]Permanent address

# The K-essence flow seen from the preferred frame $S_{V}$. A scalar field
theory with Landau superfluid structure

Rodrigo Francisco dos Santos<EMAIL_ADDRESS>Department of Education-
Belford Roxo City, Jd. Glaucia School Town, Pericles 15, pc 26195-140, Belford
Roxo, RJ, Brazil  Luis Gustavo de Almeida<EMAIL_ADDRESS>Federal
University of Acre, Rio Branco, pc 69920-900, AC, Brazil  A. C. Amaro de Faria
Jr<EMAIL_ADDRESS>Federal Technological University of Parana,
Guarapuava, pc 85053-525, PR, Brazil  [  Advanced Studies Institute - IEAv,
Sao Jose dos Campos, pc 12220-000, SP, Brazil

###### Abstract

We study the hypothesis of deformation of the invariance of Lorentz
transformations produced by the introduction of a universal minimum velocity
relative to a preferred frame. Our goal with this job is to apply this
hypothesis to superfluids and study its consequences relating the minimum
velocity to the idea of a fluid, with superfluid properties. In previous works
we related the minimum velocity to the cosmological constant and even to
cosmic inflation. Soon we could generate a hypothetical superfluid capable of
modeling with characteristics of a cosmological fluid with dark energy
properties. The first excited state of this universal superfluid would be a
preferred frame from which all other excited states are observed and then we
would have a preferred frame $S_{V}$ associated with the critical Landau
velocity, thus implying that the universal minimum velocity coincides with the
critical Landau velocity, and the objects observed by the preferred frame are
excited states of the superfluid. This coincidence between the concepts of
minimum velocity and Landau’s critical velocity makes Landau’s critical
velocity a type of limit velocity, modifying the usual causal structure of
restricted relativity. Formulating the phenomena in this preferred frame would
have the advantage of providing a simple explanation for astrophysical and
cosmological phenomena linked to a causal structure, which emerges from this
construction and is very similar to causal structures linked to Gordon
geometry and acoustic tachyons. We build a deformed relativistic Lagrangian,
demonstrate its relation with a _k_ -essence Lagrangian and calculate the
quantities associated with that Lagrangian. We also studied an irrotational
fluid and verified the role of enthalpy associated with the minimum velocity
structure.

###### pacs:

11.30.Qc

##  I Introduction

An important line of research in cosmology has been the model entitled _k_
-essence kan . The _k_ -essence models are scalar field models k1 ; ka ; k2 ;
k3 , that alongside the quintessence models q1 ; q2 , appear as alternatives
to solve the problem of barotropic fluids with $w={p}/{\rho}<0$. Therefore,
one of the most important guidelines in the study of _k_ -essence is the study
of causal structures and their pathologies kpato . Recent developments kpato1
point to an approximation of the _k_ -essence hypothesis with superfluid
models vortex ; superverse . The line of gravitational superfluids models has
received a lot of attention in recent years, in particular using the framework
of Carter superrela ; superrela1 , known as Carter’s Multifluid Theory
supercarter , and its relation of gravity supercarter1 ; supercarter2 . This
is very similar line to the line developed by Visser v2010 . In this line,
Bacetti bacejhep2  developed a treatment for dissipation relations deformed by
a Lorentz symmetry breaking, which is very similar to the Landau formalism
used in superfluids, also applicable to Planck scale dissipations volo22 . In
2017 Nassif  _et al_ r1 , in an exotic but promising approach, starts the
study of effects of deformation of Lorentz Invariance introduced by a
universal minimum velocity relative to a preferred reference frame in an
inflationary scenario r2 ; r3  and gravitational collapse gbec . Nassif’s
works N2007 ; N2008 ; N2013 ; N2015 ; N2016 ; N2019 ; N2022  have suggested a
dispersion relation deformed by the invariant minimal speed, associated with a
preferred frame $S_{V}$. However, in 2022 Santos  _et al_ nova  propose a
totally new perspective for the deformation introduced in incipient Nassif’s
works, where they look for a hydrodynamic perspective for the minimal speed
rhidro ; rhidro1 . This new interpretation aims to adjust the idea of a
minimum velocity to a relativistic hydrodynamics scenario that approximates
the gravitational interaction of phase transition phenomena svt ; svt2 ; qgv
metamaterials meta ; meta2  to a Relativistic Bose Einstein Condensate rbec .
That said, let’s explicitly enumerate the objectives of our work

  1. 1.

The goal of this paper is to find a physical justification for the
introduction of the minimum velocityN2007 , in this case the justification is
the relationship of the Nassif minimum velocity to the Landau critical
velocity. This coincidence is very promising, since superfluids have
properties analogous to anti-gravity. We can therefore use them as analogues
to the gravitational vacuum.

  2. 2.

Demonstrate that the introduction of the minimum velocity is equivalent to a
violation of the Lorentz symmetry in an acoustic geometry

  3. 3.

To resume Nassif’s uncertainty principle proposalN2013 , but with a
description that makes it clear that the quantum characteristic of this
treatment is analogous to that of a superfluidsuperverse

  4. 4.

Demonstrate the Lorentz Invariance Violation in the same terms as Zloschatiev
svt2

  5. 5.

To identify points of falsifiability of the existence of the preferred-frame
$S_{V}$ N2007  we study the relation of the Einstein-Euller referential and
the preferred-frame $S_{V}$. As we try to understand the relevance of the
$S_{V}$ referential to other proposals of fluids that intend to model the dark
energyr2 , in this paper the fluid chosen is _k_ -essence

  6. 6.

Performing the first movements to understand the energy mechanism, which would
make each excited state appear in the hypothetical fluid, excited states that
we can analogously compare with subatomic matter appearing in the
gravitational vacuum on a cosmological scale could be a more complex
Structures. Since we can hypothetically extrapolate the Jeans mechanism jeans

This work is divided as follows:

In Section 2 we briefly review the _k_ -essence model and specify the kinetic
_k_ -essence.

In Section 3, we revisit the concepts associated with the introduction of
minimum velocity in Lorentz transformations.

The Section 4 shows how the minimum speed corresponds to a critical speed in
the Landau criterion, a relation to the Einstein-Euller referential, helping
to clarify the role of the preferred-frame $S_{V}$. And we construct a emphK-
Essence Lagrangian, taking into account the scalar product deformation, which
introduces the reciprocal velocity

The Section 5 presents an approach where we seek to investigate some
thermodynamic properties of an irrotational fluid considering the Shutz
formalism sch ; sch2 . And rehearsing a discussion about the jumps of energy
levels for the appearance of each excited state with the increment of the
velocity $v$

In section 7 we present our conclusions and future perspectives.

##  II _K_ -essence framework

The _k_ -essence model is a theoretical framework used in hydrodynamics to
describe the dynamics of a scalar field with a non-canonical kinetic term.
This model introduces a new degree of freedom, which modifies the equation of
state of the fluid and allows for the possibility of an accelerated expansion
of the universe. The _k_ -essence model has been used to study a wide range of
phenomena, including cosmic inflation kNojiri2019 ; kOikonomou2021 , dark
energy kOdintsov2020 ; kOdintsov2021 ; kOikonomou2020 ; kOikonomou2021 , and
the dynamics of cosmic structure formation.

We can define a _k_ -essence model k1 ; ka  with the action

$\mathcal{S}:=\int{d^{4}x}\sqrt{g}\mathcal{L}(\mathcal{X},\Phi).$ (1)

The action (1) is associated to the lagrangian density
$\mathcal{L}(\mathcal{X},\Phi)$, where $\mathcal{X}$ is the scalar product
defined as

$\mathcal{X}:=g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi.$ (2)

Here $\Phi$ is a scalar field and $\nabla$ is a covariant derivative operator
linked to $g^{\mu\nu}$. It is very important notice that $\mathcal{X}$ is a
scalar product between two non-normalized four-vectors, with the presence of
the normalized factor $\nabla_{\mu}\Phi$. The conserved current associeted to
(1) is obtained in k2

$\left(\mathcal{L}_{,\mathcal{X}}g^{\alpha\beta}-\mathcal{L}_{,\mathcal{XX}}\nabla^{\alpha}\Phi\nabla^{\beta}\Phi\right)\nabla_{\alpha}\nabla_{\beta}\Phi+\mathcal{L}_{,\Phi\mathcal{X}}g^{\alpha\beta}\nabla_{\alpha}\Phi\nabla_{\beta}\Phi=-\mathcal{L}_{\Phi},$
(3)

where $\mathcal{L}_{,\mathcal{X}}$ is the derivative for parameter
$\mathcal{X}$, $\mathcal{L}_{,\mathcal{XX}}$ is second derivative with
$\mathcal{X}$. Also, this $\mathcal{L}_{,\Phi}$ is the derivative with $\Phi$
and $\mathcal{L}_{,\Phi\mathcal{X}}$ is a second derivative. In the spatially
homogeneous, case $\mathcal{X}>0$, it reduces to

$\left(2\mathcal{X}\mathcal{L}_{,\mathcal{XX}}+\mathcal{L}_{,\mathcal{X}}\right)\ddot{\Phi}+\mathcal{L}_{,\mathcal{X}}(3H\dot{\Phi})+\mathcal{L}_{,\Phi\mathcal{X}}\dot{\Phi}^{2}-\mathcal{L}_{,\Phi}=0.$
(4)

To ensure solubility, we take
$\left(2\mathcal{X}\mathcal{L}_{,\mathcal{XX}}+\mathcal{L}_{,\mathcal{X}}\right)\neq
0$. We can further restrict this condition, calling it a condition of
“hyperbolicity”,

$1+2\frac{\mathcal{X}\mathcal{L}_{,\mathcal{XX}}}{\mathcal{L}_{,\mathcal{X}}}\geq
0,$ (5)

thus imposing a condition on $\mathcal{L}_{,\mathcal{X}}$ and
$\mathcal{L}_{,\mathcal{XX}}$.

###  II.1 _K_ -essence purely kinetic: Effective Hydrodynamics Approach

The lagrangean $\mathcal{L}(\mathcal{X},\Phi)$ eventuality showed a dependency
only of $\mathcal{X}$, so we can write

$\mathcal{L}(\mathcal{X},\Phi)\equiv\mathcal{L}(\mathcal{X}).$ (6)

This is a particular case called “purelly kinetic _k_ -essence” and imply in
an isotropic and homogeneous preferred frame backgroud k1 ; ka . In this
particular case we have $\mathcal{L}_{\Phi\mathcal{X}}=\mathcal{L}_{\Phi}=0$.
The lagrangean (1) allow evaluation of momentum-energy tensor,
$T_{\mu\nu}:=\frac{2}{\sqrt{g}}\frac{\delta\mathcal{L}}{\delta{g}_{\mu\nu}}$.
We also can write the momentum-energy tensor as we did in r2 :

$T_{\mu\nu}=\mathcal{L}_{,\mathcal{X}}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\mathcal{L}(\mathcal{X})g_{\mu\nu}.$
(7)

The simple comparation of equation (7) with energy-momentum tensor for a
perfect fluid suggests the identification of several hydrodynamics variables.
The pressure can be written as

$p=\mathcal{L}(\mathcal{X}).$ (8)

The kinetic term requires more work. We define the four-velocity

$u_{\mu}:=\frac{\nabla_{\mu}\Phi}{\sqrt{2\mathcal{X}}},$ (9)

replacing (8) and (9) in (7), we have density of energy

$\rho=\mathcal{X}\mathcal{L}_{,\mathcal{X}}-\mathcal{L}(\mathcal{X}).$ (10)

We write now the energy-momentum tensor with _k_ -essence as source:

$T_{\mu\nu}=2\mathcal{X}\mathcal{L}_{,\mathcal{X}}u_{\mu}u_{\nu}-\mathcal{L}(\mathcal{X})g_{\mu\nu}.$
(11)

For (11) exists a rest-frame $u_{i}=0$ where the scalar field is locally
isotropic.

Now we are going to discuss the energy conditions in a similar way of the work
k2 . The _null energy condition_ (NEC), is respected

$\mathcal{L}_{,\mathcal{X}}>0,$ (12)

the _weak energy condition_ (WEC), establishes

$\mathcal{X}\mathcal{L}_{,\mathcal{X}}-\mathcal{L}(\mathcal{X})>0,$ (13)

and the (12) with $\mathcal{L}>0$. Both conditions in force guarantee that we
deal only with positive energy states. In the event of a violation, negative
energy states arise.

The expression

$\frac{\mathcal{X}\mathcal{L}_{,\mathcal{X}}-\mathcal{L}(\mathcal{X})}{\mathcal{L}(\mathcal{X})}>0,$
(14)

is associated with the _dominate energy condition_ (DEC), and (14) with (12),
generate the _null dominant energy condition_ (NDEC). The violation of those
conditions can introduce tachyons.

We define the barotropic parameter $w$ according to Visser et-al v2010 , and
establish criteria for a barotropic fluid:

  * •

Any zero-temperature fluid is automatically barotropic;

  * •

Any non-zero temperature but isothermal fluid is automatically barotropic;

  * •

Any zero-entropy fluid (superfluid) is automatically barotropic;

  * •

Any isentropic fluid is automatically barotropic.

We write, then

$w:=\frac{p}{\rho}=\frac{\mathcal{L}(\mathcal{X})}{\mathcal{X}\mathcal{L}_{,\mathcal{X}}-\mathcal{L}},$
(15)

and get the sound propagation, according to k3

$c^{2}_{s}:=\left(1+2\frac{\mathcal{X}\mathcal{L}_{,\mathcal{XX}}}{\mathcal{L}_{,\mathcal{X}}}\right)^{-1},$
(16)

with is the inverse of the expression showed in (5).

Two other hydrodynamic quantities still can be defined, the concentration of
particles

$n\equiv\exp\left[\int\frac{d\rho}{\rho+p(\rho)}\right]=\sqrt{\mathcal{X}}\mathcal{L}_{,\mathcal{X}},$
(17)

and enthalpy

$h:=2\sqrt{\mathcal{X}}.$ (18)

We then link the kinetic structure to enthalpy, which is a quantity associated
with the capacity of absorption and emission of energy. Enthalpy is a crucial
element in theoretical modeling and numerical simulation of astrophysical
systems such as neutron stars and black holes, allowing for a deeper
understanding of the physical processes that occur in these extreme
environments.

###  II.2 _K_ -essence purely kinetic: Effective Geometric Approach

In many situations in astrophysics and cosmology, the complete spacetime
metrics are extremely complex and difficult to manipulate, which makes it
difficult to understand and analyze the physical properties of the system. In
such cases, it can be useful to use a simplified space-time description, which
can be derived from the full metric, but which takes into account only the
properties most relevant to the situation at hand.

The conserved current (3) allows us to define an effective metric of
$\tilde{\mathcal{G}}^{\mu\nu}$,

$\tilde{\mathcal{G}}^{\mu\nu}=\mathcal{L}_{,\mathcal{X}}g^{\mu\nu}+\mathcal{L}_{,\mathcal{X}\mathcal{X}}\nabla_{\mu}\Phi\nabla_{\nu}\Phi.$
(19)

The effetive metric (19) has Lorentzian signature and hence describes the time
evolution of the system provided and obeys the condition of hyperbolicity (5).
If we have $\mathcal{L}_{,\mathcal{XX}}=0$, the (19) reduce to
$\tilde{\mathcal{G}}^{\mu\nu}=\mathcal{L}_{,\mathcal{X}}g^{\mu\nu}$. Very
similar to a conformal transformation, we therefore establish the criteria for
the existence of a transformation according to $\omega^{2}$. It is a scale
transformation on the metric of a metric space, which generates a new metric,
associated with another metric space as
$\mathcal{G}_{\mu\nu}=\omega^{2}{g}_{\mu\nu}$, and which obeys the following
requirements:

  * •

$\omega$ has an inverse $\omega^{-1}$ and is smooth. It has all the
derivatives. Transforms the vicinity from a $p$ point, in a topological sense,
leading to the vicinity of a $p^{\prime}$ point in the transformed metric
space. That is loosely equivalent to the properties of an isometry. The
expression (5) obeys this requeriment;

  * •

The $\omega$ transformation respects the null geodesic

$g_{\mu\nu}X^{\mu}X^{\nu}=g_{\mu\nu}\omega{x^{\mu}}\omega{x^{\nu}}=\omega^{2}g_{\mu\nu}x^{\mu}x^{\nu}=\mathcal{G}_{\mu\nu}x^{\mu}x^{\nu}=0,$
(20)

which implies that the transformation $X^{\mu}=\omega{x^{\mu}}$ preserves the
vector type (time, space and null). This also implies that $\omega>0$. For
simple inspection we notice that $\mathcal{L}$ and (12) ensure this
requirement;

  * •

$\omega$ respects the angles

$\frac{1}{\sqrt{|v||u|}}g_{\mu\nu}v^{\mu}u^{\nu}=\frac{1}{\sqrt{|V||U|}}\mathcal{G}_{\mu\nu}V^{\mu}U^{\nu}.$
(21)

The construction of Equation (2) agrees to the request;

  * •

Since $\omega(a)$ has the $a$ parameter, it must belong to the original metric
space. The (2) is a scalar product and defines a manifold $\mathcal{M}$ with
metric $g_{\mu\nu}$.

.

The criteria established in works r1 ; r2  were respected in this case. A
second criterion deducted by Kunhnel and Radmacher kunh  has shown a relation
between metrics and the Ricci tensor of a space-time, as follows

$\tilde{R}_{ab}-R_{ab}=\frac{1}{\Omega^{2}}[2\Omega\partial_{\mu}\partial_{\nu}\Omega+(\Omega\partial^{\mu}\partial_{\nu}\Omega-3\partial^{\mu}\Omega\partial_{\mu}\Omega)g_{ab}],$
(22)

where

$\tilde{g}_{ab}=\omega^{2}g_{ab}=\Omega^{-2}g_{ab},$

so that $\omega=\Omega^{-1}$. We can see here the difference between the
conformal Ricci tensor $\tilde{R}_{\mu\nu}$ and the usual one $R_{\mu\nu}$,
related to $\partial_{\mu}\Omega$, $\partial^{\mu}\partial_{\mu}\Omega$ and
$\partial_{\mu}\partial_{\nu}\Omega$. Those operators are Gradient, Laplacian
and Hessian of $\Omega$ respectively. The result is that the difference
between Ricci tensors after and before the conformal transformation is
conserved and proportional to the metric. We write explicitly

$\tilde{R}_{\mu\nu}-R_{\mu\nu}\propto{g_{\mu\nu}}\propto\tilde{g}_{\mu\nu},$
(23)

if, and only if,

$\partial_{\mu}\partial_{\nu}\Omega=(\partial^{\alpha}\partial_{\alpha}\Omega)g_{\mu\nu},$
(24)

where $g_{\mu\nu}$ has zero curvature.

The Equation (24) introduced optimization proprieties for
$\mathcal{L}_{\mathcal{X}}$. A very interesting approach can be seen in works
gwp; k1 , to discuss fluctuations in _k_ -essence background where the eikonal
is writen as

$G^{\mu\nu}=\left(\frac{c_{s}}{\mathcal{L}_{,\mathcal{X}}}\right)^{{\frac{1}{\frac{D}{2}-1}}}g^{\mu\nu}.$
(25)

We consider this approach as a good perspective for future research.

###  II.3 The metric of Refractive index perturbation (R.I.P.) frame

We are going to use works rip ; rip2  as a guide to write this section. As a
direct application of acoustic formalism, the R.I.P. is a treatment for small
fluctuations propagated in a Kerr medium or a dielectric medium. The
fluctuations can be characteristic of locally superluminal propagation and can
be described by an eikonal approximation in a stationary metric. We introduce
the specific analogue model that we have analyzed. As we need to work both in
the laboratory frame as in an inertial reference frame, which is moving at
relativistic speed relative to the original frame, it is convenient to adopt a
covariant formalism. Then consider a reference frame in which the dielectric
medium is moving with four-velocity $u^{\mu}$rip . The medium in question can
have permittivity and permeability given respectively by

$\epsilon^{\alpha\beta}:=\epsilon(E)\left(\eta^{\alpha\beta}-u^{\alpha}u^{\beta}\right),$
(26)
$\mu^{\alpha\beta}:=\mu_{0}\mu_{r}\left(\eta^{\alpha\beta}-u^{\alpha}u^{\beta}\right).$
(27)

The variable $E:=\sqrt{E_{\mu}E^{\nu}}$ is an electric field, $\mu_{r}$ is the
relativity permeability and $\mu_{0}$ is the vacuum permeability. Likewise we
have

$\epsilon(E):=\epsilon_{0}\left(1+\mathcal{X}^{(1)}+\mathcal{X}^{(3)}E^{2}\right).$
(28)

The small fluctuation “fills” an effective space-time with two polarization
possibilities

$g^{+}_{\mu\nu}=\eta_{\mu\nu}-u_{\mu}u_{\nu}\left(1-\frac{1}{n}\right),$ (29)
$g^{-}_{\mu\nu}=\eta^{\mu\nu}-u_{\mu}u_{\nu}\left(1-\frac{1}{n^{2}(1+\mathcal{X})}\right)+\frac{\mathcal{X}}{\mathcal{X}+1}l_{\mu}l_{\nu},$
(30)

where the vector $l_{\mu}:=\frac{E_{\mu}}{E}$ is unitary in direction of
$E_{\mu}$. The parameter $\mathcal{X}$ have an important role since the usual
electromagnetism has non generated space-time curvature, because it’s a linear
field theory. The shift of linearity is measured by parameter $\mathcal{X}$.
This parameter is called _nolinearity parameter_

$\mathcal{X}:=\frac{E}{\epsilon}\frac{\partial\epsilon}{\partial{E}}.$ (31)

Therefore we have a link with non-linear electrodynamics. When
$\mathcal{X}\equiv 0$, we have linear electrodynamics regime, making the
polarization (29) and ( 30) match.

After defining those tools, we have to define the refractive index
perturbation.

$n:=n_{0}+\delta{n}(x^{\prime},\rho),$ (32)

and the line-element measured of co-moving frame

$\displaystyle ds^{2}:=$ $\displaystyle
c^{2}\frac{\gamma^{2}}{n^{2}}\left(1+\frac{nv}{c}\right)\left(1-\frac{nv}{c}\right)dt^{\prime}+2\gamma^{2}\frac{v}{n^{2}}\left(1-n^{2}\right)dt^{\prime}dx^{\prime}+$
$\displaystyle-\gamma^{2}\left(1+\frac{v}{nc}\right)(dx^{\prime})^{2}-d\rho^{2}-\rho^{2}d\varphi^{2}.$
(33)

The (33) have a cylindrical symmetry, $\rho=\sqrt{y^{2}+z^{2}}$.

Any specific choice for the function $\delta{n}$ give rising to a specific
metric in the aforementioned class. We point out that an isotropic refractive
index in the laboratory frame corresponds to an anisotropic refractive index
in the pulse frame due to length contraction associated with a boost in the
x-direction. If the refractive index depends explicitly on $\rho$, the metric
is stationary but not static. It is implied in emergence of the ergoregion,
the ergosurface and event-horizon. In this last case, we have a more complex
relation, the event-horizon is characterized by $g_{00}=0$, with
$v=\frac{c}{n}$. The event-horizon in these conditions admit solution in a
range

$\frac{1}{n_{0}+n}<\frac{v}{c}<\frac{1}{n_{0}}.$ (34)

It is necessary highlight a very-important question, in stationary but not
static geometry, there is a difference between event-horizon and ergo-surface,
i.e. they do not coincide. The event-horizon in such geometry is in the
antipodal points which solve

$\delta{n}(x^{\prime},\rho=0):=\frac{c-n_{0}v}{v},$ (35)

and ergo-surface is associated with the vanishing of Killing vector
$\partial_{t}$.

###  II.4 Formulation of Einstein-Euller equations

Following Rezzolla rezz , we shall consider a foliation perpendicular to the
timeline $\Sigma_{t}$, i.e. a foliation where we can define a perpendicular
unitary vector given as follows

$\Omega_{\mu}:=\nabla_{\mu}t.$ (36)

We then can write the vector defined in (36) as proportional to another vector

$\eta_{\mu}=A\nabla_{\mu}t,$ (37)

and we can write the tensor $\eta_{\mu}=\left(A,0,0,0\right)$, so we calculate
the scalar product $\eta_{\mu}\eta^{\nu}$, where we found

$\eta_{\mu}\eta^{\mu}=g^{\mu\nu}\eta_{\mu}\eta_{\nu}=g^{tt}A^{2}.$ (38)

The variable $\eta_{\mu}$ corresponds to an observer measuring a four-speed,
the timelike condition requires $\eta_{\mu}\eta^{\mu}=-1$, therefore
normalization is required. This condition when applied in (38) gives us

$g^{tt}A^{2}=-1,$ (39)

where we chose $\alpha:=-\frac{1}{g^{tt}}$, and for a $\eta^{\mu}$ future we
have $A=-\alpha$ . Once the vector $\eta_{\mu}$ is specified, we can define a
metric for each hypersurface

$\gamma_{\mu\nu}=g_{\mu\nu}+\eta_{\mu}\eta_{\nu}$ (40)

and

$\gamma^{\mu\nu}=g^{\mu\nu}+\eta^{\mu}\eta^{\nu}.$ (41)

The metric $\gamma_{ij}$ corresponds to the spatial part of $\gamma_{\mu\nu}$,
we require that $\gamma^{ik}\gamma_{kj}=\delta^{i}_{j}$ to ensure that
$\gamma_{\mu\nu}$ and $\gamma^{\mu\nu}$ correspond to inverse metrics of each
other. This construction allows $\eta_{\mu}$ and $\gamma_{\mu\nu}$ to be two
useful tools in the description of any four-vector.

Now we are going to set the projectors

$\gamma^{\mu}_{\nu}:=g^{\mu\nu}\gamma_{\alpha\nu}.$ (42)

When we set (42), we can resume the definition of (40) and then we write
$g^{\mu}_{\nu}+\mathcal{N}^{\mu}_{\nu}$. A simple inspection ensures that
$g^{\mu}_{\nu}+\mathcal{N}^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\mathcal{N}^{\mu}_{\nu}$.
Where we set

$\mathcal{N}^{\mu}_{\nu}:=\eta^{\mu}\eta_{\nu},$ (43)

and the projector acting on (43)

$\gamma^{\alpha}_{\mu}\mathcal{N}^{\mu}_{\nu}=0,$ (44)

where a generic four-vector can be written as

$U^{\mu}=\gamma^{\mu}_{\nu}U^{\nu}+\mathcal{N}^{\mu}_{\nu}U^{\nu}.$ (45)

One can easily notice that $\gamma^{\mu}_{\nu}U^{\nu}=V^{\mu}$ has the
contravariant component $V^{t}=0$. On the other hand
$V_{t}=g_{t\mu}V^{\mu}\neq 0$, as well as the scalar product is given by
$\eta^{\mu}\Omega_{\mu}=\frac{1}{A}\eta^{\mu}\eta_{\mu}=\alpha^{-1}\neq 1$,
agreeing to the unitary vector defined in (36). Perpendicular to a
hypersurface spacelike $\Sigma_{t}$, it does not represent changes along the
temporal coordinate, and it is not the direction of the time derivative. The
vector that represents the temporal direction is

$\textbf{t}=\textbf{e}_{t}=\alpha\vec{\eta}+\vec{\beta},$ (46)

where the vector $\vec{\beta}$ is purely spatial, commonly called the vector
shift and with the composition

$t^{\mu}\Omega_{\mu}=\alpha\eta^{\mu}\Omega_{\mu}+\beta^{\mu}\Omega_{\mu}=\frac{\alpha}{\alpha}=1.$
(47)

The relation above implies the explicit construction of the Eulerian base

$\eta_{\mu}=\left(-\alpha,0,0,0\right)$ (48)

and

$\eta^{\mu}=\frac{1}{\alpha}\left(1,\beta^{i}\right).$ (49)

By Rezzolla rezz  we write the acoustic line element

$ds^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)dt^{2}+2\beta_{i}dx^{i}dt+\gamma_{ij}dx^{i}dx^{j},$
(50)

which allows us to identify the vector $\vec{\beta}$ with the velocity of a
fluid, calculating therefore the product to scale with the equations (48) and
( 49),

$\eta_{\mu}\eta_{\nu}\eta^{\mu\nu}=\alpha^{2}$ (51)
$\eta^{\mu}\eta^{\nu}\eta_{\mu\nu}=\frac{1-\vec{\beta}\cdot\vec{\beta}}{\alpha^{2}}.$
(52)

We realize then that $\alpha$ is associated with the spread of signals and
$\vec{\beta}$ to a fluid seeping. Finally we write a four-speed vector using
the Eulerian elements written above (42), (48)

$\gamma^{i}_{\mu}u^{\mu}=u^{i},$ (53)

where we have the spatial component of a four-speed and

$-\eta_{\mu}u^{\mu}=\alpha{u}^{t},$ (54)

the temporal component, wich can be written, in the case of $\beta_{i}=0$, as

$d\tau^{2}=\alpha^{2}dt^{2}.$ (55)

what allows us us to call the function $\alpha$ time-lapse function rezz .

##  III The acoustic stationary but non static geometry linked to a preferred
frame $S_{V}$

###  III.1 An acoustic geometry

Figure 1: The deviation of a photon in reference frames $S^{\prime}$ and
$S_{0}$ with non null velocities relative to a preferred frame $S_{V}$,
measured by three clocks.

Now we are going to analyze the Figure (1) from a kinematic point of view. It
was showed that N2013 , from a point  _A_ located on the axis $y^{\prime}$ of
the referential frame $S^{\prime}$, a photon is emitted, in principle, in the
direction $\overline{AO^{\prime}}$, as expected for those in the $S^{\prime}$
referential frame. However, to our surprise the photon deviates in the
direction $\overline{AO^{\prime}}$. This occurs due to the existence of an
unreachable minimum speed $V$, relative to the ultra-reference frame $S_{V}$.
The emitting electron of the photon has an uncertainty in its position in
$S^{\prime}$ due to this minimum velocity, since we cannot say that it is at
complete rest. This uncertainty generates a “delocalization” $\Delta
x^{\prime}=\overline{O^{\prime}C}$. Hence, instead of only the segment
$\overline{AO^{\prime}}$, a rectangle triangle $\overline{AO^{\prime}C}$ is
formed at the proper frame $S^{\prime}$ where it is not possible to find a set
of points at rest. For more details of this formalism see also Refs. N2007 ;
N2008 ; N2015 ; N2016 ; N2019 ; N2022 .

From the hydrodynamic perspective we also observe that the trajectory
(direction) of the excited state seen from $S^{\prime}$ is deviated, because
in fact there is no possibility of centering a rigid $S^{\prime}$ coordinate
system, whose origin $O^{\prime}$ is fixed precisely on the quasi-particle,
since it cannot be located at the origin. Therefore, there is a
$\Delta{x^{\prime}}=\overline{O^{\prime}C}>0$ “delocation” from the origin,
and this delocation depends on the speed of $S^{\prime}$ relative to the
referential frame $S_{V}$, i.e,
$\Delta{x^{\prime}}=\Delta{x^{\prime}}(v_{S^{\prime}}/S_{V})=\Delta{x^{\prime}(v_{S^{\prime}})}=\Delta{x^{\prime}_{v}}$.
There is no possibility to undo $\Delta{x^{\prime}_{v}}$. We can show that
$\Delta{x^{\prime}}\rightarrow\Delta{x^{\prime}_{min}}$ when $v\rightarrow{c}$
and $\Delta{P}$ increases since $\Delta{v}\rightarrow{\Delta{v_{max}}}=c$ and
that $\Delta{x_{1}}\rightarrow\Delta{x^{\prime}_{max}}$ maximum delocation,
when $v\rightarrow{V}$ ($\Delta{v}\rightarrow{(\Delta{v}_{min})=V)}$ in such a
way that a certain uncertainty relation
$\Delta{x^{\prime}_{v}}\Delta{v^{\prime}}$ is maintained for the particle.
Thus, it is clear that this space-time in the hydrodynamic perspective already
has the fundamental ingredients to propose a relation with a quantum
uncertainty. Within an objective framework of reality, now in the
Hydrodynamics perspective, essentially quantum.

In Figure (1), as we have $\Delta{x^{\prime}}_{v}>0$, we will see two
rectangular triangles of a cone, as in relativity. They are:

  1. 1.

$\blacktriangle BO^{\prime}A$, seen from the external $S_{0}$ referential
frame, which is already separated;

  2. 2.

$\blacktriangle BO^{\prime}C$, obtained in the particle’s own $S^{\prime}$
referential frame, ${O^{\prime}C}$ would be an uncertainty in location
(delocation) of it in its own referential $S^{\prime}$. As already said, we
have ${O^{\prime}C}=\Delta{x^{\prime}}(v)$, where

$v=v_{S^{\prime}/S_{V}}.$ (56)

The equation (56) represents the speed of a usual reference, measure of the
preferred frame $S_{V}$.To be built

Now, let’s extract the following relations from two rectangular triangles:

  1. 1.

$\blacktriangle
BO^{\prime}A:({AB})^{2}=({O^{\prime}B})^{2}+({O^{\prime}A})^{2}\Rightarrow
c^{2}\Delta{t^{2}_{0}}=({O^{\prime}B})^{2}+v^{2}\Delta{t^{2}_{0}}\Rightarrow$

$({O^{\prime}B})^{2}=c^{2}\Delta{t^{2}_{0}}-v^{2}\Delta{t^{2}_{0}};$ (57)

  2. 2.

$\blacktriangle
BO^{\prime}C:({CB})^{2}=({O^{\prime}B})^{2}+({O^{\prime}C})^{2}\Rightarrow
c^{2}\Delta{t^{\prime}}^{2}=({O^{\prime}B})^{2}+[\Delta{x^{\prime}(v)}]^{2}\Rightarrow$

${O^{\prime}B}^{2}=(c\Delta{t^{\prime}})^{2}-[\Delta{x^{\prime}_{v}}]^{2}.$
(58)

Comparing the relations (57) and (58), follows

$c^{2}\Delta{t}^{2}-v^{2}\Delta{t^{2}}=c^{2}\Delta{t^{\prime
2}}-\Delta{x^{\prime 2}_{v}}.$ (59)

where $\Delta\tau=\Delta{t^{\prime}},t_{0}=t$, we have

$c^{2}\Delta{t}^{2}-v^{2}\Delta{t^{2}}=c^{2}\Delta{\tau^{2}}-\Delta{x^{\prime
2}_{v}}.$ (60)

In Lorentz Space-time, we are (proper) at the origin $O^{\prime}$ of
$S^{\prime}$ so that we have $\Delta{x^{\prime}}=0$, lose meaning, so we have
$S$ and $S^{\prime}$. Therefore, it comes that

$c^{2}\Delta{t}^{2}-v^{2}\Delta{t}^{2}=c^{2}\Delta{\tau^{2}}=\Delta{S^{\prime
2}},$ (61)

where $\Delta{\tau}$ is the proper time and $\Delta{t}$ is the improper one.
In this sense, we define this fundamental parameter, this speed $v$, expressed
in the equation (56).

$v:=\sqrt{c^{2}-\frac{c^{2}(\Delta\tau)^{2}-(\Delta{x}^{\prime})^{2}}{(\Delta{t})^{2}}}.$
(62)

We take again the (61) and notice that

$\left(c^{2}-v^{2}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}-c^{2}+\left(\frac{\Delta{x}^{\prime}}{\Delta\tau}\right)^{2}=0,$
(63)

The equation (63) is very similar to an acoustic geodesic an ; enos

$\left(c^{2}_{s}-v^{2}\right)\left(\frac{dt}{ds}\right)^{2}-2v_{i}\left(\frac{dx^{i}}{ds}\right)\left(\frac{dt}{ds}\right)+\left(\frac{dx}{ds}\right)^{2}=0.$
(64)

When we compare (63) with (64), we get the following coincidence

$2v\frac{\Delta{x}^{\prime}}{\Delta\tau}\frac{\Delta{t}}{\Delta\tau}\equiv{c^{2}}.$
(65)

Returning to Equation (63)

$\left(c^{2}-v^{2}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}-2v\frac{\Delta{x}^{\prime}}{\Delta\tau}\frac{\Delta{t}}{\Delta\tau}+\left(\frac{\Delta{x}^{\prime}}{\Delta\tau}\right)^{2}=0.$
(66)

The Equations (66) and (64) have identical formats, then we deduce for
relation (65)

$c^{2}=\left(\frac{c^{2}}{v}\frac{\Delta\tau}{\Delta{t}}-v\right)^{2},$ (67)

wich can be considered a reissue of the equation (62), but one that allows a
better interpretation. Considering the References mg ; enos, we interpret the
equations (66) and (67) as the eikonal of a particle, the speed $v$ is a flux
of fluid and $\frac{\Delta{x}}{\Delta\tau}$ as the vector tangent of the
geodesic. Also, the flux and the geometry stay connected.

The next step in procedure is consider a four-vector

$\frac{\Delta{x^{\prime}}^{\mu}}{\Delta\tau}=\left(-c\frac{\Delta{t}}{\Delta\tau},\frac{c^{2}}{v}\frac{\Delta\tau}{\Delta{t}}\right).$
(68)

Thus we have an apparent violation of Lorentz symmetry, which actually is a
deformation of Lorentz invariance, but is recovered by doing $\Delta{x}=0$.
The equations (67) and (68) indicate a prohibition, the speed $v$ can not be
null. Otherwise we would have a singularity and an inconsistency, the speed
$\frac{\Delta{x}}{\Delta\tau}$ may be greater than the light for certain
values of $v$, forced the speed to assume a minimum value under certain
conditions, this mooring, between the two speeds
$\frac{\Delta{x}}{\Delta\tau}$ and $v$ is a real novelty.

To establish a causal relation between the two speeds, it is necessary to
resort to the concept of R.I.P. (32), where the equation (35) represents the
relation between event-horizon and R.I.P.

Replacing (35) in (65),

$\frac{\Delta{x^{\prime}}}{\Delta\tau}=\frac{c}{2}\frac{\Delta\tau}{\Delta{t}}\left(\eta(x^{\prime},\rho=0)-n_{0}\right).$
(69)

Therefore, if the R.I.P. at the origin of the cylindrical coordinate is equal
to the refraction index of the medium, we have
$\frac{\Delta{x^{\prime}}}{\Delta\tau}=0$ and we return to an acoustic
geometry of Lorentz. This result allows us to interpret the geometric
structure we are dealing with as a kind of granular geometric structure,
something similar to a vortex because we have cylindrical geometry. So we have
more than particles but quasi-particles. We can understand these little
fluctuations as phonons. A causal link between the two speeds is established,
because we associate discrepancy with a quantity that involves a horizon of
events.

We write the new eikonal

$h^{\mu\nu}\frac{{\Delta{x^{\prime}}_{\mu}}}{\Delta\tau}\frac{{\Delta{x^{\prime}}_{\nu}}}{\Delta\tau}=0,$
(70)

We have an acoustic geometry in particular, associated with a tachionic causal
structure.

###  III.2 Two-fluid Acoustic Geometry

Considering the equations (64) and (63). We will now subtract (64) from (63):

$\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}+1-\frac{1}{c^{2}}\left(\frac{\Delta{x^{\prime}}}{\Delta\tau}\right)^{2}-\frac{c^{2}_{s}}{c^{2}}=\frac{1}{c^{2}}||\frac{d\vec{x}}{dt}-\vec{u}||^{2}$
(71)

Here we have that the speed $\vec{v}$ is the velocity of the hypothetical
superfluid, measured and compared to the preferred-frame $S_{V}$. The speed
$\vec{u}$ is measured in laboratory $\frac{d\vec{x}}{dt}$ is the vector
parallel to the flow line, $c_{s}$ is the speed of sound in an acoustic
geometry that respects Lorentz symmetry. So the left side contains the sound
modified by the presence of the quantities measured by the preferred-frame,
the right side the quantities measured in a laboratory. The left side like
this, is a sound propagating in this exotic medium. If on the right side we
have $\frac{d\vec{x}}{dt}=\vec{u}$, results

$\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}+1-\frac{1}{c^{2}}\left(\frac{\Delta{x^{\prime}}}{\Delta\tau}\right)^{2}=\frac{c^{2}_{s}}{c^{2}}$
(72)

the presence of the if the factor , induces sound propagation in a naturally
flowing medium , in this case, $c_{s}<c$, if the factor
$\frac{\Delta{x^{\prime}}}{\Delta\tau}=0\rightarrow{c_{s}}=c$. So the waves
are electromagnetic. Replacing (65) in (72), we have

$\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}+1-\frac{1}{c}\left(\frac{\Delta\tau}{\Delta{t}}\right)^{2}\left(n(x^{\prime},\rho=0)-n_{0}\right)^{2}=\frac{c^{2}_{s}}{c^{2}}.$
(73)

If $c_{s}=c$, we have $n(x^{\prime},\rho=0)=n_{0}$, which implies that the
medium does not change, we have a continuous medium, with the same refractive
index value $n_{0}$. So here we have laminar flow. Therefore, we have the
Minkowski space reestablished. Replacing (35) in

$\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}+1-\frac{c^{2}}{v^{2}}\left(\frac{\Delta\tau}{\Delta{t}}\right)^{2}=\frac{c^{2}_{s}}{c^{2}}.$
(74)

The equation (74) tells us that values of $v=0$, can be problematic, because
we would have a singularity and at the same time a break in causality. We then
understand that the terms associated with $\frac{\Delta{x}}{\Delta\tau}$,
needs to have material characteristics, and specific material characteristics,
which can be related to an index of refraction. Another consequence to be
emphasized is that low velocity values imply negative values for the speed of
sound, which implies the Jeans’ collapse condition jeans

###  III.3 The Lorentz symmetry violation in an acoustic geometry

The tachikonic causal structure usualy correspond to particles more rapid as
light. But, we intent to generate a causal structure similar to acoustic
tachkion, particles faster than sound. In (63) we have that

$\Delta{x^{\prime}}(v)=[w(v)]\Delta\tau.$ (75)

The $w(v)$ function introduced in Ref. N2013  has speed dimension. We
interpret $\Delta{x^{\prime}}_{v}$ as an internal degree of freedom, then it
is better understood as an quasi-particle. The function $w(v)$ need some
proprieties:

  1. 1.

dimension of speed;

  2. 2.

$w(v)<c$;

  3. 3.

$w(v)$ is linked to equation (69).

In physics, the search for fundamental principles is essential, as
conservation laws and symmetries. The reciprocity valentina ; mg ; sismo  is a
fundamental symmetry for randomic systems and seismology. Surveys in
seismology define a quantity called _slowness_ $\left(\frac{1}{v}\right)$, see
Ref. sismo , associated to granularity of the medium of propagation p-wave. In
quantum turbulence superverse  exists a similar conception, the reconnection
of two vortexes is associated with speed of flux line
$\delta(v)\approx\frac{1}{v}$. But, $w(v)$ have speed dimension, so a better
propouse is $w(v)=\frac{a}{v}$. In this case $a$ have dimension of quadratic
speed, we write $a=v^{2}_{0}$. Furthermore, the $w(v)$ is a opportunity to
introduce reciprocal symmetry $w(v)=\frac{v^{2}_{0}}{v}$ to obey the causal
structure $v^{2}_{0}=bc$. Where $b$ is a speed, we chose $b=V$ according to
Equation (63), so we have

$w(v)=\frac{cV}{v}.$ (76)

We return 76 in 71

$\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}+1-\frac{V^{2}}{v^{2}}-\frac{c^{2}_{s}}{c^{2}}=\frac{1}{c^{2}}||\frac{d\vec{x}}{dt}-\vec{u}||^{2}$
(77)

Considering once again $\frac{d\vec{x}}{dt}=\vec{u}$.We must now interpret
each of the terms in the acoustic geometry presented, in the equation (77).

  1. 1.

The term
$\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{\Delta{t}}{\Delta\tau}\right)^{2}+1$corresponds
to a usual Lorentz factor, including the speed of the wave represented there
is the speed of light $c$.

  2. 2.

The term
$\frac{c^{2}_{s}}{c^{2}}=\frac{1}{c^{2}}||\frac{d\vec{x}}{dt}-\vec{u}||^{2}$
corresponds to an acoustic Lorentz symmetry, in the sense proposed by
Unhurunhur  and Visservis . In this geometry we have acoustic tackyons, but,
these acoustic tackyons still correspond to time-like shifts for the usual
Lorentz geometry. So we have a double cone causal structure. An acoustic cone
and a light cone

  3. 3.

The third term $1-\frac{V^{2}}{v^{2}}$ which was introduced in equation (76).
This term is associated with the speed parameter (62). This term is associated
with the parameter velocity (62), so (62) is the velocity seen from the
referential $S_{V}$, which corresponds to a referential attached to the first
excited state of a superfluid, at the moment this excited state appears, in
this sense the velocity $v$ is the flow velocity of the hypothetical
superfluid A superfluid is characterized by maintaining a flow of energy in
the direction of the capillary, but when we deal with the universe there is no
capillary, so this velocity corresponds to a rate of departure, between the
hypothetical states that would arise, with increasing $v$.We can thus consider
the recession velocity of Galaxies deduced by Hubble $v=HX$, where $X$ would
be the Galaxies’ position Hubble . Therefore, when we apply this set of ideas,
we imagine that galaxies are analogous to excited states and that the vacuum
is a superfluid

  4. 4.

A totally new and different causal structure emerges here, just as particles
faster than light describe space-like vectors. Here we have a situation, where
two events need to be connected by a vector, which corresponds to a velocity
$v>V$, so we have a new range for the validity of the causal structure. For
velocities less than $V$, the concept of an event is not defined by the simple
laminar flow of this cosmological fluid. So we write the validity range of the
causal structure

$V<v<c.$ (78)

It is necessary to comment on the similarity of this causal structure, with
the idea of RIP, described in the equation (32). By pointing out that space-
time can go through phase transitions like a fluid, we are not bringing back
the idea of a luminiferous aether, but rather conceiving of a model of space-
time that would allow us to have different material properties throughout the
cosmological ages. Indeed, such a hypothesis is supported by the studies of
running constantrvm

  5. 5.

The equation allows us to calculate a new time-lapse function, considering the
speed of sound of the acoustic causal structure $c_{s}$ and the granular
structure associated with $V$.

$\frac{\Delta{t}}{\Delta\tau}=\frac{1}{c}\sqrt{\frac{c^{2}\left(1-\frac{V^{2}}{v^{2}}\right)-c^{2}_{s}}{1-\frac{v^{2}}{c^{2}}}}$
(79)

This result 79 indicates the interaction of three geometries, one the usual
geometry of Minkowski space, but compared to the AETHER flow with velocity
$v$, a Lorentzian acoustic geometry in the terms the works unhur  and vis
represented by $c_{s}$ and the granular term represented by the critical
velocity, which is also compared to the AETHER flow with velocity $v$. In the
limit where the aether flows with $v=V$, we have a situation where
$c^{2}\left(\Delta{t}\right)\equiv-\frac{c^{2}_{s}}{1-\xi^{2}}\left(\Delta\tau\right)^{2}$.
It must be said, that the result generalized the works of Unhur unhur  and
Visser vis , constituting a more complex acoustic geometry than the acoustic
geometry proposed by the unhur ,vis . So here we have the Lorentz Symmetry
Violation.

The introduction of the term

$\theta(v):=\sqrt{1-\frac{V^{2}}{v^{2}}}$ (80)

modifies the geometry, introducing granularity, being granularity, which is
understood as the first excited state, so a very slow particle, a particle
that resisted the drag of the cosmic fluid and became very slow, would be
dissolved .

When we have $c_{s}=0$ the equation 79 we get the factor

$\psi(v)=\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}},$ (81)

$\psi(v)$ is a deformed Lorentz factor, taking the fluid flow as the velocity
parameter $v$. So when we eliminate the critical velocity, we get the usual
Lorentz factor, see the N2007 , N2008 ,N2013 ,N2015 ,N2016 .

$\psi(v)=\frac{\Delta{t}}{\Delta\tau}.$ (82)

We get:

  1. 1.

$v\rightarrow{c}\Rightarrow\Delta{t_{0}}>>\Delta\tau$(time dilation: the time
march in $S^{\prime}$ is very slow compared to the time march in $S_{0}$);

  2. 2.

$v=v_{0}\Rightarrow\Delta{t_{0}}=\Delta\tau$, here

$v_{0}=\sqrt{Vc}$ (83)

  3. 3.

Unprecedented $v\rightarrow{V}\Rightarrow\Delta\tau>>\Delta{t}_{0}$. Time
contraction: the time-march in $S^{\prime}$ is much faster compared to the
march in $S_{0}$.

Figure 2:  We have $c=\sqrt{v^{2}_{t}+v^{2}}$ – see (77) and (85) – which
represents the space-temporal speed of any particle (hypothenuse of the
triangle$=c$. The novelty here is that such a structure of space-time
implements the preferred frame $S_{V}$ from Ref. N2013

.

###  III.4 A clock hypothesis associated with the introduction of the
privileged reference $S_{V}$

Once we deduce (81), we can introduce a deformation into Lorentz (82) and
thus, we see the space-time interval considering the existence of the
privileged referential of this deformation, we introduce (80) and we rewrite
the space-time interval:

$c^{2}\Delta\tau^{2}=\frac{1}{1-\frac{V^{2}}{v^{2}}}\left[c^{2}\Delta{t}-v^{2}\Delta{t}^{2}\right].$
(84)

In Ref. N2013 ; N2019  we see the space-time interval considering the
existence of the preferred frame $S_{V}$.

Manipulating (84) and following N2013 , we get

$c^{2}\left(1-\frac{V^{2}}{v^{2}}\right)\left(\frac{d\tau}{dt}\right)^{2}+v^{2}=c^{2}.$
(85)

We assume then that the Equation (85) represents the march of time, where the
parameter $v$ is the measured speed relative to the preferred frame $S_{V}$.

Figure 3:  The four cones proposed by Ref. N2013  ilustrate an unification of
Landau superfluid criticial speed (167), the uncertainty-priniple and an
acustic causal structure. We see in these figures the causal structures
associated with momentum-position and energy-time uncertainty.

We’ve rearranged the Equation (84)

$c^{2}\Delta{t}^{2}-v^{2}\Delta{t}^{2}=c^{2}\Delta\tau^{2}-\frac{c^{2}V^{2}}{v^{2}}\Delta\tau^{2}$
(86)

where we can rewrite the Equation (83). We can then reverse (86), writing as

$c^{2}\left(1-\frac{v^{2}}{c^{2}}\right)\left(\frac{dt}{d\tau}\right)^{2}+\frac{v^{4}_{0}}{v^{2}}=c^{2}$
(87)

Here we define a speed which we call the _march of space_ , as opposed to the
march of time $v_{t}$ in Equation (87)

$v_{trec}=c\sqrt{1-\frac{v^{2}}{c^{2}}}\frac{dt}{d\tau}.$ (88)

Although the name marches from space may seem unusual, the idea is quite
understandable when we think that, in our case, space corresponds to a fluid.
We understand that the Equation (88) can be interpreted as the propagation
speed of a fluid. We can imagine that (87) is the module of the tangent vector
the flow line, parameterized by $\tau$. The Equation (88), would be a
complementary fluid to that, analogous to what is done in superfluids. We
write then

$v^{2}_{trec}+\frac{v^{4}_{0}}{v^{2}}=c^{2}.$ (89)

Therefore, associating the Equations (85) and (87), we can write a new
equation

$v^{2}_{trec}+v^{2}_{rec}=c^{2}.$ (90)

Now we get a line element, which at the same time is related to $d\tau$ and to
$dt$,

$ds_{5}=cd\tau\sqrt{1-\frac{V^{2}}{v^{2}}}=cdt\sqrt{1-\frac{v^{2}}{c^{2}}}.$
(91)

The Equation (91) is a new hypothesis of the deformed clock. Starting from
(91) we can write

$\frac{ds_{5}}{d\tau}=c\sqrt{1-\frac{V^{2}}{v^{2}}},$ (92)

and

$\frac{ds_{5}}{dt}=c\sqrt{1-\frac{v^{2}}{c^{2}}}.$ (93)

The interpretation suggested in our previous work nova  for (92) and ( 93)
discusses the possibility of the two line elements corresponding to a Weyl
geometry. And adressing the clock-hypothesis of Weyl for a new perspective
nova . This kinematics, proposed orginally by Ref. N2007  is totally new,
because it binds the causal sectors in order to be reciprocal of each other.
This reciprocity establishes a structure very similar to wave-particulate
duality (see Figure 3).

If we do $\frac{ds_{5}}{dt}\frac{d\tau}{ds_{5}}$, we get

$\psi=\frac{dt}{d\tau}=\frac{v_{trec}}{v_{t}}\propto{E}.$ (94)

In this case $v=\frac{dx}{dt}$ in Equation (85) and (76) corresponding to an
internal structure of the quase-particle. Such internal structure, like the
external structure, can be understood as a fluid, also endowed with causal
structure as well as the outer fluid. In practice, both external and internal
causal structures are associated by (92) and (93). We call (76) reciprocal
speed.Alternatively, if we choose to do
$\frac{ds_{5}}{d\tau}\frac{dt}{ds_{5}}$, we get:

$\psi^{-1}=\frac{d\tau}{dt}=\frac{v_{t}}{v_{trec}}\propto{time}.$ (95)

In this case $v_{t}$ is given for (85), $v_{trec}$ for (88). The Equations
(94) and (95) indicate the presence of a reciprocal symmetry between the
dispersion relation and the principle of energy-time uncertainty. We will
study further this issue in the next section.

###  III.5  Principle of Uncertainty in the presence of an hypothetical
superfluid

The momentum of a massive quasi-particle with respect to $S_{V}$, with respect
to the world line of the particle parameterized by $\tau$ is given by:

$P:=m_{0}c\left[\frac{dt}{d\tau}+\Sigma_{i=1}^{3}\frac{dx^{\prime}_{i}}{d\tau}\right].$
(96)

Where $P$ is the current moment relative to the preferred frame $S_{V}$. The
Equation (96), when compared to the Equation (82), suggests a dispersion
relation

$P^{2}=\frac{m^{2}_{0}v^{2}-m^{2}_{0}V^{2}}{1-\frac{v^{2}}{c^{2}}},$ (97)

very similar to a roton superverse . As (94), already indicates, the temporal
composition of the moment is related to reciprocal symmetry with energy mg ,
we have a strong indication that we are dealing with excited states of a
superfluid. That indication would give a new meaning to the privileged
reference frame $S_{V}$. Therefore, the relation (97) is associated with a
differential momentum between the first excited state $P_{V}=m_{0}V$ of a
superfluid and the other states $P_{+}=m_{0}v$. This perspective consists of a
different interpretation of Nassif’s works N2008 ; N2013 . We are considering
that the reference $S_{V}$ is a reference associated with the speed of the
first excited state, being dredged by the superfluid. When this reference
arises, the fluid itself shifts with minimal velocity, the next states will
arise, when the fluid exceeds other speed limits. Thus the difference
$\Delta{P}:=\sqrt{P^{2}_{+}-P^{2}_{V}}$ corresponds to the variation of the
velocity of the fluid itself in relation to the critical velocity of the
fluid, which generates the first excited state. In this perspective, we write:

${P}^{2}=\frac{\Delta{P}^{2}}{1-\frac{v^{2}}{c^{2}}}.$ (98)

As example, if $v{\rightarrow}V$ implies $(P\rightarrow 0)$ then we have
${\Delta}x^{\prime}_{v}\rightarrow\infty$. The new hydrodynamic interpretation
makes it possible to understand the delocation of the quasi-particle. When the
quasi-particle is slow, it simply dismounts in the superfluid. This leads the
understanding of the preferred reference frame $S_{V}$ more palpable, because
actually the reference $S_{V}$ becomes a conceptual reference frame. It is
physical in the meaning of corresponding to a reference with speed where the
first excited state arises, then to a virtual observer, which propagates with
the emergency speed of the first excited state even if in the system this
state is not present.

Now we can calculate the following quantity

${\Delta}x^{\prime}_{v}P=\frac{v^{2}_{0}}{v}{\Delta}t\Psi^{-1}m_{0}v=(m_{0}v_{0})(v_{0}{\Delta}t).$
(99)

We can now consider the relation (76), (82) and (96). In addotion to
$v_{0}=\sqrt{cV}$. Naturally, we have ${\Delta}x^{\prime}_{v}P=0$ in special
relativity. In Minkowsky space-time $V=0$ so $v_{0}=0$ too. As an analogy, a
non-inercial reference frame has the same state of motion given by any
acceleration relative to a galinean reference frame, thus here we define the
flux-line of the hypothetic superfluid (78) relative to a preferred frame
$S_{V}$.

In other words, we say that every flux-line is a frame system, these frames
systems agree with each other as to the given speed of hypothetical superfluid
with respect to preferred frame $S_{V}$. In this meaning, the critical speed
of the superfluid is an invariant. Since the speed $V$ is inaccessible for any
outher flux-line in a minkowski space-time we have to ${\Delta}v$ is the width
of a quasi-particle moving with speed $v$ measurement of a reference
associated with a fluid $S_{V}$.

Let’s consider a particle that has wavelength $\lambda$ relative to a
preferred frame $S_{V}$ associated with a hypothetical fluid. It would be
natural to think that the delocation $({\Delta}x^{\prime}_{v})_{0}$, would be
given by

$\lambda\sim({\Delta}x^{\prime}_{v})_{0}=\frac{v^{2}}{v_{0}}({\Delta}\tau)_{0}=\frac{v^{2}_{0}}{v}({\Delta}t)_{0}\Psi^{-1},$
(100)

where $({\Delta}t)$ will be calculated. We have (76) using the wavelength
definition,

$\lambda=\frac{h}{P}=\frac{h}{m_{0}v}\sqrt{\frac{1-\frac{v^{2}}{c^{2}}}{1-\frac{V^{2}}{v^{2}}}}$
(101)

and with the Equation (96) we write:

$\lambda=\frac{h}{m_{0}v}\Psi^{-1}\sim\frac{v^{2}_{0}}{v}({\Delta}t)_{0}\Psi^{-1}.$
(102)

From that we get

$m_{0}v^{2}_{0}({\Delta}t)_{0}{\sim}h.$ (103)

Finally we compare the equation above (103) with the Equation (99) and we have

$({\Delta}x^{\prime}_{v})_{0}P=m_{0}v^{2}_{0}({\Delta}t)_{0}{\sim}h.$ (104)

Alternatively

$({\Delta}x^{\prime}_{v})_{0}{\Delta}P=m_{0}v_{0}\lambda_{0}{\sim}h,$ (105)

where we have (98) for any flux-line. The Equation (103) is the uncertainty
relation of momentum emerging from a space-time with properties of a
hypothetically superfluid.

Now in order to get the relation of uncertainty to energy

$m_{0}cV({\Delta}t)_{0}{\sim}h.$ (106)

We can write Equation (103) as

$m_{0}c^{2}\Psi({\Delta}t)\Psi^{-1}{\sim}h.$ (107)

From Equation $m_{0}c^{2}\Psi({\Delta}t)\Psi^{-1}{\sim}h$ into Equation (95)

$E=m_{0}c^{2}\Psi.$ (108)

Thus we write the relation of energy-time uncertainty

$E({\Delta}\tau)_{c}=m_{0}c^{2}({\Delta}t){\sim}h,$ (109)

or alternatively

$E({\Delta}\tau)_{c}=m_{0}c^{2}\lambda{\sim}h,$ (110)

where $\lambda=c({\Delta}t)$.

We have considered (98) for any flux line. We can consider the energy $E$
relative to a preferred frame $S_{V}$, we expect uncertainty as far as
${\Delta}E$ for any flux line $E\equiv{\Delta}E$, in same meaning of (98).

According to the Equation (110), which is the energy-time uncertainty
principle, we rewrite the Equation (95) as

$\psi^{-1}(v)=\frac{\Delta\tau}{\Delta{t}},$ (111)

therefore a relation of reciprocity between the clock hypothesis and the
principle of energy-time uncertainty is established. The relation of
uncertainty becomes an interpretation associated with causality, this
causality, which is associated with the emergence of a first excited state,
breaks with the laminar flow of a superfluid at the limit where we are close
to critical speed. Thus it occurs that the first excited state breaks the
laminar flow, but there is still a drag effect associated with the original
laminar flow. We can say on limit $v{\rightarrow}V$, it implies
${\Delta}E\rightarrow 0$, where

$\psi^{-1}_{v=V}(v)\rightarrow\infty.$ (112)

The relathion present on Equation (110) is an uncertainty time-energy emergent
of space-time. In this case the space-time correspond to a hypothetical
superfluid vortex .

###  III.6 Dispersion Relation in presence of the preferred frame $S_{V}$

Revisiting the Equations (82) and (95), we notice an interesting symmetry

$\psi^{-1}(v)=\psi(w),$ (113)

where $w(v)$ is given for Equation (76), it is an indication that $w(v)$ plays
the role of the velocity of counterflow vortex . Considering this perspective
the Equation (87) also gains a hydrodynamic interpretation, as a typical
superfluid structure, where the superfluid is modeled as two fluids. However,
here the two fluids are complementary, in terms of their causal structures.

The appearance of this new structure and its effects on the causal structure
have implications, one of these implications concerns the Lorentz symmetry.
Revisiting (108) we have a relation between energy and causal strucuture, this
relation was adressed initially by Equations (95) and (94), knowing that
$\Delta{E}\equiv{E}|_{S_{V}}$. Thus we can write

$\frac{\Delta{t}}{\Delta\tau}=\frac{E|_{S_{V}}-E_{0}}{m_{0}c^{2}}=\frac{E|_{S_{V}}-E_{0}}{E_{0}}.$
(114)

The meaning of the Equation (114) is very important because it is a virtually
identical result obtained by Zloshchatiev kgz2010 . According to Zloschatiev
work kgz2010  we have a Lorentz Invariant Violation (LIV), expressed by the
line element being multiplied by a function of the energy in Equations (92)
and (93). The difference from the work of the Zloschatiev is he considers
$E_{0}$ as the scale of energy of quantum-gravitational interactions and we
consider $E_{0}$ as the resting energy. It was demonstrated that the
introduction of a minimal speed breaks the symmetry of Lorentz r1 ; r3 ; r2 ;
gbec . However we have not yet established the size of the Lorentz symmetry
shift, certainly the energy function (108), is a conformal function function
as already was showed in r1 ; r3 .

Now, we consider the Equations (97) and (108) and we write

$P^{2}-\frac{E^{2}}{c^{2}}=\frac{p^{2}-m^{2}_{0}V^{2}}{1-\frac{v^{2}}{c^{2}}}-m^{2}_{0}c^{2}\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}.$
(115)

A transformation in relativistic energy $E$ and relativsitic momentum $P$ can
be made $E\rightarrow\hbar\omega$ and $P\rightarrow\hbar\omega$. We then
rewrite (115)

$P^{2}-\frac{E^{2}}{c^{2}}=\hbar^{2}\kappa^{2}-\hbar^{2}\omega^{2}\left(1+\frac{c^{2}V^{2}}{v^{2}}\right)-\frac{m^{2}_{0}V^{2}}{1-\frac{v^{2}}{c^{2}}},$
(116)

where (32), is an refractive index perturbation rip .

We call the Equation (69) and we relate to (32)

$\frac{cV}{v}=\frac{c}{2}\frac{\Delta\tau}{\Delta{t}}\left(\eta(x^{\prime},\rho=0)-n_{0}\right).$
(117)

We have the establishment of a lump, a sector where small fluctuations
differentiate the fluid from inside the pit, in relation to the outer fluid.

The term $-\frac{m^{2}_{0}V^{2}}{1-\frac{v^{2}}{c^{2}}}$ is a part of the
maxon (97), the important to be highleted is that
$\frac{m^{2}_{0}V^{2}}{1-\frac{v^{2}}{c^{2}}}$ can be infinite. Baceti et-al
valentina  survey a dispersion relation. Baceti _et-al_ valentina  build a
kinematic relation which they call an _isotropic Aether-Frame_. We have

$E^{2}-||\vec{p}||^{2}c^{2}=E^{2}_{0}(\tilde{F}),$ (118)

here $\tilde{F}$ is an inertial frame. For Bacetti _et-al_ we are working an
“absolute moving”, where right-side of (118) is given for $E^{2}_{0}(v)$ and
$v$ is measured from frame of aether. The energy $E(v)$ is an usual function
$\eta(V_{aether},V)$. Constituting a minimalist violation of Lorentz symmetry,
actually a “deformation” of Lorentz symmetry.

We compare (118) and (116) and visualize an equivalence

$E^{2}_{0}\equiv\frac{m^{2}_{0}V^{2}}{1-\frac{v^{2}}{c^{2}}}.$ (119)

Therefore, we understand the sector associated with (119) as a kind of aether,
a kind of energy for a privileged reference frame like what we’ve been calling
reference frame $S_{V}$ in the above sections.

##  IV A two-fluid model in presence of preferred frame $S_{V}$

###  IV.1 The Lorentz Invariance Violations and deformed kinematics
transformations

After the study of the previous sections and the deduction of the deformed
Lorentz factor (82), we are allowed to write the kinematic transformations,
similar to Lorentz transformations, which relate the references as follows
$S^{\prime}\rightarrow{S}_{V}$:

$dt^{\prime}=\psi(v)\left[dt+\left(V-v\right)\frac{dx}{c^{2}}\right],$ (120)

and

$dx^{\prime}=\Psi(v)\left[dx+\left(V-v\right)dt\right].$ (121)

Being $S^{\prime}$ any reference and $S_{V}$ a privileged reference proposed
by Nassif in a series of works that begin in N2007  being the latest N2022 ,
all of them already cited in the above sections. So we also have inverse
transformations $S_{V}\rightarrow{S}^{\prime}$

$dx=\Psi(v)\left[dt^{\prime}+\left(v-V\right)dt\right],$ (122)
$dt=\Psi(v)\left[dt^{\prime}+\left(v-V\right)\frac{dx^{\prime}}{c^{2}}\right].$
(123)

The transformations (120), (121), (122) and (123) are do not conceive the
existence of a perpendicular direction. In order to proceed our study of
superfluids, we need to write broader transformations that conceive a larger
number of dimensions. The transformations deduced in N2016  satisfy our need.

Let’s therefore define the following perpendicular transformations, the one
that relates the references $S^{\prime}\rightarrow{S_{V}}$

$dx_{\perp}=\Psi^{-1}(v)dx^{\prime},$ (124)

and the reverse transformation $S_{V}\rightarrow{S^{\prime}}$

$dx^{\prime}_{\perp}=\Psi(v)Vdt.$ (125)

Once the equations have been written (120), (121), (124) and (125), we write
the matrices associated with transformations

$\left[\begin{matrix}x^{\prime}_{\perp}\\\ x^{\prime}_{\parallel}\\\
x^{\prime}_{0}\\\ \end{matrix}\right]=M_{3\times
3}(S_{V}\rightarrow{S^{\prime}})\left[\begin{matrix}x_{\perp}\\\
x_{\parallel}\\\ x_{0}\\\ \end{matrix}\right]^{S_{V}}.$ (126)

We conceive that the relation (126) can be undone, carrying an array that
would exist as

$\left[\begin{matrix}x_{\perp}\\\ x_{\parallel}\\\ x_{0}\\\
\end{matrix}\right]^{S_{V}}=M_{3\times
3}(S_{V}\rightarrow{S^{\prime}})\left[\begin{matrix}x^{\prime}_{\perp}\\\
x^{\prime}_{\parallel}\\\ x^{\prime}_{0}\\\ \end{matrix}\right].$ (127)

The matrix of transformations written by Nassif N2016

$M_{2\times
2}(S_{V}\rightarrow{S}^{\prime})=\Psi(v)\left[\begin{matrix}1&i\frac{v}{c}\left(1-\frac{V}{v}\right)\\\
-i\frac{v}{c}\left(1-\frac{V}{v}\right)&1\\\ \end{matrix}\right],$ (128)

provides an idea of what the matrix might be $M_{3\times
3}(S_{V}\rightarrow{S^{\prime}})$, as the same algebraic and kinematic
properties need to be respected. Our goal here is to include an element
perpendicular to translation, then a non-translational term. In this sense, we
have no terms outside the diagonal, apart from those present in (128), because
there is no need to represent translations like the terms outside the diagonal
$M_{12}=-M_{21}=i\frac{v}{c}\left(1-\frac{V}{v}\right)$. Thus we can build
$M_{1\times 3}=M_{3\times 1}=M_{1\times 2}=M_{2\times 1}=0$ and we write the
matrix

$M_{3\times
3}(S_{V}\rightarrow{S}^{\prime})=\Psi(v)\left[\begin{matrix}1&0&0\\\
0&1&i\frac{v}{c}\left(1-\frac{V}{v}\right)\\\
0&-i\frac{v}{c}\left(1-\frac{V}{v}\right)&1\end{matrix}\right].$ (129)

It is trivial to find the inverse of $M_{3\times
3}(S_{V}\rightarrow{S}^{\prime})$, we write

$M_{3\times
3}(S^{\prime}\rightarrow{S}_{V})=\Psi^{-1}(v)\left[\begin{matrix}1&0&0\\\
0&\frac{1}{1-\frac{v^{2}}{c^{2}}\left(1-\frac{V}{v}\right)^{2}}&i\frac{\frac{v}{c}\left(1-\frac{V}{v}\right)}{1-\frac{v^{2}}{c^{2}}\left(1-\frac{V}{v}\right)^{2}}\\\
0&-i\frac{\frac{v}{c}\left(1-\frac{V}{v}\right)}{1-\frac{v^{2}}{c^{2}}\left(1-\frac{V}{v}\right)^{2}}&\frac{1}{1-\frac{v^{2}}{c^{2}}\left(1-\frac{V}{v}\right)^{2}}\end{matrix}\right].$
(130)

The composition of relations (126) and (127) implies that

$M_{3\times 3}(S_{V}\rightarrow{S}^{\prime})M_{3\times
3}(S^{\prime}\rightarrow{S}_{V})=\textbf{I}_{3\times 3}$ (131)

where $\textbf{I}_{3\times 3}$ is the identity matrix. This implies that the
crack of matrices

$\left[\textbf{I}_{3\times 3},M_{3\times
3}(S_{V}\rightarrow{S}^{\prime}),M_{3\times
3}(S^{\prime}\rightarrow{S}_{V})\right],$ (132)

make a group, where $\textbf{I}_{3\times 3}$ is the neutral element and the
two matrices $M_{3\times 3}(S_{V}\rightarrow{S}^{\prime})$ and $M_{3\times
3}(S^{\prime}\rightarrow{S}_{V})$ are inverse to each other. Given the
simplicity of this group, it is trivial to note that it is Abelian. However,
it is necessary to comment that the frame $S_{V}$ is unique, this implies that
$X^{\prime}=M_{3\times 3}(S_{V}\rightarrow{S^{\prime}})X_{V}$ is an operation
defined in the group, but $M_{3\times
3}(S_{V}\rightarrow{S})X^{\prime}=M_{3\times 3}(S_{V}\rightarrow{S})M_{3\times
3}(S_{V}\rightarrow{S^{\prime}})X_{V}$ is not an operation defined in the
group. The existence of another equivalent reference would imply the
destruction of this structure. The same applies to a transformation that would
$S^{\prime}\rightarrow{S_{V}}\rightarrow{S_{V}}$. The only possible operation
in this case is the identity $I_{3\times 3}$ that does
$S_{V}\rightarrow{S_{V}}$. In addition, the crack of matrices (132) does not
answer the question of transformations between two distinct reference frames
$S_{V}$ and $S^{\prime}$. As Nassif proposed in N2015  and N2016  the set of
kinematic transformations would not correspond to a group. This conclusion of
Nassif is based on Bacetti _et-al_ studies mg ; bacjhep .

Relations (120), (121) and (124) allow us to write compositions of velocities
between the references $S_{V}\rightarrow{S^{\prime}}$, so we write the
parallel component

$v^{\prime}_{\parallel}=\frac{v_{\parallel}-u+V}{1+\frac{v_{\parallel}}{c^{2}}(V-u)}.$
(133)

From now on we will just write $v_{\parallel}$ as $v$.

The perpendicular component is

$v^{\prime}_{\perp}=\frac{V}{1+\frac{v}{c^{2}}(V-u)}.$ (134)

Another possible simplification is to adopt $u=v$, which is equivalent to
considering only the two reference frames: $S^{\prime}$, the reference frame
on the quasi-particle world line; and, the preferred frame, $S_{V}$. Then we
rewrite (133)

$v^{\prime}_{\parallel}=\frac{V}{1+\frac{(V-v)v}{c^{2}}},$ (135)

and (134)

$v_{\perp}=\frac{V}{1+\frac{(V-v)v}{c^{2}}}.$ (136)

The Equations (135) and (136) are the equations associated with the
propagation velocity of any excited quasi-particle in the privileged reference
frame, and we fall into the first excited state, when we do $v=V$, which means
the quasi-particle is being dragged by this exotic fluid. Besides making it
clear that this exotic fluid is isotropic, the components of the velocity are
identical in all directions, as a wave. A fluctuation that propagates in the
fluid, which serves as a privileged reference and that here, already shows a
property of superfluidity, associated with the term

$V-v.$ (137)

We write the vector $\vec{v}^{\prime}$, seen from the referential $S^{\prime}$

$\vec{v}^{\prime}=\left[\begin{matrix}v^{\prime}_{\parallel}\\\
v^{\prime}_{\perp}\\\ \end{matrix}\right].$ (138)

We can also find parallel and perpendicular velocities when viewed from the
point of view of $S_{V}$, these transformations are important because are
based on the reference frame in the superfluid where the currents are
constructed vortex ; superverse ; qgv ; an . We write below the speeds seen in
the reference $S_{V}$

$v_{\parallel}=\frac{v+u-V}{1+\frac{v}{c^{2}}(u-V)},$ (139)
$v_{\perp}=\frac{\left[1-\frac{v^{2}}{c^{2}}\left(1-\frac{V^{2}}{v^{2}}\right)^{2}\right]V}{1+\frac{v}{c^{2}}(u-V)},$
(140)

where

$u=v_{S/S_{V}}.$ (141)

Now we write the vector

$\vec{v}_{S_{V}}=\left[\begin{matrix}v_{\parallel}\\\
v_{\perp}\end{matrix}\right]_{S_{V}}.$ (142)

###  IV.2 A concrete example: Vortex

The Visser work ergovisser  proposes a flow with a vortex profile

$\vec{v}=\frac{A\hat{r}+B\hat{\theta}}{r}.$ (143)

We consider (143) with relation to preferred frame $S_{V}$ starting from (139)

$\vec{v}_{\parallel}(S_{V})=\frac{\sqrt{\frac{A^{2}+B^{2}}{r^{2}}}+u-V}{1+\sqrt{A^{2}+B^{2}}\frac{(u-V)}{rc^{2}}}\textbf{e}$
(144)

where e this related to Equation (143) and follows
$\textbf{e}=\frac{\vec{v}}{||\vec{v}||}$. We write like this, for (140)

$v_{\perp}(S_{V})=\left[\frac{1-\frac{A^{2}+B^{2}}{r^{2}c^{2}}\left(1-\frac{V^{2}r^{2}}{A^{2}+B^{e}}\right)}{1+\sqrt{\frac{A^{2}+B^{2}}{r^{2}}}\frac{u-V}{c^{2}}}\right]V\textbf{e}_{\perp}$
(145)

When the flux line is given by (143). The line element associated with (143)
is

$ds^{2}=-c^{2}_{som}dt^{2}+\left(dr-\frac{A}{r}dt\right)^{2}+\left(rd\theta-\frac{B}{r}dt\right)^{2}.$
(146)

The limits of (144) and (145) are

$\lim_{r\rightarrow 0}v_{\parallel}\rightarrow\frac{c^{2}}{u-V}$ (147)

and

$\lim_{r\rightarrow 0}v_{\perp}\rightarrow\frac{c^{2}}{u-V}.$ (148)

Without incorporating the effects of preferred frame $S_{V}$, to kinematic
transformations does not clearly express the kinematic and geometric role of
the first excited state of the superfluid. This paper is clear in the
Equations (144) and (145), both have the vortex drag effect by the fluid in
laminar flow with minimal speed $V$. The two equations present a second
observer $S_{u}$, expressed by speed $u$, for simplicity we will make sure
that we have a single state excited in this case $u\equiv{v}$. We rewrite then
(144) and (145)

$\vec{v}_{\parallel}(S_{V})=\frac{2\sqrt{\frac{A^{2}+B^{2}}{r^{2}}}-V}{1+\frac{\sqrt{A^{2}+B^{2}}}{rc^{2}}\left(\frac{\sqrt{A^{2}+B^{2}}}{r}-V\right)}\textbf{e}.$
(149)

We see that in the vicinity of the center of the vortex the speed has an
isotropic behavior. The other limit $r\rightarrow\infty$ causes
$v_{\parallel}\rightarrow{u-V}$, so the parallel component with the offset
would be related to that of the $S_{u}$, already the other component
$v_{\perp}\rightarrow\xi^{2}V$. The dimensionless constant $\xi$ is defined
and its physical meaning are better explained in Ref. gbec  as

$\xi=\frac{V}{c}=\sqrt{\frac{Gm_{p}m_{e}}{4\pi\epsilon_{0}}}\frac{q_{e}}{\hbar
c},$ (150)

where $G$ is the gravitational constant, $m_{p}$ is the proton mass, $m_{e}$
and $q_{e}$ are mass and charge of electron. In the same work gbec  it is
estimated as $\xi=1.5302\times 10^{-22}$.

It is remarkable that for small values of $r$, the denominator goes to
infinity faster than the numerator, bringing the expression to zero, thus
generating a natural cutoff. Such parameters $A$ and $B$ are responsible for
the causal structure, so let’s look for a relation to the radius of the
ergoregion. Better said, a relation to which distance the effects of Aether
drag are felt

$r_{0}=2\frac{\sqrt{A^{2}+B^{2}}}{V},$ (151)

it is notable that the effect has a much greater scope than the effect
predicted by Visser ergovisser  $2\frac{\sqrt{A^{2}+B^{2}}}{c}$, curiously the
ratio between the two effects is of the order of the constant $\xi$.

Figure 4:  The Vortex Geometry and Ergoregions ergovisser  .

$v_{\perp}(S_{V})=\left[\frac{1-\frac{A^{2}+B^{2}}{r^{2}c^{2}}\left(1-\frac{V^{2}r^{2}}{A^{2}+B^{e}}\right)}{1+\frac{\sqrt{A^{2}+B^{2}}}{rc^{2}}\left(\frac{\sqrt{A^{2}+B^{2}}}{r}-V\right)}\right]V\textbf{e}_{\perp}$
(152)

Similar to (149), we see a cutoff for $r\rightarrow 0$, which indicates that
this would be a serene region, like the eye of a hurricane, but the existing
velocity would be isotropic, providing a type of expansion, linear velocity in
$r$, we then have an effect similar to the Hubble parameter vfundo .

When we do $\vec{v}_{\parallel}-\vec{v}_{\perp}$ and we consider the limit, we
have

$\lim_{r\rightarrow\infty}||\vec{v}_{\parallel}(S_{V})-\vec{v}_{\perp}(S_{V})||\rightarrow{V},$
(153)

consisting of a region where the vortex influence ceases and we have only the
laminar flow of the fluid.

###  IV.3 The usual Lorentz symmetry in a superfluid

We now write transformations of Lorentz, usually for a superfluid

$v^{\mu}=(\tilde{\mu},\vec{\nabla}\sigma)=\tilde{\mu}\left(1,\vec{v}_{s}\right),$
(154)

and a perpendicular fluid

$v^{\mu}_{n}=\tilde{s}\left(1,\vec{v}_{n}\right).$ (155)

We have established a pattern

$\tilde{\mu}=\gamma_{s}\mu,$ (156)

implying in a Lorentz factor

$\gamma_{s}=\frac{1}{\sqrt{1-\frac{v^{2}_{s}}{c^{2}}}}.$ (157)

The same for $\tilde{s}$

$\tilde{s}=\gamma_{n}s,$ (158)
$\gamma_{n}=\frac{1}{\sqrt{1-\frac{v^{2}_{n}}{c^{2}}}}.$ (159)

The scalar product between (154) and (155)

$v^{\mu}_{s}v^{n}_{\mu}=-c^{2}y=-c^{2}\tilde{\mu}\tilde{s}\left(1-\frac{\vec{v}_{s}\cdot\vec{v}_{n}}{c^{2}}\right)=-c^{2}\mu{s}\gamma_{n}\gamma_{s}\left(1-\frac{\vec{v}_{s}\cdot\vec{v}_{n}}{c^{2}}\right),$
(160)

with Lorentz transformations for the addition of $\vec{v}_{s}$ and
$\vec{v}_{n}$

$\vec{v}_{ns}=\frac{\vec{v}_{n}-\vec{v}_{s}}{1-\frac{\vec{v}_{s}\cdot\vec{v}_{n}}{c^{2}}},$
(161)

Thus the squared module $||\vec{v}_{ns}||=||\vec{v}_{s}-\vec{v}_{n}||^{2}$ is
deliberate

$\frac{v^{2}_{ns}}{c^{2}}=1-\frac{\mu^{2}s^{2}}{y^{4}}.$ (162)

Here is established a ratio of the module product of the difference between
perpendicular and circulating speeds $||\vec{v}_{s}-\vec{v}_{n}||^{2}$ with
the entropy of the system, as a Lorentzian invariant, but there is no lower
limit for the speeds of the excited states, there is no information on how
excited states behave, in a range where the superfluid begins to generate its
first turbulent states. Here is the advantage of the treatment exposed in the
Equations (139) and (140), since the transformations (139) and (140) are more
complete than the transformation (162). Although in future work we can
integrate the same conceptions set out in (162) with thermodynamic concepts,
thus leaving a thermodynamics that takes into account the existence of this
superfluid. We also add that the Equation (162) is a referential
transformation, a composition of velocities, as well as (153), whereas (153)
implies a Lorentz Invariance Violation (114).

###  IV.4 Kinematics symmetry compatible with Landau Criteria

Since we describe the dispersion relation (108), we can consider the critical
speed generated by it, applying the criterion of Landau pai , superverse  and
vortex . So we rewrite (108)

$E(v)=m_{0}c^{2}\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}}.$
(163)

Following vortex , we consider (108), which we rewrite

$E(v)=m_{0}c^{2}\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}}+\vec{p}\cdot\vec{v}_{n}.$
(164)

In (164), we have to $\vec{v}_{s},\vec{v}_{n}$ are defined as in (139) and
$v_{n}$ are defined as in vortex  like a perpendicular wind, although here it
is more like an induced velocity. The induced expression reflects an analogy
with the Faraday-Lens effect. The momentum $\vec{p}$ is given by the relation
(97). We will define another scatter relation

$\epsilon(v,p)=m_{0}c^{2}\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}}+\vec{p}\cdot\left(\vec{v}_{n}-\vec{v}_{s}\right).$
(165)

Equation (164) is associated with its excitements. Thinking of a linear
dispersion relation with momentum $\epsilon(p)\propto{p}$, we consider that
the dispersion relation (165), at the limit where $p\rightarrow 0$ in (96), we
will have $v\rightarrow{V}$. That implying that,
$\vec{v}_{n}+\vec{v}_{s}\rightarrow{V}$, so (165) will tend to

$\epsilon(p)\rightarrow{Vp}.$ (166)

That represents the structure of a phonon, as that phonon is associated with a
causal structure (85), we can call it acoustic tachyon tacos . This indicates
that, by minimizing (165), we get

$min\frac{\epsilon(p)}{p}=V,$ (167)

and we find a critical speed superverse ; vortex  for the hypothetical
gravitational superfluid. Nassif in previous works N2015 ; N2016  had reached
a result, associating the limit (166), to the cosmological constant. We find
this new kinematic structure, established by Nassif in N2008 ; N2013  and
other works, that the cosmologic constant corresponds to the first excited
state of a gravitational vacuum. Corresponding directly to Nassif’s result in
N2016 .

###  IV.5 Conversion between a Eulerian observer and a preferred observer
$S_{V}$

Comparing with the equations of the preferred frame $S_{V}$, the first
conclusion we get is

$\alpha\equiv\psi(v),$ (168)

the second is the immediate comparison between (54) and (85), that reaffirms
(168). Understanding the role of $\psi(v)$ as a lapse function, we can write
the Eulerian base in terms of this new lapse function, via Equations (48) and
(49)

$\tilde{\eta}_{\mu}=\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}}\left(-1,0,0,0\right),$
(169)

and

$\tilde{\eta}^{\mu}=\sqrt{\frac{1-\frac{v^{2}}{c^{2}}}{1-\frac{V^{2}}{v^{2}}}}\left(1,\beta^{i}\right).$
(170)

It is easy to notice that the scalar product
$\tilde{\eta}_{\mu}\tilde{\eta}^{\mu}=-1$ implies that $\tilde{\eta}_{\mu}$
and $\tilde{\eta}^{\mu}$ continue to be unitary, thereby causing

$g^{tt}=\psi(v).$ (171)

We postulate now.

$\phi=c^{2}\left(\psi(v)-1\right).$ (172)

Building projectors

$\tilde{\gamma}_{\mu\nu}=\eta_{\mu\nu}+\tilde{\eta}_{\mu}\tilde{\eta}_{\nu},$
(173)

and

$\tilde{\gamma}^{\mu\nu}=\eta^{\mu\nu}+\tilde{\eta}^{\mu}\tilde{\eta}^{\nu},$
(174)

we have established that $\tilde{\gamma}^{ik}\tilde{\gamma}_{kj}$.

We write the new projectors

$\tilde{\gamma}^{\mu}_{\nu}:=g^{\mu\nu}\tilde{\gamma}_{\alpha\nu}.$ (175)

To determine the four-speed we apply (169) in (54) and we have

$-\tilde{\eta}_{\mu}u^{\mu}=c\sqrt{1-\frac{V^{2}}{v^{2}}}.$ (176)

Applying $\tilde{\eta}^{\mu}$ on both sides

$u^{\mu}=-c\sqrt{1-\frac{V^{2}}{v^{2}}}\tilde{\eta}^{\mu},$ (177)

remembering that $\tilde{\eta}^{\mu}$ is given in the (170).

After we write the Equation (177), we can determine by rezz , the speeds
$v^{i}:=\frac{\gamma^{i}_{\mu}u^{\mu}}{\alpha{u}^{t}}$ and
$v_{i}:=\frac{\gamma_{i\mu}u^{\mu}}{\alpha{u}^{t}}$. We do then

$\tilde{v}^{i}=-2c\psi^{-2}(v)\beta^{i},$ (178)

and

$\tilde{v}_{i}=-2c\eta_{i}=0.$ (179)

Returning again to Rezzolla rezz , using the Equations (177) and (170), we
write $\beta^{i}$, so we get the expression

$\frac{\beta^{i}}{c}=\left(\psi(v)-1\right)\eta^{i},$ (180)

bringing back the relation (172).

We also studied the condition of normalization $u^{\mu}u_{\nu}=-1$, to find
$u_{\nu}$, we do then $u_{\mu}=a(v)\eta_{\mu}$. Replacing along with (177), we
found the value of $a=\frac{1}{c\sqrt{1-\frac{V^{2}}{v^{2}}}}$. Thus

$u_{\mu}=\frac{1}{c\sqrt{1-\frac{V^{2}}{v^{2}}}}\tilde{\eta}_{\mu}.$ (181)

Studying because

$\eta^{\mu\nu}u_{\mu}u_{\nu}=\frac{1}{c^{2}}\frac{1}{1-\frac{V^{2}}{v^{2}}},$
(182)

alternatively

$\eta_{\mu\nu}u^{\mu}u^{\nu}=c^{2}\left(1-\frac{V^{2}}{v^{2}}\right).$ (183)

The Equation (183) suggest a relation with the _k_ -essence models, in the
meaning that they are defined in the Equation (2).

###  IV.6 Supefluids and _k_ -essence an unified description via preferred
frame $S_{V}$

Considering the work of Nassif et alr3 . We have the Lagrangian for a
relativistic free particle

$\mathcal{L}=-m_{0}c^{2}\sqrt{1-\frac{v^{2}}{c^{2}}},$ (184)

however, if the particle suffers effects from a conservative force, which is
independent of speed, we have $\mathcal{L}=-m_{0}c^{2}\sqrt{1-\beta^{2}}-U$,
where $U=U(r)$. If the Lagrangian $\mathcal{L}$ is a function that does not
depend on time, if there is no movement constant.

We write the Hamiltonian

$h=\dot{q}_{j}p_{j}-\mathcal{L}=\frac{m_{0}v_{j}v_{j}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}+m_{0}c^{2}\sqrt{1-\frac{v^{2}}{c^{2}}}+U,$
(185)

also

$h=\frac{m_{0}c^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}+U=E,$ (186)

where $h$ is the total energy.

If $U=0$, so $E=\gamma{m_{0}}c^{2}$, there is a constant movement of the free
particle. We just write $v^{2}=v_{j}v^{j}$ for single particle.

Now, by using analog procedures to obtain a relativistic Lagrangian, affected
by the existence of a preferred frame $S_{V}$ associated with the critical
velocity of a superfluid (167). We consider to obtain first Lagrangian of an
ideal flow. With $h=E$, where $E$ is get from (95), the constant movement for
a flow-line, knowing the momentum (97) is a momentum for the hypothetical
superfluid. We write then

$E=m_{0}c^{2}\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}}=h=m_{0}v^{2}\sqrt{\frac{1-\frac{V^{2}}{v^{2}}}{1-\frac{v^{2}}{c^{2}}}}-\mathcal{L},$
(187)

where $v^{2}=v_{j}v^{j}$ and $\mathcal{L}$ is the free particle Lagrangian.

Starting from the Equation (187), we extract:

$\mathcal{L}=-m_{0}c^{2}\theta\sqrt{1-\frac{v^{2}}{c^{2}}}=-m_{0}c^{2}\sqrt{\left(1-\frac{V^{2}}{v^{2}}\right)\left(1-\frac{v^{2}}{c^{2}}\right)}.$
(188)

If we do $V\rightarrow 0$, or equivalently $v>>V$, in Equation (188), we
recover the Lagrangian (184).

If we take Equation (183) and compare with (2), we recover the Equation (80).
In (183) making a change of variable, we get

$\mathcal{K}=1-\frac{V^{2}}{v^{2}},$ (189)

it is necessary to comment on the range of values that the variable
$\mathcal{K}$ can assume

$0<\mathcal{K}<1-\xi^{2},$ (190)

A small manipulation in (189) and we have the relation between (62) and (189)

$v^{2}=\frac{V^{2}}{1-\mathcal{K}}.$ (191)

The Equation (191) represents the relation between a preferred frame $S_{V}$
and the _k_ -essence typical parameter $\mathcal{K}$, generating an observer
coupled to groud-state. It is an imposition for _k_ -essence parameter, which
is limited to varying within the range, thus constituting a timelike region
associated with the causal structure with minimal speed $V$, which
simultaneously is the speed criticized according to the criteria of Landau
(167).

We introduce (191) in (188) and we obtain

$\mathcal{L}(\mathcal{K})=-m_{0}c^{2}\sqrt{\mathcal{K}\left(1-\frac{\xi^{2}}{1-\mathcal{K}}\right)}.$
(192)

The Lagrangian (192) is a _k_ -essence purely kinetic Lagrangian just as (6)
and same for the Lagrangian of a superfluid. The consequences and implications
of (188) are implications of generating this unified scenario for cosmology
and astrophysics. The _k_ -essence was proposed as a hydrodynamic description
for cosmological constant k2 ; kpato , pathological causal structures in
scalar models are mapped kpato1  in several works. But (192) is a different
approach. The causal structure of scalar model match with critical speed,
building a preferred frame. We write

$\mathcal{L}_{\mathcal{K}}=-\frac{m_{0}c^{2}}{2\sqrt{\mathcal{K}(1-\frac{\xi^{2}}{1-\mathcal{K}})}}\left(1-\frac{\xi^{2}}{1-\mathcal{K}}-\frac{\xi^{2}\mathcal{K}}{(1-\mathcal{K})^{2}}\right),$
(193)

and

$\displaystyle\mathcal{L}_{,\mathcal{KK}}=$
$\displaystyle-\frac{3m_{0}c^{2}}{4\left[\mathcal{K}\left(1-\frac{\xi^{2}}{1-\mathcal{K}}\right)\right]^{\frac{3}{2}}}\left[1-\frac{\xi^{2}}{1-\mathcal{K}}-\frac{\xi^{2}\mathcal{K}}{(1-\mathcal{K})^{2}}\right]+$
$\displaystyle-\frac{m_{0}c^{2}}{2\sqrt{\mathcal{K}(1-\frac{\xi^{2}}{1-\mathcal{K}})}}\left(-\frac{\xi^{2}}{(1-\mathcal{K})^{2}}-\frac{1-2\mathcal{K}}{(1-\mathcal{K})^{3}}\right).$
(194)

The equations permit to write an hydrodynamic approach. The pressure

$\mathcal{P}=\mathcal{L}(\mathcal{K}),$ (195)

and density (10), using (189), (192) and (193)

$\rho=\mathcal{K}\mathcal{L}_{,\mathcal{K}}-\mathcal{L}(\mathcal{K}),$ (196)

or even

$\rho=-m_{0}c^{2}\sqrt{\mathcal{K}}\left[\frac{1-\frac{\xi^{2}}{1-\mathcal{K}}\left(1-\frac{\mathcal{K}}{1-\mathcal{K}}\right)}{2\sqrt{1-\frac{\xi^{2}}{1-\mathcal{K}}}}-\sqrt{\left(1-\frac{\xi}{1-\mathcal{K}}\right)}\right].$
(197)

Therefore, Equation (197) reproduces a vast amount of state equations, with
the barotropic parameter

$w=\frac{\sqrt{\left(1-\frac{\xi}{1-\mathcal{K}}\right)}}{\frac{1-\frac{\xi^{2}}{1-\mathcal{K}}\left(1-\frac{\mathcal{K}}{1-\mathcal{K}}\right)}{2\sqrt{1-\frac{\xi^{2}}{1-\mathcal{K}}}}-\sqrt{\left(1-\frac{\xi}{1-\mathcal{K}}\right)}}.$
(198)

The speed of sound is calculated

$\displaystyle c^{2}_{s}=$
$\displaystyle\left[1-\frac{\left[-\frac{3}{4}\left[1-\frac{\xi^{2}}{1-\mathcal{K}}\left(1-\frac{\mathcal{K}}{(1-\mathcal{K})^{2}}\right)+\frac{1}{2(1-\mathcal{K})^{2}}\left(\xi^{2}-\frac{1-2\mathcal{K}}{1-\mathcal{K}}\right)\right]\right]\sqrt{\frac{\mathcal{K}}{1-\frac{\xi^{2}}{1-\mathcal{K}}}}}{\frac{1}{2\sqrt{\mathcal{K}\left(1-\frac{\xi^{2}}{1-\mathcal{K}}\right)}}\left[1-\frac{\xi^{2}}{1-\mathcal{K}}\left(1-\frac{\mathcal{K}}{1-\mathcal{K}}\right)\right]}\right]^{-1}.$
(199)

The Equations (198) and (199) correspond to the values of the barotropic
factor and the sound propagation within the range of values that $\mathcal{K}$
can assume.

##  V A velocity-Potential approach

Inspired by Rezzola (see details section 3.9 of Ref. rezz ), we are going to
make a very brief approximation of works sch ; sch2 . Velocity-potential
approach allows the definition of some thermodynamic quantities associated
with large kinematics. In this work we will not define the four-speed in terms
of the 6 potential scalings of schutz sch ; sch2  in formalism, because we
intend to address the temperature problem associated with event horizons in
future work. But the discussion about enthalpy gives back a clarification
about the role of the preferred frame $S_{V}$.

###  V.1 The Irrotational perfect fluid

The hydrodynamic four-speed (9)

$\mathcal{U}_{\mu}:=\frac{\partial_{\mu}\Phi}{\sqrt{2\left(1-\frac{V^{2}}{v^{2}}\right)}}.$
(200)

The quantity $\mathcal{U}_{\mu}$ represents an irrotational fluid written in
Schutz formalism, sch ; sch2 , with normalization

$\mathcal{U}^{\mu}\mathcal{U}_{\mu}=-1,$ (201)

which implies

$\mathcal{U}^{\mu}:=\sqrt{2\left(1-\frac{V^{2}}{v^{2}}\right)}\partial^{\mu}\Phi.$
(202)

Performing a contraction with Minkowski metric

$\eta_{\mu\nu}\mathcal{U}^{\mu}\mathcal{U}^{\nu}=2\left(1-\frac{V^{2}}{v^{2}}\right),$
(203)

as in the Schutz formalismsch , is given for enthalpy

$h(v)=\sqrt{2\left(1-\frac{V^{2}}{v^{2}}\right)},$ (204)

a thermodynamic quantity associated the capacity of absorb or emit heat, the
simplest substance have enthalpy $h=0$. In this case, the more simple state
have $h(v=V)=0$. This results confirm a propriety of ground state, the
enthalpy have a relation with pressure $h=\frac{\rho+P}{n}$, so $h=0$ implies

$P=-\rho.$ (205)

We have indication that the cosmological constant is a ground-state of this
cosmological superfluid. The density of particle is related with the
quantities in this description. The concept of preferred frame $S_{V}$ then
makes it possible to introduce the cosmological constant naturally into a
hydrodynamic structure.

$n=\sqrt{1-\frac{V^{2}}{v^{2}}}\mathcal{P}_{,\mathcal{K}}$ (206)

where $\mathcal{P}_{,\mathcal{K}}=\mathcal{L}_{,\mathcal{K}}$. Using (193), we
rewrite so

$n(\mathcal{K})=-\frac{m_{0}c^{2}}{2\sqrt{(1-\frac{\xi^{2}}{1-\mathcal{K}})}}\left(1-\frac{\xi^{2}}{1-\mathcal{K}}-\frac{\xi^{2}\mathcal{K}}{(1-\mathcal{K})^{2}}\right).$
(207)

We have the limits of $n(\mathcal{K}\rightarrow 0)\rightarrow{m_{0}}c^{2}$,
the $n(\mathcal{K}\rightarrow 1-\xi^{2})\rightarrow\infty$. In the first one
we recover the classical resting energy, in the second one there is
predominance of the _k_ -essence and tending to ground-state. The energy-
momentum tensor (7) is written in terms of thermodynamics quantities

$T_{\mu\nu}=nh\mathcal{U}_{\mu}\mathcal{U}_{\nu}-P(\mathcal{K})\eta_{\mu\nu}$
(208)

Also, we have a hydrodynamic description of _k_ -essence superfluid. Thus the
Equation (202) is an enthalpy current.

###  V.2 Conserved current of enthalpy

We reconsider the Equation (202), and we study its conservation

$\partial_{\mu}\mathcal{U}^{\mu}=\frac{\sqrt{2}}{\sqrt{1-\frac{V^{2}}{v^{2}}}}\frac{V^{2}}{v^{3}}\partial_{\mu}v\partial^{\mu}\Phi+\sqrt{2\left(1-\frac{V^{2}}{v^{2}}\right)}\partial_{\mu}\partial^{\mu}\Phi$
(209)

We consider as Rezzola rezz

$U=U^{\mu}e_{\mu}$ (210)

where $e_{\mu}$ is a vectorial base. Let us now make a derivation in (210)

$\partial_{\nu}U=\partial_{\nu}U^{\nu}e_{\mu}+U^{\mu}\partial_{\nu}e_{\mu}$
(211)

when we compare (211) with (209). We identify

$\partial_{\nu}e_{\mu}\mathcal{U}^{\mu}\equiv\sqrt{\frac{2}{1-\frac{V^{2}}{v^{2}}}}\frac{V^{2}}{v^{3}}\partial_{\mu}v\partial^{\mu}\Phi.$
(212)

Again by Rezzola rezz , we write

$\partial_{\nu}e_{\mu}=\Gamma^{\kappa}_{\mu\nu}e_{\kappa}.$ (213)

Therefore

$\Gamma^{\alpha}_{\alpha\nu}=\sqrt{\frac{2}{1-\frac{V^{2}}{v^{2}}}}\frac{V^{2}}{v^{3}}\partial_{\nu}v,$
(214)

we thus constitute the hydrodynamic equivalent of a connection. The
consequences regarding curvature will be investigated in another work, for now
the term is immediately identified with some type of shear between flow lines.
The appearance of this term in this hypothetical superfluid model is quite
interesting, as it allows connections between vortices and intense
gravitational fields, a very interesting line of research with many open
questions.

When we consider the conservation of current (209), we obtain

$\sqrt{\frac{2}{1-\frac{V^{2}}{v^{2}}}}\frac{V^{2}}{v^{3}}\partial_{\mu}v\partial^{\mu}\Phi+\sqrt{2\left(1-\frac{V^{2}}{v^{2}}\right)}\partial_{\mu}\partial^{\mu}\Phi=0$
(215)

we see a wave equation for the scalar field, but the wave equation is
increased by a term in (214), in classical terms, this term can be compared to
shear, in quantum terms the reconnection between vortices. The hypothesis of a
type of sink or source is also not ruled out and should be investigated in
detail in future works.

##  VI Conclusions and perspectives

We revisited the basic of _k_ -essence fluid formalism and build a relation of
the proposal of a universal minimum velocity under a preferred reference frame
$S_{V}$ with the Landau criterion for superfluids. We did this unification of
disjoint concepts to the structure of a _k_ -essence fluid. We also identified
the enthalpy of a fluid in the Schutz formalism, the function that deforms the
Lorentz transforms, introducing the concept of minimum velocity. In this way,
we were able to construct an observed _k_ -essence fluid from the first
excited state of a superfluid. This implies a reinterpretation of the _k_
-essence term, thus allowing us to understand a causal structure with the
presence of sonic tachyons.

We have found that the hydrodynamic equivalent of a relativistic connection.
Such thing could bring light to the discovery of constraints between
hydrodynamis of dark superfluids and space-time curvature. The calculation on
curvature tensor on a preferred reference frame is something new under the _k_
-essence formalism of superfluids.

We also intend to address the problem of constraining the thermodynamics with
the preferred frame and associate its quantities with event horizons in future
work. The discussion about enthalpy made here gives back the question about
the role of the preferred frame $S_{V}$.

This theoretical construction allows an association of thermodynamic concepts
with causal structures in a very simple way. Using mathematical concepts
common in the academic community, such as conformal transformations.

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