# On non-Euclidean Newtonian theories and their cosmological backreaction

Quentin Vigneron1 1 Institute of Astronomy, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Toruń,
Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Constructing an extension of Newton’s theory which is defined on a non-
Euclidean topology (in the sense of Thurston’s decomposition), called a non-
Euclidean Newtonian theory, corresponding to the zeroth order of a non-
relativistic limit of general relativity is an important step in the study of
the backreaction problem in cosmology and might be a powerful tool to study
the influence of global topology on structure formation. After giving a
precise mathematical definition of such a theory, based on the concept of
Galilean manifolds, we propose two such extensions, for spherical or
hyperbolic topologies, using a minimal modification of the Newton-Cartan
equations. However as for now we do not seek to justify this modification from
general relativity. The first proposition features a non-zero cosmological
backreaction, but the presence of gravitomagnetism and the impossibility of
performing exact $N$-body calculations make this theory difficult to be
interpreted as a Newtonian-like theory. The second proposition features no
backreaction, exact $N$-body calculation is possible and no gravitomagnetism
appears. In absence of a justification from general relativity, we argue that
this non-Euclidean Newtonian theory should be the one to be considered, and
could be used to study the influence of topology on structure formation via
$N$-body simulations. For this purpose we give the mass point gravitational
field in $\mathbb{S}^{3}$.

  *

##  1 Introduction

The cosmological backreaction corresponds to the effect of small scale
inhomogeneities on the global expansion of the Universe. The main question
surrounding this phenomenon is whether or not it can produce observable
effects on the scale factor evolution, in particular an acceleration of its
evolution Buchert (2008) or an effect which might explain the discrepancy in
the measure of the Hubble constant Heinesen & Buchert (2020). One of the main
result obtained about backreaction is the Buchert-Ehlers theorem Buchert &
Ehlers (1997); Vigneron (2021), stating that this phenomenon is exactly zero
when calculated using Newton’s theory of gravitation, which may not be the
case in general relativity. This seems to indicate that in a universe whose
structure formation is well described by Newton’s theory, backreaction should
be negligible.

As pointed out by Vigneron (2022b), this statement is valid only if that
universe has a Euclidean topology (we define this term in section 2), i.e. the
topology on which Newton’s theory is defined, implying that the Buchert-Ehlers
theorem needs to be generalised for non-Euclidean topologies. This
generalisation requires an extension of Newton’s theory to such topologies,
called a non-Euclidean Newtonian theory (hereafter NEN theory). As defined in
Vigneron (2022b), the equations in this theory should reduce to the classical
Newton’s equations on small scales (locally Newtonian theory) but be defined
on a non-Euclidean topology.

We stress that the purpose of this modification of Newton’s theory is not to
take into account relativistic effects of spatial curvature that would come
from general relativity (as in e.g. (Abramowicz et al., 2014)), but effects of
the global topology. This latter aspect of cosmology can indeed not only be
described by a relativistic theory, i.e. invariant under Lorentzian
transformations, but also by a Newtonian theory, i.e. invariant under (local)
Galilean transformations.

Another motivation than the study of backreaction for the construction of a
NEN theory is the study of structure formation. As for now most of the
simulations of this phenomenon are done using Newton’s theory, mostly for its
simplicity with respect to general relativity. These simulations are thus
limited to model universes with a (closed) Euclidean topology, which is
generally the 3-torus $\mathbb{T}^{3}$. A NEN theory would be a powerful tool
for cosmology as it would keep the simplicity of Newton’s theory, but would
allow for the study of structure formation in non-Euclidean topologies, in a
sense allowing for other types of periodic boundary conditions than the usual
periodic cube. In particular, such a theory would assess whether or not
observable effects of topology could be visible on the dynamics and morphology
of structures, which as for now, has never been studied to our knowledge. This
might provide us with a new method to detect the precise topology of our
Universe, other than the existing cosmic crystallography Lehoucq et al. (1996)
and “circle in the sky” (e.g. Luminet & Roukema (1999) and Luminet et al.
(2003)) methods.

To our knowledge, only one NEN theory has been proposed to date, by Roukema &
Różański (2009); Barrow (2020). As we will see in the following sections, this
theory is unphysical and cannot be used to compute the backreaction. The goal
of this paper is to propose two NEN theories based on a minimal modification
of the Newton-Cartan equation (proposed in Künzle (1976)) and calculate the
backreaction in both cases. This approach makes use of the results of Vigneron
(2021) on the ability of the Newton-Cartan formalism to describe expansion as
a fundamental field in Newton’s theory. Especially, this approach is based on
the 1+3-Newton-Cartan equations developed in that paper. We will not seek to
justify one of these NEN theories with a non-relativistic limit of general
relativity, which is the ultimate goal but is let for a futur work.

The preliminary section 2 defines the notion of “(non)-Euclidean topology”
used in the present paper and some properties related to this notion. In
section 3, we introduce the strategy we use to define a NEN theory. Section 4
reviews the NEN theory of Roukema & Różański (2009); Barrow (2020). We give a
precise mathematical definition of a NEN theory in section 5 based on the
concept of Galilean manifolds. From this definition we propose two theories
and calculate the backreaction in each case, along with discussing others of
their properties. These results are summarised in table 1.

##  2 What we mean by (non)-Euclidean topology

In the literature on cosmology and general relativity, the terms “Euclidean”
and “non-Euclidean” associated to “geometry” generally refer to the presence
or not of a curvature. Our notion of Euclidean and non-Euclidean is different
(more general) as it is associated to the topology of a manifold.

The topology of closed 3-manifolds is classified by Thurston’s decomposition
into eight distinct classes, each corresponding to a set of different (multi-
connected) topologies having the same covering space. A 3-manifold $\Sigma$
has a Euclidean topology if the covering space $\tilde{\Sigma}$ of $\Sigma$ is
$\mathbb{E}^{3}$. The topology in called non-Euclidean if
$\tilde{\Sigma}\not=\mathbb{E}^{3}$. The most common non-Euclidean topologies,
considered in the $\Lambda$CDM model, are the spherical and hyperbolic
topologies having respectively $\tilde{\Sigma}=\mathbb{S}^{3}$ and
$\tilde{\Sigma}=\mathbb{H}^{3}$. For the description of the other five types
of topologies, see (e.g. Lachieze-Rey & Luminet, 1995).

An important property relating the type of topology and the Ricci curvature
tensor $\boldsymbol{\mathcal{R}}$ associated to any Riemannian metric
$\boldsymbol{h}$ defined on $\Sigma$ is:

$\displaystyle\boldsymbol{\mathcal{R}}=0\ \Rightarrow\ \textrm{Euclidean
topology},$ (1)

or equivalently,

$\displaystyle\textrm{non-Euclidean topology}\ \Rightarrow\
\boldsymbol{\mathcal{R}}\not=0.$

This is the reason why Newton’s theory is defined on a Euclidean topology.
This is also why a NEN theory requires a non-zero spatial Ricci tensor as
discussed in the next section.

However, because the term “non-Euclidean” in the name “non-Euclidean Newtonian
theory” refers first to the topology, the purpose of this modification of
Newton’s theory is not to take into account spatial curvature effects coming
from general relativity, but effects of the global topology.

##  3 The strategy: introduction of a spatial Ricci curvature

The NEN theories we present in this paper are all constructed using the same
strategy: we algebraically keep Newton’s equations and we add a non-zero
spatial Ricci curvature in the spatial connection such that the spatial
topology is non-Euclidean. This is a heuristic approach as it is not yet
justified from general relativity. Consequently there is a freedom on the
choice of curvature, that we restrict with the following argument: the spatial
topology in Newton’s theory is Euclidean because the spatial Ricci curvature
$\boldsymbol{\mathcal{R}}$ is zero. However, we see from property (1) that
such a curvature tensor is not the only possible one in Euclidean topologies,
i.e. in a 3-manifold with a Euclidean topology it is possible to define a
metric whose Ricci curvature tensor is non-zero. Nevertheless,
$\boldsymbol{\mathcal{R}}=0$ is the simplest Ricci tensor we can define on a
Euclidean topology. Therefore, it is natural to expect this tensor, in the
case of a NEN theory, to be also the simplest one for the considered topology.

In this paper, we will be interested in either spherical or hyperbolic
topologies. We will not consider the other five classes of topologies of the
Thurston decomposition along with combinations of these topologies. Therefore,
the simplest Ricci tensor is

$\boldsymbol{\mathcal{R}}=\frac{\mathcal{R}(t)}{3}\boldsymbol{h},$ (2)

where $\boldsymbol{h}$ is the spatial metric. We can see with the above
formula that there is only one degree of freedom in $\boldsymbol{\mathcal{R}}$
given by the scalar curvature $\mathcal{R}$, and because of the contracted
Bianchi identity, $\mathcal{R}$ is a spatial constant. Therefore, the only
parameter on which $\mathcal{R}$ can depend is the time $t$ present in the
Newtonian theory. In the NEN theory of (Roukema & Różański, 2009; Barrow,
2020) (section 4), $t$ will parametrise a single 3-manifold (the spatial
manifold) as in classical Newton’s theory; in the NEN theories presented in
section 5, $t$ will have a geometrical origin as it will be linked to a
foliation of a 4-manifold.

The main reasons to consider closed topologies either spherical or hyperbolic
are:

  1. 1.

The closeness is the most physical choice of boundary conditions in a
cosmological context. As we will see in section 5.5.2, the closedness is also
required in the calculation of the backreaction in our proposed NEN theories,
along with the derivation of the Poisson equation.

  2. 2.

These topologies are globally isotropic, hence respecting the cosmological
principle.

Another problem that we face with this heuristic strategy is related to the
choice of equations on which we must add the curvature. Indeed, in Newton’s
theory the Poisson and the Raychaudhuri equations are equivalent. However, as
pointed out by (Vigneron, 2022b), once we introduce a non-zero curvature in
the spatial connection present in these equations, they are not equivalent
anymore and therefore define two different NEN theories. In Roukema & Różański
(2009); Barrow (2020), the authors use the Poisson equation. We detail this
approach in the next section and we will see that physical inconsistencies
appear. In section 5, we will propose NEN theories based on the Newton-Cartan
equation.

##  4 Poisson equation approach

The NEN theory of Roukema & Różański (2009); Barrow (2020) is defined with the
following equations:

$\displaystyle D_{c}g^{c}=-4\pi G\rho\quad;\quad
D_{[a}g_{b]}=0\quad;\quad\mathcal{R}_{ab}=\frac{\mathcal{R}}{3}h_{ab},$ (3)
$\displaystyle\dot{\boldsymbol{v}}=\boldsymbol{g}+\boldsymbol{a}_{\not=\mathrm{grav}}\quad;\quad\dot{\rho}+\rho
D_{c}v^{c}=0,$ (4)

where $\boldsymbol{D}$ is the spatial connection related to the metric with
the above Ricci curvature, $\boldsymbol{g}$ is the gravitational field, $\rho$
the mass density, $\boldsymbol{v}$ the velocity of the cosmic fluid, the dot
derivative is the Lagrangian derivative with respect to $\boldsymbol{v}$, and
$\boldsymbol{a}_{\not=\mathrm{grav}}$ is the non-gravitational spatial
acceleration.

Throughout this paper, we denote indices running from 0 to 3 by Greek letters
and indices running from 1 to 3 by Roman letters.

###  4.1 The mass point gravitational field

The authors in Roukema & Różański (2009); Barrow (2020) use this theory to
compute the gravitational field created by one mass point. For the purpose of
this section we only give the solution in the case where the topology111In the
cases where the topology is multi-connected spherical, a sum over the holonomy
group of the solution in $\mathbb{S}^{3}$ gives the mass point solution (see
Roukema & Różański (2009) for details). is $\mathbb{S}^{3}$ which can be found
when solving equation (3) in spherical symmetry with $\rho=0$. Then the
gravitational field $\boldsymbol{g}$ takes the form
$g^{a}=\left(g^{\xi},0,0\right)$ when using hyperspherical coordinates, in
which the line element is

$\mathrm{d}l^{2}=R_{\rm
c}^{2}\left[\mathrm{d}\xi^{2}+\sin^{2}\left(\xi\right)\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi\right)\right],$
(5)

where $\xi\in[0,\pi]$, and $R_{\rm c}\coloneqq\sqrt{6/\mathcal{R}}$ is the
curvature radius of the spherical space. In these coordinates, $r\coloneqq
R_{\rm c}\,\xi$ corresponds to the distance to the origin $(0,0,0)$. Then we
have

$g^{\xi}=\frac{A}{R_{\rm c}^{2}\sin^{2}\left(\xi\right)},$ (6)

with $A$ a constant. To be compatible with Newton’s theory close to the
center, i.e. close to the mass point, one must retrieve $g^{\xi}(\xi\sim
0)\sim-GM/r^{2}$. Thus we need to take $A=-GM$. We see that in the case of a
spherical topology, the gravitational field diverges at the mass point
position ($\xi=0$) as expected, but also at the pole opposed to this position
($\xi=\pi$). The field around this pole is given by:

$g^{\xi}(\xi\sim\pi)=-\frac{GM}{R_{\rm
c}^{2}\left(\pi-\xi\right)^{2}}+\mathcal{O}(1).$ (7)

This field points towards the origin, i.e. outwards from the pole $\xi=\pi$,
which means that the latter pole behaves as a white hole.

###  4.2 Problems of the approach

The NEN theory defined above has two major problems which make it unsuited for
studying the backreaction problem or structure formation:

  1. 1.

The Poisson equation (3) requires that the total mass must be zero in a closed
space. This can be derived when averaging the equation over the whole space,
and using Stoke’s theorem. In other words, the equation we solved is not the
one with “$\rho=M\delta(\xi)$”, but with
$\rho=M[\delta(\xi)-\delta(\xi-\pi)]$. This explains why the mass point
solution in spherical topology has two diverging points: one corresponding to
an attractive (positive) mass and one to a repulsive (negative) mass, such
that the sum of these two masses is zero. This solution is not physical as a
spherical universe described by this NEN theory would be full of white holes.

  2. 2.

The proposed NEN theory cannot account for the spatial expansion, which, as
shown in Vigneron (2021), should be a fundamental field present in the
equations. Furthermore introducing an expansion velocity of the form
${v}^{a}_{\mathrm{H}}=H(t)x^{a}$ as in Peebles (1980), is not possible because
this construction requires the use of Cartesian coordinates, only defined for
Euclidean topologies.

In the approach of Roukema & Różański (2009); Barrow (2020), the Poisson
equation is algebraically kept and thus is considered as a fundamental feature
of a Newtonian theory regardless of the topology. Another approach would be to
consider that the fundamental feature which should be common to Newton’s and
NEN theories is the Galilean invariance. This is the approach we followed and
which is detailed in the next section.

##  5 The Newton-Cartan approach

###  5.1 Definition of a NEN theory

In Vigneron (2022b) and in the introduction of the present paper we defined a
NEN theory as a theory whose equations should reduce to Newton’s equations on
scales small with respect to the size of the closed space, which corresponds
to the spatial curvature scale in our strategy. However it is not clear to us
how to define the mathematical procedure that would test if the theory is
“locally Newtonian” or not. For the NEN theory proposed in section 4, we can
only say that if we assume $\boldsymbol{\mathcal{R}}\sim\boldsymbol{0}$, we
retrieve Newton’s equations. In what follows, we propose a better definition
which is based on a mathematical framework that keeps the essence of Newton’s
theory, the Galilean invariance.

In Vigneron (2021), we showed that Newton’s theory is best formulated, and
argued that it must be, in its geometrised form, i.e. the Newton-Cartan
theory. In particular, we showed that from this formulation, the expansion
arises as an emerging field of the theory. In this formulation, the equations
are defined on a 4-manifold whose structure is invariant under local Galilean
transformations (contrary to general relativity where the structure is
invariant under local Lorentzian transformations). Therefore, in this
formulation the Galilean invariance of spacetime is taken as a fundamental
principle. We can therefore redefine a NEN theory as follows:  Definition. A
non-Euclidean Newtonian theory is a theory defined on a Galilean manifold
whose spatial sections have a non-Euclidean topology. It is given by equations
relating the Riemann tensor of the Galilean structure to the energy content of
the manifold.

From this definition and the strategy proposed in section 3, it seems natural
that to define a NEN theory, one should add the non-zero spatial curvature
directly in the Newton-Cartan equations. This was actually proposed by Künzle
(1976), however the expansion law, along with the full 3-dimensional system of
equations, were not derived in that paper. This is done in the next sections,
but we first recall the formalism behind the Newton-Cartan theory.

###  5.2 The Newton-Cartan theory

A Galilean manifold is a differentiable 4-manifold $\mathcal{M}$ equipped with
a Galilean structure $(\boldsymbol{\tau},\boldsymbol{h},\boldsymbol{\nabla})$,
where $\boldsymbol{\tau}$ is an exact 1-form, $\boldsymbol{h}$ is a symmetric
(2,0)-tensor of rank 3, with $h^{\alpha\mu}\tau_{\mu}=0$, and
$\boldsymbol{\nabla}$ is a connection compatible with $\boldsymbol{\tau}$ and
$\boldsymbol{h}$, called a Galilean connection:

$\nabla_{\alpha}\tau_{\beta}=0\quad;\quad\nabla_{\gamma}h^{\alpha\beta}=0.$
(8)

A vector $\boldsymbol{u}$ is called a unit timelike vector if
$u^{\mu}\tau_{\mu}=1$, and an (n,0)-tensor $\boldsymbol{T}$ is called spatial
if $\tau_{\mu}{T^{...\overset{\overset{\alpha}{\downarrow}}{\mu}...}}=0$ for
all $\alpha\in\llbracket 1,n\rrbracket$. The exact 1-form $\boldsymbol{\tau}$
defines a foliation $\\{\Sigma_{t}\\}_{t\in\mathbb{R}}$ in $\mathcal{M}$,
where $\Sigma_{t}$ are spatial hypersurfaces in $\mathcal{M}$ defined as the
level surfaces of the scalar field $t$, with
$\boldsymbol{\tau}=\boldsymbol{\mathrm{d}}t$. The time metric
$\boldsymbol{\tau}$ and the space metric $\boldsymbol{h}$ define what we can
call a preferred foliation in $\mathcal{M}$, i.e. preferred (or absolute)
space and time. This is the Newtonian picture of spacetime. However, from the
definition alone of the Galilean structure, no constraint exists on the
spatial sections $\Sigma_{t}$, especially on the topology of these sections
and their curvature related to the spatial metric $\boldsymbol{h}$. Such
constraints will only come from the physical equations relating the geometry
of the Galilean manifold with its energy content.

From the coefficients $\Gamma^{\gamma}_{\alpha\beta}$ of the Galilean
connection, one can define a Riemann tensor as
${R^{\sigma}}_{\alpha\beta\gamma}\coloneqq
2\,\partial_{[\beta}\Gamma^{\sigma}_{\gamma]\alpha}+2\,\Gamma^{\sigma}_{\mu[\beta}\Gamma^{\mu}_{\gamma]\alpha}$.
Then the Newton-Cartan equations are:

$\displaystyle\nabla_{\mu}T^{\mu\alpha}$ $\displaystyle=0,$ (9) $\displaystyle
R_{\alpha\beta}$ $\displaystyle=\tau_{\alpha}\tau_{\beta}\left(4\pi
G\tau_{\mu}\tau_{\nu}T^{\mu\nu}-\Lambda\right),$ (10) $\displaystyle
h^{\mu[\alpha}{R^{\beta]}}_{(\gamma\sigma)\mu}$ $\displaystyle=0,$ (11)

where $\Lambda$ is the cosmological constant and $\boldsymbol{T}$ is symmetric
and corresponds to the energy-momentum tensor of the matter.

Projecting twice the second equation along the spatial metric leads to
$\boldsymbol{\mathcal{R}}=0$, where $\boldsymbol{\mathcal{R}}$ is the Ricci
tensor related to the spatial metric. This means that the Newton-Cartan
equations impose the topology of the sections $\Sigma_{t}$ to be Euclidean.
Therefore allowing for non-Euclidean topologies requires a modification of
these equations.

###  5.3 Some remarks about Galilean structures

An important technical point about Galilean structures is that that they do
not introduce any object, like a Lorentzian metric, that would induce an
isomorphism between forms and vectors on $\mathcal{M}$. Therefore, raising or
lowering indices is not possible. That is why all the spacetime tensors we
introduce in this paper will always be written with the same types of indices
i.e. contravariant/covariant. For more details regarding mathematical
properties of Galilean structures see (Künzle, 1972; Vigneron, 2021).

It is still possible to relate Galilean structures with Lorentzian structures
used in general relativity. This has first been done in (Künzle, 1976), where
the author defines the Newtonian limit using these structures: in this limit,
the Lorentzian structure of general relativity tends to a Galilean structure.
In other words, the Newtonian limit transforms a Lorentz invariant theory into
a Galilean invariant theory, and for this reason is also called Galilean limit
(see (Vigneron, 2022a) for a detailed study of the relation between this limit
of global topology).

Finally, the Galilean invariance at the origin of the definition of Galilean
structures is defined as a local invariance between frames (see Künzle
(1972)), whereas a global invariance would necessarily require a flat
Euclidean space. That is why Galilean manifolds can have non-zero spatial
curvature and non-Euclidean topologies for their spatial sections.

###  5.4 A curvature in Newton-Cartan

To allow for $\boldsymbol{\mathcal{R}}\not=0$, Künzle (1976) proposed222Künzle
claims to give references for the modification (13), but these are unrelated
to this equation. We can therefore consider Künzle (1976) to be the first
occurrence of this modification. to modify the system (9)-(11) by adding a
‘curvature term’ in the second equation, leading to the system:

$\displaystyle\nabla_{\mu}T^{\mu\alpha}$ $\displaystyle=0,$ (12)
$\displaystyle
R_{\alpha\beta}-\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{B}}]{b}{}_{\alpha\beta}$
$\displaystyle=\tau_{\alpha}\tau_{\beta}\left(4\pi
G\tau_{\mu}\tau_{\nu}T^{\mu\nu}-\Lambda\right),$ (13) $\displaystyle
h^{\mu[\alpha}{R^{\beta]}}_{(\gamma\sigma)\mu}$ $\displaystyle=0,$ (14)

where $\boldsymbol{B}$ is a unit timelike vector and
$\tensor[^{\boldsymbol{B}}]{b}{}_{\alpha\beta}$ is the projector orthogonal to
this vector, defined as:

$\tensor[^{\boldsymbol{B}}]{b}{}_{\alpha\mu}B^{\mu}\coloneqq
0\quad;\quad\tensor[^{\boldsymbol{B}}]{b}{}_{\alpha\mu}h^{\mu\beta}\coloneqq{\delta_{\alpha}}^{\beta}-\tau_{\alpha}B^{\beta}.$
(15)

From now we also assume that $\boldsymbol{T}$ corresponds to the energy-
momentum tensor of a general fluid with

$T^{\alpha\beta}\coloneqq\rho
u^{\alpha}u^{\beta}+ph^{\alpha\beta}+2q^{(\alpha}u^{\beta)}+\pi^{\alpha\beta},$
(16)

where $\boldsymbol{u}$ is timelike and is the 4-velocity of the fluid, $p$ its
pressure, $\boldsymbol{q}$ its heat flux and $\boldsymbol{\pi}$ its
anisotropic stress.

When projected twice along $h^{ab}$, equation (13) indeed leads to
$\mathcal{R}^{ab}=\frac{\mathcal{R}}{3}h^{ab}$. This modification of the
Newton-Cartan equations is the simplest one allowing for a non-zero spatial
Ricci curvature, and in the sense given in section 5.1, the system (12)-(14)
defines a NEN theory.

In the framework of Galilean manifolds, any orthogonal projector
$\tensor[^{\boldsymbol{B}}]{b}{}_{\alpha\beta}$ can play the role of a twice
covariant spatial metric. This is because in any coordinate system adapted to
the foliation $\\{\Sigma_{t}\\}_{t\in\mathbb{R}}$ we have
$\tensor[^{\boldsymbol{B}}]{b}{}_{ab}=h_{ab}$, with $h_{ab}$ the inverse
matrix of $h^{ab}$. That is why the additional term in equation (13) features
$\tensor[^{\boldsymbol{B}}]{b}{}_{\alpha\beta}$. As a consequence there is a
freedom on the choice of the vector $\boldsymbol{B}$, i.e. on the choice of
observer to which the curvature term is related.

We can choose $\boldsymbol{B}$ to be a ‘preferred’ unit timelike vector.
Following Vigneron (2021), two such vectors can be chosen:

  1. 1.

$\boldsymbol{B}=\boldsymbol{u}$: the curvature term is related to the fluid
observer.

  2. 2.

$\boldsymbol{B}=\boldsymbol{G}$: the curvature term is related to the Galilean
observer. It is defined such that its tilt with respect to the fluid
$\boldsymbol{v}=\boldsymbol{u}-\boldsymbol{G}$ is the one present in the
Scalar-Vector-Tensor (SVT) decomposition of the expansion tensor
$\boldsymbol{\Theta}$ of $\boldsymbol{u}$, defined in equation (28) (see
Vigneron (2021) for a detailed description of the Galilean vector). In section
5.6.8 we will provide a better definition for $\boldsymbol{G}$ based on the
results of section 5.6.5.

In the following sections we will develop the NEN theory in both cases: we
will derive the 1+3-system, or kinematical system (similarly as in Vigneron
(2021)), and the gravitational system, i.e. featuring the gravitational field
and the expansion law. The cosmological backreaction will be given, thus
generalising the Buchert-Ehlers theorem in both cases. Finally we will show
whether or not each theory is suited for $N$-body calculations.

###  5.5 NEN theory I: Fluid observer curvature term

In this section, we consider $\boldsymbol{B}\coloneqq\boldsymbol{u}$.

####  5.5.1 The system of equations.

The derivation of the 1+3-system of equations is done by projecting the
modified Newton-Cartan equations with respect to $\boldsymbol{u}$ and
$\boldsymbol{h}$. We will not detail its full derivation, which is similar to
the one made by Vigneron (2021). The main difference comes from the
projections of the additional curvature term in the Newton-Cartan equation. We
have

$\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{u}}]{b}{}_{\mu\nu}h^{a\mu}h^{b\nu}=\frac{\mathcal{R}}{3}h^{ab}\quad;\quad\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{u}}]{b}{}_{\mu\nu}u^{\mu}h^{a\nu}=0\quad;\quad\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{u}}]{b}{}_{\mu\nu}u^{\mu}u^{\nu}=0.$
(17)

Then when introducing the expansion tensor $\Theta^{\alpha\beta}\coloneqq
h^{\mu(\alpha}\nabla_{\mu}u^{\beta)}$ with its trace
${\theta\coloneqq\Theta^{\mu\nu}\tensor[^{\boldsymbol{u}}]{b}{}_{\mu\nu}}=\nabla_{\mu}u^{\mu}$
and the vorticity tensor $\Omega^{\alpha\beta}\coloneqq
h^{\mu[\alpha}\nabla_{\mu}u^{\beta]}$ of the fluid, one gets the following
1+3-system, or kinematical system, composed of: evolution equations

$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\rho=-\rho\theta-
D_{c}q^{c},$ (18)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)h^{ab}=-2\Theta^{ab},$
(19)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\theta=-4\pi
G\rho+\Lambda+D_{c}\tensor[^{\boldsymbol{u}}]{a}{}^{c}-\Theta^{cd}\Theta_{cd}+\Omega^{cd}\Omega^{cd},$
(20)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\Omega_{ab}=D_{[a}\tensor[^{\boldsymbol{u}}]{a}{}_{b]},$
(21)

and constraint equations

$\displaystyle D_{c}\left(\Theta^{ac}+\Omega^{ac}\right)-D^{a}\theta=0,$ (22)
$\displaystyle D_{[a}\Omega_{bc]}=0,$ (23)
$\displaystyle{\mathcal{R}^{ab}=\color[rgb]{0.7,0,0}\frac{\mathcal{R}}{3}h^{ab}},\color[rgb]{0,0,0}$
(24) $\displaystyle\mathcal{R}^{d[abc]}=0,$ (25)

with

$\displaystyle\rho\,\tensor[^{\boldsymbol{u}}]{a}{}^{a}$
$\displaystyle=-D^{a}P-D_{c}\pi^{ca}-\left[\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)q^{a}+q^{a}\theta+2q_{c}\left(\Theta^{ca}+\Omega^{ca}\right)\right].$
(26)

$\boldsymbol{\beta}$ is the shift vector, it is spatial and corresponds to the
choice of adapted coordinate system with
$\boldsymbol{\beta}\coloneqq\boldsymbol{\partial}_{t}-\boldsymbol{u}$, and
$(\boldsymbol{\partial}_{t})^{\mu}\tau_{\mu}\coloneqq 1$ (see section II.C.3
in Vigneron (2021) for more details on coordinate choices in the Newton-Cartan
formalism). The 4-acceleration $\tensor[^{\boldsymbol{u}}]{\boldsymbol{a}}{}$
of the fluid plays the role of a non-gravitational 3-acceleration, and from
now on will be denoted as $\boldsymbol{a}_{\not=\rm grav}$. Note that
centrifugal and Coriolis accelerations are not part of
$\boldsymbol{a}_{\not=\rm grav}$. These accelerations arise from a non-
Galilean choice of coordinates and are contained in the Lie derivative term
$\mathcal{L}_{\boldsymbol{\beta}}v^{a}$ in equation (34) (also see section II.
in Vigneron (2020) for details on how these accelerations can be described by
the shift vector).

The only difference of the system (18)-(26) with respect to the 1+3-Newton-
Cartan system (equations (36)-(44) in Vigneron (2021)) is the curvature
equation (24).

Remark. If we want to retrieve the mass conservation equation we need to
assume $\boldsymbol{q}=0$ in equation (18). This was also the case in Newton-
Cartan as shown by Künzle (1976).

####  5.5.2 Solving the constraint equations.

A key step in the derivation of the gravitational system, is the resolution of
the constraint equations (22) and (23). But while these equations are
algebraically the same as in the Euclidean case, because the spatial curvature
is not zero anymore, their solution is different.

The constraint (23) implies that $\boldsymbol{\Omega}$ can be written as

$\Omega_{ab}=D_{[a}w_{b]}+\omega_{ab},$ (27)

with $\boldsymbol{\omega}$ a harmonic 2-form and $\boldsymbol{w}$ a spatial
vector. In the case of a spherical topology, $\boldsymbol{\omega}$ is
necessarily zero. Because we consider a closed 3-manifold, we can decompose
the expansion tensor $\boldsymbol{\Theta}$ into scalar, vector and tensor
parts (York, 1973):

$\Theta_{ab}=\chi h_{ab}+D_{(a}v_{b)}+\Xi_{ab},$ (28)

with $\chi$ a scalar field, $\boldsymbol{v}$ a spatial vector and
$\boldsymbol{\Xi}$ is a transverse-traceless tensor , i.e.
${\Xi_{c}}^{c}\coloneqq 0$ and $D_{c}\Xi^{ca}\coloneqq 0$. Then, from equation
(28) we have $\theta=3\chi+D_{c}v^{c}$, and the momentum constraint (22)
becomes

$-2D_{a}\chi+D^{c}\tensor[^{\Delta}]{\Omega}{{}_{a}{}_{c}}+v_{a}\frac{\mathcal{R}}{3}=0,$
(29)

where $\tensor[^{\Delta}]{\Omega}{{}_{a}{}_{b}}\coloneqq
D_{[a}w_{b]}-D_{[a}v_{b]}$. We introduce the Hodge decomposition on
$\boldsymbol{v}$ as
${\boldsymbol{v}\eqqcolon\boldsymbol{D}\Phi_{v}+\textrm{curl}\boldsymbol{A}+\boldsymbol{\lambda}}$,
where $\Phi_{v}$ and $\boldsymbol{A}$ are respectively the scalar potential
and the vector potential of $\boldsymbol{v}$, and $\boldsymbol{\lambda}$ is a
harmonic 1-form. As the term $D_{a}\chi$ is vorticity-free and the term
$D^{c}\tensor[^{\Delta}]{\Omega}{{}_{c}{}_{a}}$ is divergence-free, we have

$\displaystyle D_{a}\chi=D_{a}\left(\frac{\mathcal{R}}{6}\Phi_{v}\right),$
(30) $\displaystyle
D^{c}\tensor[^{\Delta}]{\Omega}{{}_{c}{}_{a}}=\frac{\mathcal{R}}{3}\left(\epsilon_{acd}D^{c}A^{d}+\lambda_{a}\right).$
(31)

We see that, contrary to the Euclidean case in Vigneron (2021), the first
equation (30) does not imply anymore that $\chi$ is only a function of time.
Instead we have

$\chi(t,x^{i})=cst(t)+\frac{\mathcal{R}(t)}{6}\Phi_{v}(t,x^{i}).$ (32)

The second equation (31) shows that $\boldsymbol{v}\not=\boldsymbol{w}$ in
general. This is a radical difference with the Euclidean case: the vectors
inside the expansion and vorticity tensors are different. We finally have

$\Theta_{ab}=\chi(t,x^{i})h_{ab}+D_{(a}v_{b)}+\Xi_{ab}\quad;\quad\Omega_{ab}=D_{[c}v_{a]}+\tensor[^{\Delta}]{\Omega}{{}_{a}{}_{b}}+\omega_{ab}.$
(33)

####  5.5.3 The gravitational system.

The system (18)–(26) of the present NEN theory can be rewritten to feature the
gravitational field and the analogue to the Poisson equation. The
gravitational field is defined as the opposite to the 4-acceleration of the
Galilean observer (see section V in Vigneron (2021)) and is given by

$g^{a}\coloneqq\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)v^{a}+2v^{c}\left({\Theta_{c}}^{a}+{\Omega_{c}}^{a}\right)-v^{c}D_{c}v^{a}-\tensor[^{\boldsymbol{u}}]{a}{}^{a}.$
(34)

Then, accounting for the expression of $\boldsymbol{\Theta}$ and
$\boldsymbol{\Omega}$ in equation (33) and defining the tensor
$\boldsymbol{\mathcal{T}}\coloneqq\tensor[^{\Delta}]{\boldsymbol{\Omega}}{}+\boldsymbol{\omega}$,
the gravitational system takes the form:

Non-Euclidean Newtonian theory I: The gravitational field is given by
$\displaystyle g^{a}$
$\displaystyle=\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)v^{a}+v^{c}D_{c}v^{a}+2v^{c}\left(\chi{\delta_{c}}^{a}+{\Xi_{c}}^{a}+{\mathcal{T}_{c}}^{a}\right)-(a_{\not=\rm
grav})^{a},$ (35) $\displaystyle D_{c}g^{c}$ $\displaystyle=-4\pi
G{\rho}-{\Xi_{cd}\Xi^{cd}}+{\mathcal{T}_{cd}\mathcal{T}^{cd}}-\frac{\mathcal{R}}{3}{v_{c}v^{c}}$
(36)
$\displaystyle\qquad-3\left[{\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\chi}+{\chi^{2}}-{v^{c}D_{c}\chi}\right],$
$\displaystyle D_{[a}g_{b]}$
$\displaystyle=-\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}+\boldsymbol{v}}\right)\mathcal{T}_{ab},$
(37) with $\displaystyle\chi$
$\displaystyle=cst(t)+\frac{\mathcal{R}(t)}{6}\Phi_{v},$ (38) $\displaystyle
D^{c}\mathcal{T}_{ca}$
$\displaystyle=\frac{\mathcal{R}}{3}\left(V_{a}-D_{a}\Phi_{v}\right).$ (39)
where $\Phi_{v}$ is the scalar potential of $\boldsymbol{v}$, and
$\boldsymbol{\mathcal{T}}$ a closed 2-form. These equations are completed by
$\displaystyle\mathcal{R}_{ab}$
$\displaystyle=\frac{\mathcal{R}(t)}{3}h_{ab},$ (40)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)h_{ab}$
$\displaystyle=2\left(\chi h_{ab}+D_{(a}v_{b)}+\Xi_{ab}\right),$ (41)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\rho$
$\displaystyle=-\rho\left(3H+D_{c}v^{c}\right),$ (42) and the averaged
equations $\displaystyle 3\left(\dot{H}+H^{2}\right)+4\pi
G\left\langle\rho\right\rangle_{\Sigma}-\Lambda=\mathcal{Q}_{\Sigma}\quad;\quad\left\langle\rho\right\rangle_{\Sigma}=\frac{M_{\rm
tot}}{V_{\Sigma}(t)},$ (43) where $V_{\Sigma}$ is the volume of $\Sigma$ and
$H(t)\coloneqq\dot{V}_{\Sigma}/3V_{\Sigma}=\left\langle\chi\right\rangle_{\Sigma}$
its global expansion rate, and where
$\displaystyle\mathcal{Q}_{\Sigma}=-\left\langle\Xi_{cd}\Xi^{cd}\right\rangle_{\Sigma}+\left\langle\mathcal{T}_{cd}\mathcal{T}^{cd}\right\rangle_{\Sigma}-\frac{\mathcal{R}}{3}\left\langle
v_{c}v^{c}\right\rangle_{\Sigma}+\frac{\mathcal{R}}{6}\left(\left\langle\Phi_{v}^{2}\right\rangle_{\Sigma}-\left\langle\Phi_{v}\right\rangle_{\Sigma}^{2}\right),$
(44) where the average on a scalar $\psi$ over the whole spatial manifold is
defined as
$\left\langle\psi\right\rangle_{\Sigma}(t)\coloneqq\frac{1}{V_{\Sigma}}\int_{\Sigma}\psi\sqrt{\mathrm{det}(h_{ab})}\mathrm{d}^{3}x$.

####  5.5.4 Gravitomagnetism.

In the above system we introduced the closed 2-form
$\boldsymbol{\mathcal{T}}\coloneqq\tensor[^{\Delta}]{\boldsymbol{\Omega}}{}+\boldsymbol{\omega}$.
This field corresponds to a gravitomagnetic field whose Newton-Faraday
equation is equation (37) and whose Newton-Ampère equation is equation (39).
The later does not feature any displacement current. The analogue to the
current density is
$J_{a}\coloneqq\frac{\mathcal{R}}{3}\left(V_{a}-D_{a}\Phi_{v}\right)$.

We recall that the “non-Euclidean” in the name NEN theory refers to the
topology and not the spatial curvature. Therefore the NEN theory is an
extension of Newton’s theory whose purpose is not to take into account post-
Newtonian, i.e. relativistic, effects coming from spatial curvature, but
effects coming from the global topology and which should be present at the
zeroth order of a non-relativistic limit of general relativity. In this sense,
we expect, as for classical Newton’s theory, that no-gravitomagnetism should
be present in the “right” NEN theory, i.e. $D_{[a}g_{b]}=0$. This phenomenon
is a relativistic effect that should appear only when considering the first
order of the limit (e.g. Kofman & Pogosyan (1995)). Gravitomagnetism will not
be present in the second NEN theory presented in section 5.6.

The reason why topology is not something relativistic is because this is a
property of the 4-manifold $\mathcal{M}$ we consider for our Universe, which
does not depend on the structure we define on this manifold: i.e. whether
$\mathcal{M}$ is equipped with a Lorentzian or a Galilean structure. This
implies that topology should not only be considered under the framework of a
relativistic theory, but also under the framework of a Newtonian (i.e.
Galilean invariant) theory, as done in the present paper.

####  5.5.5 Expansion law and generalised Buchert-Ehlers theorem.

The term $\mathcal{Q}_{\Sigma}$ in the expansion law (43) measures the
difference of the global expansion of the manifold $\Sigma$ with respect to
the Friedmanian case. It is called the cosmological backreaction. With respect
to the Euclidean case (derived in Buchert & Ehlers (1997); Vigneron (2021))
where
‘$\mathcal{Q}_{\Sigma}=-\left\langle\Xi_{cd}\Xi^{cd}\right\rangle_{\Sigma}+\left\langle\omega_{cd}\omega^{cd}\right\rangle_{\Sigma}$’,
equation (44) features additional terms. These terms depend on the scalar
curvature, the velocity $\boldsymbol{v}$, and the gravitomagnetic field
$\boldsymbol{\mathcal{T}}$. This shows that the backreaction depends on the
class of topology, via the sign of $\mathcal{R}$, but also on the dynamical
properties of the fluid, via $\boldsymbol{v}$ and $\boldsymbol{\mathcal{T}}$.
Especially the term ‘$-\frac{\mathcal{R}(t)}{3}\left\langle
v^{c}v_{c}\right\rangle_{\Sigma}$’ shows that the mean specific energy of the
fluid directly affects the backreaction. We expect this specific energy to be
more important in the late Universe when structure formation increases. Then,
formula (44) suggests that the structure formation might play a major role on
the global expansion of our Universe if the topology is non-Euclidean. This
behaviour is also present in general relativity Brunswic & Buchert (2020), but
in our case it arises at a non-relativistic level, as it shows that the
Universe might have a local Newtonian dynamics, but a global dynamics which
differs from the one of a homogeneous Universe. However, quantifying the mean
specific energy remains difficult because it depends on the mesoscopic scale,
which is not well defined in cosmology.

####  5.5.6 Space–time separation of the spatial metric?

In important feature of Newton’s theory is the fact that the spatial metric
can be separated in space and time (if the expansion is isotropic) with
“$h_{ab}(t,x^{i})=a^{2}(t)\tilde{h}_{ab}(x^{i})$”, where $\bar{h}_{ab}(x^{i})$
is a flat metric. This separation ensures that the spatial metric has no local
dynamics and that the resolution of Newton’s equations is “simple”. Such a
separation is not possible in the proposed NEN theory because $\chi$ depends
on time and space. Instead, when solving equation (41) with
$\boldsymbol{\Xi}=0$ we can only have the following conformal form (which is
valid in the Galilean coordinates: $\boldsymbol{\beta}=-\boldsymbol{v}$):

$h_{ab}(t,x^{i})=\psi^{2}(t,x^{i})\bar{h}_{ab}(x^{i}),$ (45)

where $\partial_{t}\psi/\psi\coloneqq\chi$. We can still define a background
metric $\bar{\boldsymbol{h}}$ independent of time (as in the Euclidean case),
but its Ricci curvature tensor $\boldsymbol{\bar{\mathcal{R}}}$ has a
traceless part, and its scalar curvature ${\bar{\mathcal{R}}}$ is not
spatially constant anymore (see the conformal relation (7.42) in Gourgoulhon
(2012)). Such a feature makes the equations hardly tractable, which is
something we do not expect for a Newtonian-like theory. The only way to have a
complete separation is to suppose $\boldsymbol{D}\Phi_{v}=0$. But in that case
the vector $\boldsymbol{v}$, which is interpreted as the spatial velocity of
the fluid, is purely rotational.

In conclusion, it is difficult to consider that the proposed NEN theory is
physically relevant.

####  5.5.7 Limitations of this NEN theory.

We list here the features of this theory which, to our opinion, make it
unphysical as a Newtonian theory:

  1. 1.

There is a gravitomagnetic field which cannot be taken to be zero without loss
of generality on $\boldsymbol{v}$.

  2. 2.

The vectors inside $\boldsymbol{\Theta}$ and $\boldsymbol{\Omega}$ are
different, i.e. $\tensor[^{\Delta}]{\Omega}{{}_{a}{}_{b}}\not=0$.

  3. 3.

The spatial metric cannot be separated in space and time without loss of
generality on $\boldsymbol{v}$. No “simple” background metric can be defined,
and the system of equations is hardly tractable.

  4. 4.

There is no exact $N$-body description because the “Poisson equation” (36) is
not linear and does not only feature the gravitational field, but also the
velocity. In other words, the gravitational field created by two mass points
(described by two Dirac fields in the density field) does not corresponds to
the sum of the gravitational field solution of (36) for a single Dirac field,
and this mainly because of the presence of
$\frac{\mathcal{R}}{3}{v_{c}v^{c}}$.

Furthermore, as the curvature term in the modified Newton-Cartan equations is
added for topological reasons, we do not want to interpret it as a curvature-
fluid coupling term. Therefore, in the absence of a fluid we would expect this
term to be still present, which seems difficult to concile with the fact that
$\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{u}}]{b}{}_{\alpha\beta}$
explicitly depends on the presence of a fluid.

###  5.6 NEN theory II: Galilean observer curvature term

In this section, we consider $\boldsymbol{B}\coloneqq\boldsymbol{G}$.

####  5.6.1 The system of equations.

The curvature term projected with respect to $\boldsymbol{u}$ and
$\boldsymbol{h}$ gives

$\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{G}}]{b}{}_{\mu\nu}h^{a\mu}h^{b\nu}=\frac{\mathcal{R}}{3}h^{ab}\quad;\quad\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{G}}]{b}{}_{\mu\nu}u^{\mu}h^{a\nu}=\frac{\mathcal{R}}{3}v^{a}\quad;\quad\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{G}}]{b}{}_{\mu\nu}u^{\mu}u^{\nu}=\frac{\mathcal{R}}{3}v^{\mu}v^{\nu}\tensor[^{\boldsymbol{G}}]{b}{}_{\mu\nu}.$
(46)

Then the 1+3-system of equations is composed of: evolution equations

$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\rho=-\rho\theta-
D_{c}q^{c},$ (47)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)h^{ab}=-2\Theta^{ab},$
(48)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\theta=-4\pi
G\rho+\Lambda+D_{c}\tensor[^{\boldsymbol{u}}]{a}{}^{c}-\Theta^{cd}\Theta_{cd}+\Omega^{cd}\Omega^{cd}\,{\color[rgb]{0.7,0,0}-\frac{\mathcal{R}}{3}v^{c}v_{c}},$
(49)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\Omega_{ab}=D_{[a}\tensor[^{\boldsymbol{u}}]{a}{}_{b]},$
(50)

and constraint equations

$\displaystyle{D_{c}\left(\Theta^{ac}+\Omega^{ac}\right)-D^{a}\theta=\color[rgb]{0.7,0,0}\frac{\mathcal{R}}{3}v_{a}},$
(51) $\displaystyle D_{[a}\Omega_{bc]}=0,$ (52)
$\displaystyle{\mathcal{R}^{ab}=\color[rgb]{0.7,0,0}\frac{\mathcal{R}}{3}h^{ab}},$
(53) $\displaystyle\mathcal{R}^{d[abc]}=0,$ (54)

with

$\displaystyle\rho\,\tensor[^{\boldsymbol{u}}]{a}{}^{a}$
$\displaystyle=-D^{a}P-D_{c}\pi^{ca}-\left[\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)q^{a}+q^{a}\theta+2q_{c}\left(\Theta^{ca}+\Omega^{ca}\right)\right].$
(55)

With respect to the first proposed NEN theory in section 5.5, this system of
equations features additional terms in the Raychaudhuri equation (49) and in
the momentum constraint (51). These terms will radically change the solutions
to the constraint equations and the expansion law, as presented in the
following sections.

####  5.6.2 Solving the constraint equations.

Following the same logic than in section 5.5.2 for the first NEN theory, the
momentum constraint (51) becomes

$-2D_{a}\chi+D^{c}\tensor[^{\Delta}]{\Omega}{{}_{a}{}_{c}}+v_{a}\frac{\mathcal{R}}{3}=v_{a}\frac{\mathcal{R}}{3},$
(56)

which implies through uniqueness of the Hodge decomposition

$\displaystyle D_{a}\chi=0,$ (57) $\displaystyle
D^{c}\tensor[^{\Delta}]{\Omega}{{}_{c}{}_{a}}=0.$ (58)

This is equivalent to the Euclidean case in Vigneron (2021). Therefore $\chi$
depends only on time, and $\tensor[^{\Delta}]{\boldsymbol{\Omega}}{}=0$. We
finally have

$\Theta_{ab}=H(t)h_{ab}+D_{(a}v_{b)}+\Xi_{ab}\quad;\quad\Omega_{ab}=D_{[c}v_{a]}+\omega_{ab},$
(59)

where
$H\coloneqq\dot{V}_{\Sigma}/3V_{\Sigma}=\left\langle\chi\right\rangle_{\Sigma}=\chi$
is the expansion rate of the total volume $V_{\Sigma}$ of the spatial manifold
$\Sigma$.

####  5.6.3 The gravitational system.

The gravitational system is obtained when rewriting the system (47)–(55) as
function of the gravitational field, and accounting for the expressions of
$\boldsymbol{\Theta}$ and $\boldsymbol{\Omega}$ in equation (59):

Non-Euclidean Newtonian theory II: The gravitational field is given by
$\displaystyle g^{a}$
$\displaystyle=\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)v^{a}+v^{c}D_{c}v^{a}+2v^{c}\left(H{\delta_{c}}^{a}+{\Xi_{c}}^{a}+{\omega_{c}}^{a}\right)-(a_{\not=\rm
grav})^{a},$ (60) $\displaystyle D_{c}g^{c}$ $\displaystyle=-4\pi
G\widehat{\rho}-\widehat{\Xi_{cd}\Xi^{cd}}+\widehat{\omega_{cd}\omega^{cd}},$
(61) $\displaystyle D_{[a}g_{b]}$
$\displaystyle=-\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}+\boldsymbol{v}}\right)\omega_{ab},$
(62) where
$\widehat{\psi}\coloneqq\psi-\left\langle\psi\right\rangle_{\Sigma}$. These
equations are completed by $\displaystyle\mathcal{R}_{ab}$
$\displaystyle=\frac{\mathcal{R}(t)}{3}h_{ab},$ (63)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)h_{ab}$
$\displaystyle=2\left(Hh_{ab}+D_{(a}v_{b)}+\Xi_{ab}\right),$ (64)
$\displaystyle\left(\partial_{t}-\mathcal{L}_{\boldsymbol{\beta}}\right)\rho$
$\displaystyle=-\rho\left(3H+D_{c}v^{c}\right),$ (65) and the averaged
equations $\displaystyle 3\left(\dot{H}+H^{2}\right)+4\pi
G\left\langle\rho\right\rangle_{\Sigma}-\Lambda=-\left\langle\Xi_{cd}\Xi^{cd}\right\rangle_{\Sigma}+\left\langle\omega_{cd}\omega^{cd}\right\rangle_{\Sigma}\quad;\quad\left\langle\rho\right\rangle_{\Sigma}=\frac{M_{\rm
tot}}{V_{\Sigma}(t)}.$ (66)

A remarkable thing is that the above system of equations is algebraically
equivalent to Newton’s gravitational system of equations, but with a non-zero
curvature in the spatial connection. Especially, contrary to the 1+3-system
(47)-(55) and the spacetime system (12)-(14), the gravitational system
(60)-(66) does not feature any additional curvature terms with respect to the
Euclidean case.

Remark. This system of equations is similar to the one proposed by Roukema &
Różański (2009); Barrow (2020), the main difference being that expansion is
present and that the Poisson equation (61) appears in its cosmological form,
i.e. featuring $\widehat{\rho}$ instead of $\rho$. Actually, this form should
be considered as the right form of the Poisson equation, the one with just
$\rho$ being a restricted case when we consider an isolated system in an
infinite space, implying $\left\langle\rho\right\rangle_{\Sigma}=0$ and
$\widehat{\rho}=\rho$.

####  5.6.4 Expansion law or generalised Buchert-Ehlers theorem.

The expansion law is the same as for Newton’s theory. Therefore, the same
conclusions apply: the expansion law for
${\boldsymbol{\Xi}=0=\boldsymbol{\omega}}$ is given by the Friedmann equation
for the acceleration of the scale factor, and this for any inhomogeneous
solutions. This means that there is no global backreaction of the
inhomogeneities on the isotropic expansion.

If $\boldsymbol{\Xi}\not=0$ and $\boldsymbol{\omega}\not=0$, the backreaction
is

$\mathcal{Q}_{\Sigma}=-\left\langle\Xi_{cd}\Xi^{cd}\right\rangle_{\Sigma}+\left\langle\omega_{cd}\omega^{cd}\right\rangle_{\Sigma}.$
(67)

The fields $\boldsymbol{\Xi}$ and $\boldsymbol{\omega}$ are free, and
especially they are decoupled from the fluid spatial velocity
$\boldsymbol{v}$. When deriving the Newton-Cartan equations as a limit of
general relativity, we showed in Vigneron (2021) that $\boldsymbol{\omega}$
must vanish due to geometrical constraints. Therefore we also expect this
tensor to vanish if this NEN theory can be retrieved with a non-relativistic
limit. We are let with the traceless-transverse tensor $\boldsymbol{\Xi}$
which describes an anisotropic expansion and which is not constrained by the
system (12)-(14). So unless there is a hidden condition relating this tensor
to $\boldsymbol{v}$, coming from general relativity, it is not clear to us if
we can consider that equation (67) corresponds to a physical backreaction.

If this NEN theory is the right one, then a Euclidean, spherical or hyperbolic
closed universe, whose dynamics is locally Newtonian and the global expansion
isotropic, has no global backreaction. For such a universe, no effects of the
inhomogeneities can be observed on the evolution of the scale factor.

Remark. In equation (67), the backreaction $\mathcal{Q}_{\Sigma}$ is defined
as the deviation with respect to the Friedmann equation: i.e.
$\mathcal{Q}_{\Sigma}\coloneqq 3\left(\dot{H}+H^{2}\right)+4\pi
G\left\langle\rho\right\rangle_{\Sigma}-\Lambda$. Because the Raychaudhuri
equation (49) features an additional term, it is not algebraically equivalent
to the Raychaudhuri equation of Newton’s theory, and thus the usual formula
“$\mathcal{Q}_{\mathcal{D}}\coloneqq\frac{2}{3}\left(\left\langle\theta^{2}\right\rangle_{\cal
D}-\left\langle\theta\right\rangle_{\cal
D}^{2}\right)-\left\langle\sigma_{cd}\sigma^{cd}\right\rangle_{\cal
D}+\left\langle\Omega_{cd}\Omega^{cd}\right\rangle_{\cal D}$” is not valid
anymore. Instead the definition $\mathcal{Q}_{\Sigma}\coloneqq
3\left(\dot{H}+H^{2}\right)+4\pi
G\left\langle\rho\right\rangle_{\Sigma}-\Lambda$ leads to

$\mathcal{Q}_{\Sigma}=\frac{2}{3}\left(\left\langle\theta^{2}\right\rangle_{\Sigma}-\left\langle\theta\right\rangle_{\Sigma}^{2}\right)-\left\langle\sigma_{cd}\sigma^{cd}\right\rangle_{\Sigma}+\left\langle\Omega_{cd}\Omega^{cd}\right\rangle_{\Sigma}-\frac{\mathcal{R}}{3}\left\langle
v^{c}v_{c}\right\rangle_{\Sigma}.$

####  5.6.5 Space–time separation of the spatial metric.

Taking equation (64) in the class of coordinates where
$\boldsymbol{\beta}=-\boldsymbol{v}$ (Galilean coordinates), it becomes

$\tensor[]{\partial}{{}_{t|_{-\boldsymbol{v}}}}h_{ab}+2H(t)h_{ab}=\Xi_{ab},$
(68)

which, in the case $\boldsymbol{\Xi}=0$, leads to

$h_{ab}(t,x^{i})=a^{2}(t)\bar{h}_{ab}(x^{i}),$ (69)

where the Ricci curvature tensor $\bar{\boldsymbol{\mathcal{R}}}$ related to
the metric $\bar{\boldsymbol{h}}$ is
$\displaystyle\bar{\boldsymbol{\mathcal{R}}}=\frac{\mathcal{R}_{\rm\bf
i}}{3}\bar{\boldsymbol{h}}$, with $\mathcal{R}_{\rm\bf i}$ the initial scalar
curvature of $\boldsymbol{h}$.

Equation (69) shows that the spatial metric components can be separated into
space and time dependence, which was also the case in Newton’s theory. This is
a major property as it ensures that in Galilean coordinates, we can solve the
system (63)–(66) assuming a time-independent background metric
$\bar{\boldsymbol{h}}$.

In the case $\boldsymbol{\Xi}\not=0$, the equation (68) does not lead to the
separation of the metric components. From the uniqueness of the SVT
decomposition, because $\boldsymbol{\Xi}$ is a symmetric traceless-transverse
tensor, it cannot be written as the symmetric gradient of a vector. This means
that there is no coordinate system in which equation (64) becomes
$\tensor[]{\partial}{{}_{t|_{\boldsymbol{\beta}}}}h_{ab}+2H(t)h_{ab}=0$.
Therefore, the space and time separation of $\boldsymbol{h}$ is only possible
if $\boldsymbol{\Xi}=0$.

####  5.6.6 $N$-body description.

A major strength of Newton’s theory is the fact that its system of equations
can be written either as a partial differential system (fluid description), or
as an ordinary differential system for the evolution equations (particle
description). This results from the linearity of the Poisson equation. In the
present proposed NEN theory, the same feature is present: the Green function
of the Poisson equation (61) is sufficient to describe the gravitational field
of any distribution of matter. Therefore, we have an exact $N$-body
description:

  1. 1.

the gravitational field $\boldsymbol{g}_{\rm tot}$, i.e. the solution of
equation (61), for $N$ mass points described by Dirac fields
$\\{M_{i}\delta(\boldsymbol{x}-\boldsymbol{x}_{i})\\}_{i\in\llbracket
1,N\rrbracket}$, with $M_{i}$ and $\boldsymbol{x}_{i}$ the mass and position
of the $i^{\rm th}$ particle, is

$\boldsymbol{g}_{\rm
tot}(\boldsymbol{x})=\sum_{i}^{N}\boldsymbol{g}_{\Sigma}(\boldsymbol{x}-\boldsymbol{x}_{i}),$
(70)

where $\boldsymbol{g}_{\Sigma}$ is the solution of (61) for a single Dirac
field in $\Sigma$. We calculate it in the case $\Sigma=\mathbb{S}^{3}$ in the
next section.

  2. 2.

The trajectory of each point mass is given (for isotropic expansion) by
Newton’s second law (60):

$\ddot{x_{i}}^{a}+2H\dot{x_{i}}^{a}=g_{\rm tot}^{a}+(a_{\not=\rm grav})^{a}.$
(71)

This is a major strength of the present NEN theory as $N$-body numerical
simulations can be performed without any approximation with respect to the
equations, something not possible with the first theory we proposed in section
5.5 which a priori cannot be reduced to a system of ordinary differential
equations for the evolution equations.

####  5.6.7 The mass point gravitational field.

The $N$-body description requires the mass point solution of the Poisson
equation (61) which corresponds to the solution of

$D_{c}g^{c}=-4\pi G\left[M\delta_{\Sigma}-\frac{M}{V_{\Sigma}}\right],$ (72)

where $\delta_{\Sigma}$ is the Dirac field of the manifold $\Sigma$. For
simplicity, we consider only the case where $\Sigma=\mathbb{S}^{3}$, i.e. the
3-sphere. Then, choosing the hyperspherical coordinates (5), and the particle
placed at the origin, we obtain

$\displaystyle g^{\xi}=-\frac{GM}{R_{\rm
c}^{3}\sin^{2}\xi}\left[1-\frac{\xi}{\pi}+\frac{\cos\xi\sin\xi}{\pi}\right],$
(73)

where $R_{\rm c}(t)=a(t)R_{c,\rm\bf i}=\sqrt{6/\mathcal{R}}$ is the curvature
radius of the 3-sphere and $R_{c,\rm\bf i}$ its initial value. Note that
because we do not use normalised coordinate basis vectors, we have
$\boldsymbol{g}\cdot\boldsymbol{g}=R_{\rm c}^{2}(g^{\xi})^{2}$, which means
that $R_{\rm c}g^{\xi}$ corresponds to the measured gravitational field
strength.

The behaviour of the gravitational field at the poles is ($\xi=0$ is the
particle position and $\xi=\pi$ is the opposite pole)

$\displaystyle R_{\rm c}g^{\xi}(\xi\sim 0)$
$\displaystyle=-\frac{GM}{r_{0}^{2}}-\frac{GM}{3R_{\rm
c}^{2}}+\mathcal{O}(r_{0}),$ (74) $\displaystyle R_{\rm c}g^{\xi}(\xi\sim\pi)$
$\displaystyle=-\frac{2GM}{3\pi R_{\rm
c}^{3}}r_{\pi}+\mathcal{O}\left(r_{\pi}^{3}\right),$ (75)

where $r_{0}:=R_{\rm c}\xi$ is the distance to the origin and $r_{\pi}:=R_{\rm
c}(\pi-\xi)$ the distance to the opposite pole. The first problem of the NEN
theory proposed by Roukema & Różański (2009); Barrow (2020) is solved: the
field at the opposite pole $\xi=\pi$ of a point mass particle does not diverge
anymore. It is a first order repulsive field, and therefore tends to zero when
$\xi\rightarrow\pi$. This is physically coherent.

The second term in the right-hand-side of equation (74) is an additional term
with respect to Newton’s gravitational law. It is called topological
acceleration. Such a zeroth order term is also present in Newton’s theory (but
in a different form) in the case $\Sigma$ is a 3-torus Roukema et al. (2007),
instead of $\mathbb{R}^{3}$ with fall-off conditions at infinity. Therefore it
is not necessarily linked to non-Euclidean topologies, but also to
non-$\mathbb{R}^{3}$ topologies.

####  5.6.8 Discussion.

This NEN theory has none of the limitations we listed for the first one in
section 5.5.7: the spatial metric is separable in space and time (if
$\boldsymbol{\Xi}=0$ which is the case if the expansion is isotropic); no
gravitomagnetic field is present (if $\boldsymbol{\omega}=0$ which is expected
from general relativity); $N$-body description is possible. Furthermore, it
solves the first problem of the NEN theory proposed by Roukema & Różański
(2009); Barrow (2020).

However, the curvature term in the modified Newton-Cartan equations still
requires the presence of a fluid to be defined. This contrasts with the
gravitational system (60)-(66) resulting from these equations, which is
defined even in the absence of a fluid. This suggests that we could define a
Galilean observer independently of a fluid. We propose the following covariant
definition:

Definition. A Galilean observer (in a 4-manifold equipped with a Galilean
structure) is define by a timelike vector $\boldsymbol{G}$ constrained by the
following equations:
$\displaystyle\begin{cases}h^{\mu[\alpha}\nabla_{\mu}G^{\beta]}=0,\\\
\nabla_{\mu}\left(h^{\nu(\mu}\nabla_{\nu}G^{\alpha)}-\frac{1}{3}\nabla_{\nu}G^{\nu}h^{\mu\alpha}\right)=0,\\\
h^{\alpha\mu}R_{\mu\nu}G^{\nu}=0.\end{cases}$ (76)

The second condition is equivalent to saying that the shear tensor
$\tensor[^{\boldsymbol{G}}]{\sigma}{}^{ab}\coloneqq\tensor[^{\boldsymbol{G}}]{\Theta}{}^{ab}-\frac{1}{3}\tensor[^{\boldsymbol{G}}]{\theta}{}h^{ab}$
of the Galilean observer is transverse, i.e.
$D_{c}\tensor[^{\boldsymbol{G}}]{\sigma}{}^{ca}=0$, where
$\tensor[^{\boldsymbol{G}}]{\Theta}{}^{ab}$ is the expansion tensor of
$\boldsymbol{G}$, and $\tensor[^{\boldsymbol{G}}]{\theta}{}$ its trace,
defined equivalently as for $\boldsymbol{u}$ in section (5.5.1). This
condition implies that in a SVT decomposition of
$\tensor[^{\boldsymbol{G}}]{\Theta}{}_{ab}$, the term $D_{(a}v_{b)}$ (see
section (28)) is zero.

(76) is an explicitly covariant definition of a Galilean observer. It is
equivalent to the following two conditions: (i) in coordinates adapted to the
foliation defined by $\boldsymbol{\tau}$ and such that
$\boldsymbol{\partial}_{t}=\boldsymbol{G}$, the time derivative of the spatial
components of the space metric have the form

$\frac{1}{2}\partial_{t}h^{ij}=-H(t)h^{ij}-\Xi^{ij},$ (77)

with $\Xi^{ij}$ being traceless-transverse and (ii) $\boldsymbol{G}$ is
vorticity free with respect to the Galilean structure, i.e.

$h^{\mu[\alpha}\nabla_{\mu}G^{\beta]}=0.$ (78)

This definition is close to the one of “Newtonian observers” in van Elst &
Ellis (1998), where the translation of their definition in the framework of
Galilean manifolds would give
“$h^{\mu[\alpha}\nabla_{\mu}G^{\beta]}=0=h^{\mu(\alpha}\nabla_{\mu}G^{\beta)}-h^{\alpha\beta}\nabla_{\mu}G^{\mu}/3$”.
In other words, the last conditions in (76) are replaced by a shear free
condition. This definition is a bite stronger than ours as it imposes
$\boldsymbol{\Xi}=0$ implying that expansion in a Newtonian-like theory should
be isotropic. Therefore, in absence of a condition obtained from a limit of
general relativity that would impose $\boldsymbol{\Xi}=0$, formula (76) is the
definition of a Galilean observer to use.

As for now we have not found a more compact equation which would be equivalent
to the three conditions (76). With this definition, we retrieve the same
gravitational system (60)-(66) but the spacetime equations of the theory, and
in particular the term
$\frac{\mathcal{R}}{3}\tensor[^{\boldsymbol{G}}]{b}{}_{\alpha\beta}$, are well
defined in vacuum with $\boldsymbol{u}=0$. The only requirement of the theory
is the existence of such a Galilean observer, which must be justified from
general relativity or be an additional physical property in the definition of
Galilean structures.

Remark. In order for $\Xi_{ij}$ to describe only anisotropic expansion, we
expect this tensor to be gradient free, i.e. $D_{k}\Xi_{ij}=0$. Having just a
divergence free condition would allow $\boldsymbol{\Xi}$ to have local
dynamics. Therefore, a better definition for a Galilean observer might be to
impose the full expansion tensor of $\boldsymbol{G}$ to have a zero spatial
gradient:
$h^{\gamma\mu}\nabla_{\mu}\left[h^{\nu(\alpha}\nabla_{\nu}G^{\beta)}\right]=0$.

##  6 Conclusion

In this paper we presented two novel approaches to define a non-Euclidean
Newtonian theory in either spherical or hyperbolic classes of topologies.
These approaches consider the Galilean invariance as a fundamental principle
of a Newtonian-like theory, regardless of the topology. They are based on a
minimal modification of the Newton-Cartan equation allowing for a non-
Euclidean topology. We derived the expansion law in each case, thus
generalising the Buchert-Ehlers theorem. The main properties of these two
theories are summarised in table 1.

In the first approach, a backreaction appears which depends, in particular, on
the averaged specific energy of the fluid (an expected behaviour of
backreaction), and also on the type of topology. However because the spatial
metric is non-separable, no $N$-body calculation is possible and a
gravitomagnetic field is present, this theory is far from what we a priori
expect of a Newtonian-like theory.

The second approach has the same expansion law as in Newton’s theory. Then in
the assumption of an isotropic global expansion, there is no cosmological
backreaction. This extends the conclusions of the Buchert-Ehlers theorem to
spherical and hyperbolic topologies. In this theory, $N$-body calculation can
be performed, along with the possibility of separating the spatial metric in
space and time, i.e. having a background description.

Without a justification from a non-relativistic limit of general relativity,
the question of the “right” non-Euclidean Newtonian theory remains open, as
well as the generalisation of the Buchert-Ehlers theorem. Because of the many
similarities of the second proposed theory in section 5 with the (Euclidean)
Newtonian theory, we expect it to be the “right” one, and its equations
[equations (60)-(66)] should be used to perform $N$-body simulations of
structure formation in a universe with a non-Euclidean topology. For this
purpose, in (Vigneron & Roukema, 2022) we calculate the gravitational
potential in different spherical topologies using this NEN theory.

Follow-up work is dedicated to further study the second NEN theory and to
develop a limit of general relativity allowing us to derive the “right” NEN
theory (Vigneron, 2022a).

## Acknowledgements

This work has received funding from the Center of Excellence of Nicolaus
Copernicus University in Toruǹ. I would like to thank Boudewijn Roukema for
making me discover the world of non-Euclidean Newtonian theories during my
master thesis in Toruń. I also thank the referees for their very constructive
remarks.

NEN theory |  $3\left(\dot{H}+H^{2}\right)+4\pi G\left\langle\rho\right\rangle_{\Sigma}-\Lambda=$ |  $N$-body |  Separable spatial metric |  Gravitomagnetism
---|---|---|---|---
Based on the NC equation: $R_{ab}-\tensor[^{\boldsymbol{u}}]{b}{}_{ab}\frac{\mathcal{R}}{3}=4\pi G\rho\,\tau_{a}\tau_{b}$ (Fluid observer term) |  $-\left\langle\Xi_{cd}\Xi^{cd}\right\rangle_{\Sigma}+\left\langle\omega_{cd}\omega^{cd}\right\rangle_{\Sigma}$ $\displaystyle-\frac{\mathcal{R}}{3}\left\langle v_{c}v^{c}\right\rangle_{\Sigma}+\frac{\mathcal{R}}{6}\left(\left\langle\Phi_{v}^{2}\right\rangle_{\Sigma}-\left\langle\Phi_{v}\right\rangle_{\Sigma}^{2}\right)$ $+\left\langle\tensor[^{\Delta}]{\Omega}{{}_{c}{}_{d}}\tensor[^{\Delta}]{\Omega}{{}^{c}{}^{d}}\right\rangle_{\Sigma}$ with $v_{a}=D_{a}\Phi_{v}+\epsilon_{acd}D^{c}A^{d}$ |  No |  No, unless $\boldsymbol{\Xi}=0$ and $\Phi_{v}=0$ |  Yes: $\mathcal{T}_{ab}\coloneqq\tensor[^{\Delta}]{\Omega}{{}_{a}{}_{b}}+\omega_{ab}$ $D_{[a}g_{b]}=-\tilde{\partial}_{t}\mathcal{T}_{ab}$ $D^{c}\mathcal{T}_{ca}=J_{a}$ with $J_{a}\coloneqq\frac{\mathcal{R}}{3}\epsilon_{acd}D^{c}A^{d}$ and no displacement current
Based on the NC equation: $R_{ab}-\tensor[^{\boldsymbol{G}}]{b}{}_{ab}\frac{\mathcal{R}}{3}=4\pi G\rho\,\tau_{a}\tau_{b}$ (Galilean observer term) |  $-\left\langle\Xi_{cd}\Xi^{cd}\right\rangle_{\Sigma}+\left\langle\omega_{cd}\omega^{cd}\right\rangle_{\Sigma}$ |  Yes, gravitational field given by (73) in $\mathbb{S}^{3}$ |  Yes, if $\boldsymbol{\Xi}=0$ |  No, if $\boldsymbol{\omega}=0$
Table 1: Summary of the proposed NEN theories. The second column quantifies
the difference with an FLRW expansion. For the sake of generality we kept the
fields $\boldsymbol{\Xi}$ and $\boldsymbol{\omega}$ non zero. However, as in
Newton’s theory (see appendix B in Vigneron (2021)) we expect
$\boldsymbol{\omega}=0$ if the NEN theory is derived from a non-relativistic
Galilean limit. For an isotropic expansion, we also expect
$\boldsymbol{\Xi}=0$. In that case, only the first NEN theory features un non-
zero cosmological backreaction and a departure from the $\Lambda$CDM scenario.

## References

When available the bibliography style features three different links
associated with three different colors: links to the journal/editor website or
to a numerical version of the paper are in red, links to the ADS website are
in blue and links to the arXiv website are in green.

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$\Delta_{H}^{1/2}T\Delta_{K}^{-1/2}$ is defined on
${\mathcal{D}}(\Delta^{-1/2}_{K})$ and its closure coincides with
$J_{H}T^{\prime}J_{K}$.

  4. 4.

The map
$T(s):=\Delta_{H}^{-\mathrm{i}s}T\Delta_{K}^{\mathrm{i}s}\in{\mathcal{B}}({\mathcal{K}},{\mathcal{H}})$,
$s\in{\mathbb{R}}$ extends to a bounded weakly continuous map on
${\mathbb{R}}+\mathrm{i}[0,1/2]$ analytic in ${\mathbb{R}}+\mathrm{i}(0,1/2)$
and satisfying $T(\mathrm{i}/2)=J_{H}T^{\prime}J_{K}$.

###### Proof.

To see 2 $\Leftrightarrow$ 3 we note that $J_{\bullet}$ is an involution and
$S_{\bullet}=\Delta_{\bullet}^{-1/2}J_{\bullet}$ for $\bullet=H,K$. Therefore,
$TS_{H}\subset S_{H}T^{\prime}$ is equivalent to
$T\Delta_{K}^{-1/2}\subset\Delta_{H}^{-1/2}J_{H}T^{\prime}J_{K}$. This is
equivalent to $\Delta_{K}^{1/2}T\Delta_{H}^{-1/2}\xi=J_{H}T^{\prime}J_{K}\xi$
for $\xi\in{\mathcal{D}}(\Delta_{K}^{-1/2})$.

For 3 $\Rightarrow$ 4 let $\xi\in{\mathcal{H}}$ and $\eta\in{\mathcal{K}}$ be
entire vectors of exponential growth for $\Delta_{H}$ and $\Delta_{K}$,
respectively. We define

$f_{\xi,\eta}(z):=\langle\xi,\Delta_{H}^{-\mathrm{i}z}T\Delta_{K}^{\mathrm{i}z}\eta\rangle\equiv\langle\Delta_{H}^{\overline{-\mathrm{i}z}}\xi,T\Delta_{K}^{\mathrm{i}z}\eta\rangle$

which is an entire function with
$f_{\xi,\eta}(t+\mathrm{i}/2)=\langle\Delta_{H}^{1/2}\Delta_{H}^{\mathrm{i}t}\xi,T\Delta_{K}^{-1/2}\Delta^{\mathrm{i}t}_{K}\eta\rangle$,
which equals
$\langle\Delta_{H}^{\mathrm{i}t}\xi,J_{H}T^{\prime}J_{K}\Delta^{\mathrm{i}t}_{K}\eta\rangle$
by assuming 3. A priori one has the estimate

$\|f_{\xi,\eta}(z)\|\leq\max\\{\|T\|,\|T^{\prime}\|\\}\left\|({\mathbbm{1}}+\Delta_{H}^{-1/2})\xi\right\|\left\|({\mathbbm{1}}+\Delta_{K}^{-1/2})\eta\right\|$

and this can be improved to
$\|f_{\xi,\eta}(z)\|\leq\max\\{\|T\|,\|T^{\prime}\|\\}\|\xi\|\,\|\eta\|$ by
the three-line theorem (e.g. [35]). By density of $\xi$’s in $H$ and $\eta$’s
in $K$, 4 follows.

4 $\Rightarrow$ 3: Assuming 4 as in the step before $\langle
T\Delta_{K}^{-1/2}\eta,\Delta_{H}^{1/2}\xi\rangle=\langle
J_{H}T^{\prime}J_{K}\eta,\xi\rangle$ holds and the $\eta$’s and $\xi$’s form a
core for $\Delta_{K}^{-1/2}$ and $\Delta_{H}^{-1/2}$, respectively. It follows
that the equation holds for all $\eta\in{\mathcal{D}}(S_{K})$ and
$\xi\in{\mathcal{D}}(S_{H})$. This implies
$\Delta_{H}^{1/2}T\Delta_{K}^{-1/2}\eta=J_{H}T^{\prime}J_{K}\eta$ for all
$\eta\in{\mathcal{D}}(S_{K})$, namely 3 holds.

To see that 2 implies 1 we calculate for $g^{\prime}\in H^{\prime}$ and $f\in
K$

$\displaystyle\langle g^{\prime},Tf\rangle$ $\displaystyle=\langle
g^{\prime},TS_{K}f\rangle$ $\displaystyle=\langle
g^{\prime},S_{H}T^{\prime}f\rangle$ $\displaystyle=\langle
T^{\prime}f,S_{H}^{\ast}g^{\prime}\rangle$ $\displaystyle=\langle
T^{\prime}f,g^{\prime}\rangle\,\text{.}$

By assuming $\langle g^{\prime},Tf\rangle=\langle
T^{\prime}f,g^{\prime}\rangle$ for all $f\in K$ and $g^{\prime}\in H^{\prime}$
we get

$\displaystyle\langle g^{\prime},S_{H}T^{\prime}f\rangle$
$\displaystyle=\langle T^{\prime}f,S_{H}^{\ast}g^{\prime}\rangle=\langle
T^{\prime}f,g^{\prime}\rangle=\langle g^{\prime},Tf\rangle$

and because $H^{\prime}+\mathrm{i}H^{\prime}$ is dense in ${\mathcal{H}}$ it
holds $S_{H}T^{\prime}f=Tf$ for all $f\in K$. Then we have for $f,g\in K$

$\displaystyle TS_{K}(f+\mathrm{i}g)$
$\displaystyle=Tf-\mathrm{i}Tg=S_{H}T^{\prime}f-\mathrm{i}S_{H}T^{\prime}g=S_{H}T^{\prime}(f+\mathrm{i}g)$

and in particular $TS_{K}\subset S_{H}T^{\prime}$ and we showed 1 implies 2. ∎

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