# Phase space of static wormholes sustained by an isotropic perfect fluid

Stéphane<EMAIL_ADDRESS>Palais de la Découverte
Astronomy Department
Avenue Franklin Roosevelt
75008 Paris
France

###### Abstract

A phase space is built that allows to study, classify and compare easily large
classes of static spherically symmetric wormholes solutions, sustained by an
isotropic perfect fluid in General Relativity. We determine the possible
locations of equilibrium points, throats and curvature singularities in this
phase space. Throats locations show that the spatial variation of the
gravitational redshift at the throat of a static spherically symmetric
wormhole sustained by an isotropic perfect fluid is always diverging,
generalising the result that there is no such wormhole with zero-tidal force.
Several specific static spherically symmetric wormholes models are studied. A
vanishing density model leads to an exact solution of the field equation
allowing to test our dynamical system formalism. It also shows how to extend
it to the description of static black holes. Hence, the trajectory of the
Schwarzschild black hole is determined. The static spherically symmetric
wormhole solutions of several usual isotropic dark energy (generalised
Chaplygin gas, constant, linear and Chevallier-Polarski-Linder equations of
state) and dark matter (Navarro-Frenk-White profile) models are considered.
They show various behaviours far from the throat: singularities, spatial
flatness, cyclic behaviours, etc. None of them is asymptotically Minkowski
flat. This discards the natural formation of static spherically symmetric and
isotropic wormholes from these dark fluids. Last we consider a toy model of an
asymptotically Minkowski flat wormhole that is a counterexample to a recent
theorem claiming that a static wormhole sustained by an isotropic fluid cannot
be asymptotically flat on both sides of its throat.

###### pacs:

95.30.Sf, 98.80.Jk

##  I Introduction

In this paper, we define a phase space to study, classify and compare large
classes of static spherically symmetric wormholes solutions sustained by a
perfect fluid in General Relativity and derive various physical results.
Such wormholes are defined by two free functions. We choose the first one by
relating the radial pressure $p$ to the tangential pressure $\tau$ such as
$p=-\tau$. The perfect fluid is then isotropic as a cosmological fluid (other
choices of perfect fluid are possible with $p\not=-\tau$ as in Kuh17 ; Sus05 ;
Lob06 ; Kuh16 ). Such fluids are interesting for several reasons. In
particular, they are generally (not always) physically motivated and
constrained by observations such as the supernovae and the CMB. Another reason
is the possibility that a ghost dark energy could both rules the Universe
expansion and forms static spherically symmetric wormholes naturally, a
possibility that we discard in this paper for some of the most common dark
fluids used in the literature. The second free function can be the wormhole
energy density, its pressure, its spatial shape, its gravitational redshift,
etc. We let it free when studying general properties of the phase space and we
specify it when examining some particular wormhole models.
Dynamical system analysis is often used in cosmologyWai05 . If it is not new
for inhomogeneous spherical symmetric solutions, few of them have been studied
that way. See for instance Mig89 ; Mig92 ; Pol95 ; Gan15 ; Cru17 ; Sus08 ;
MonZan09 . Due to the complexity of the field equations, it is generally
difficult to obtain exact wormhole solutions or study extensively some
numerical ones. Some wormhole models are easier to analyse than others, as the
ones with zero tidal-force contrary to those for which the form of the exotic
fluid is specified (for instance by its equation of state). The dynamical
system analysis is then particularly useful. It allows representing in a
finite phase space, with an appropriate choice of normalised variables, all
the solutions of a first order equations system describing a static
spherically symmetric wormhole without having to solve them. Beyond the
mathematical analysis, classification and comparison of wormhole models in a
common framework, the dynamical system formalism also allows to get general
physical results thanks to the determination of specific points in the phase
space that can be common to any model. Among them, the equilibrium points
representing transient or asymptotically behaviours like the Minkowski
solutions, but also some non equilibrium points like those standing for the
wormhole throats.
To find these points, we need to define a finite phase space by a set of
normalised variables independent from the choice of the above mentioned free
function and to rewrite the field equations. Then, in this general framework,
we look for all the possible locations of throats, equilibrium points and
curvature singularities in the phase space. To check the presence of
singularities, we also calculate the geodesic equation for spacetimes
associated to a phase space trajectory. We note that the points of the phase
space corresponding to throats are not equilibrium points despite they are
common to all phase space trajectories of wormhole models. After these
mathematical general properties, we study several static spherically symmetric
wormhole models in this unified framework and get some physical results.
A first model considers a vanishing density. Its exact solution is given. It
allows checking the consistency of our dynamical system of equations with the
original wormhole field equations systemMorTho88 . From a physical viewpoint,
it also shows how our static wormhole phase space can be used to represent
static black holes as well although we did not pursue further in this paper
dedicated to wormholes. We just determine the trajectory of the Schwarzschild
black hole. Other models we consider are some usual cosmological dark fluids
as some dark energy equations of state (generalised Chaplygin gasBen02 ,
constantRah07 , linear redshift parameterisationHut01 ; Wel02 , Chevallier-
Polarski-Linder parameterisationChe01 ; Lin03 ) and a Navarro-Frenk-WhiteNav96
profile dark matter distribution. We classify their wormhole solutions and
show that none of them is asymptotically Minkowski flat that disagrees with
their natural formation from these dark fluids. Last, we consider a toy model
of static isotropic wormhole that can be asymptotically Minkowski flat. We
show that it is a counter-example to a recent theorem claiming that a static
wormhole sustained by an isotropic fluid cannot be asymptotically flat on both
sides of its throat.
Plan of the paper is as follows. In section II, we write the static
spherically symmetric wormhole field equations as an autonomous system of
normalised variables. We relate some of its phase space geometrical properties
to some wormholes properties. In section III, we determine the possible
locations of equilibrium points in the phase space. In section IV, we do the
same for the possible locations of curvature singularities. In section V, we
study several static spherically symmetric wormhole models phase space and
find the above mentioned physical results. We conclude in section VI.

##  II Phase space

In the first subsection, we cast the wormhole field equations MorTho88  into
an autonomous dynamical system. We show how some large classes of wormholes
can be studied in this unified framework. In the second subsection, we relate
some phase space geometrical properties to some physical wormholes properties.

###  II.1 Dynamical system

The metric for a static spherically symmetric wormhole writes

$ds^{2}=-e^{2\Phi}dt^{2}+\frac{dr^{2}}{1-b/r}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (1)

$b(r)$ determines its spatial shape and $\Phi(r)$, its gravitational redshift.
Considering General Relativity and a stress energy tensor for a perfect fluid
defined by a density $\rho$, a pressure $p$ and a tension per unit area
$\tau$, the field equations areMorTho88

$\dot{b}=8\pi Gc^{-2}r^{2}\rho$ (2) $\dot{\Phi}=(-8\pi Gc^{-4}\tau
r^{3}+b)/\left[2r(r-b)\right]$ (3) $\dot{\tau}=(\rho
c^{2}-\tau)\dot{\Phi}-2(p+\tau)/r$ (4)

A dot means a derivative with respect to $r$. In the rest of the paper, we
choose $8\pi G=1$ and $c=1$. To rewrite the field equations as an autonomous
dynamical system for a finite phase space, we define the following variables

$u=\ln r$ (5) $P=pr^{2}$ (6) $T=\tau r^{2}$ (7) $\mu=\rho r^{2}$ (8)
$\beta=\tanh(1-b/r)$ (9) $\theta=\tanh\Phi^{\prime}$ (10)

A prime means a derivative with respect to $u$, an increasing function of $r$.
Moreover, the proper radial distance from the throat is

$l=\pm\int^{r}_{r_{0}}\frac{dr}{\sqrt{1-b/r}}=\pm\int^{r}_{r_{0}}\frac{dr}{\sqrt{\operatorname{arctanh}{\beta}}}$
(11)

with $r_{0}$, the throat size such as $\beta(r_{0})=0$. By definition, $l=0$
at the throat, the $+$ sign stands for above it and the $-$ sign, below it. As
shown below, $P$ and $T$ can be expressed with $\beta$ and $\theta$ but it is
not always possible explicitly for $\mu$ or other free functions. For this
reason, we then define

$\zeta=\tanh(du/dl)=\pm\tanh(\sqrt{\operatorname{arctanh}\beta}e^{-u})$ (12)

$\zeta=0$ at the throat. It is positive or negative just above or below it.
But the sign of $\zeta$ does not necessarily stays constant above or below the
throat. It changes when it crosses the plane $\zeta=0$ (that is not always at
a throat, see V.2.1). Obviously the sign of $\zeta$ is the same as this of
$dl/dr$. Note that $\beta$ and $\theta$ are two independent variables but not
$\beta$ and $\zeta$. Indeed, when we define a trajectory in the phase space by
some initial conditions, this on $\zeta$ depends on the initial condition on
$\beta$ or vice-versa (see subsection II.2). $\beta$, $\theta$ and $\zeta$ are
three normalised variables in the range from $-1$ to $1$. They allow to
rewrite the field equations as an autonomous dynamical system for numerous
forms of $\mu$ or other free functions. It comes:

$\frac{d\beta}{dl}=\operatorname{arctanh}\zeta\beta^{\prime}=\operatorname{arctanh}\zeta(1-\beta^{2})(1-\operatorname{arctanh}\beta-\mu)$
(13) $\displaystyle\frac{d\theta}{dl}$
$\displaystyle=\operatorname{arctanh}\zeta\theta^{\prime}=$
$\displaystyle\operatorname{arctanh}\zeta\frac{(\theta^{2}-1)}{2\operatorname{arctanh}\beta}\mbox{[}(1-\mu)(\operatorname{arctanh}\theta+1)+\operatorname{arctanh}\beta(-1-\operatorname{arctanh}\theta+$
(14) $\displaystyle 2\operatorname{arctanh}^{2}\theta)-2P\mbox{]}$
$\frac{d\zeta}{dl}=\operatorname{arctanh}\zeta\zeta^{\prime}=-\frac{(1-\zeta^{2})\operatorname{arctanh}^{2}\zeta(-1+\mu+3\operatorname{arctanh}\beta)}{2\operatorname{arctanh}\beta}$
(15) $-T=-1+\operatorname{arctanh}\beta(1+2\operatorname{arctanh}\theta)$ (16)

Equations (13-15) form a dynamical system. It is autonomous since $\mu$,
$\beta$, $\theta$ and $\zeta$ are some functions of $l$ and it is always
possible, at least formally, to write $\mu=\mu(\beta,\theta,\zeta)$. We now
assume that the perfect fluid is isotropic, i.e. $p=-\tau$ and define its
equation of state $w(r)=p/\rho$. This is the usual case of a cosmological
perfect fluid (other choices of perfect fluid are possible with $p\not=-\tau$,
see for instance Kuh17  where $w=\tau/\rho$). Then, the equations (14) and
(16) rewrite

$\displaystyle\frac{d\theta}{dl}$ $\displaystyle=$
$\displaystyle\operatorname{arctanh}\zeta\frac{(\theta^{2}-1)}{2\operatorname{arctanh}\beta}\mbox{[}3+\operatorname{arctanh}\theta-\mu(1+\operatorname{arctanh}\theta)+$
(17)
$\displaystyle\operatorname{arctanh}\beta(-3-5\operatorname{arctanh}\theta+2\operatorname{arctanh}^{2}\theta)\mbox{]}$
$P=-1+\operatorname{arctanh}\beta(1+2\operatorname{arctanh}\theta)$ (18)

When $\mu(\zeta)\not=\mu(-\zeta)$, phase space and thus spacetime are
different on both sides of the throat. We can then either work with the three
equations system $(d\beta/dl,d\theta/dl,d\zeta/dl)$ that allows to plot the
trajectories as a function of $l$ in the phase space with $-1<\zeta<1$ or with
the three equations system
$(\beta^{\prime},\theta^{\prime},\zeta^{\prime})=(d\beta/dl,d\theta/dl,d\zeta/dl)/\operatorname{arctanh}\zeta$
that allows to plot the trajectories as a function of $u$ in two separated
phase space, one with $-1<\zeta<0$ and the other with $0<\zeta<1$. In some
cases, the dynamical system can be simplified. In particular, when
$\mu(\zeta)=\mu(-\zeta)$, phase space trajectories are symmetric with respect
to the plane $\zeta=0$. We can then only consider the range $0<\zeta<1$ and
the dynamical system $(\beta^{\prime},\theta^{\prime},\zeta^{\prime})$. If
moreover $\mu=\mu(\beta,\theta)$, the dynamical system reduces to a set of two
equations for $\beta^{\prime}$ and $\theta^{\prime}$.
Let us give some examples of wormholes models defined by the form of their
free function and that can be studied with the above general dynamical system
or its simplified forms.

  * •

If the free function is $b(r)$Lob05 ; Cha07 , then from (2) we derive $\mu(r)$
and from (12), $\mu(\beta,\zeta)$. If moreover $b(r)$ is invertible to get
$r(b)$, then from (2), one gets $\mu(r)=\mu(b)=\mu(\beta)$. Then, the
dynamical system reduces to a set of two equations for $\beta^{\prime}$ and
$\theta^{\prime}$.

  * •

If the free function is $\Phi(r)$Cat15 ; Bro03 ; Kuh06 , we calculate
$\Phi^{\prime}(r)$ and then derive $\theta(\beta,\zeta)$ and
$\frac{d\theta}{dl}(\beta,\zeta)$ with (12). We then deduce $\mu(\beta,\zeta)$
from (17) with $\theta(\beta,\zeta)$ a constraint between the three variables.

  * •

If the free function is $\rho(r)$Lob13 , one gets $\mu(\beta,\zeta)$ from
(12).

  * •

If the free function is $p(r)$Lob13 ; Rah07 , from (3) and (4) we derive the
Tolman-Oppenheimer-Volkoff equation for static wormhole Gor08

$\dot{p}=-\frac{(b+r^{3}p)(p+\rho)}{2r(r-b)}$

from which we get $\mu(\beta,\zeta)$ with (9) and (12). If moreover $r(P)$ can
be calculated, one derives $\zeta(\beta,\theta)$ from (18) and thus
$\mu(\beta,\theta)$ from $\mu(\beta,\zeta)$.

  * •

If the free function is $w(r)$, we get
$\mu(\beta,\theta,\zeta)=P(\beta,\theta)/w(\zeta,\beta)$ using (12) and (18).

  * •

If the free function is $w(z)$ with $z$ the redshift and $w$ is invertible,
and if one can calculate $\rho(z)$, then one gets $\rho(w)=\rho(P/\mu)$. Since
$r=r(\beta,\zeta)$, then from (12) it is possible to calculate a relation
between $\mu=\rho r^{2}$ and a function of $(\mu,\beta,\theta,\zeta)$. This
relation with the equations for $\beta^{\prime}$, $\theta^{\prime}$ and
$\zeta^{\prime}$ allow to study the phase space of static wormhole sustained
by an equation of state initially given as a redshift function (see subsection
V.2.3).

The above list of model classes that can be studied with the formalism of this
paper is not exhaustive. Among others possibilities, one can also derive
$\mu(\beta,\theta,\zeta)$ for important barotropic equation of state whose
dependence on $r$ is unknown like a constant equation of state (see subsection
V.2.2)Rah06 , the generalised Chaplygin gas (see subsection V.2.1)Ben02 , the
Van der Waals equation of state ($p=\frac{8w\rho}{3-\rho}-3\rho^{2}$ with $w$
a constant)Kre03 , the quadratic equation of state
($p=\alpha\rho+\beta\rho^{2}$ with $\alpha$ and $\beta$ some constants)Ana06 ,
etc. Some wormhole models are studied in section V.

###  II.2 Relations between some phase space geometrical properties and some
wormhole properties

Some geometrical properties of the phase space $(\beta,\theta,\zeta)$ are
related to some physical properties of the wormholes.

Hence, from the definition of $l$, following a trajectory in the phase space
$(\beta,\theta,\zeta)$ from a throat means physically to roll away from this
throat. Concerning the plane $\beta=\tanh 1$, it defines the phase space
points where the metric becomes spatially flat. A throat in $b=r$ lies on the
phase space on the line $(\beta,\zeta)=(0,0)$. Let us show that a throat can
only be on this line when $\theta=\pm 1$. Indeed, when $\beta=0$,
$d\theta/d\beta$ is diverging, but possibly in $\theta=\pm 1$. Hence the line
$(\beta,\zeta)=(0,0)$ cannot be reached by a trajectory but in $\theta=\pm 1$.
The points $(\beta,\theta,\zeta)=(0,\pm 1,0)$ are thus the only ones where
stand the wormholes throats. From a physical viewpoint, this means that the
variation of the gravitational redshift at the throat of a static spherically
symmetric wormhole sustained by an isotropic fluid is always diverging (that
is not in disagreement with a finite gravitational redshift $\Phi(r_{0})$).
This generalises the recent result in Cat16  showing that there is no zero-
tidal force static spherically symmetric wormhole ($\theta=0$) sustained by an
isotropic perfect fluid. With a non isotropic fluid, things would be
different. There are then (among others) zero-tidal force solutionsMorTho88
for which the throat would be in $\theta=0$.
Another condition to have a throat at the points $(\beta,\theta,\zeta)=(0,\pm
1,0)$ is given by the flaring-out conditionMorTho88 . It writes for a static
wormhole in $r=r_{0}$

$\frac{b-b^{\prime}}{2b^{2}}=\frac{e^{-u}\beta^{\prime}}{2(\operatorname{arctanh}\beta-1)^{2}(1-\beta)^{2}}>0$

Hence a phase space trajectory is a wormhole solution if
$\beta(r_{0})^{\prime}>0$ at the throat, implying that $\beta>0$ in its
neighbourhood. Moreover, we also have
$\operatorname{arctanh}^{2}\zeta/\operatorname{arctanh}\beta=r_{0}^{-2}$ and
thus $d\zeta/dl\not=0$ at the throat. Hence, all the phase space trajectories
describing the wormhole solutions pass through a throat at some special points
in the phase space but these points are not equilibrium points. Note also that
at a throat $d\beta/dl<0$ in $\zeta\rightarrow 0^{-}$ but $d\beta/dl>0$ in
$\zeta\rightarrow 0^{+}$. These inequalities simply describe a trajectory
crossing a throat from $\zeta<0$ to $\zeta>0$ with $\beta\geq 0$. From a
physical viewpoint, the flaring-out condition $\beta(r_{0})^{\prime}>0$ also
implies from (13) that $\mu<1$ since at a throat $\beta=0$. Hence we must have
$\rho(r_{0})<r_{0}^{-2}$: the larger the throat, the smaller the density to
have a wormhole.

The curve $P=0$ corresponds to the vanishing of the pressure $p$. It is
plotted in the phase space on figure 1. It splits the phase space in several
parts $P<0$ and $P>0$, whatever $\zeta$. This curve passes through
$(\beta,\theta)=(\tanh 1,0)$ where stands the Minkowski spacetime when
$\zeta=0$ (see section III). Its ends for $\beta>0$ come close to
$(\beta,\theta)=(1,\tanh(-1/2)^{+})$ and $(\beta,\theta)=(0,1)$ and for
$\beta<0$ (this part of the space phase shelters the Schwarzschild black hole
singularity, see subsection V.1), close to
$(\beta,\theta)=(-1,\tanh(-1/2)^{-})$ and $(\beta,\theta)=(0,-1)$. The exact
value of $P$ as well as the behaviours of the trajectories at the throats
$(\beta,\theta)=(0,\pm 1)$ depend on the value of
$\operatorname{arctanh}\beta\operatorname{arctanh}\theta$ in the neighbourhood
of the throats where we have the approximations

$\operatorname{arctanh}\beta\rightarrow\beta\mbox{ when }\beta\rightarrow 0$
$\operatorname{arctanh}\theta\rightarrow\mp 1/2\log\epsilon\mbox{ when
}\theta\rightarrow\pm 1\mp\epsilon\mbox{ and }\epsilon\rightarrow 0$
$\operatorname{arctanh}\zeta\rightarrow\zeta\mbox{ when }\zeta\rightarrow 0$

Assuming that $\mu$ does not diverge near the throat where no singularity
should stand, it comes

$\frac{d\beta}{dl}=\zeta(1-\mu)$ (19)
$\frac{d\epsilon}{dl}=-\zeta\frac{(1-\mu\mp\beta\log\epsilon)\epsilon\log\epsilon}{2\beta}$
(20) $\frac{d\zeta}{dl}=\zeta^{2}\frac{(1-\mu)}{2\beta}$ (21)

From equations (19) and (21) we recover at the throat that

$\zeta\simeq\pm r_{0}^{-1}\sqrt{\beta}$

in agreement with the definition (12) for $\zeta$. If we exclude the
possibility of a singularity at the throat, $\beta\log\epsilon$ cannot diverge
(see subsection IV.1). Then $\beta\log\epsilon$ tends to a constant $C$ when
approaching the throat where, from (18), $P\rightarrow-1\mp C$. When $C=0$, we
derive from (19) and (20) that

$\epsilon\simeq e^{const\beta^{-1/2}}$

where $const$ is a negative constant (since at the throat $\beta\rightarrow
0^{+}$) depending on the wormhole model integration constants and parameters.
When $C\not=0$, it comes

$\epsilon\simeq e^{C\beta^{-1}}$

with $C<0$ such as $\epsilon\rightarrow 0$ when $\beta\rightarrow 0$. These
relations allow approximating the behaviour of the trajectories around a
throat and can be useful for numerical calculations in its neighbourhood.

A last remark is about initial conditions. When solving the original field
equations (2-4), we need three initial conditions corresponding to the three
functions $b$, $\Phi$ and $p$. However, when solving the dynamical system for
$(\beta,\theta,\zeta)$, we only need two initial conditions for $\beta$ and
$\theta$, the initial condition on $\zeta$ being fixed by the one on $\beta$.
This is due to the definition of the variable $\theta$. It casts the
differential equation (3) for $\Phi$ into an algebraic equation (18) for $P$,
$\beta$ and $\theta$. But a third integration constant is recovered if we
integrate $\theta$ to get $\Phi$. Moreover, when one chooses some initial
conditions $(\beta,\theta,\zeta)=(\beta_{1},\theta_{1},\zeta_{1})$ in
$l=l_{1}\not=0$, $l_{1}$ must be such as $\beta=0$ (and thus $\zeta=0$) when
$l=0$. This fixes the value of the throat size $r_{0}$ in (11). This choice of
initial conditions in $l_{1}\not=0$ corresponds to a unique trajectory in the
phase space since there is only one trajectory that contains the point
$(\beta_{1},\theta_{1},\zeta_{1})$. However, if one chooses some initial
conditions $(\beta_{0},\theta_{0},\zeta_{0})=(0,\pm 1,0)$, i.e. at a throat in
$l=0$, there are an infinite number of trajectories that passes through it.
Then a unique trajectory is selected by a given value of $r_{0}$ or any other
integration constant.

##  III Equilibrium points

In this section, we look for the possible equilibrium points of the dynamical
system (13-15), i.e. $(d\beta/dl,d\theta/dl,d\zeta/dl)$. For the reader who
wants to avoid this mathematical section, its results are summarised on figure
1. We do not determine the equilibrium points stability that can only be found
once the dynamical system is fully specified. However, some general remarks
can be made about source and sink points. Hence, a trajectory has not always a
source or a sink point. Some of them can begin or end at a non equilibrium
point, like some singularities (see an example in subsection V.1). Note also
that the physical meaning of a source or a sink point for a static wormhole
trajectory is different from the one of an isotropic and homogeneous
cosmology. When in cosmology some phase space trajectories depend on the
proper time parameter $t$, a source or a sink point then takes place
respectively at early or late times. Physically, an observer can thus only
follow a trajectory from this source to this sink to respect the causality.
For a static wormhole, the parameters $l$ and $r$ of the trajectories, are
related to the distance to a throat and its axis. An observer can thus
physically go to a sink as well as to a source point on a wormhole trajectory.
In the following, the three sets of equilibrium points for the equations
(13-15) correspond to the three possibilities such as $d\beta/dl=0$ in (13),
respectively $\beta=\tanh(1-\mu)$, $\beta=\pm 1$ or $\zeta=0$. Note that the
first equality is an equation $\beta=\tanh(1-\mu(\beta,\theta,\zeta))$ that
can correspond to several values of $\beta$.

###  III.1 Set 1: $\beta=\tanh(1-\mu)$

In this subsection, we consider that $d\beta/dl=0$ when $\beta=\tanh(1-\mu)$.
Then from (15), $d\zeta/dl=0$ implies that the equilibrium points are on the
plane $\zeta=0$ (then, from the definition (12) for $\zeta$, $\beta=0$ with
$\mu=1$ or/and $r$ is diverging) or $\zeta=\pm 1$ (then $\beta=\pm 1$ with
$\mu=0$ or/and $r=0$. We have a singularity (see section IV.1)). Last, from
(17), $d\theta/dl=0$ when $\zeta=0$ but also if

$\theta=\pm 1$

or

$\theta=\tanh(1\pm\frac{\sqrt{2\mu^{2}-3\mu+1}}{\mu-1})$ (22)

The first subset with $\theta=\pm 1$ can correspond to some singularities (see
section IV). The second subset (22) of equilibrium points is real when
$\mu\not\in\left[1/2,1\right]$. It defines three dotted curves in the plane
$(\beta,\theta)$ (shown on figure 1) where these points can be located. More
specifically, when the density tends to vanish ($\mu\rightarrow 0$) or
equivalently $\beta\rightarrow\tanh 1$, spacetime becomes spatially flat and
we get the following values for $\theta$:

  * •

$\theta=0$. This is the Minkowski solution when $\Phi$ tends asymptotically
($r\rightarrow+\infty$) to a constant and $\zeta$ to $0$. Since in
$(\beta,\theta)=(\tanh 1,0)$, $\mu$ and $P$ are vanishing, this point is
reached if the density $p$ and the pressure $\rho$ vanish faster than $r^{-2}$
as indicated by the definitions (6) and (8) for $P$ and $\mu$. See subsection
V.3 for a wormhole model tending asymptotically to the Minkowski spacetime.
Let us remark that the Minkowski solution is reached when $d\Phi/du\rightarrow
0$ and not simply $d\Phi/dr\rightarrow 0$. This is because $d\Phi/dr$ have to
decrease faster than $1/r$ such as $\Phi(r)$ tends to a constant when $r$ is
diverging. This implies $rd\Phi/dr=d\Phi/du\to 0$.

  * •

$\theta=\tanh 2$. This point corresponds to a spatially flat Universe
different from the Minkowski one. Especially, $\Phi^{\prime}\rightarrow 2$
implies that the gravitational redshift $\Phi$ diverges as $2\ln r$ far from
the throat or equivalently, the metric function $e^{2\Phi}$ as $r^{4}$.

  * •

$\theta=\pm 1$. As shown on section IV, these points can correspond to
singularities when they are reached at finite $u$. Such a singularity in a
spatially flat spacetime is not a curvature singularity (see subsection IV.1)
produced by a strong gravity field. But note that at these points the
variation of the gravitational redshift diverges.

###  III.2 Set 2: $\beta=\pm 1$

In this subsection, we consider that $d\beta/dl=0$ when $\beta=\pm 1$. As
shown in section IV.1, this set can correspond to some singularities when
$\beta=\pm 1$ is reached at finite $u$. Following the definition (12) for
$\zeta$, $d\zeta/dl=0$ implies $\zeta=0$ (with then $r$ diverging) or
$\zeta=\pm 1$. Moreover, $d\theta/dl=0$ when $\zeta=0$ but also when

$\theta=\pm 1$

or

$\theta=\tanh\frac{\mu-1+5\operatorname{arctanh}\pm
1+\sqrt{(\mu-1)^{2}+2(9\mu-17)\operatorname{arctanh}\pm
1+49\operatorname{arctanh}^{2}1}}{4\operatorname{arctanh}\pm 1}$

or

$\theta=\tanh\frac{\mu-1+5\operatorname{arctanh}\pm
1-\sqrt{(\mu-1)^{2}+2(9\mu-17)\operatorname{arctanh}\pm
1+49\operatorname{arctanh}^{2}1}}{4\operatorname{arctanh}\pm 1}$

In the above expressions, we leave the diverging value
$\operatorname{arctanh}\pm 1$ since to evaluate the corresponding limit for
$\theta$, the exact form of $\mu$ is necessary. In the particular case
$\mu<<\operatorname{arctanh}1$, we get the equilibrium points

$(\beta,\theta)=(\pm 1,-\tanh\frac{1}{2})$ (23) $(\beta,\theta)=(\pm 1,\tanh
3)$ (24)

Figure 1:  Some properties of the phase space $(\beta,\theta,\zeta)$ projected
on the surface $(\beta,\theta)$ for a static wormhole sustained by a perfect
fluid in General Relativity. The dark (white) area is such that $P<0$
(respectively $P>0$). The dotted curves and dashed lines contain the phase
space points that can be some equilibrium points (on the planes $\zeta=0$ or
$\pm 1$ if $\mu$ depends on $\zeta$). The second figure shows these possible
equilibrium points near $\theta=1$ when $\beta<0$. The large point in
$(\beta,\theta)=(\tanh 1,0)$ stands for the Minkowski spacetime. Singularities
can be found wherever $\mu$ diverges and, independently on $\mu$, on the
dashed lines $\theta=\pm 1$ and $\beta=\pm 1$. On the vertical line $\beta=0$,
the slope $d\beta/d\theta$ diverges but not necessarily in $\theta=\pm 1$
where stand throats, singularities or the Schwarzschild event horizon.

###  III.3 Set 3: $\zeta=0$

In general, $\zeta=0$ implies $d\beta/dl=0$, $d\theta/dl=0$ and $d\zeta/dl=0$
but at the points $(\beta,\theta,\zeta)=(0,\pm 1,0)$ (that are not necessarily
some throats when $\beta^{\prime}<0$) since then
$\operatorname{arctanh}\zeta/\sqrt{\operatorname{arctanh}\beta}=r_{0}^{-1}$
(see the definition (12) of $\zeta$) and thus $d\zeta/dl\not=0$. There is also
no equilibrium when $\mu$ is diverging but there is then a singularity.
Consequently, no wormhole trajectory can cross the plane $\zeta=0$ but in
$(\beta,\theta,\zeta)=(0,\pm 1,0)$.

The positions of all the possible equilibrium points projected on the surface
$(\beta,\theta)$ are shown on figure 1. They all stand on the planes $\zeta=0$
or $\pm 1$.

##  IV Singularity

In this section, we determine the location of (curvature) singularities in the
phase space and the geodesic equation for spacetimes described by these
trajectories. For readers who want to jump this section, its results are
summarised on figure 1.

###  IV.1 Singularity

We are interested by curvature singularities, i.e. Ricci and Weyl
singularities. To find them, we consider the Ricci scalar

$R=g_{\mu\nu}R^{\mu\nu}=e^{-2u}\left[3+\mu-3\operatorname{arctanh}\beta(1+2\operatorname{arctanh}\theta)\right]$

but also

$\displaystyle R_{\gamma\delta}R^{\gamma\delta}$ $\displaystyle=$
$\displaystyle
e^{-4u}\mbox{[}3+\mu^{2}-6\operatorname{arctanh}\beta(1+2\operatorname{arctanh}\theta)+$
$\displaystyle
3\operatorname{arctanh}^{2}\beta(1+2\operatorname{arctanh}\theta)^{2}\mbox{]}$

and the Kretschman scalar

$\displaystyle R_{\gamma\delta\nu\xi}R^{\gamma\delta\nu\xi}$ $\displaystyle=$
$\displaystyle
e^{-4u}(15+\mu(3\mu-10)+2(\mu-3)\operatorname{arctanh}\beta(5+2\operatorname{arctanh}\theta)+$
$\displaystyle
3\operatorname{arctanh}^{2}\beta\left[5+4\operatorname{arctanh}\theta(1+\operatorname{arctanh}\theta)\right])$

When $\mu$ diverges at finite $u$, we have a Ricci singularity. But the
scalars can also diverge, whatever $\mu$, when $\beta=\pm 1$ and/or
$\theta=\pm 1$ (these values resulting from the compactification) but in
$(\beta,\theta)=(0,\pm 1)$ if we have a throat. If this occurs at finite $u$
and $\mu$, we have a Weyl singularity for which the tidal forces become
infinitely largeUgg06 . Note that even if a special form of $\mu$ is chosen to
cancel the divergence of $R$, it will not avoid this of the other scalars and
vice-versa.

Since the $(t,r,\theta,\phi)$ coordinates of the metric (1) we use in this
paper do not cover any horizon, we do not look for their possible positions in
the phase space. However, in section V.1, we show how to extend this work to
the representation of black holes in the phase space by examining the
Schwarzschild black hole. Then, we can locate its event horizon in the phase
space since it has been shown in other coordinates (Kruskal-Szekeres for
instance) that it stands in $\Phi\rightarrow-\infty$. In a future work, we
hope to describe black holes representation in a phase space where the notion
of horizon could be appropriately treated.

###  IV.2 Geodesic equation

To confirm the presence of a singularity when some scalars diverge, we want to
check when a spacetime is geodesically incomplete. We thus calculate the
geodesic equation giving $u$ as a function of the proper time $\tau$ for a
test particle in a spacetime described by a phase space trajectory. Since we
are interested by static spherically symmetric wormhole, we only consider the
equatorial plane $\theta=\pi/2$ (the $\theta$ of the spacetime metric, not the
variable of the dynamical system). From the geodesic equations, we get the
conserved quantitiesCat17

$\frac{dt}{d\tau}=\frac{E}{m}e^{-2\Phi}$ $\frac{d\phi}{d\tau}=Ae^{-2u}$

with $E$ and $A$, respectively the conserved energy and angular momentum of
the test particle with mass $m$. From the relation for four-velocity $U$ when
considering time-like geodesics, we have $U^{\mu}U_{\mu}=-1$ from which we
derive

$A^{2}e^{-2u}+\frac{E^{2}}{m^{2}}e^{-4\Phi(u)}\left[-e^{2\Phi(u)}+\frac{e^{2u}(du/d\tau)^{2}}{\operatorname{arctanh}\beta(u)}\right]=-1$

For sake of simplicity, we now choose $E=A=1$ (and thus non radial geodesics
since $A\not=0$). Then, from this last equation, we deduce

$e^{2\Phi(u)}=\frac{2e^{2u}(du/d\tau)^{2}}{\operatorname{arctanh}\beta(u)\pm\sqrt{\operatorname{arctanh}\beta(u)\left[\operatorname{arctanh}\beta(u)-4(1+e^{2u})(du/d\tau)^{2}\right)]}}$
(25)

Equation (25) contains two solutions for $u(\tau)$, depending on the $\pm$
sign. This is due to the terms $e^{-2\Phi}$ and $e^{-4\Phi}$ in the previous
equation. The solution with the plus (minus) sign describes trajectories with
$\beta>0$ (respectively $\beta<0$) and allows to calculate $u(\tau)$ for phase
space trajectories.
From a technical viewpoint, to calculate $u(\tau)$, we first get a (generally)
numerical solution for $(\beta(u),\theta(u))$ from the dynamical system
(13-15). Since $\theta$ depends on $\Phi^{\prime}$, we differentiate (25),
introduce the solution for $(\beta(u),\theta(u))$ in this equation for
$\Phi^{\prime}$, choose some initial conditions for $u(\tau)$ and then solve
the differential equation for $u(\tau)$. This method is applied in the next
section when we need to confirm the presence of some singularities.

##  V Some wormholes phase spaces

In this section, we present the phase space of several wormhole models:

  * •

a vanishing density

  * •

a generalised Chaplygin gas

  * •

a constant equation of state

  * •

a linear equation of state

  * •

a Chevallier-Polarski-Linder equation of state

  * •

a Navarro-Frenk-White profile density

  * •

a toy model with an asymptotically vanishing pressure

From a mathematical viewpoint, we show how to apply the dynamical system
formalism to get and classify the solutions of the above models. All these
solutions are summarised in table 1. From a physical viewpoint, the vanishing
density model (see subsection V.1) shows how to possibly extend this work to
consider static black hole trajectories. Dark energy and matter models (see
subsection V.2) show that usual models of dark fluids cannot form wormholes
naturally since they cannot be asymptotically Minkowski flat. Finally, a toy
model with an asymptotically vanishing pressure (see subsection V.3) shows
that a static wormhole can be flat on both sides of its throat contrary to
some recent resultsBro17 .

Model | Parameter range | Throat | End point
---|---|---|---
$\mu=0$ |  | $(0,1)$ |  $\theta=-1$ with $P<0$
if $P<0$ |  |  | singularity
Chaplygin | $(A,\alpha)>0$ | $(0,1,0)$ |  $(0,-1,0)$ with $P<0$
if $A<r_{0}^{-2(1+\alpha)}$ |  |  | Cyclic
Constant eos | $w<-3-2\sqrt{2}$ | $(0,1)$ |  $(0,-1)$ with $P<0$, $r\rightarrow\infty$
if $w\not\in\left[-1,0\right]$ |  | $(0,1)$ |  $(\tanh\frac{(1+w)^{2}}{w^{2}+6w+1},\tanh\frac{2w}{1+w})$ with $P>0$, $r\rightarrow\infty$
| $-3-2\sqrt{2}<w<-1$ | $(0,1)$ |  $(0,-1)$ with $P<0$, $r\rightarrow\infty$
|  | $(0,1)$ |  $(1,\tanh\frac{1+3w}{w-1})$ with $P>0$, $r\rightarrow\infty$
| $0<w<1$ | $(0,1)$ |  $(1,\tanh\frac{1+3w}{w-1})$ with $P<0$, $r\rightarrow\infty$
|  | $(0,-1)$ |  $(1,\tanh\frac{1+3w}{w-1})$ with $P<0$, $r\rightarrow\infty$
| $1<w$ | $(0,1)$ |  $(1,-1)$ with $P<0$, $r\rightarrow\infty$
Linear eos | $-1.5<w_{0}<-1,-0.5<w_{1}<0$ | $(0,1,0)$ |  $(0,-1,0)$ with $P<0$, $r\rightarrow\infty$
NA |  |  |
CPL eos | $-1.5<w_{0}<-1,-0.5<w_{1}<0$ | $(0,1,0)$ |  $(0,-1,0)$ with $P<0$, $r\rightarrow\infty$
NA |  |  |
NFW | $(\rho_{r},r_{s})>0$ | $(0,1,0)$ |  $(\tanh 1,1,0)$ with $P>0$, $r\rightarrow\infty$, flat
if $1-\frac{\pm r_{0}r_{s}^{3}\rho_{s}}{\pm r_{0}+r_{s}}^{2}>0$ |  | $(0,1,0)$ |  $(\tanh 1,-1,0)$ with $P<0$, $r\rightarrow\infty$, flat
$p=p_{0}r^{-n}$ | $n>2$ | $(0,1)$ |  $\theta=0$ with $P<0$
if $p_{0}r_{0}^{2-n}>-1$ |  |  | singularity
|  | $(0,1)$ |  $(\tanh 1,0)$ with $P>0$, $r\rightarrow\infty$
|  |  | Minkowski
Table 1: The first column gives the model and conditions to have a wormhole.
The second column gives the range of parameters we have considered. The third
column gives the location of the throats in the 2 or 3 dimensional space phase
$(\beta,\theta,\zeta)$. The fourth column gives the behaviour far from the
throat of the trajectories. None of the usual dark energy and dark matter
models produces an asymptotically flat Minkowski wormhole.

###  V.1 Vanishing density: a wormhole exact solution and a Schwarzschild
black hole trajectory

The first model we choose to study is defined by a vanishing density $\mu=0$.
From a mathematical viewpoint, it can be solved exactly with the original
equations system (2-4). This allows to check its consistency with the
dynamical system (13-15). From a physical viewpoint, the $\mu=0$ model
contains as a particular case ($P=0$) the Schwarzschild black hole. This gives
us the opportunity to examine how a phase space describing a static
spherically symmetric wormhole could also be used to study a static
spherically symmetric black hole for future work and to better understand the
relations between these two types of objects.

The exact solution of the equations system (2-4) is:

$b=r_{0}$ $p=p_{1}e^{-\Phi}$ $\displaystyle e^{-\Phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2c^{4}\sqrt{r}\sqrt{-r_{0}+r}}(-30r_{0}^{2}G\pi\sqrt{r}\sqrt{-r_{0}+r}p_{1}+4G\pi
r^{5/2}\sqrt{-r_{0}+r}p_{1}$ (26) $\displaystyle-r\Phi_{1}+r_{0}(10G\pi
r^{3/2}\sqrt{-r_{0}+r}p_{1}+\Phi_{1})-30r_{0}^{2}G\pi(r_{0}-r)p_{1}\times$
$\displaystyle\log\left[\sqrt{r}+\sqrt{-r_{0}+r}\right])$

with $r_{0}$ the throat size, $p_{1}$ and $\Phi_{1}$, three integration
constants. At the throat, we have
$e^{-\Phi}\rightarrow-(r_{0}^{2}p_{1})^{-1}$. The special case $p_{1}=0$
corresponds to the Schwarzschild black hole (see the end of this subsection)
with an event horizon in $\Phi\rightarrow-\infty$. Let us first consider that
$p_{1}\not=0$. A solution defined by $r_{0}=0.5$, $p_{1}=1$ and $\Phi_{1}=1$
is plotted in the phase space on the first graph of figure 2. The comparison
between this solution and the phase space trajectories shows the consistency
of the dynamical system approach with respect to the above exact solution
calculated with equations (2-4).
The phase space trajectories representing wormhole solutions are such as
$\beta<\tanh 1$ since on the line $\beta=\tanh 1$, $d\theta/d\beta$ diverges.
Hence, trajectories coming from the throat cannot cross this line. The
dynamical system has four equilibrium points, all on the line $\beta=\tanh 1$.
They thus correspond to spatially flat solutions. They are

  * •

$(\beta,\theta)=(\tanh 1,-1)$ that is a sink.

  * •

$(\beta,\theta)=(\tanh 1,1)$ that is a saddle.

  * •

$(\beta,\theta)=(\tanh 1,0)$ that is a saddle Minkowski point.

  * •

$(\beta,\theta)=(\tanh 1,\tanh 2)$ that is a sink.

At the throat in $(\beta,\theta)=(0,1)$, $\beta^{\prime}\rightarrow 1>0$ in
agreement with the flaring-out condition, and the absence of equilibrium point
there. We also have that $P=-1$ and thus the equation of state
$p/\rho\rightarrow-\infty$, in agreement with violation of the weak energy
condition to have a wormhole.

Figure 2: First graph: Phase space of a wormhole with vanishing density. The
equilibrium points are in bold on the line $\beta=\tanh 1$. The thick
trajectory is defined by $r_{0}=0.5$, $p_{1}=1$ and $\Phi_{1}=1$. There are
two different parts. The first one describes a wormhole: the throat is in
$(\beta,\theta)=(0,1)$ and it ends at a finite $r$ and $\tau$ in
$(\beta,\theta)=(\beta_{1}^{-},-1)$ with a pressure that tends to $-\infty$.
The second one, that does not describe a wormhole, starts in
$(\beta,\theta)=(\beta_{1}^{+},+1)$ where the pressure tends to $+\infty$ and
then goes to $(\beta,\theta)=(\tanh 1,\tanh 2)$ at infinite $r$. Second graph:
$u(\tau)$, $\beta(\tau)$ and $\theta(\tau)$ for the left part of the thick
trajectory representing a wormhole on the first graph. The gray points
indicate the ends of the curves $u(\tau)$ , $\beta(\tau)$ and $\theta(\tau)$
in $\tau=25.5$ and $u=0.08$.

A wormhole trajectory behaves in the following way. It starts at a throat in
$(\beta,\theta)=(0,1)$. Further from the throat, at a finite coordinate
$r=r_{1}>r_{0}$, the trajectory reaches the line $\theta\rightarrow-1$ in
$r_{1}^{-}$. The pressure and the Ricci scalar $R$ diverge. The geodesic
equation (25) shows that the coordinate $r$ ends in $r_{1}^{-}$ at a finite
time $\tau$. There is thus a singularity at finite distance and time from the
wormhole throat. A geodesic solution $u(\tau)$ for the special case
$r_{0}=0.5$, $p_{1}=1$ and $\Phi_{1}=1$ is plotted on the second graph of
figure 2 with a singularity occurring in $\tau=25.5$ and $u=0.08$. In
$r_{1}^{+}$, the trajectory restarts with the same $\beta$ as in $r_{1}^{-}$
but in $\theta\rightarrow+1$. At this point, anew, the pressure and $R$ are
diverging. We thus also have a singularity in $r_{1}^{+}$ that is confirmed,
as for $r_{1}^{-}$, by the geodesic equation (25). For larger $r$, the
trajectory goes to the saddle points and then to the sink point
$(\beta,\theta)=(\tanh 1,\tanh 2)$ with $r\rightarrow\infty$, where the metric
becomes spatially flat. This second part of the trajectory with $r>r_{1}^{+}$
does not belong to a wormhole solution since it is disconnected from the
trajectory with $r_{0}<r<r_{1}^{-}$. Physically the $\mu=0$ model thus
generate a wormhole that relates two singularities.
Although this paper is dedicated to wormholes, let us have a look now on the
representation of the Schwarzschild black hole defined by $\mu=P=0$ (i.e.
$p_{1}=0$) in such a phase space. Its singularity is in $r=0$, i.e.
$\beta=-1$. As is well known, its event horizon occurs in
$\Phi\rightarrow-\infty$ at finite $u$, that means
$\Phi^{\prime}\rightarrow\pm\infty$ or $\theta\rightarrow\pm 1$. But we know
that on these lines we also have a singularity but possibly in $\beta=0$.
Hence, the event horizon of the Schwarzschild black hole can only take place
in $(\beta,\theta,\zeta)=(0,\pm 1,0)$. It follows that we have to consider the
whole phase space with $-1<\beta<1$ to plot the trajectory of the
Schwarzschild black hole (see Dea99  for a dynamical system analysis of
Schwarzschild orbital dynamics). It is represented in bold on figure 3 and can
be described as follows. It starts at a singularity in
$(\beta,\theta)=(-1,-\tanh 1/2)$, which is a source equilibrium point defined
in (23), and simply follows the curve $P=0$. The trajectory is discontinued
(as the curve $P=0$) when crossing the event horizon in $(\beta,\theta)=(0,\pm
1)$. This reflects the usual discontinuity in $r=2M$ (with $M$ the black hole
mass) of the Schwarzschild metric in $(r,t)$ coordinates. The trajectory then
continues until reaching the Minkowski equilibrium point. Others trajectories
with $P\not=0$, starting at the same singularity, are not black holes. They
stay in the area $\beta<0$ and end in another singularity in $\theta=\pm 1$ at
a constant value of $\beta<0$ (in a similar way to the wormhole trajectories
with $\beta>0$ and $\theta=-1$). Hence, the only type of black hole that can
exist with a fluid having a vanishing density is the Schwarzschild black hole.
We will not go further in this paper about considerations on black holes but
it could be interesting to use this phase space to determine the possible
formation of black holes with various dark fluids.

Figure 3: phase space for $\mu=0$ but with $-1<\beta<1$. The Schwarzschild
black hole trajectory is plotted as a bold trajectory. This is the only one
reaching asymptotically the Minkowski equilibrium point. The trajectory is
discontinued at the event horizon, in $(\beta,\theta)=(0,\pm 1)$.

###  V.2 Usual cosmological dark fluids: no asymptotically Minkowski flat
wormhole

In this subsection, we analyse some static spherically symmetric wormholes
sustained by models of dark fluids usually used to describe the effects of
dark energy and dark matter on Universe. These models are:

  * •

a generalised Chaplygin gas (dark energy, three dimensional phase space)

  * •

a constant equation of state (dark energy, two dimensional phase space)

  * •

a linear equation of state (dark energy, usually given as a redshift function,
three dimensional phase space)

  * •

a Chevallier-Polarski-Linder equation of state (dark energy, usually given as
a redshift function, three dimensional phase space)

  * •

a Navarro-Frenk-White profile density (dark matter, three dimensional phase
space)

From a mathematical viewpoint, these models illustrate how to apply the
dynamical system formalism of this paper to large classes of wormholes, even
when their equation of state is known as a redshift function. From a physical
viewpoint, we classify the wormhole solutions of these dark fluids and show
that none of them is asymptotically Minkowski flat. These isotropic dark
fluids should thus not form static wormholes naturally.

####  V.2.1 Generalised Chaplygin gas

The pressure of a generalised Chaplygin gasBen02  is given by

$p=-\frac{A}{\rho^{\alpha}}$

with $A$ and $\alpha$ two positive constants. The pressure is thus negative.
We deduce that

$\mu=(\frac{\operatorname{arctanh}\beta}{\operatorname{arctanh}^{2}\zeta})^{\frac{1+\alpha}{\alpha}}(\frac{A}{1-\operatorname{arctanh}{\beta}(1+2\operatorname{arctanh}{\theta})})^{\frac{1}{\alpha}}$

The density is thus positive. Approximating $\operatorname{arctanh}\beta$ and
$\operatorname{arctanh}\theta$ by respectively $\beta$ and $\mp
1/2\log\epsilon$ near the throat (see subsection II.2), we derive that
$\beta^{\prime}>0$ implies $A<r_{0}^{-2(1+\alpha)}$ in agreement with Lob06A .
Since $\mu$ depends on $\zeta$, we have to use the three dimensional dynamical
system for $(\beta,\theta,\zeta)$ to study the phase space. However, since
$\mu(\zeta)=\mu(-\zeta)$, the phase space for both sides of the throat is
symmetric with respect to the plane $\zeta=0$ and we thus only consider the
range $0<\zeta<1$. Moreover, when $\beta=\tanh 1$, $d\beta/dl<0$ and thus
$\beta$ is decreasing for this special value. This implies that the wormhole
trajectories coming from the throat in $\beta=0$ cannot cross the plan
$\beta=\tanh 1$. Numerical simulations show then that the throat is in
$(\beta,\theta,\zeta)=(0,1,0)$ and the trajectories go to
$(\beta,\theta,\zeta)=(0,-1,0)$ at finite $l$, density and curvature as shown
on figure 4. By symmetry with respect to the $\zeta=0$ plane, we thus get a
closed trajectory. This describes a succession of identical wormholes
connected by their throats, i.e. a cyclic structure repeating infinitely.

Figure 4: Some wormhole trajectories in the $(\beta,\theta,\zeta)$ phase
space, sustained by a Chaplygin gas with $A=0.1$ and $\alpha=1$. Each curve
ends in $(\beta,\theta,\zeta)=(0,-1,0)$ at finite $l$ and has a symmetrical
part with respect to the $\zeta=0$ plane. We get similar figure with other
values of $A$ and $\alpha$.

####  V.2.2 Constant equation of state

A constant equation of state $w=p/\rho=P/\mu$ has observational preferred
value around $w=-1$, i.e. near the value of a cosmological constant (see for
instance Sta17 ). Wormholes with similar equation of state have been studied
in Rah06 ; Cat15 , with $p\not=-\tau$ in Sus05 ; Lob06 ; Kuh16  or with
$\tau=w\rho$ in Kuh17 . From the pressure (18), we derive that a constant
equation of state $w$ is obtained when

$\mu=\left[-1+\operatorname{arctanh}\beta(1+2\operatorname{arctanh}\theta)\right]/w$

$\mu$ does not depend on $\zeta$ and the dynamical system thus reduces to the
differential equations for $\beta^{\prime}$ and $\theta^{\prime}$. Wormhole
throats for this model are in $(\beta,\theta)=(0,\pm 1)$ depending on $w$
values (see below or table 1). Considering that $\operatorname{arctanh}\beta$
and $\operatorname{arctanh}\theta$ respectively tend to $\beta$ and $\mp
1/2\log\epsilon$ near the throat, we derive that there
$\beta^{\prime}\rightarrow 1+1/w$, $\mu\rightarrow-1/w$ and $P\rightarrow-1$
if we also assume that $\beta\log\epsilon$ is vanishing. This assumption is
numerically checked by wormhole solutions presented below. It follows, as is
well known, that the weak energy condition cannot be violated in
$(\beta,\theta)=(0,\pm 1)$ when $-1<w<0$ since then $\mu>0$ and $w$ is not
smaller than $-1$ (or from a geometrical viewpoint, the flaring-out condition
$\beta^{\prime}(r_{0})>0$ is not respected). There is thus no wormhole for
this range of values of $w$.

Figure 5: Phase space of wormholes defined by a constant equation of state $w=-6.2$, $w=-2.3$, $w=0.2$ and $w=1.2$ (respectively the first, second, third and fourth graph). Thick trajectories and dots represent respectively some wormhole trajectories and their equilibrium points. $(\beta,\theta)$ | $-3-2\sqrt{2}$ | $-1...0$ | $1$
---|---|---|---
$(\tanh 1,0)$ | sa | sa |  | sa | sa
$(\tanh\frac{(1+w)^{2}}{w^{2}+6w+1},\tanh\frac{2w}{1+w})$ | si | NE |  | si | si
$(1,-1)$ | sa | sa |  | sa | si
$(1,1)$ | sa | sa |  | sa | so
$(0,-1)$ | si | si |  | NE | NE
$(0,1)$ | NE | NE |  | sa | sa
$(1,\tanh(-1/2))$ | so | so |  | so | so
$(1,\tanh\frac{1+3w}{w-1})$ | sa | si |  | si | sa
Table 2: This table shows the $8$ equilibrium points of the phase space for a
constant equation of state such as $w\not\in\left[-1,0^{-}\right]$. ”sa”, ”si”
and ”so” stand for ”saddle”, ”sink” and ”source”. ”NE” stands for ”No
Equilibrium”. Those in bold belong to wormhole trajectories.

Consequently, we look for the phase space equilibrium points and their
stabilities when $w\not\in\left[-1,0\right]$. They are summarised in table 2.
All these points do not necessarily belong to trajectories describing
wormholes. However, using numerical calculations, they allow to find the
interesting intervals of $w$ that define the various families of wormhole
trajectories. We then get

  * •

When $w<-3-2\sqrt{2}$, a wormhole throat is in $(\beta,\theta)=(0,1)$. The
trajectories end in $u\rightarrow\infty$, either in $(\beta,\theta)=(0,-1)$
and such as $P<0$ or in
$(\beta,\theta)=(\tanh\frac{(1+w)^{2}}{w^{2}+6w+1},\tanh\frac{2w}{1+w})$ with
$P>0$.

  * •

When $-3-2\sqrt{2}<w<-1$, a wormhole throat is in $(\beta,\theta)=(0,1)$. We
recover the previous trajectories that end asymptotically in
$(\beta,\theta)=(0,-1)$ such as $P<0$. A new set of trajectories ends
asymptotically in $(\beta,\theta)=(1,\tanh\frac{1+3w}{w-1})$ with $P>0$.

  * •

When $0<w<1$, some wormholes trajectories start with a throat in
$(\beta,\theta)=(0,1)$ and end in $(\beta,\theta)=(1,\tanh\frac{1+3w}{w-1})$,
like above but with $P<0$ whatever $u$. A second set of wormhole trajectories
starts in $(\beta,\theta)=(0,-1)$ and also ends asymptotically in
$(\beta,\theta)=(1,\tanh\frac{1+3w}{w-1})$.

  * •

When $1<w$, wormholes trajectories have their throat in $(\beta,\theta)=(0,1)$
and end asymptotically in $(\beta,\theta)=(1,-1)$.

All these wormhole trajectories are plotted on figure 5. We checked
numerically that at a throat, $P\rightarrow-1$ and $\beta^{\prime}\rightarrow
1+1/w$. None of them tends to the Minkowski spacetimeLob05  since it is a
saddle point whatever $w$, as indicated in table 2. Concerning the presence of
singularities, since $P=-1$ at the throat,
$\operatorname{arctanh}{\beta}\operatorname{arctanh}{\theta}$ is vanishing and
there is none there. All the trajectories end in $u\rightarrow\infty$, without
reaching the lines $\theta=\pm 1$ or $\beta=\pm 1$ at finite $u$. The density
and the scalars are thus finite everywhere at finite $u$ and these wormhole
solutions are free from singularity.

####  V.2.3 Equation of state as a redshift function

Two equations of state are widely used in cosmology

  * •

a linear equation of state $w=w_{0}+w_{1}z$

  * •

the Chevallier-Polarski-Linder parameterisation $w=w_{0}+\frac{w_{1}z}{1+z}$

$z$ being the redshift. To study the wormholes solutions of these dark energy,
we look for a relation of the form $\mu=f(\mu,\beta,\theta,\zeta)$. For the
linear equation of state, the conservation of the energy density for a FLRW
cosmology gives

$\rho=3e^{-3w_{1}}H_{0}^{2}\Omega_{0}e^{3w_{1}(1+z)}(1+z)^{3+3w_{0}-3w_{1}}$

with $H_{0}$ and $\Omega_{0}$ the Hubble constant and the dark matter energy
parameter. The equation of state being invertible, we get $z(w)=z(P/\mu)$ and

$\rho=3e^{-3w_{1}}H_{0}^{2}\Omega_{0}e^{3(-w_{0}+w_{1}+\frac{P}{\mu})}(1+\frac{P-w_{0}\mu}{w_{1}\mu})^{3(1+w_{0}-w_{1})}$

Multiplying this last expression by $r^{2}$, we find

$\mu=3e^{-3w_{1}}H_{0}^{2}\Omega_{0}\frac{e^{3(-w_{0}+w_{1}+\frac{-1+\operatorname{arctanh}{\beta}(1+2\operatorname{arctanh}{\theta})}{\mu})}\operatorname{arctanh}{\beta}(1+\frac{-1-w_{0}\mu+\operatorname{arctanh}{\beta}(1+2\operatorname{arctanh}{\theta})}{w_{1}\mu})^{3(1+w_{0}-w_{1})}}{\operatorname{arctanh}{\zeta}^{2}}$

With this expression, it is now possible to look for wormholes in the phase
space $(\beta,\theta,\zeta)$. Since $\mu(\zeta)=\mu(-\zeta)$, their
trajectories are symmetric with respect to the plane $\zeta=0$. We choose to
consider a dark energy with $-1.5<w_{0}<-1,-0.5<w_{1}<0$ and cosmological
parameters $H_{0}=70$, $\Omega_{m0}=0.27$ in agreement with the
observationsRud15 ; Jas10 ; Tsu09 . We get the figure 6. All the trajectories
have their throat in $(\beta,\theta,\zeta)=(0,1,0)$ and end in
$(\beta,\theta,\zeta)=(0,-1,0)$ with diverging $l$ and density. Hence, they do
not have any singularity and are not asymptotically Minkowski flat.
The same calculations can be made with the Chevallier-Polarski-Linder
parameterisation with then

$\mu=3e^{-3w_{1}}H_{0}^{2}\Omega_{0}\frac{e^{3\left(w_{0}+w_{1}-\frac{-1+\operatorname{arctanh}{\beta}(1+2\operatorname{arctanh}{\theta})}{\mu}\right)}w_{1}^{3}\mu^{3}\operatorname{arctanh}{\beta}\left(\frac{w_{1}\mu}{1+\mu(w_{0}+w_{1})-\operatorname{arctanh}{\beta}(1+2\operatorname{arctanh}{\theta})}\right)^{3(w_{0}+w_{1})}}{\operatorname{arctanh}{\zeta}^{2}(1+\mu(w_{0}+w_{1})-\operatorname{arctanh}{\beta}(1+2\operatorname{arctanh}{\theta}))^{3}}$

and the same ranges for the parameters $w_{0}$ and $w_{1}$. Once again,
$\mu(\zeta)=\mu(-\zeta)$. We get a figure similar to figure 6. The linear and
Chevallier-Polarski-Linder dark energy thus lead to the same type of wormholes
solutions.

Figure 6: Wormhole phase space for the cosmological equation of state
$w=w_{0}+w_{1}z$ with $w_{0}=-1.1$ and $w_{1}=-0.2$.

####  V.2.4 Navarro-Frenk-White

A Navarro-Frenk-White profileNav96  is generally used to fit dark matter
halos. It has also been shown Rah14  that a galactic halo with such a profile
and observed rotation curves has the properties for supporting traversable
wormhole sustained by this fluid in the anisotropic case $p\not=\tau$. Also
the NFW profile is now considered in the wormhole literatureIsl18 ; Rah16 . In
this subsection, we determine the different families of wormhole solutions for
this dark matter in the isotropic case, showing that none of them is
asymptotically Minkowski flat. The NFW density writes as

$\rho=\frac{\rho_{s}}{\frac{r}{r_{s}}(1+\frac{r}{r_{s}})^{2}}$

with $\rho_{s}$ and $r_{s}$ some positive constants. Using the definition
(12), we get

$\mu=\frac{\pm
r_{s}^{3}\rho_{s}\sqrt{\operatorname{arctanh}\beta}\operatorname{arctanh}\zeta}{(\pm\sqrt{\operatorname{arctanh}\beta}+r_{s}\operatorname{arctanh}\zeta)^{2}}$

with $\pm$ corresponding to the upper (lower) part of the wormhole. Once again
$\mu(\zeta)=\mu(-\zeta)$ and we limit the phase space study to $0<\zeta<1$,
the phase space trajectories being symmetric with respect to the plane
$\zeta=0$. At the throat, it is easy to show that $\beta^{\prime}>0$ if
$1-\frac{\pm r_{0}r_{s}^{3}\rho_{s}}{\pm r_{0}+r_{s}}^{2}>0$. Moreover, when
$\beta=\tanh 1$, $\beta^{\prime}<0$ indicating that the wormhole trajectories
coming from the throat are such as $\beta<\tanh 1$. The equilibrium points
respecting these limits for $\beta$ and $\zeta$ and intervening in the
wormholes trajectories are:

  * •

$(\beta,\theta,\zeta)=(\tanh 1,\tanh 2,0)$ which is a sink when
$\zeta\rightarrow 0^{+}$

  * •

$(\beta,\theta,\zeta)=(\tanh 1,-1,0)$ which is a sink when $\zeta\rightarrow
0^{+}$

We have then two families of wormhole trajectories, symmetric with respect to
the $\zeta=0$ plane. They are separated by the surface $P=0$ for $0<\zeta<1$
(that is such as $\operatorname{arctanh}\beta=1-\mu$ and thus
$\beta^{\prime}=0$) but in the neighbourhood of the throat in
$(\beta,\theta,\zeta)=(0,1,0)$ where $P=-1$. The first family ends in
$(\beta,\theta,\zeta)=(\tanh 1,\tanh 2,0^{+})$ with $P>0$ and the second one
in $(\beta,\theta,\zeta)=(\tanh 1,-1,0^{+})$ with $P<0$, both when
$r\rightarrow+\infty$. These non singular trajectories thus tend to a
spatially flat metric but not to a flat Minkowski spacetime. Some examples of
these curves are plotted on figure 7.

Figure 7: The two families of wormhole trajectories for a Navarro-Frenk-White
distribution of matter when $\rho_{s}=1$ and $r_{s}=1$. Each curve is
symmetric with respect to the $\zeta=0$ plane. The two wormhole families are
separated by the sign of $P$ which is projected on the plane $\zeta=0$ but
near the throat where they both are such as $P=-1$.

###  V.3 A toy model of asymptotically flat wormhole on both side of its
throat

In this last subsection, we consider a toy model of static wormhole that is
asymptotically Minkowski flat222We do not consider the Bronnikov-Ellis
modelBro73 ; Ell73  which is not based on an isotropic perfect fluid.. As a
physical result, we show that such a wormhole can be asymptotically flat on
both sides of its throat. This is thus a counterexample to a recent
theoremBro17  that excludes such a possibility for a static wormhole sustained
by an isotropic fluid.

We consider the toy model defining by the following pressure

$p=p_{0}r^{-n}$

with $n>2$ since then, when $r$ is diverging, the pressure vanishes faster
than $r^{-2}$ as required to get a Minkowski equilibrium point as a sink (see
subsection III.1). Such a model has already been considered in Lob13 ; Rah07 .
From equation (4) and the fact that
$d\Phi/dr=e^{-u}\operatorname{arctanh}\theta$, we get

$w=p/\rho=P/\mu=\frac{\operatorname{arctanh}\theta}{n-\operatorname{arctanh}\theta}$

This equation of state is independent from $p_{0}$ and tends to $-1^{-}$ in
$\theta\rightarrow+1$. The weak energy condition is thus violated that
explains the presence of a throat in $(\beta,\theta)=(0,1)$. Then using (18),
it comes that

$\mu=\frac{(n-\operatorname{arctanh}\theta)(-1+\operatorname{arctanh}\beta+2\operatorname{arctanh}\beta\operatorname{arctanh}\theta)}{\operatorname{arctanh}\theta}$

$\mu$ is independent on $\zeta$ and the dynamical system thus reduces to two
differential equations for $\beta^{\prime}$ and $\theta^{\prime}$. The
flaring-out condition $\beta^{\prime}(r_{0})>0$ implies $p_{0}r_{0}^{2-n}>-1$.

Figure 8: Phase space representations for $n=3$. The first graph shows the
different parts of the phase space, separated by some gray curves
corresponding to $\theta=0$ and $\theta^{\prime}=0$. Only the part on the
upper left corner contains wormhole trajectories which are shown on the second
graph with the three equilibrium points. Two numerical wormhole solutions are
plotted for $(r_{0},p_{0})=(0.1,-0.5)$ and $(r_{0},p_{0})=(0.02,0.1)$ (black
bold curves, the right one ending at the Minkowski equilibrium point). The
third and fourth graphs show the signs of $P$ and $\mu$ in the phase space.
The fourth graph is an enlargement of the third one for $\theta\simeq 1$ and
shows parts of the numerical solutions of the second graph

We then get the phase space trajectories plotted on the first graph of figure
8 for $n=3$ (similar figures are obtained for other values of $n$). The line
$\theta=0$ and the curve $\theta^{\prime}=0$ (in gray on the mentioned first
graph) split the trajectories in four sets. Only the one with $\theta>0$ and
$\theta^{\prime}<0$, shown on the second graph of figure 8, contains the
wormhole trajectories. Some numerical solutions are plotted in bold (with
enlargement for $\theta\simeq 1$ on the fourth graph). The signs of $\mu$ and
$P$ are also shown on the third and fourth graphs on figure 8. Equilibrium
points of wormholes trajectories are the following

  * •

$(\beta,\theta)=(\tanh 1,0)$. This is the Minkowski equilibrium point whose
stability is confirmed by numerical simulations in $\zeta=0^{+}$ for $P>0$.

  * •

$(\beta,\theta)=(\tanh 1,1)$ is a saddle.

  * •

$(\beta,\theta)=(1,1)$ is also a saddle.

We have then two sets of phase space trajectories representing wormhole
solutions, that both have their throats in $(\beta,\theta)=(0,1)$

  * •

For the first set, a trajectory starts with a negative pressure (i.e.
$p_{0}<0$) and positive density and ends with a singularity on the line
$\theta=0$ with $\beta<\tanh 1$. At the singularity, the density and the
scalars diverge whereas the pressure is finite. We checked the presence of
this singularity with the geodesic equation for $u(\tau)$ whose solution for a
wormhole trajectory defined by $r_{0}=1$ and $p_{0}=-0.5$ is plotted on figure
9, showing that $u$ and $\tau$ end when $\theta$ reaches zero.

  * •

For the second set, a trajectory starts with a positive pressure (i.e.
$p_{0}>0$) and negative density and it goes asymptotically to the Minkowski
equilibrium point where the pressure, the density and $w$ tend to vanish. We
thus recover an empty Universe far from the throat. Such a trajectories is
plotted in the white area of the second graph on figure 8. Numerical
simulations show that the larger $n$, the smaller the area where stand these
trajectories.

Since $\mu$ does not depend on $\zeta$, we should thus have a flat spacetime
on both sides of the wormhole. Let us show that this is a counterexample to
the no-go theorem of Bro17  that excludes such a behaviour of the metric in
presence of an isotropic perfect fluid. To show this no-go theorem, the
authors used the form of the metric
$ds^{2}=A(x)dt^{2}-dx^{2}/A(x)-r^{2}(x)d\Omega^{2}$. The no-go theorem then
rests upon the fact that the quantity $D=Ar^{2}>0$ should go to $+\infty$ on
both sides of the throat when spacetime becomes flat with $A\rightarrow 1$ and
$r\rightarrow+\infty$. $D$ should thus possess a minimum at the throat such as
$dD/dx=0$. But the behaviour of $D$ is given by the equation (14) in Bro17
that writes

$\frac{d^{2}D}{dx^{2}}-4\frac{1}{r}\frac{dD}{dx}\frac{dr}{dx}+4\frac{D}{r^{2}}(\frac{dr}{dx})^{2}+2=0$

Hence, when $dD/dx=0$, $d^{2}D/dx^{2}<-2$, that implies a maximum for $D$ and
not a minimum, thus leading to the no-go theorem. Let us show that the model
of the present subsection is a counterexample to this theorem. Adopting the
same metric signature, we have $D=e^{2\Phi}r^{2}$ and
$dr/dx=\mp\sqrt{\operatorname{arctanh}\beta}e^{-\Phi}$. $\mp$ stands for above
or below the throat where $dr/dx$ vanishes, $r$ reaching the minimum value
$r_{0}$. We also have that $P\rightarrow p_{0}r_{0}^{2-n}$ and thus
$\operatorname{arctanh}\beta\operatorname{arctanh}\theta\rightarrow
1/2(p_{0}r_{0}^{2-n}+1)$. Consequently,
$\frac{dD}{dx}\frac{dr}{dx}=2r(\operatorname{arctanh}\theta+1)\operatorname{arctanh}\beta$
tends to the non vanishing constant $r_{0}(p_{0}r_{0}^{2-n}+1)$ at the throat
and $\frac{dD}{dx}$ tends to $\mp\infty$ on both sides of it, $D$ being a
finite quantity (since in $r\rightarrow r_{0}^{+}$, $\theta\rightarrow 1$,
$\Phi^{\prime}\rightarrow+\infty$ and thus $\Phi$ cannot diverge positively).
An extremum value of $D(x)$ is thus reached at the throat but with
$dD/dx\not=0$, a case not taken into account in Bro17 . This extremum is also
a minimum since $4\frac{D}{r^{2}}(\frac{dr}{dx})^{2}=0$, $p_{0}$ is positive
for the trajectories ending at the Minkowski sink point and thus
$\frac{d^{2}D}{dx^{2}}=2+4p_{0}r_{0}^{2-n}>0$. Hence, the no-go theorem of
Bro17  does not apply to the model of this subsection and we have an
asymptotic Minkowski flat spacetime on both sides of the throat.

Figure 9: $u(\tau)$, $\beta(\tau)$ and $\theta(\tau)$ for a wormhole
trajectory defined by $n=3$, $r_{0}=1$ and $p_{0}=-0.5$. This trajectory is
plotted in bold in the phase space on figure 8. The gray points indicate the
ends of the curves $(u,\beta,\theta)$ in $\tau=551$ and $u=0.14$.

##  VI Conclusion

In this paper we rewrote the field equations of static spherically symmetric
wormholes sustained by an isotropic perfect fluid as a dynamical system with
normalised variables. It allows displaying at a glance all the solutions of
the field equations in a finite phase space without having to solve them. It
offers a unified framework to compare different wormhole models and to study
their global properties. We determined the possible locations of throats,
equilibrium points as well as curvature singularities in the phase space. We
also associated a geodesic equation to spacetime that describes a phase space
trajectory. The wormhole throats stand in $(\beta,\theta,\zeta)=(0,\pm 1,0)$.
They are not equilibrium points despite all the wormhole trajectories pass
through them. Their location allow to prove a first general physical result,
i.e. that the variation of the gravitational redshift at the throat of a
static and isotropic wormhole is always diverging (but $\Phi$ can be finite),
generalising the recent result that there is no zero-tidal force static
spherically symmetric wormhole sustained by an isotropic perfect fluid. The
equilibrium point corresponding to the Minkowski solution is in
$(\beta,\theta,\zeta)=(\tanh 1,0,0)$ and can only be reached by solutions
whose pressure and density of the fluid decrease faster than $r^{-2}$. We then
used this formalism to study the dynamical systems of several static
spherically symmetric wormholes models, getting other physical results.

The first model we studied is defined by a vanishing density $\rho=0$. From a
mathematical viewpoint, its exact solution allows checking the agreement of
our dynamical system of equations with the usual wormhole field equations
(2-4). This model has only one family of wormhole solutions in the two
dimensional phase space $(\beta,\theta)$. Its trajectories reach a singularity
at finite distance and time from the throat. From a physical viewpoint, this
model contains as a special case ($P=0$) the Schwarzschild black hole. We have
plotted it in the phase space to show what looks like its trajectory. It is
discontinued at the horizon in $(\beta,\theta,\zeta)=(0,\pm 1,0)$, reflecting
the discontinuity of the Schwarzschild metric in $r=2M$ with $(r,t)$
coordinates that are not appropriate to study horizons. This is the only type
of black hole such a fluid admits. Since this paper is dedicated to static
spherically symmetric wormholes, we did not pursue further in the study of
static spherically symmetric black holes, nor horizons that would need a
different set of coordinates to be clearly identified in the phase space
$(\beta,\theta,\zeta)$. However, this shows that the dynamical system analysis
of this paper could also be used to consider these black holes and their
horizons.
We then look for some wormholes that could be generated by some usual dark
energy and dark matter models. Hence, the generalised Chaplygin gas has
wormhole solutions when $A<r_{0}^{-2(1+\alpha)}$. There is only one family of
trajectories describing identical wormholes connected the ones with the others
and thus repeating cyclically.
A constant equation of state $p/\rho=w$ has a richer dynamics. It has five
families of wormhole solutions. As is well known, we have no wormhole for
$-1<w<0$ since then the weak energy condition cannot be violated. Wormholes
exist for a ghost dark energy with $w<-1$ and a positive density or for $w>0$
with a negative density at the throat. All the trajectories end asymptotically
with a diverging $r$ at some equilibrium points. None of them possesses a
singularity (at finite $r$). Some of them are asymptotically spatially flat
but do not tend to a Minkowski spacetime.
We also look for the wormhole solutions that can be generated by the linear
and CPL equations of state (depending on the redshift $z$) widely used in the
literature to study dark energy. Taking the range of their parameters in
agreement with observations and a ghost dark energy ($w<-1$), we found that
they possess only one family of solutions, describing some wormholes without
singularity at finite $l$ but that are not asymptotically flat.
Last, the static wormhole model defined by a Navarro-Frenck-White density
profile has wormhole solutions when $1-\frac{\pm r_{0}r_{s}^{3}\rho_{s}}{\pm
r_{0}+r_{s}}^{2}>0$. Two wormhole families are found with trajectories that
both end in $r\rightarrow+\infty$ with a spatially flat spacetime different
from the Minkowski one.
The main physical result about these five dark energy and dark matter models
that are often used to describe our Universe is that none of them can sustain
a static wormhole that could be asymptotically Minkowski flat. This tends to
show that they should not form naturally some static wormholes.

Finally, we consider a toy model such that the pressure vanishes as $r^{-n}$
with $n>2$. There are then two families of trajectories describing wormholes.
For one of them, the trajectories end at a singularity at finite $r$ and time.
For the other one, the trajectories asymptotically tend to the Minkowski
equilibrium point far from the throat. The physical result here, is that this
flatness occur for both sides of the wormhole. This is thus a counterexample
to a theorem claiming that a static wormhole sustained by a perfect fluid
could not be flat on both sides of its throat. This is interesting since it
has not to be glued to a vacuum exterior spacetime to be physically
acceptableLem03 ; Hos18 .

Let us conclude with some possible extension of this work. Dynamical system
equations (13-15) with the constraint on $T$ allow generalising the framework
we develop to non-isotropic forms of perfect fluid. Hence, it could be
interesting to study the cases when $p/\rho$ and $\tau/\rho$ are some
constants for instance. The dynamical system could also be used to study
relativistic star models defined by their barotropic equation of state as in
Nil01 ; Nil01A ; Hei03  or static black holes as shown with the Schwarzschild
one.

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