are considered relevant for quantum gravity in the
quantum gravity community? The answer is that we know—independently from these
specific experiments—that in the real world gravity is _not_ an instantaneous
interaction at a distance, and—in fact—that there is no instantaneous
interaction at a distance. It is only by folding this additional knowledge in,
that the experiment becomes informative for quantum gravity.

In other words, while the experiment alone might not have strong implications,
the experiment in conjunction with the fact that we have good reasons to
believe, from a hundred years of experiments, that gravity is mediated by a
field that propagates information at finite speeds does have strong
implications. If the information between the two particles is mediated by a
field, and if the two particles get entangled, this does definitely count as
evidence that the field cannot be a classical field. Therefore the detection
of the GME effect leaves us with an alternative: either there is no such thing
as the gravitational field, or this field cannot be a classical.

##  IV Longitudinal and transverse components of the field

The GME experiments are in the linear (weak field) regime. In this regime, the
field can be usefully decomposed into a sum of components that can be treated
differently. There are several ways to implement this decomposition. To
illustrate the fact, its implications, and also some common misunderstandings
it generates, it is easier to discuss the electromagnetic version of the
field-mediated entanglement effect, which is parallel but simpler Chen  _et
al._ (2022b). In this case the masses have charges and the mediator is the
electromagnetic field.

A vector field $\vec{A}_{\perp}$ is called transverse if

$\vec{\nabla}\cdot\vec{A}_{\perp}=0,$ (6)

while a vector field $A_{\parallel}$ is called longitudinal if it can be
written as the gradient of a scalar function

$\vec{A}_{\parallel}=\vec{\nabla}f.$ (7)

If $f$ is harmonic, namely it satisfies the Poisson equation $\Delta f=0$, the
field $\vec{A}=\vec{\nabla}f$ is, according to these definitions, both
longitudinal and transverse, but it is conventional to call it longitudinal.
If boundary conditions are fixed, the Poisson equation has a unique solution.

In 1930, Enrico Fermi showed that the four dimensional Maxwell potential
$A_{\mu}$ can be decomposed into two parts Fermi (1930, 1932). The first part,
called longitudinal, is formed by the time component $A_{0}$ and the
longitudinal part of the vector potential $\vec{A}_{\parallel}$; the second
part, called radiative, is formed by the transverse (and non longitudinal)
part of the vector potential $A_{\perp}$. Fermi showed that the first part can
be seen as giving rise to Coulomb interactions between particles, while the
second part gives rise to the electromagnetic radiation from moving charges.
With this separation, one can use a non-relativistic form of the Hamiltonian
for charged particles and fields (with self explanatory notation)

$H=\sum_{n}\frac{(\vec{p}_{n}-e_{n}\vec{A}_{\perp}(\vec{x}_{n}))^{2}}{2m_{n}}+\sum_{n>m}\frac{e_{n}e_{m}}{|\vec{x}_{n}-\vec{x}_{m}|}+H_{\mathrm{rad}}[A_{\perp}].$
(8)

where $H_{\mathrm{rad}}$ is the free field Hamiltonian that only depends on
the transverse part of the field. For a derivation, see for example Cohen-
Tannoudji  _et al._ (1989) chapter IB.

This decomposition and similar ones are useful and widely used in
electromagnetism and in gravity, but it is important not to misinterpret them.
The decomposition into longitudinal and transverse parts (while gauge
invariant) is not Lorentz invariant: it depends on the frame in which it is
done. Hence it cannot correspond to a physical distinction between different
kinds of degrees of freedom of the theory.

In gravity the decomposition is even less well defined, because it can be done
only in the vicinity of flat space.

In general, there is no invariant sense in which a field splits into a
radiative and a non radiative part: the split is always conventional.

To understand why the longitudinal/transverse split is unphysical, let us
consider the following situation. A charge is kept at the origin until the
time $t=0$. The field –say– is the static Coulomb field centered on the
charge. During a finite time interval $\Delta t$ the charge is smoothly
displaced from the origin to the new location $\vec{x}(t)\neq 0$, and then it
is brought back to the origin. (That is $x(t)=0$ for all $t\notin[0,\Delta t]$
and $\vec{x}(t)\neq 0$ for some $t\in[0,\Delta t]$.) Consider the value of the
electric field during the interval $\Delta t$ at a location $\vec{y}$ such
that $|\vec{y}|\gg c\Delta t$. Since the Maxwell equations are relativistic,
the electric field $\vec{E}(\vec{y})$ is obviously still the Coulomb field
centered at the origin

$\vec{E}(\vec{y})=-\vec{\nabla}\frac{e}{|\vec{y}|}.$ (9)

The decomposition of the field into a longitudinal and a radiative part splits
it as

$\vec{E}(\vec{y})=-\vec{\nabla}\frac{e}{|\vec{y}-\vec{x}(t)|}+\vec{E}_{\perp}(\vec{y}).$
(10)

The first term is the Coulomb field of the displaced source. The second term,
namely

$\vec{E}_{\perp}(\vec{y})\equiv-\vec{\nabla}\left(\frac{e}{|\vec{y}|}-\frac{e}{|\vec{y}-\vec{x}(t)|}\right),$
(11)

is a non-vanishing radiative part, which has magically appeared very far away
from the charge, even if nothing is radiating there from the charge! Yet, it
is necessary to account for the fact that the information about the change of
location of the charge has not yet arrived at $\vec{y}$.

To interpret this decomposition as a decomposition into physically distinct
degrees of freedom is obviously a physical nonsense: nothing happens at
$\vec{y}$ when the charge at the origin is moved, because no information has
had the time to get from the origin to $\vec{y}$. It is absurd to imagine that
what happens in reality is that a physical part of the field instantaneously
determined by the charges changes, and a radiative part magically appears to
make this variation undetectable.

This example also shows that setting the transverse modes to zero means
breaking the Maxwell equations even where there is no radiation actually
emitted by the charges. Also, notice that in the decomposition, not only the
longitudinal part, but also the radiative part of the field has a badly non
local dynamics. On this, and on analogous confusions permeating the
gravitational waves literature, see also Ashtekar and Bonga (2017a, b).

The same argument applies in the gravity case. Imagine that a fast neutron
star arrives at extremely high speed from the sky and wipes away our sun. For
eight minutes, the Earth will continue to follow its curved orbit, being
attracted by the sun, even if (in our Lorentz frame) the sun is not anymore at
its place. Since the Sun is eight light-minutes away, no physical information
about its disappearance can affect the Earth’s motion. This is natural in
field theory: the degrees of freedom of the field are local and they change at
finite speed. If we artificially separate the Newtonian from the radiative
field, something absurd happens: the Newtonian field of the sun disappears
instantaneously (in our Lorentz frame) as soon as the sun is wiped away by the
neutron star, and a radiative mode appears, equally instantaneously, to
compensate for its disappearance and make it undetectable for eight minutes.

The split into radiative and longitudinal modes is a computational trick that
rearranges field components globally and does not respect the local nature of
the physical degrees of freedom of the field.

The moral is that Fermi’s distinction between the Coulomb or Newton field
components, and the radiative components of the field is a convenient
mathematical trick that one can use in a frame and not a separation between
physically distinct degrees of freedom. This conclusion is important because
it has been argued that the component of the field involved in the Coulomb
interaction is not a genuine field degree of freedom. This is incorrect. In
the limit $c\rightarrow\infty$, the notion of field may be discarded
altogether, but when $c$ is kept finite the appropriate account of the
phenomena involves a propagating field and there is no physical distinction
between genuine and not genuine parts of the field.

##  V Can the radiative modes be quantum and the longitudinal modes be
classical?

Let us disregard the conclusion of the previous section and ignore the fact
that the distinction between longitudinal and radiative modes of the field is
not Lorentz invariant and not physically meaningful. Can we nevertheless say
that the radiative parts of the field are affected by quantum theory, while
the longitudinal parts are not? A moment of reflection shows that this would
lead to contradictions. See also Belenchia  _et al._ (2018, 2019).

In the GME experiments, the effect of the radiative part of the field is
negligible and the field is always well approximated by its longitudinal part.
If this remains classical, the field would be entirely classical in the
experiment. But if it is classical, it cannot be in a state which is a quantum
superposition of classical configurations. It must be in a single classical
configuration, the same in all branches. Which one? Any such configuration
would violate the Gauss law in at least one branch. That is, for the effect to
happen, it must be possible to have quantum superpositions of classical field
configurations for its longitudinal parts as well. In this sense the
longitudinal part is equally quantum, and this is precisely what the GME
experiments test.

Let us see this in detail. In the Coulomb gauge, the longitudinal component
$\vec{A}_{\parallel}$ of the vector potential $\vec{A}$ is set to $0$. The
longitudinal part of the electric field is

$\vec{E}_{\parallel}=-\vec{\nabla}A_{0}$ (12)

and the transverse part is

$\vec{E}_{\perp}=-\partial_{t}\vec{A}.$ (13)

Thanks to the Gauss law, $A_{0}$ can be expressed completely in terms of
variables of the matter sources. For a set of particles, for instance

$A_{0}(x,t)=\sum_{n}\frac{q_{n}}{|\vec{x}_{n}(t)-\vec{x}|}.$ (14)

In the quantum theory in this gauge, the $\vec{A}(x)$ are operators on a Fock
space, but also the $A_{0}(x)$ are operators: they act on the Hilbert space of
the matter sources. From (12) and (13) it then follows that
$\hat{\vec{E}}_{\parallel}$ acts on the Hilbert space of matter while
$\hat{\vec{E}}_{\perp}$ acts on the Fock space of the field. In this gauge
(and in a fixed frame), we can say that the longitudinal field is ‘enslaved to
matter’. Nevertheless, this does not mean that it is classical:
mathematically, $\hat{\vec{E}}_{\parallel}$ is very much a quantum field: a
family of operators (more precisely, operator distributions) labeled by points
in spacetime. Physically, precisely because matter is quantum, so is the
longitudinal field: as matter can be in quantum superposition, so can the
longitudinal field. Concretely, the split between parallel and longitudinal
fields is unobservable: the electric field’s observables are local quantities,
sum of the two:

$\hat{\vec{E}}=\hat{\vec{E}}_{\perp}+\hat{\vec{E}}_{\parallel}.$ (15)

Therefore even disregarding the transverse modes, we can still conclude that
the field is in a quantum superposition during the experiment.

In gravity, as we have seen by changing reference frame, even the distinction
between the positions of the particles and the degrees of freedom of the field
looses meaning, because general coordinate invariance (or, equivalently,
because of presence of the diffeomorphism constraint) implies that the
physical quantity relevant in the experiment mixes the two.

##  VI The Quantum Information perspective

A different source of confusion in the discussion on the physical relevance of
the GME is rooted in the different interests and the different perspectives of
the two main research communities concerned by this phenomenon. In this
regard, the history of the GME is peculiar. Although motivated by quantum
gravity, the idea of GME is rooted in the field of quantum information theory.
The idea of the experiment is in fact a beautiful case of cross fertilization
between distant fields. But the two communities work on the basis of different
assumptions and different scientific objectives.

For the quantum gravity community, the momentous relevance of the GME
experiments is, as mentioned, that that they can detect an effect of quantum
superposition of geometry Christodoulou and Rovelli (2019). The connection
between GME detection and geometry superposition gives for granted two basic
facts: (a) the gravitational interaction is mediated by a local field, namely,
a field that does not allow action at a distance, and (b) this field
determines the spacetime geometry. These two ideas played a pivotal role in
the development of our current theory of gravity (and modern physics in
general) and are generally taken for granted by the majority of quantum
gravity researchers.

The quantum information and quantum foundations community, on the other hand,
is rooted in non-relativistic physics and is interested in particular in
theory-independent results: results that one can derive directly from the
measurement, with minimal additional theoretical inputs. In this logic, a
number of assumptions considered obvious in the high energy community are not
taken for granted. In this perspective, a relevant question is what does the
detection of GME imply by itself, without additional knowledge about the
nature of the gravitational interaction.

For the vast majority of the important physical experiments, the result is
meaningful in the context of additional information and acquired knowledge
about the world. This is also a general point of philosophy of science of the
last decades: there are no pure observations. An observation makes sense only
within a rich context of assumptions, and is interpreted as supporting, or
questioning this context, as a whole. Without additional assumptions, it is
rare to be able to extract useful information from a single observation.

This fact has been explicit since the first proposal of GME. The seminal paper
Bose  _et al._ (2017) states:

> Our proposal relies on a simple assumption: the gravitational interaction
> between two masses is mediated by a gravitational field (in other words, it
> is not a direct interaction-at-a-distance).

On the other hand, however, the same paper continues:

> Once we make this assumption, we use a central principle of quantum
> information theory: entanglement between two systems cannot be created by
> Local Operations and Classical Communication (LOCC).

Let us see in more detail if this statement can be made precise. In the LOCC
setup Horodecki  _et al._ (2009), one imagines two experimenters, each with
full quantum control on a quantum system in their laboratory, that can
coordinate their behaviour only by exchange of classical bits. Alice performs
a generic operation on one system, communicates the classical outcome to Bob,
who can use that information to pick what operation to perform on his system,
send the outcome to Alice and so on. The relevant theorem states that
entanglement is not generated, in this way.

Let us state the theorem more precisely. Call a local operation on a bipartite
system a separable quantum channel, that is, a channel that takes the form
$T_{A}\otimes T_{B}$ for some quantum channels $T_{A}$ and $T_{B}$. Here
$T_{A}$ are Alice operations on one particle in her lab and $T_{B}$ are Bob
operations on the other particle in his lab. A forward round of local
operations and classical communication is represented by a map of the form

$\rho\longmapsto\sum_{i}(\mathbb{I}\otimes
T^{i})\big{[}M_{i}(\rho)\otimes\mathbb{I}\big{]},$ (16)

where each $T^{i}$ is a quantum channel and the $\\{M_{i}\\}$ form an
instrument. Each $M_{i}$ is a completely positive map and
$\tr\sum_{i}M_{i}[\rho]=\tr\rho$. Here the label $i$ includes any signal or
information transmitted by the field. Bob’s quantum channel depends on the
field, and this in turn is affected by Alice’s operation. We can define a
backward round of local operations and classical communication in total
analogy. Two rounds will look like

$\rho\longmapsto\sum_{ij}(T^{j}\otimes\mathbb{I})\left[(\mathbb{I}\otimes\tilde{M}_{j}^{i})\big{[}M_{i}(\rho)\otimes\mathbb{I}\big{]}\right],$
(17)

where each of the $T^{j}$ are channels, $\\{M_{i}\\}$ is an instrument, and
for each $i$, $\\{\tilde{M}^{i}_{j}\\}$ is an instrument. We can keep defining
$n$ rounds this way Donald  _et al._ (2002). The LOCC theorems show that the
class of maps representing such exchanges is a subset of the separable
operations, those that can be represented with separable Kraus operators

$\rho\longmapsto\sum_{i}(A_{i}\otimes B_{i})\rho(A_{i}\otimes
B_{i})^{\dagger},$ (18)

where $\sum_{i}A_{i}^{\dagger}A_{i}\otimes B_{i}^{\dagger}B_{i}=\mathbb{I}$.
These cannot entangle the quantum systems if they were originally in a tensor
state.

Notice that the notion of locality employed in this theorem is only loosely
connected with relativistic locality. It refers simply to the split between
the two subsystems and the fact Alice and Bob can each act only on one of the
two. To apply the theorem to the GME effect, we have to identify the field
with the classical communication channel, namely with any classical
information exchanged between Alice and Bob’s labs. We can thus model the
experiments as two quantum systems, namely the two particles, on which Alice
and Bob intervene and that they can measure.

Notice the subtle shift with respect to the usual quantum information setting.
What is meant here by (classical) information is any gravitational influence
between the two labs. This information affects any operation or measurement
Alice and Bob perform. In other words, the configuration of the gravitational
field is in the index $i$ in the above formulas. Then the result follows: if
any reciprocal gravitational influence of Alice’s and Bob’s operations is via
the effect of gravity and this is a classical communication channel, then the
GME effect does not happen.

Of course by relaxing the definition of terms such as local, classical, and
communication, entanglement can be generated in special situations. In exotic
quantum-classical hybrid models for instance Hall and Reginatto (2018), it
might in principle be possible for an evidently classical field to mediate
entanglement between two quantum systems. Even in standard quantum theory one
can turn a separable state into an entangled state333Prepare the quantum
systems in either of two entangled states, and store the information of which
state was prepared in a classical system. Then the state or the quantum
systems alone is a separable state, but it is entangled when conditioning on
the classical system. by the exchange of a classical bit, if the bit is
correlated with the preparation of the quantum systems Krisnanda  _et al._
(2017). Theory-independent claims about the experiment have to be understood
in their own terms. The GME cannot be explained by a local classical mediator,
under _specific_ definitions of these terms _and_ given extra assumptions
(about the interactions, about superdeterminism etc). This is nothing new. The
violation of the Bell inequalities show that the world is non-local, by a
specific definition of non-local and assuming no-superdeterminism, the
existence of a single world etc. Theory-independent results about experimental
observations limit the space of possible theories. They can only “prove
something” given extra assumptions.

From the quantum gravity perspective, the full force of the GME experiments,
as they have been conceived so far, is not in ‘proving’ something: it is in
confirming a general prediction that in quantum gravity comes about as a
consequence of the fact that spacetime can be in superposition. In spite of
its roots in quantum information, the main interest of the experiment for
quantum gravity research is not so much in the context of a theory independent
framework. The experiment is important because, given the current theoretical
knowledge that we have about the world, GME is a direct consequence of quantum
superposition of geometries.

##  VII Difference from previous experiments

A number of past experiments have involved quantum matter and geometry. For
instance the COW experiment Colella  _et al._ (1975), neutron bouncing Jenke
_et al._ (2011) and atomic fountain experiments can be interpreted
relativistically as the measurement of the interference due to the different
time dilation at different altitudes Roura (2022); Wootters (2003). None of
these experiments, however, involve actual quantum properties of geometry
itself. They are all compatible with a world where quantum matter moves in a
classical geometry described by classical general relativity; for instance
with the semiclassical theory defined by the Einstein equations coupled with
the expectation value of the matter’s energy momentum tensor $G=8\pi\langle
T\rangle$ Wallace (2021).

Not so the GME. In a world described by quantum matter on a classical geometry
obeying general relativity, GME would not happen. In this sense, detecting
this effect can be the first direct evidence of a quantum gravity phenomenon.

The fact that there is a regime in nature that is outside semiclassical
gravity and yet gives an effect that survives in the $c\to\infty$ limit is
perhaps obvious a posteriori, but it is surprising and interesting. The GME
effect is accounted for by a non-relativistic theory of the world where there
is action at a distance and no quantum geometry, but this theory of the world
is falsified by a wide variety of other experiments. Hence, among the known
viable theoretical alternatives, the GME effect can only be interpreted as a
manifestation of quantum geometry.

The true reason for the extreme interest raised by the prospect of GME
detection is therefore subtle. The GME effect is not predicted by
semiclassical gravity, nor by most gravity induced collapse theories. Hence it
counts as evidence against these exotic alternatives. But this is not the main
reason of interest of the experiment. After all, most scientists expect the
effect to be real. The reason of the interest is that as far as we can see
today, the GME effect has only two possible explanations: (i) the world is
genuinely non-relativistic or governed by something totally unknown, and this
would be astonishing, or (ii) the effect is a direct consequence of quantum
geometry: seeing it could be our first direct glimpse into quantum gravity.

###### Acknowledgements.

Acknowledgments– This work was supported by the QISS (The Quantum Information
Structure of Spacetime) John Templeton Foundation Grant No.61466 (qiss.fr). CR
thanks Nick Huggett for interesting inputs and Flaminia Giacomini and Lin-
Quing Chen for many discussions.

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# A Hamiltonian description of finite-time singularity in Euler’s fluid
equations

Philip J. Morrison<EMAIL_ADDRESS>Department of Physics and
Institute for Fusion Studies, The University of Texas at Austin, Austin, TX,
78712, USA  Yoshifumi Kimura  Graduate School of Mathematics, Nagoya
University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan

###### Abstract

The recently proposed low degree-of-freedom model of Moffat and Kimura MK19 ;
MK19b  for describing the approach to finite-time singularity of the
incompressible Euler fluid equations is investigated. The model assumes an
initial finite-energy configuration of two vortex rings placed symmetrically
on two tilted planes. The Hamiltonian structure of the inviscid limit of the
model is obtained. The associated noncanonical Poisson bracket pjm98  and two
invariants, one that serves as the Hamiltonian and the other a Casimir
invariant, are discovered. It is shown that the system is integrable with a
solution that lies on the intersection for the two invariants, just as for the
free rigid body of mechanics whose solution lies on the intersection of the
kinetic energy and angular momentum surfaces. Also, a direct quadrature is
given and used to demonstrate the Leray form for finite-time singularity in
the model. To the extent the Moffat and Kimura model accurately represents
Euler’s ideal fluid equations of motion, we have shown the existence of
finite-time singularity.

###### Contents

  1. I Introduction
  2. II The Moffatt-Kimura system
  3. III Hamiltonian Structure
    1. III.1 Generalities
    2. III.2 The Hamiltonian and Poisson bracket
    3. III.3 The Casimir Invariant
  4. IV The nature of the solution, reduction to quadrature, and analysis
    1. IV.1 Geometrical solution
    2. IV.2 Reduction to quadrature
    3. IV.3 Exact solutions with Leray scaling
    4. IV.4 General analysis
  5. V Conclusions
  6. A Non-Hamiltonian spectrum
  7. B The Leray Hamiltonian
  8. C Using $\kappa$ as a clock

##  I Introduction

One approach to the pedigreed quest for determining the existence or
nonexistence of finite-time singularity in the Euler and Navier-Stokes
equations of fluid dynamics (see, e.g., doering  for an overview) is to
analyze them with very specialized initial conditions. Because the vorticity
at a singularity must diverge B_etal94 , effort has been spent on
understanding the behavior of interacting localized de-singularized vortex
tubes, leading to the study of vortex reconnection and its role in turbulence.
Various configurations have been proposed and investigated numerically for
reconnection in classical turbulence (e.g., Kt94 ; K05 ; HD11 ), quantum
turbulence (e.g., B_etal08 ; Z_etal12 ; VPK17 ), and for the existence of
singularity (e.g., SP85 ; PGGB ; BMH16 ; KM18 ; K18 ).

The present work investigates the reduced model of Moffatt and Kimura MK19 ;
MK19b  that describes the interaction of two circular vortex rings. To the
extent this model accurately represents Euler’s ideal fluid equations of
motion, we have shown the existence of finite-time singularity. Explicitly, we
have shown that within the proposed limits of applicability of this model
there exist solutions that blow up in finite time.

Section II describes the Moffatt and Kimura (MK) model. This is followed by
Sec. III where the Hamiltonian structure of the MK model is given, which is
essential for our analysis. Here we discover two constants of motion for the
MK model. One invariant serves as the Hamiltonian for its noncanonical
Hamiltonian formulation (flow on a Poisson manifold; see pjm98 ), while the
other turns out to be a Casimir invariant. The Hamiltonian formulation allows
us, in Sec. IV, to obtain geometrical intuition about the solution space by
examining the intersection of the level sets of the two invariants, akin to
the visualization afforded by the constancy of the energy and angular momentum
magnitude for the Euler equations that describe the free rigid body. Also in
this section we show how to reduce the MK system to quadrature and obtain for
special initial conditions explicit solutions that have exact Leray scaling,
which is representative of the finite-time singularity. Conditions for
singularity within the range of applicability of the derivation given in Refs.
MK19 ; MK19b  are presented. In Sec. V we summarize our results and mention
some future avenues. Appendices are provided that exhibit additional features
of our results.

##  II The Moffatt-Kimura system

The MK system is a three-dimensional system of ordinary differential equations
that describes the evolution of two initially circular vortices of radius $R$
and circulations $\pm\Gamma$, located symmetrically on planes $x=\pm
z\tan\alpha$, with pitch angle $\alpha$, in an $(x,y,z)$ Cartesian coordinate
system. The system is written in dimensionless form using the space scale $R$,
time scale $R^{2}/\Gamma$, and effective Reynolds number
$R_{\Gamma}=\Gamma/\nu=:1/\varepsilon>>1$. Here $\nu$ is the usual kinematic
viscosity of the fluid. It is assumed that the vortices have Gaussian cores of
radius measured by $\delta$, a separation measured by $s$, and a curvature
given by $\kappa$, which evolve in terms of the dimensionless time according
to

$\displaystyle\dot{(\delta^{2})}$ $\displaystyle=\varepsilon-
c_{2}\frac{\kappa\delta^{2}}{s}\,,$ (1) $\displaystyle\dot{s}$
$\displaystyle=-c_{2}\kappa\left[\ln\left(\frac{s}{\delta}\right)+\beta_{1}\right]$
(2) $\displaystyle\dot{\kappa}$ $\displaystyle=c_{1}\frac{\kappa}{s^{2}}\,,$
(3)

subject to the inequality constraints for applicability

$\delta<s<1/{\kappa}\,,$ (4)

which are assumed in the derivation. Roughly speaking, these inequalities
assure that the vortex cores, with size measured by $\delta$, are small enough
so that cores do not overlap as the rings merge, i.e., as their separation
given by $s$ decreases, and that portions of the rings are sufficiently far
away, as measured by $1/\kappa$, to allow far field expansion. The constants
appearing in Eqs. (1), (2), and (3) are given by

$c_{1}=\frac{\cos\alpha\sin\alpha}{4\pi}\qquad\mathrm{and}\qquad
c_{2}=\frac{\cos\alpha}{4\pi}\,,$ (5)

and the parameter $\beta_{1}$ depends on the vortex core, with the value
$\beta_{1}=0.4417$ for a gaussian core profile.

If the MK system is to be a reduction that inherits the noncanonical
Hamiltonian structure of Euler’s fluid equations (see e.g. pjm98 ) then it
will have a Hamiltonian form upon setting the $\varepsilon=0$. Thus we
investigate

$\displaystyle\dot{\delta}$ $\displaystyle=$
$\displaystyle-\frac{c_{2}}{2}\,\frac{\kappa\delta}{s}$ (6)
$\displaystyle\dot{s}$ $\displaystyle=$ $\displaystyle-
c_{2}\,\kappa\left[\ln\left(\frac{s}{\delta}\right)+\beta_{1}\right]$ (7)
$\displaystyle\dot{\kappa}$ $\displaystyle=$ $\displaystyle
c_{1}\,\frac{\kappa}{s^{2}}\,.$ (8)

For later use we record here the equation of motion for $x:=s/\delta$,

$\dot{x}=-\frac{c_{2}\kappa}{\delta}(\Lambda-1/2)\,,$ (9)

which is easily verified. Here, for convenience, we have defined

$\Lambda:=\ln\left(\frac{s}{\delta}\right)+\beta_{1}=\ln(x)+\beta_{1}\,.$ (10)

The smallest value of $\Lambda$ consistent with inequality (4) occurs when
$\delta=s$; thus,

$\Lambda\geq\beta_{1}\qquad\mathrm{or}\qquad\Lambda-1/2\geq\beta_{1}-1/2=-0.0583\,,$
(11)

where the gaussian value of $\beta_{1}=0.4417$ is used in the second
expression. We note here that

$\Lambda=1/2\qquad\mathrm{at}\qquad x^{*}=e^{1/2-\beta_{1}}\approx 1.0600\,.$
(12)

##  III Hamiltonian Structure

###  III.1 Generalities

Hamiltonian systems are usually written in terms of canonically conjugate sets
of variables, a configuration space coordinate and its conjugate momentum. The
noncanonical Hamiltonian description is one where the form in terms of
canonical variables is not necessary and replaced by algebraic properties of
the Poisson bracket. The terminology noncanonical Hamiltonian was introduced
in the context of the ideal fluid and magnetohydrodynamics in pjmG80 , but the
ideas date back to the work of Sophus Lie. (See, e.g., pjm98  for review.)

Given an $n$-dimensional phase space with coordinates
$z=(z^{1},z^{2},\dots,z^{n})$, a system of ordinary differential equations has
noncanonical Hamiltonian form if there exists a Poisson bivector $J$, an
antisymmetric second rank contravariant tensor, and a conserved phase space
function $H(z)$ such that the equations can be written as follows:

$\dot{z}^{i}=\\{z^{i},H\\}=J^{ij}\frac{\partial H}{\partial z^{j}}\qquad
i,j=1,2,\dots n\,,$ (13)

where repeated indices are summed and the Poisson bracket defined on functions
of the coordinate $z$,

$\\{f,g\\}=\frac{\partial f}{\partial z^{i}}J^{ij}\frac{\partial g}{\partial
z^{i}}\,$ (14)

is bilinear, antisymmetric, and most importantly satisfies the Jacobi
identity,

$\\{f,\\{g,h\\}\\}+\\{f,\\{g,h\\}\\}+\\{f,\\{g,h\\}\\}=0\,,$ (15)

for all functions $f,g,h$. Unlike for the canonical description, the tensor
$J$, the Poisson tensor, may depend on the coordinate $z$. It generates the
Hamiltonian vector field as depicted on the righthand side of the second
equality of (13). In coordinates, (15) is equivalent to the vanishing of the
following purely antisymmetric three-tensor:

$S^{ijk}=J^{i\ell}\frac{\partial J^{jk}}{\partial
z^{\ell}}+J^{j\ell}\frac{\partial J^{ki}}{\partial
z^{\ell}}+J^{k\ell}\frac{\partial J^{ij}}{\partial z^{\ell}}\equiv 0\,,$ (16)

a quantity that is checked in practice.

When $\det J\neq 0$, an old theorem of Darboux, based on the algebraic
properties of the Poisson bracket, implies that there is a coordinate change
from the noncanonical coordinates $z$ to a set of canonically conjugate
coordinates. However, when $\det J=0$ this is not possible because of
degeneracy, i.e., the existence of special functions $C$, called Casimir
invariants, that satisfy $\\{C,f\\}=0$ for all phase space functions $f$.
Thus, Casimir invariants are built-in to the phase space, for they will be
conserved by a system generated by any Hamiltonian function. In the
coordinates $z$, $C$ is a Casimir invariant if it satisfies

$J^{ij}\frac{\partial C}{\partial z^{j}}=0\,.$ (17)

Noncanonical Hamiltonian systems possess the rich geometrical structure of so-
called Poisson manifolds, where through every point of the phase space
manifold is a conserved canonical Hamiltonian subspace, i.e., the manifold is
foliated by symplectic leaves (see, e.g., the seminal reference weinstein  and
the recent contribution pjmY20 ). We will see in practical terms how an
interesting Poisson manifold emerges from the MK system.

For three-dimensional systems, like the MK system, the Poisson tensor $J$ has
the form

$J=\begin{bmatrix}0&V_{3}&-V_{2}\\\ -V_{3}&0&V_{1}\\\ V_{2}&-V_{1}&0\\\
\end{bmatrix}$ (18)

for some vector $\mathbf{V}(z)=(V_{1},V_{2},V_{3})$, and it can be shown
easily that (15), for the Jacobi identity, is satisfied if $\mathbf{V}(z)$
satisfies

$\mathbf{V}\cdot\nabla\times\mathbf{V}=0\,.$ (19)

Thus, for three-dimensional systems there is a convenient way to check the
Jacobi identity. Because antisymmetric matrices have even rank, $J$ must have
rank 2 or 0. The latter of course would generate trivial dynamics – of
interest is the case of rank 2 where there is a single Casimir invariant, and
condition (17) can be written compactly as

$\mathbf{V}\times\nabla C=0\,.$ (20)

Relations (19) and (20) play central roles in our discovery of the Hamiltonian
structure of the MK system.

###  III.2 The Hamiltonian and Poisson bracket

Given the equations of motion of a system, like (6), (7), and (8), and a
constant of motion, one can seek a Poisson tensor by matching to the equations
of motion while enforcing the Jacobi identity. Thus we seek a suitable
invariant, one that physically we expect to be an energy-like quantity. Using
the structure of the MK system and some insight we find (6), (7), and (8)
conserve the following:

$H=\frac{1}{\delta^{2}}\left[\ln\left(\frac{s}{\delta}\right)+\beta_{1}-\frac{1}{2}\right]=\frac{1}{\delta^{2}}\big{[}\Lambda-1/2\big{]}\,,$
(21)

which can be shown directly. From (21) we obtain

$s=\delta\,e^{\delta^{2}{H}-\beta_{1}+1/2}\,.$ (22)

Again, as with (11), the threshold for inequality (4) occurs when $\delta=s$,
yielding

$H>(\beta_{1}-1/2)/\delta^{2}=-0.0583/\delta^{2}\,,$ (23)

where again the gaussian value of $\beta_{1}$ is used in the equality.

In terms of the coordinates $z=(\delta,s,\kappa)$, the analog of equation (13)
for the MK system takes the form

$\begin{bmatrix}\dot{\delta}\\\ \dot{s}\\\ \dot{\kappa}\\\
\end{bmatrix}=\begin{bmatrix}0&V_{3}&-V_{2}\\\ -V_{3}&0&V_{1}\\\
V_{2}&-V_{1}&0\\\ \end{bmatrix}\begin{bmatrix}\partial H/\partial\delta\\\
\partial H/\partial s\\\ \partial H/\partial\kappa\\\ \end{bmatrix}\,,$ (24)

which upon making use of the MK equations of motion (6), (7), and (8), and the
candidate Hamiltonian (21), we obtain the following equation for the Poisson
tensor:

$\begin{bmatrix}-{c_{2}}\,{\delta\kappa}/{(2s)}\\\ -c_{2}\,\kappa\,\Lambda\\\
c_{1}\,{\kappa}/{s^{2}}\\\ \end{bmatrix}=\begin{bmatrix}0&V_{3}&-V_{2}\\\
-V_{3}&0&V_{1}\\\ V_{2}&-V_{1}&0\\\
\end{bmatrix}\begin{bmatrix}-2\Lambda/\delta^{3}\\\ 1/(s\delta^{2})\\\ 0\\\
\end{bmatrix}\,.$ (25)

Thus, the goal is to solve (25) for a $\mathbf{V}=(V_{1},V_{2},V_{3})$ that
satisfies (19).

It follows immediately from (25) that

$\dot{\delta}=-{c_{2}}\,{\delta\kappa}/{(2s)}=V_{3}/(s\delta^{2})\quad\quad\mathrm{and}\quad\quad\dot{s}=-c_{2}\,\kappa\Lambda=2V_{3}\Lambda/\delta^{3}\,;$
(26)

thus,

$V_{3}=-c_{2}\kappa\delta^{3}/2$ (27)

works for both equations of (26). The remaining equation yields the
expression,

$\dot{\kappa}=c_{1}\kappa/s^{2}=-2V_{2}\Lambda/\delta^{3}-V_{1}/(s\delta^{2})\,.$
(28)

Thus, it appears there is freedom in the choices of $V_{1}$ and $V_{2}$ to
satisfy (19), the Jacobi identity. Upon setting

$V_{1}=-c_{1}\kappa\delta^{2}/s-2V_{2}\Lambda s/\delta\,.$ (29)

we seek to find a $V_{2}$ that ensures (19) is satisfied. A direct calculation
implies

$\displaystyle\mathbf{V}\cdot\nabla\times\mathbf{V}$
$\displaystyle=-\frac{c_{2}}{2}\,\kappa\delta^{3}\,\frac{\partial
V_{2}}{\partial\delta}-c_{2}\,\kappa s\delta^{2}\Lambda\,\frac{\partial
V_{2}}{\partial s}+c_{1}\,\frac{\kappa\delta^{2}}{s}\,\frac{\partial
V_{2}}{\partial\kappa}$
$\displaystyle+V_{2}\left(-c_{1}\,\frac{\delta^{2}}{s}+\frac{c_{2}}{2}\,\kappa\delta^{2}-c_{2}\,\kappa\delta^{2}\Lambda\right)+\frac{c_{1}c_{2}}{2}\,\frac{\delta^{5}\kappa^{2}}{s^{2}}\,.$
(30)

and the goal is to find a $V_{2}$ such that (30) vanishes.

Upon inserting the following into (30),

$V_{2}=\kappa\delta A(x)\,,$ (31)

where recall $x:=s/\delta$ and using

$\partial_{k}V_{2}=\delta A(x)\,,\qquad\partial_{s}V_{2}=\kappa
A^{\prime}(x)\,,\qquad\partial_{\delta}V_{2}=\kappa A(x)-\kappa
xA^{\prime}(x)$ (32)

(30) becomes

$\mathbf{V}\cdot\nabla\times\mathbf{V}=-c_{2}\kappa^{2}\delta^{3}\left[xA^{\prime}\left(\Lambda-\frac{1}{2}\right)+\Lambda
A-\frac{c_{1}}{2x^{2}}\right]\,.$ (33)

Thus, the Jacobi identity is satisfied provided we can find a solution $A(x)$
to

$xA^{\prime}\left(\Lambda-\frac{1}{2}\right)+\Lambda
A-\frac{c_{1}}{2x^{2}}=0\,,$ (34)

where recall $\Lambda(x)$ is given by (10). If such a function $A$ is found,
then the MK system is Hamiltonian with a bracket defined by

$V_{1}=-c_{1}\frac{\kappa\delta^{2}}{s}-2\kappa\Lambda s\,A(x)\,,\quad
V_{2}=\kappa\delta\,A(x)\,,\quad\mathrm{and}\quad
V_{3}=-\frac{c_{2}}{2}\kappa\delta^{3}\,.$ (35)

Thus, we proceed to solve for $A$. In the light of the inequality (4),
$\Lambda$ is seen to be a positive monotonic function with the inverse

$x=e^{\Lambda-\beta_{1}}\,.$ (36)

Consequently, we can use $\Lambda$ as the independent variable and rewrite
(34) as

$\frac{d}{d\Lambda}\left(e^{\Lambda}\sqrt{\Lambda-1/2\,}\,A\right)=\frac{c_{1}}{2x^{2}}\frac{e^{\Lambda}}{\sqrt{\Lambda-1/2\,}}=\frac{c_{1}e^{2\beta_{1}}}{2}\frac{e^{-\Lambda}}{\sqrt{\Lambda-1/2\,}}\,.$
(37)

Here we have assumed $\Lambda-1/2\geq 0$. In light of (11) this may not be
true; thus, we will return and consider the case where $\Lambda-1/2\leq 0$.

Antidifferentiating both sides gives

$A=\frac{e^{-\Lambda}}{\sqrt{\Lambda-1/2\,}}\frac{c_{1}e^{2\beta_{1}-1/2}}{2}\int^{\Lambda-1/2}\frac{e^{-\ell}}{\sqrt{\ell\,}}\,\,d\ell\,.$
(38)

Then, with the definition of the incomplete gamma function

$\gamma(\sigma,u)=\int_{0}^{u}u^{\prime\sigma-1}e^{-u^{\prime}}du^{\prime}\,,$
(39)

and the identity

$\gamma(1/2,u)=\sqrt{\pi}\mathrm{erf}(\sqrt{u})=2\int_{0}^{\sqrt{u}}e^{-u^{\prime
2}}du^{\prime}\,,$ (40)

with erf being the error function, we obtain

$A=c_{0}\,\frac{e^{-\Lambda}}{\sqrt{\Lambda-1/2\,}}+\frac{\sqrt{\pi}c_{1}\,}{2}\,e^{2\beta_{1}-1/2}\,\frac{e^{-\Lambda}}{\sqrt{\Lambda-1/2\,}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)\,,$
(41)

where $c_{0}\in\mathbb{R}$ is the integration constant. Upon inserting (41)
into the expressions of (35), we have a one-parameter family of Poisson
brackets. It is convenient to choose $c_{0}=0$, giving

$A=\frac{\sqrt{\pi}c_{1}\,}{2\,x}\,\frac{e^{\beta_{1}-1/2}}{\sqrt{\Lambda-1/2\,}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)\,,$
(42)

which is valid for $\Lambda-1/2\geq 0$, but we will see it is also valid for
$\Lambda-1/2<0$. Observe, the choice $c_{0}=0$ gives us regularity at
$\Lambda=1/2$. Inserting (42) into the equations of (35) defines the
noncanonical Poisson bracket that we will use for $\Lambda\geq 1/2$.

Now consider the case where $\Lambda\leq 1/2$, which occurs for

$0\leq 1/2-\Lambda\leq 1/2-\beta_{1}-\ln x\leq 1/2-\beta_{1}\,,$ (43)

where we assume $x\geq 1$ consistent again with (4). Instead of (37), consider

$\frac{d}{d\Lambda}\left(e^{\Lambda}\sqrt{1/2-\Lambda\,}\,A\right)=-\frac{c_{1}e^{2\beta_{1}}}{2}\,\frac{e^{-\Lambda}}{\sqrt{1/2-\Lambda}}\,,$
(44)

where (34) has been used. Integrating both sides from $\Lambda=1/2$ leads to

$A={c_{1}\,e^{2\beta_{1}-1/2}\,e^{-\Lambda}}\,\,M(1/2,3/2,1/2-\Lambda)\,,\qquad\Lambda\leq
1/2\,,$ (45)

where

$M(1/2,3/2,z)=\frac{1}{2}\int_{0}^{1}\frac{e^{zt}}{\sqrt{t}}\,dt=\frac{1}{2\sqrt{z}}\int_{0}^{z}\frac{e^{u}}{\sqrt{u}}\,du\,,$
(46)

is the Kummer function a&s , which is sometimes called the confluent
hypergeometric function of the first kind and denoted by
${}_{1}F_{1}(1/2;3/2;z)$. Inserting (45) into the equations of (35) defines
the noncanonical Poisson bracket defined for $\beta_{1}\leq\Lambda\leq 1/2$.

A comparison of (42) and (45) follows from the identity

$\mathrm{erf}(z)=\frac{2z}{\sqrt{\pi}}\,M(1/2,3/2,-z^{2})\,,$ (47)

which implies

$\frac{\sqrt{\pi}}{2}\frac{1}{\sqrt{\Lambda-1/2}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)=M(1/2,3/2,1/2-\Lambda))\,.$
(48)

Because $\mathrm{erf}(ix)$ for $x\in\mathbb{R}$ is pure imaginary, like
$\sqrt{\Lambda-1/2}$ for $\Lambda<1/2$, we can analytically continue the
expression in terms of the error function through zero to pure imaginary
values yielding a real quantity. Thus, expression (42) can be used for both
$\Lambda\geq 0$ and $\Lambda<0$.

###  III.3 The Casimir Invariant

Given the Hamiltonian structure of Sec. III.2 we know immediately that the MK
system must possess another constant of motion, the Casimir invariant. A
little thought reveals the Casimir must have the form

$C=\frac{c_{2}}{2}\kappa+\frac{\lambda(x)}{\delta}\,.$ (49)

In the present context, (20) is equivalent to

$\begin{bmatrix}0&V_{3}&-V_{2}\\\ -V_{3}&0&V_{1}\\\ V_{2}&-V_{1}&0\\\
\end{bmatrix}\begin{bmatrix}\partial C/\partial\delta\\\ \partial C/\partial
s\\\ \partial C/\partial\kappa\\\
\end{bmatrix}=\begin{bmatrix}0&V_{3}&-V_{2}\\\ -V_{3}&0&V_{1}\\\
V_{2}&-V_{1}&0\\\
\end{bmatrix}\begin{bmatrix}-\lambda/\delta^{2}-s\lambda^{\prime}/\delta^{3}\\\
\lambda^{\prime}/\delta^{2}\\\ c_{2}/2\\\ \end{bmatrix}=0$ (50)

The first equation gives

$0=-\frac{c_{2}}{2}\kappa{\delta^{3}}\frac{\lambda^{\prime}}{\delta^{2}}-\kappa\delta
A\frac{c_{2}}{2}\quad\Rightarrow\quad\lambda^{\prime}=-A\,,$ (51)

while the second gives after some manipulation

$0=-\frac{c_{2}\kappa\delta}{2}\left(\lambda-
xA+\frac{c_{1}}{x}+2Ax\Lambda\right)\,.$ (52)

Because of the Jacobi identity, the third equation must automatically be
solved by the above, which can be verified directly. Thus we have

$\displaystyle\lambda^{\prime}(x)$ $\displaystyle=-A(x)$ (53)
$\displaystyle\lambda(x)$
$\displaystyle=Ax(1-2\Lambda)-\frac{c_{1}}{x}=-2Ax(\Lambda-1/2)-\frac{c_{1}}{x}\,,$
(54)

where $A$ solves (34). Upon differentiating (54) we see that if $A$ satisfies
(34), then (53) is automatic for a solution of (54). In particular, using (42)
we obtain

$\lambda(x)=-\sqrt{\pi}c_{1}\,e^{\beta_{1}-1/2}\,{\sqrt{\Lambda-1/2\,}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)-\frac{c_{1}}{x}\,,$
(55)

where use has been made of $xe^{-\Lambda}=e^{-\beta_{1}}$. Figure 1 displays
plots of both $A/c_{1}$ of (42) (upper) and $\lambda/c_{1}$ (55) (lower) as
functions of $x$, beginning with its critical value of $x=s/\delta=1$,
revealing that both are relatively simple monotonic functions consistent with
inequality (4).

Figure 1: Plots of $A/c_{1}$ (blue) and $\lambda/c_{1}$ (orange) vs.
$x=s/\delta$. Recall $\lambda^{\prime}=-A(x)$ [cf. Eq. (53)].

Thus to summarize, the Casimir, a second invariant, is given by

$\displaystyle C$
$\displaystyle=\frac{c_{2}}{2}\kappa+\frac{\lambda(x)}{\delta}$
$\displaystyle=\frac{c_{2}}{2}\kappa-\sqrt{\pi}c_{1}\,e^{\beta_{1}-1/2}\,\frac{1}{\delta}{\sqrt{\Lambda-1/2\,}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)-\frac{c_{1}}{s}\,,$
(56)

where we have inserted $\lambda$ from (55) in (56) and used $x=s/\delta$. We
remind the reader that (56) is valid for $\Lambda<1/2$ as well as for
$\Lambda\geq 1/2$.

Because of inequality (4), not all values of (56) are permissible. To
understand the permissible range, we rewrite (56) as follows:

$\frac{s}{c_{1}}C-\frac{c_{2}}{2c_{1}}\kappa s=F(\Lambda)\,,$ (57)

where

$F(\Lambda)=\sqrt{\pi}\,e^{\Lambda-1/2}\,{\sqrt{\Lambda-1/2\,}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)+1>0\,.$
(58)

As noted in (11), the smallest allowable value of $\Lambda$ is $\beta_{1}$,
and it is not hard to show that $F(\Lambda)$ obtains its minimum value at
$\beta_{1}$, at which it is positive. Thus we see, the Casimir must satisfy

$C<\frac{c_{2}}{2}\kappa$ (59)

to be consistent with the inequality (4).

##  IV The nature of the solution, reduction to quadrature, and analysis

###  IV.1 Geometrical solution

Given that we have a three-dimensional system with two constants of motion,
the solution space can be visualized by examining the intersection of the
level sets of the Hamiltonian $H$ of (21) with those of the Casimir $C$ of
(56). This is a direct analog of how the stable and unstable trajectories of
the the free rigid body, as governed by Euler’s equations, are understood in
terms of the intersection of the angular momentum spheres with the energy
ellipsoids. The same situation occurs for other noncanonical Hamiltonian
systems such as the Kida problem of fluid mechanics pjmMF97  and the
rattleback toy pjmYT17 , and indeed a large class of flows on Poisson
manifolds pjmY20 . Thus, by plotting level sets of (21) and (56) the nature of
trajectories is revealed and, in addition, one can delineate the accessible
phase space consistent with (4).

Figure 2 displays contours of the Hamiltonian $H$ of (21). Since $H$ is
independent of the variable $\kappa$ it has translational symmetry along the
$\kappa$ axis and, as can be seen in Fig. 2(a), has a sheet-like topology for
positive values of the variables. Level sets corresponding to different signs
of $H$ have opposite curvature in the $s-\delta$ plane, with there being a
region of negative values of $H$ consistent with (23) and (4). This ‘negaive
energy’ interval is shown in Fig. 2(b) for $\delta<1$. For all values of $H$,
the sheets become tangent to the $H=0$ plane with a slope given by $s=\delta
e^{1/2-\beta_{1}}$. Consequently, if $\delta$ and $s$ approach zero, they do
so in a clear and universal way independent of $\mathring{H}$, the initial
value of the Hamiltonian constant of motion.

(a)

20pt5pt

(b)

Figure 2: Contour plots of the Hamiltonian $H$ of (21). (a)
$\mathring{H}={-.50}$ (orange), $\mathring{H}={0}$ (blue), $\mathring{H}={1}$
(green), $\mathring{H}={10}$ (magenta), and $\mathring{H}={100}$ (lavender).
(b) Closeup plots of $s$ vs. $\delta$ at arbitrary $\kappa$. Note the region
to the left of $\delta=1$ for the case where $\mathring{H}=-0.0583$ (orange)
compared to the case $s=\delta$ (blue) with the inset showing the difference
(orange-blue), showing that $s>\delta$ to the left. The other curve with
$\mathring{H}=0.005$ (green) is added for comparison.

In Figs. 3, 4, and 5 we show two views of contour plots of the Casimir of (56)
for three values of the vortex ring tilt angle
$\alpha\in\\{\pi/4,\pi/6,\pi/90\\}$. Observe that these surfaces are again
sheet like but with more interesting structure, no longer having the $\kappa$
independence of the $H$-surfaces.

(a)

(b)

Figure 3: (a) View of the level sets of the Casimir $C$ for $\alpha=\pi/4$.
(b) Same as (a) but rotated.

(a)

(b)

Figure 4: (a) View of the level sets of the Casimir $C$ for $\alpha=\pi/9$.
(b) Same as (a) but rotated.

(a)

(b)

Figure 5: (a) View of the level sets of the Casimir $C$ for $\alpha=\pi/90$.
(b) Same as (a) but rotated.

The case where $\alpha=\pi/90$ corresponds to nearly parallel vortices. For
this case $c_{1}\approx 5\times 10^{-5}<<1$ and (3) implies $\kappa$ is nearly
constant. For this case the trajectory lingers in the flat regions of Fig. 5
and the dynamics is approximately governed by

$\displaystyle\dot{\delta}$
$\displaystyle=-\frac{c_{2}}{2}\,\frac{\mathring{\kappa}\delta}{s}$ (60)
$\displaystyle\dot{s}$
$\displaystyle=-c_{2}\,\mathring{\kappa}\left[\ln\left(\frac{s}{\delta}\right)+\beta_{1}\right]\,,$
(61)

with $c_{2}\approx 0.0796$. Because $H$ is independent of $\kappa$, it is not
a surprise that (21) is still conserved by (60) and (61). This leads to the
quadrature discussed in Appendix C, where the ‘$\kappa$-clock’ is proportional
to ordinary time. As $s$ gets small, $\kappa$ becomes activated, demonstrating
the importance of the local induction velocity for developing curvature.

Now consider the roles played by the $H$ and $C$ surfaces. As a specific
example, consider the case with the initial conditions

$\mathring{\delta}=0.01<\mathring{s}=0.10<\mathring{\kappa}=1\,.$ (62)

With the values of (62) and a choice for the vortex ring tilt angle $\alpha$,
the solution lies on the intersection of the level sets with

$\mathring{H}=22442.9\quad\quad\mathrm{and}\quad\quad\mathring{C}=-6.40628\,,\qquad\mathrm{for}\quad\alpha=\pi/9\,,$
(63)

where again we use $\beta_{1}=0.4417$. Thus, these initial conditions and
corresponding initial values of $\mathring{H}$ and $\mathring{C}$ are
consistent wtih (4). This particular intersection is displayed in Fig. 6.

(a)

(b)

Figure 6: Plot of the intersection of the level set $\mathring{H}=22442.9$
(orange) with that of $\mathring{C}=-6.40628$ (blue) for $\alpha=\pi/9$. These
values correspond to the initial conditions of (62). In (a) $\kappa$ ranges to
100, while in (b) we zoom out to $\kappa$ ranging to 1000 to give a more
global perspective on the shape of the surfaces.

Observe how the curve of intersection approaches increasingly large values of
$\kappa$ as both $\delta$ and $s$ approach zero.

In Fig. 7 we plot multiple intersections of the $H$ and $C$ contours.

(a)

(b)

Figure 7: Intersections of the level sets of $H$ and $C$ for the initial
conditions of 62 with $\alpha=\pi/9$. In (a) we set $\mathring{C}=-6.40628$
and show contours for
$\mathring{H}\in\\{5,000;15,000;20,100;30,000;45,000\\}$, while in (b) we set
$\mathring{H}=22442.9$ and show contours for
$\mathring{C}\in\\{-10,-9,-8,-7,-6,-4\\}$.

From this figure we see how $\kappa$ diverges for a variety of initial
conditions, behavior that is in fact generic.

Yet another picture of the singularity emerges if we eliminate $\delta$
between (21) and (56), giving the expression

$\displaystyle\frac{c_{2}\kappa}{2C}$
$\displaystyle=1-\frac{\sqrt{H}}{C}\frac{\lambda(x)}{\sqrt{\Lambda(x)-1/2}}$
$\displaystyle=1+\frac{c_{1}\sqrt{H}/{C}}{\sqrt{\Lambda(x)-1/2}}\left(\sqrt{\pi}\,e^{\beta_{1}-1/2}\,{\sqrt{\Lambda-1/2\,}}\,\mathrm{erf}\left(\sqrt{\Lambda-1/2}\right)+\frac{1}{x}\right)$
$\displaystyle=:1+\frac{c_{1}\sqrt{H}}{C}\,K[x]\,.$ (64)

This formula is well defined for $H<0$ because
$\delta=\sqrt{H/(\Lambda-1/2)}>0$. Figure 8 shows the function $K$ with the
singularity occurring at $x^{*}\approx 1.0600$.

Figure 8: Plots of the function $K$ of (64) vs. $x$, depicting the singularity
that occurs for $\kappa$ at $x^{*}\approx 1.0600$.

The reader may wonder what would happen to the Casimir if we retained the term
of (41) with the integration constant $c_{0}$. It turns out that this merely
adds a term proportional to $\sqrt{H}$ to the Casimir and thus just shifts the
value of $C$, and thus has no consequence other than changing the numerical
value of $C$ for the same plots.

###  IV.2 Reduction to quadrature

Let us now consider exact integration of the system. Suppose
$(\mathring{\delta},\mathring{s},\mathring{\kappa})$ represent an arbitrary
initial condition at time $\mathring{t}$ with corresponding initial values of
our two invariants
$\mathring{H}=H(\mathring{\delta},\mathring{s},\mathring{\kappa})$ and
$\mathring{C}=C(\mathring{\delta},\mathring{s},\mathring{\kappa})$. Because
$H=\mathring{H}$ for all time, (21) can be used in (56) to obtain

$\mathring{C}=\frac{c_{2}}{2}\kappa-
c_{1}e^{\beta_{1}-1/2}\sqrt{\pi\mathring{H}}\,\,\mathrm{erf}\left(\delta\sqrt{\mathring{H}}\right)-\frac{c_{1}}{s}\,,$
(65)

where recall this formula is well-defined when $\mathring{H}<0$ because it
analytically continues to the expression of (45). Next, using (22) to
eliminate $s$ we obtain

$\mathring{C}=\frac{c_{2}}{2}\kappa-
c_{1}e^{\beta_{1}-1/2}\sqrt{\pi\mathring{H}}\,\,\mathrm{erf}\left(\delta\sqrt{\mathring{H}}\right)-\frac{c_{1}}{\delta}\,e^{-\delta^{2}\mathring{H}+\beta_{1}-1/2}\,.$
(66)

Solving (66) for $\kappa$ is immediate

$\frac{c_{2}}{2}\kappa=\mathring{C}+c_{1}e^{\beta_{1}-1/2}\sqrt{\pi\mathring{H}}\,\,\mathrm{erf}\left(\delta\sqrt{\mathring{H}}\right)+\frac{c_{1}}{\delta}\,e^{-\delta^{2}\mathring{H}+\beta_{1}-1/2}\,.$
(67)

With (22) and (67), we obtain from (6), the following:

$\displaystyle\dot{\delta}$
$\displaystyle=-\frac{c_{2}}{2}\frac{\kappa\delta}{s}=-\frac{c_{2}}{2}\kappa\,e^{-\delta^{2}\mathring{H}+\beta_{1}-1/2}$
$\displaystyle=-\left(\mathring{C}+c_{1}e^{\beta_{1}-1/2}\sqrt{\pi\mathring{H}}\,\,\mathrm{erf}\left(\delta\sqrt{\mathring{H}}\right)+\frac{c_{1}}{\delta}\,e^{-\delta^{2}\mathring{H}+\beta_{1}-1/2}\right)\,e^{-\delta^{2}\mathring{H}+\beta_{1}-1/2}\,,$
(68)

which leads immediately to the quadrature

$\mathring{t}-t=e^{-2\beta_{1}+1}\,\,\int_{\mathring{\delta}}^{\delta}\frac{e^{\delta^{\prime
2}\mathring{H}}\,\,\delta^{\prime}\,d\delta^{\prime}}{\delta^{\prime}\,e^{-\beta_{1}+1/2}\mathring{C}+c_{1}\sqrt{\pi\mathring{H}}\delta^{\prime}\,\,\mathrm{erf}\left(\delta^{\prime}\sqrt{\mathring{H}}\right)+c_{1}e^{-\delta^{\prime
2}\mathring{H}}}\,.$ (69)

Although unwieldy, integration of (69) gives $\delta$ as a function of time,
and via (22) and (67) we obtain $s$ and $\kappa$ as functions of time, with
the latter diverging.

From (69) we see that if there is a finite-time singularity where
$\delta\rightarrow 0$, then it occurs at a time $t_{\infty}$ in accordance
with the following formula:

$t_{\infty}=\mathring{t}+e^{-2\beta_{1}+1}\,\,\int^{\mathring{\delta}}_{0}\frac{e^{\delta^{\prime
2}\mathring{H}}\,\,\delta^{\prime}\,d\delta^{\prime}}{\delta^{\prime}\,e^{-\beta_{1}+1/2}\mathring{C}+c_{1}\sqrt{\pi\mathring{H}}\delta^{\prime}\,\,\mathrm{erf}\left(\delta^{\prime}\sqrt{\mathring{H}}\right)+c_{1}e^{-\delta^{\prime
2}\mathring{H}}}\,.$ (70)

For $\mathring{H}\neq 0$, we see from (21) that at a singularity where
$\delta\rightarrow 0$, we must have $\Lambda-1/2\rightarrow 0$. As noted in
(23), the smallest initial value of $H$ that satisfies (4) is
$\mathring{H}^{*}=(\beta_{1}-1/2)/\mathring{\delta}^{2}$, which is negative;
if $\mathring{H}>\mathring{H}^{*}$, then (4) is satisfied initially. If
$\delta$ decreases and it initially satisfies $s>\delta$, then it will satisfy
it throughout its evolution because
$\delta=s\,e^{-\delta^{2}\mathring{H}+\beta_{1}-1/2}$.

###  IV.3 Exact solutions with Leray scaling

Evidently, the integral of (69) is dramatically simplified with the choice
$\mathring{H}=0$. With this choice, (22) implies

$s=\delta e^{-\beta_{1}+1/2}$ (71)

and equations (6) and (7) become identical upon setting

$\Lambda=\ln(s/\delta)+\beta_{1}=1/2\,,$

a choice consistent with the inequality of (4). Given that for gaussian core
profiles $\beta_{1}=0.4417$ (see MK19 ), we have

$e^{-\beta_{1}+1/2}=1.0600\quad\Rightarrow\quad s>\delta\,.$ (72)

Thus inequality (4) is satisfied, although only barely. Proceeding, the MK
system reduces for $\mathring{H}=0$ to

$\dot{s}=-\frac{c_{2}}{2}\,\kappa\quad\quad\mathrm{and}\quad\quad\dot{\kappa}=c_{1}\,\frac{\kappa}{s^{2}}\,,$
(73)

with the Casimir becoming

$C=\frac{c_{2}}{2}\kappa-\frac{c_{1}}{s}\,,$ (74)

which follows from (65).

Before considering arbitrary initial $\mathring{C}$, we consider the easily
tractable case where both $\mathring{H}=0$ and $\mathring{C}=0$. This implies

$s\kappa=2c_{1}/c_{2}=2\sin\alpha\,.$ (75)

Therefore if we choose

$2\sin\alpha<1\qquad\mathrm{or}\qquad\alpha<\frac{\pi}{6}\,,$ (76)

then the other part of the inequality (4) is satisfied, viz. $s<1/\kappa$.
Note, the value of $\alpha=\pi/4$ chosen in MK19 ; MK19b  does not satisfy
this inequality. Using (75) in the $\dot{\kappa}$ equation of (73) gives

$\dot{\kappa}=\frac{c_{2}^{2}}{4c_{1}}\kappa^{3}\,,$ (77)

which is easily integrated to obtain

$-\frac{1}{2\kappa^{2}}+\frac{1}{2\mathring{\kappa}^{2}}=\frac{c_{2}^{2}}{4c_{1}}(t-\mathring{t})$
(78)

and the exact solution

$\kappa^{-1}=\sqrt{\frac{c_{2}^{2}}{2c_{1}}(t_{\infty}-t)}=\sqrt{\frac{\mathrm{cot}\alpha}{8\pi}(t_{\infty}-t)}\,,$
(79)

where

$t_{\infty}=\mathring{t}+\frac{2c_{1}}{c_{2}^{2}\mathring{\kappa}^{2}}=\mathring{t}+\frac{8\pi\tan\alpha}{\mathring{\kappa}^{2}}\,.$
(80)

As expected, smaller values of $\kappa$ take longer to diverge.

The rest of the solution is obtained from (75) and (71), i.e.,

$\displaystyle s$
$\displaystyle=2\kappa^{-1}\,\sin\alpha=2\sin\alpha\sqrt{\frac{\mathrm{cot}\alpha}{8\pi}(t_{\infty}-t)}$
$\displaystyle=\sqrt{\frac{\sin(2\alpha)}{4\pi}(t_{\infty}-t)}=\delta
e^{1/2-\beta_{1}}\,.$ (81)

It is a simple matter to insert the solutions of (79) and (81) into (6), (7),
and (8) to verify directly that they are indeed an exact solution, one that
satisfies the inequalities of (4).

Now consider the more general case $\mathring{C}\neq 0$. Solving (74) for $s$
and inserting into (73) gives

$\dot{\kappa}=\frac{\kappa}{c_{1}}\left(\frac{c_{2}}{2}\kappa-\mathring{C}\right)^{2}=\frac{c_{2}^{2}}{4c_{1}}\kappa\left(\kappa-2\mathring{C}/c_{2}\right)^{2}\,,$
(82)

which is easily integrated to obtain the exact solution,

$\ln\Big{(}1-\frac{2\mathring{C}/c_{2}}{\kappa}\Big{)}+\frac{2\mathring{C}/c_{2}}{\kappa\,(1-\frac{2\mathring{C}/c_{2}}{\kappa})}-\ln\Big{(}1-\frac{2\mathring{C}/c_{2}}{\mathring{\kappa}}\Big{)}-\frac{2\mathring{C}/c_{2}}{\mathring{\kappa}\,(1-\frac{2\mathring{C}/c_{2}}{\mathring{\kappa}})}=-\frac{\mathring{C}^{2}}{c_{1}}\,(t-\mathring{t})\,.$
(83)

Because the model only makes sense if $\delta,s$, and $\kappa$ are all greater
than or equal to zero, we must have $c_{1}/s=\kappa{c_{2}}/{2}-C\geq 0$ or
$1-2{C}/({\kappa}c_{2})\geq 0$, which in fact according to (59) is true for
all allowable values of $\mathring{H}$.

Assuming the physical initial conditions satisfy
$\mathring{\kappa}>2\mathring{C}/c_{2}$, we see the $\dot{\kappa}>0$, so
$\kappa$ continues to grow, with divergence occurring at the finite time

$t_{\infty}=\mathring{t}+\frac{c_{1}}{\mathring{C}^{2}}\left(\ln\Big{(}1-\frac{2\mathring{C}/c_{2}}{\mathring{\kappa}}\Big{)}+\frac{2\mathring{C}/c_{2}}{\mathring{\kappa}\,(1-\frac{2\mathring{C}/c_{2}}{\mathring{\kappa}})}\right)\,,$
(84)

an approximation to (70) but a generalization to (80). This is depicted in
Fig. 9(a),

(a)

(b)

Figure 9: Plots of the blowup times for $\mathring{H}=0$, $\mathring{C}\neq
0$, (a) according to Eq. (84) where
$\tau=(t_{\infty}-\mathring{t})\mathring{C}^{2}/c_{1}$ vs.
$2\mathring{C}/(c_{2}\mathring{\kappa})$ and (b) according to Eq. (84) where
$\tau=(t_{\infty}-\mathring{t})\mathring{C}^{2}/c_{1}$ vs.
$X={2\sin\alpha}/({\mathring{\kappa}\mathring{s}})$.

The quantity $t_{\infty}$ can also be written as a function of $\kappa s$,
which makes it convenient for assessing initial conditions compatible with
(4). Using (74) we obtain

$t_{\infty}=\mathring{t}+\frac{c_{1}}{\mathring{C}^{2}}\left(\ln(X)+\frac{1}{X}-1\right)\,,$
(85)

where

$X=\frac{2c_{1}}{c_{2}\mathring{\kappa}\mathring{s}}=\frac{2\sin\alpha}{\mathring{\kappa}\mathring{s}}\,.$
(86)

Observe from Fig. 9(a) there is blow up for all values of $\mathring{C}$, but
only values consistent with $\mathring{s}<1/\mathring{\kappa}$ are acceptable.
Thus,

$\quad\frac{1}{\mathring{s}\mathring{\kappa}}>1\quad\Rightarrow\quad\frac{2\sin\alpha}{\mathring{s}\mathring{\kappa}}>2\sin\alpha\,.$
(87)

Therefore after choosing $2\sin\alpha<1$, only blowup times to the right of
this value in Fig. 9(b) are consistent with (4).

As for the case $\mathring{C}=0$, given the solution of (83) for $\kappa$, we
immediately obtain the solutions for $s$ and $\delta$ from (74) and (71),
respectively,

$s=\frac{2c_{1}/c_{2}}{\kappa-2\mathring{C}/c_{2}}=\delta\,e^{-\beta_{1}+1/2}\,.$
(88)

At late times, expansion of (83) again gives

$\kappa^{-1}\sim\sqrt{\frac{c_{2}^{2}}{2c_{1}}(t_{\infty}-t)}\,,$ (89)

in agreement with the Leray scaling suggested in MK19  and proven above in
(79) and (81). Note, in (79), the constant $\mathring{C}$ only appears via
$t_{\infty}$. The zero energy case is special in that it is tractable, but the
Leray scaling is ubiquitous. Inserting (79) into (88) we obtain for late times

$s=\delta\,e^{-\beta_{1}+1/2}\sim\sqrt{2c_{1}(t_{\infty}-t)}$ (90)

and so $\kappa s\sim 2c_{2}/c_{1}$ as for the case with $\mathring{C}=0$.

###  IV.4 General analysis

Let us now return to (69) and consider the general solution. Because
$\mathring{C}$ is arbitrary, it follows immediately from (68) that there is a
family of equilibrium points of $\delta$ when the righthand side vanishes.
However, by (67), all of these correspond to $\kappa=0$ (see Appendix A for an
analysis). Also note, for $\kappa>0$, $\dot{\delta}<0$ and so even when
$\mathring{H}\neq 0$, we expect a solution where $\delta$ approaches zero. If
indeed $\delta\rightarrow 0$, then (22) implies

$s\approx\delta e^{-\beta_{1}+1/2}+\mathcal{O}(\delta^{3})$ (91)

for all $\mathring{H}$; i.e., as for $\mathring{H}=0$, $\delta$ must go to
zero with $s$ along the line

$s=\delta e^{-\beta_{1}+1/2}\,,$ (92)

while from (67) $\kappa$ must diverge as

$\frac{c_{2}}{2}\kappa=\mathring{C}+\frac{c_{1}}{\delta}\,e^{\beta_{1}-1/2}\,.$
(93)

Note, the sign of the righthand side of (68) is definite provided
$\mathring{C}>0$ and $\delta>0$.

Let us explore further the behavior for small $\delta$. To this end we use

$\mathrm{erf}(\sqrt{y})=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\,y^{n+1/2}}{n!\,(2n+1)}\,,$

for small $y\in\mathbb{R}_{+}$, $\sqrt{y}\,\mathrm{erf}(\sqrt{y})\in
C^{\omega}$, and the identities are

$\frac{d}{dx}\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\,e^{-x^{2}}\quad\mathrm{and}\quad\mathrm{erf}(x)=\frac{d}{dx}\left(x\,\mathrm{erf}(x)+\frac{e^{-x^{2}}}{\sqrt{\pi}}\right)\,.$
(94)

Upon multiplying (68) by $\delta$ and defining $u=\delta^{2}/2$, it takes the
form

$\dot{u}=u^{\frac{1}{2}}g(u)-f(u)$ (95)

where $f,g\in C^{\omega}(\mathbb{R})$ are defined by

$\displaystyle f(u)$
$\displaystyle=e^{-2\mathring{H}u+2\beta_{1}-1}c_{1}\left(\sqrt{2\pi\mathring{H}u\,}\,\mathrm{erf}\left(\sqrt{2\mathring{H}u\,}\right)+e^{-2\mathring{H}u}\right)$
(96) $\displaystyle g(u)$
$\displaystyle=-e^{-2\mathring{H}u+\beta_{1}-1/2}\sqrt{2}\,\mathring{C}\,,$
(97)

and satisfy

$f(0)=c_{1}e^{2\beta_{1}-1}\qquad\mathrm{and}\qquad
g(0)=-\sqrt{2}\,\mathring{C}\,e^{\beta_{1}-1/2}\,.$ (98)

Using (94) we see that the function $f(u)$ is positive and monotonically
decreasing, while the function $g(u)$ has a sign determined by the sign of
$\mathring{C}$, and is at least monotonic for small argument.

The functions $f$ and $g$ are depicted in Fig. 10.

Figure 10: Plots of $\bar{f}=e^{-x}(\sqrt{\pi
x}\,\mathrm{erf}(\sqrt{x})+e^{-x})$ (blue), $\bar{g}=0.10\times e^{-x}$
(orange), and $f/g=\sqrt{\pi x}\,\mathrm{erf}(\sqrt{x})+e^{-x}$ (green) vs.
$x=2\mathring{H}u$. Note, $\sqrt{u_{\infty}}\sim f(u_{\infty})/g(u_{\infty})$
are the equilibria points corresponding to $\kappa=0$.

Thus, near $u=0$ the MK system behaves as

$\dot{u}=u^{\frac{1}{2}}g(0)-f(0)$ (99)

Because $u^{\frac{1}{2}}g(0)-f(0)$ is not Lipschitz on $u\in[0,\epsilon]$ with
$\epsilon>0$, we do not have the usual ODE existence theorem to rely on (see
e.g. c&l ). If $u(0)=\mathring{u}>0$ is small, then
$\dot{u}(0)=\mathring{u}^{\frac{1}{2}}g(0)-f(0)<0$, because $c_{1}>0$, and $u$
should decrease. However, the Lipschitz condition for uniqueness is a
sufficient but not a necessary condition. Thus, further analysis is necessary,
but indeed (99) does have a unique solution. To see this let $u_{1}$ and
$u_{2}$ be two solutions that coincide at some time. A measure of their
difference

$D=\left(\sqrt{u_{1}(t)}-\sqrt{u_{2}(t)}\right)^{2}$ (100)

satisfies $\dot{D}=0$ for all time, so they must coincide.

From (99) we see there is a family of equilibrium points given by

$\sqrt{u_{0}}=\frac{f(0)}{g(0)}=\frac{c_{1}e^{\beta_{1}-1/2}}{\sqrt{2}\,\mathring{C}}$
(101)

which only exists for $\mathring{C}>0$. However, these are merely the
$\kappa=0$ solutions mentioned above. In Appendix A it is seen that
linearization about any of these equilibrium solutions yields a spectrum with
two zero eigenvalues and one unstable (positive) eigenvalue. This implies
$\kappa$ will grow so as to decrease the radius of curvature.

In any event, it is a simple matter to integrate (99), yielding

$\frac{2}{g(0)}\left[u^{\frac{1}{2}}+\frac{f(0)}{g(0)}\ln\left(1-{u^{\frac{1}{2}}g(0)}/{f(0)}\right)\right]=t+\mathrm{const}\,,$
(102)

which, upon expanding, gives for small $u$

$\delta\sim\sqrt{2f(0)({t_{\infty}-t)}}=e^{\beta_{1}-1/2}\sqrt{2c_{1}(t_{\infty}-t)}\,.$
(103)

the solution with Leray scaling consistent with (90) and (79).

##  V Conclusions

We have shown that the system put forth by Moffat and Kimura MK19 ; MK19b  for
the interaction of two tilted vortex rings has solutions with finite-time
singularity. This was achieved by finding the noncanonical Hamiltonian
structure of the equations, which naturally led to a geometrical depiction and
explicit forms for the solutions by making use of the newly discovered
Hamiltonian and Casimir invariants. Exact Leray divergence was demonstrated,
within the inequalities proposed in MK19 ; MK19b  for the model.

Future publications will consider the effect of viscous dissipation, further
physical interpretation of the results, and various bounds and perturbation
expansions.

## Acknowledgment

PJM was supported by U.S. Dept. of Energy Contract # DE-FG05-80ET-53088 and a
Forschungspreis from the Alexander von Humboldt Foundation. YK acknowledges
support from JSPS KAKENHI grant # JP19122056 and # JP16747321.

##  Appendix A Non-Hamiltonian spectrum

Usually one studies the equilibrium points of a dynamical system in order to
get a view into the nature of trajectories in phase space. Thus we study the
only equilibrium point of (1), (2), and (3), viz. $\kappa=0$, for any values
of $\delta$ and $s$, which corresponds to the vortex rings having infinite
radii. Interestingly, although our system of equations (1), (2), and (3) is a
Hamiltonian system, it has an associated singular Poisson tensor at this
equilibrium point. As discussed in pjmYT17 ; pjmY20 , such systems may not
have the usual Hamiltonian spectra when expanded about an equilibrium state,
i.e., the symmetry of growing and decaying eigenvalues having the same
magnitude.

The singularity occurs at $\kappa=0$ for any values of $\delta$ and $s$. Such
co-dimension one singularities may have the peculiar spectra. This kind of
singularity follows because the Poisson tensor $J$, as given by $\mathbf{V}$,
vanishes identically for $\kappa=0$. This is seen because the components
$V_{i}$ for $i=1,2,3$ are all proportional to $\kappa$. At other points of
phase space it has rank 2, while along this line rank zero.

The only equilibria of our system occur along the line $\kappa=0$, while any
values of $\delta$ and $s$ are allowed. Thus expanding as

$s=s_{0}+\tilde{s}\,,\qquad\delta=\delta_{0}+\tilde{\delta}\,,\qquad\mathrm{and}\qquad\kappa=\tilde{\kappa}$
(104)

we obtain a simple eigenvalue problem

$\begin{bmatrix}\dot{\tilde{\delta}}\\\ \dot{\tilde{s}}\\\
\dot{\tilde{\kappa}}\\\ \end{bmatrix}=\begin{bmatrix}\,0&\ 0\,&a\,\\\ \,0&\
0\,&b\,\\\ \,0&\ 0\,&L\,\\\ \end{bmatrix}\begin{bmatrix}\tilde{\delta}\\\
\tilde{s}\\\ \tilde{\kappa}\\\ \end{bmatrix}\,,$ (105)

where

$a=-\frac{c_{2}\delta_{0}}{2s_{0}}\,,\qquad
b=-c_{2}(\ln(s_{0}/\delta_{0})+\beta_{1})\,,\qquad\mathrm{and}\qquad
L=\frac{c_{1}}{s_{0}^{2}}\,,$

with the matrix

$\mathbb{M}:=\begin{bmatrix}\,0&\ 0\,&a\,\\\ \,0&\ 0\,&b\,\\\ \,0&\ 0\,&L\,\\\
\end{bmatrix}$

giving rise to following characteristic polynomial by assuming temporal
behavior of $e^{\gamma t}$

$\gamma^{2}(\gamma-c_{1}/s_{0}^{2})=0\,.$ (106)

Thus the spectrum of $M$ is $\\{0,0,c_{1}/s_{0}^{2}\\}$.

In the right coordinates $\mathbb{M}$ is the direct sum of commutating
diagonal (semisimple) and nilpotent pieces. To this end we change coordinates
by replacing $\tilde{\delta}$ and $\tilde{s}$, while retaining
$\tilde{\kappa}$, as follows;

$\bar{\delta}=\tilde{\delta}-\frac{a}{L}\tilde{\kappa}\qquad\mathrm{and}\qquad\bar{s}:=\tilde{s}-\frac{b}{L}\tilde{\kappa}$

in which case $\mathbb{M}$ is replaced by

$\bar{\mathbb{M}}:=\begin{bmatrix}\,0&\ 0\,&0\,\\\ \,0&\ 0\,&0\,\\\ \,0&\
0\,&L\,\\\ \end{bmatrix}$

and the linear dynamics is trivial. With $\tilde{\kappa}$ exponentiating away
at fixed initial $\bar{\delta}$ and $\bar{s}$. From this we conclude that the
rings will initially exponentially decrease their radii of curvature while
$\delta$ and $s$ decrease.

##  Appendix B The Leray Hamiltonian

In light of Sec. IV.4, we observed that for small $\delta$ the system exhibits
Leray scaling. This behavior follows upon expanding (69) or by approximating
(1), (2) , and (3). Here we follow the second route, using in the vicinity of
the singularity $s\sim e^{1/2-\beta_{1}}\delta=:r\delta$, to obtain the
following set of equations that describe the dynamics near the singularity:

$\displaystyle\dot{\delta}$
$\displaystyle=-\frac{c_{2}}{2r}\kappa+\frac{\varepsilon}{2\delta}$ (107)
$\displaystyle\dot{\kappa}$
$\displaystyle=\frac{c_{1}}{r^{2}}\frac{\kappa}{\delta^{2}}\,.$ (108)

Upon setting $\varepsilon=0$ (future work considers its retention), we expect
this reduced system of (107) and (108) to be Hamiltonian. Indeed, it conserves
the quantity

$\mathfrak{c}=\frac{c_{2}}{2}\kappa-\frac{c_{1}}{r\delta}\,,$ (109)

which upon using (42) and (54) can be shown to be the Casimir of (49) expanded
to leading order, and (107) and (108) can be written in the noncanonical
Hamiltonian form as

$\begin{bmatrix}\dot{\delta}\\\
\dot{\kappa}\end{bmatrix}=\begin{bmatrix}0&-{\kappa}/{r}\\\ {\kappa}/{r}\
&0\\\ \end{bmatrix}\begin{bmatrix}\partial\mathfrak{c}/{\partial\delta}\\\
\partial\mathfrak{c}/{\partial\kappa}\\\
\end{bmatrix}=\begin{bmatrix}0&-{\kappa}/{r}\\\ {\kappa}/{r}\ &0\\\
\end{bmatrix}\begin{bmatrix}{c_{1}}/({r\delta^{2})}\\\ {c_{2}}/{2}\\\
\end{bmatrix}\,.$ (110)

Upon changing variables according to

$q=r\delta\qquad\mathrm{and}\qquad p=-\ln(\kappa)$ (111)

the following canonical Hamiltonian form is obtained:

$\dot{q}=\frac{\partial\mathfrak{c}}{\partial
p}=-\frac{c_{2}}{2}e^{-p}\quad\quad\mathrm{and}\quad\quad\dot{p}=-\frac{\partial\mathfrak{c}}{\partial
q}=-\frac{c_{1}}{q^{2}}$ (112)

where the Hamiltonian in canonical coordinates is

$\mathfrak{c}=\frac{c_{2}}{2}e^{-p}-\frac{c_{1}}{q}\,.$ (113)

Various canonical coordinate changes are possible, but given that there is a
simple quadrature, they do not add insight. Clearly, upon setting
$\mathfrak{c}$ to a constant, $\mathring{\mathfrak{c}}$, and solving for $p$,
as is usual for natural Hamiltonians of the form of (113), we obtain the
quadrature

$\int\frac{dq}{\mathring{\mathfrak{c}}q+c_{1}}=-\int dt\,,$ (114)

leading, yet again, to the Leray solution

$q=r\delta\sim\sqrt{2c_{1}(t_{\infty}-t)\,}\,,$ (115)

with $\kappa$ following from $\mathfrak{c}=\mathring{\mathfrak{c}}$,

$\mathring{\mathfrak{c}}+\frac{c_{1}}{q}\sim\frac{c_{1}}{q}=\frac{c_{2}}{2}e^{-p}=\frac{c_{2}}{2}\kappa\qquad\Rightarrow\qquad\kappa\sim\sqrt{\frac{2c_{1}}{c_{2}^{2}(t_{\infty}-t)}}\,.$
(116)

##  Appendix C Using $\kappa$ as a clock

There are various paths to quadrature. Here we present one where the system is
transformed so that $\kappa$ measures time. This is done by dividing (6) and
(7) by (8), giving

$\displaystyle\frac{d\delta}{d\bar{\kappa}}$ $\displaystyle=$
$\displaystyle-s\delta=-x\delta^{2}$ (117)
$\displaystyle\frac{ds}{d\bar{\kappa}}$ $\displaystyle=$
$\displaystyle-s^{2}\left[\ln\left(\frac{s}{\delta}\right)+\beta_{1}\right]=-s^{2}\Lambda$
(118)

where $\bar{\kappa}=c_{2}\kappa/c_{1}$. Using (9) we can replace (118) by

$\displaystyle\frac{dx}{d\bar{\kappa}}=-\delta x({\Lambda}-1/2)\,.$ (119)

Next, because $H$ is $\kappa$ independent, we can use (21) in (117) to obtain
the quadrature

$\frac{d\delta}{d\bar{\kappa}}=-\frac{e^{1/2-\beta_{1}}}{2}\,\delta^{2}\,e^{\delta^{2}\mathring{H}}\quad\Rightarrow\quad\int^{\delta}_{\mathring{\delta}}\frac{e^{-\delta^{\prime
2}\mathring{H}}}{\delta^{\prime
2}}\,d\delta^{\prime}=-\frac{e^{1/2-\beta_{1}}}{2}\,\int_{\mathring{\bar{\kappa}}}^{\bar{\kappa}}\\!d\bar{\kappa}^{\prime}\,.$
(120)

With the substitution $u=\delta^{2}\mathring{H}$, (120) becomes

$\int_{\mathring{H}\mathring{\delta}^{2}}^{\mathring{H}\delta^{2}}\frac{e^{-u^{\prime}}}{u^{\prime
3/2}}\,du^{\prime}=-\frac{c_{2}}{c_{1}}\,\frac{e^{1/2-\beta_{1}}}{\sqrt{\mathring{H}}}\,(\kappa-\mathring{\kappa})\,.$
(121)

The lefthand side of (121) can be written in terms of the incomplete gamma or
error function defined by (40). Then, inverting (121) for $\delta(\kappa)$ and
inserting into (119) gives the following separable equation:

$\frac{dx}{d\kappa}=-\frac{c_{2}}{c_{1}}\,x\big{(}\Lambda(x)-1/2\big{)}\delta(\kappa)\,,$
(122)

the solution of which yields $x(\kappa)$, whence we obtain $s=x\delta$
implying $s(\kappa)$. Finally, upon inserting $s(\kappa)$ into (8), we can
obtain $\kappa(t)$, and all quantities are known as functions of time.

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