# Relativistic Hydrodynamics a brief review of classical and
quantum fluids in relativistic astrophysics

R. F. Santos  Escola Municipal Jardim Glaucia, Belford Roxo, Rua Péricles N.
15, 26195-140, Belford Roxo, RJ, Brazil<EMAIL_ADDRESS>A. C. Amaro Faria
Jr  Universidade Federal Tecnológica do Paraná, 85053-525, Guarapuava, PR,
Brazil<EMAIL_ADDRESS>L. G. Almeida  Universidade Federal do Acre,
Rodovia BR 364, Km 04, 69920-900, Rio Branco, AC, Brazil<EMAIL_ADDRESS>
###### Abstract

The objective of this work is to revisit fundamental aspects of relativistic
hydrodynamics, aiming at the construction of a first course in relativistic
hydrodynamics and its applications to astrophysics at the level of end of
undergraduate course and beginning of graduate course. We aim to introduce
more basic concepts of basic hydrodynamics, going through models analogous to
gravity to the theory of superfluids, applying mainly to astrophysics and the
cosmology of the dark universe. We review the classical hydrodynamics, Galileo
symmetry and its extension to Lorentz Symmetry applied to fluids, enabling the
analogy of fluids with space-time. We study the conservation of the momentum-
energy tensor and the energy conditions of Hawking-Ellis. In the next sections
we investigate quantum effects, in particular linked to superfluids, and we
also sketch an application to dark matter. In this study, we conclude that
superfluidity is one of the possible ways to quantize gravity.

Hydrodynamics – Dark Energy – Cosmological Parameters – Gravitation

##  I Introduction

Relativistic Fluids have received great attention from the scientific
community recently, see Rezzolla and Zanotti (2013); Andersson and Comer
(2021); Minguzzi (2019); Santos  _et al._ (2020). The works of Unruh (1981)
and Visser (1993) opened an unexpected perspective on the study of Lorentz
Symmetry in fluids. Later, the seminal work of Lebedev and Khalatnikov (1982)
in superfluid hydrodynamics, associated with geometric structures, would
generate a fruitful and still promising line of study for models analogous to
gravity, Barceló  _et al._ (2011); Baccetti  _et al._ (2012). Although the
search for privileged reference frames in general relativity, Gordon (1923),
and exotic causal structures, Recami (1987), is not new, the advent of the
work of Unruh (1981) and Visser (1993) brought a new spirit to the research,
which began to open a new frontier linked to phenomena of superfluids, Lebedev
and Khalatnikov (1982). The works of Volovik (2001) and Huang (2016) were
certainly exponents in this investigation, which is supported by verifications
of superfluid phenomena in neutron stars, see Martin and Urban (2016),
theoretical models linking superfluids to the cosmological constant by Khoury
_et al._ (2018) and detection of phenomena in galactic structure by Harko and
Madarassy (2022), which can be understood as an indication that dark matter is
related to superfluids. Some authors started to use the name Bose-Einstein
Gravitational Condensate to model the interior of ultracompact objects, as
Santos and Almeida (2022), cosmic inflation by Cruz  _et al._ (2020) and dark
matter by Levkov  _et al._ (2018) associating these phenomena to a type of
_gravitational phonon_. The hydrodynamic treatment for gravitational phenomena
is also a possibility of quantizing the gravitational field through the
introduction of quantum potentials, see Chiarelli (2019); Durr  _et al._
(2013); Falciano  _et al._ (2010).

Quantum potentials are a way of introducing quantum effects in the covariant
formalism, this perspective has been quite successful especially with the
works of Schutz (1970) and Schutz (1971), which are crucial in the debate
regarding quantum cosmology Tavakoli and Vakili (2019) introducing
thermodynamic potentials in such formalism. Another line that we want to
highlight is called Superfluid Vacuum Theory – SVT Zloshchastiev (2020a, b,
2010) – which has been quite successful especially in the description of the
dark sector, Zloshchastiev (2020a).

##  II Non-relativistic fluid

###  II.1 Continuity Equation

We will now demonstrate the most important equation of hydrodynamics, the
continuity equation. The mass of a fluid can be defined as follows

$m=\int\rho{dV}$ (1)

where $\rho$ is the volumetric density of mass and $V$ the volume occupied by
the fluid, considering the mass of a given fluid that flows through an element
of area $ds$ per unit of time, we have:

$\frac{dm}{dt}=\frac{d}{dt}\left(\int\rho{dV}\right).$ (2)

We can define the current

$\vec{J}=\rho\vec{v}$ (3)

where $\vec{v}=\vec{v}(x,y,z,t)$ is the velocity of the fluid. If we calculate
the fluid current flow through the same area element as the integral of the
dot product between the current and the normal vector to the area element

$\Phi=\int_{\mathcal{D}s}\vec{J}\cdot{d}\vec{s}.$ (4)

By the theorem of Gauss, we relate the integral in the area with the volume
integral of the current source, represented by the divergence,

$\int_{\mathcal{D}s}\vec{J}\cdot{d}\vec{s}=\int_{\mathcal{D}V}\vec{\nabla}\cdot\vec{J}dV,$
(5)

where the operator $\vec{\nabla}$ is given by

$\vec{\nabla}=e_{x}\partial_{x}+e_{y}\partial_{y}+e_{z}\partial_{z}.$ (6)

The triplet $(e_{x},e_{y},e_{z})$ is the set of unitary vectors in the
respective coordinate directions, and the operation

$\vec{\nabla}\cdot\vec{J}=\partial_{x}J_{x}+\partial_{y}J_{y}+\partial_{z}J_{z}$
(7)

is a dot product, representing the sources and drains of current. Substituting
3 in 4 we get

$\Phi=\int_{\mathcal{D}s}\rho\vec{v}\cdot{d}\vec{s}.$ (8)

Considering now the rate of mass that passes through a given unit of area per
unit of time, 2 and taking $\vec{v}$ as a constant, we write

$\frac{d\Phi}{dt}=\int_{\mathcal{D}s}\frac{d\rho}{dt}\vec{v}\cdot{d}\vec{s}.$
(9)

Analyzing the right side of 5, we have

$\int\frac{d\rho}{dt}\vec{v}\cdot\vec{ds}=\frac{d}{dt}\left[\int\vec{J}\cdot\vec{ds}\right].$
(10)

Calculating the scalar product

$\vec{v}\cdot{d}\vec{s}=\frac{dxds}{dt},$ (11)

we find

$\frac{dxds}{dt}=\frac{dV}{dt}.$ (12)

We now consider the action of the operator ${d}/{dt}$ on the right side of 5,

$\frac{d}{dt}\left[\int\vec{\nabla}\cdot\vec{J}dV\right]=\int\vec{\nabla}\cdot\vec{J}\frac{dV}{dt},$
(13)

which is equivalent to saying that the font $\vec{\nabla}\cdot\vec{J}$ is a
time-independent function. Therefore, we can write

$\frac{d\rho}{dt}=\vec{\nabla}\cdot\vec{J}.$ (14)

This is the so-called _continuity equation_ , which implies that fluid is a
conserved quantity, so the same amount of fluid that enters the pipe is the
amount that leaves, as the sources and sinks are computed in the temporal
variation of the fluid density.

###  II.2 Conservation of Momentum – The Navier Stokes equation

We are going to calculate the pressure under a fluid element. We can consider
an infinitesimal element of volume of cubic form and calculate the force on
walls that have a normal in the $x$ direction

$dF_{x}=p(x,y,z)dydz-p(x+dx,y+dy,z+dz)dydz,$ (15)

expanding $p(x+dx,y+dy,z+dz)$ in Taylor series and rewriting 15, we can
approximate

$dF(x)=p(x,y,z)dydz-p(x,y,z)dydz-\frac{\partial{p}}{\partial{x}}dxdydz.$ (16)

Considering then the force on the mass element $dm$ and remembering that

$\displaystyle\rho=\frac{dm}{dV}$ $\displaystyle dV=dxdydz,$

we can write

$(dm)\vec{a}=d\vec{F}_{x}=-\vec{\nabla}pdV,$ (17)

where

$\vec{\nabla}{p}=e_{x}\partial_{x}p+e_{y}\partial_{y}p+e_{z}\partial_{z}p$
(18)

is the pressure gradient. We know that

$\displaystyle m\frac{d\vec{v}}{dt}=\vec{F}$
$\displaystyle\frac{dm}{dV}\frac{d\vec{v}}{dt}=\frac{d\vec{F}}{dV}=-\vec{\nabla}p,$

where

$\vec{v}=\vec{v}(t,\vec{r}),$ (19)

we then write:

$\frac{d\vec{v}(t,\vec{r})}{dt}=\frac{\partial\vec{v}}{\partial{t}}dt+\frac{\partial\vec{v}(t,\vec{r})}{\partial{\vec{r}}}d\vec{r},$
(20)

where

$\frac{\partial\vec{v}(t,\vec{r})}{\partial{r}}=\frac{\partial\vec{r}}{\partial{t}}\cdot\frac{\partial\vec{v}}{\partial\vec{r}}\vec{v}=\frac{\partial\vec{v}(t,\vec{r})}{\partial{t}}+(\vec{v\cdot\vec{\nabla}})\vec{v}.$
(21)

Returning the result above in 20, we get

$\frac{\partial\vec{v}}{\partial{t}}+(\vec{v}\cdot\vec{\nabla})\vec{v}=-\frac{1}{\rho}\vec{\nabla}p.$
(22)

The equation 22 is the _Navier Stokes equation_ , which corresponds to
Newton’s second law for the fluid mass element $dm$.

###  II.3 Barotropic fluids

A fluid is defined as _barotropic_ if the energy density is a function only of
the pressure, or $\rho\equiv\rho(p)$, as seen in Visser and Molina-París
(2010). Therefore, the inverse function shall be $p=p(\rho)$ and we can define

$\mathcal{P}=\int\frac{dp}{\rho}\Rightarrow d\mathcal{P}=\frac{dp}{\rho},$
(23)

thus

$d\vec{x}\cdot\vec{\nabla}\mathcal{P}=\frac{dp}{\rho},$ (24)

and we can write

$\frac{dp}{\rho}=d\vec{x}\cdot\frac{\vec{\nabla}p}{\rho},$ (25)

so

$d\vec{x}\cdot\vec{\nabla}\mathcal{P}=d\vec{x}\cdot\frac{\vec{\nabla}p}{\rho}$
(26)

then

$d\vec{x}\cdot\left(\vec{\nabla}\mathcal{P}-\frac{\vec{\nabla}p}{\rho}\right)=0$
(27)

indicating then that

$\vec{\nabla}\mathcal{P}=\frac{\vec{\nabla}p}{\rho}.$ (28)

We then write the acceleration as the sum of the gravitational potential
gradient and the gradient of $\mathcal{P}$,

$\vec{a}=\vec{\nabla}G-\vec{\nabla}\mathcal{P}=\vec{\nabla}\left(G-\int\frac{dp}{\rho}\right).$
(29)

Returning the equation 22, we can see that

$\vec{v}\cdot\vec{\nabla}\vec{v}=\frac{1}{2}\vec{\nabla}\left(\vec{v}\cdot\vec{v}\right)-\vec{v}\times\vec{\nabla}\times\vec{v},$
(30)

we then define the vorticity of a fluid as

$\vec{w}=\vec{\nabla}\times\vec{v}.$ (31)

Replacing 31 in 22, we have

$\frac{\partial\vec{v}}{\partial{t}}-\vec{v}\times\vec{w}=\vec{\nabla}\left[G-\int\frac{dp}{\rho}-\frac{|\vec{v}|^{2}}{2}\right].$
(32)

If we do $\vec{w}=0$ in 31 we have an irrotational fluid. If we have, in
addition to an irrotational liquid, a fluid with steady flow
${\partial\vec{v}}/{\partial{t}}=0$, then we have:

$\vec{\nabla}\left[G-\int\frac{dp}{\rho}-\frac{|\vec{v}|^{2}}{2}\right]=0,$
(33)

so we have the _Bernouli equation_

$\left[G-\int\frac{dp}{\rho}-\frac{|\vec{v}|^{2}}{2}\right]=constant.$ (34)

Considering, therefore, a steady state

$f(t)=G-\int\frac{dp}{\rho}-\frac{|\vec{v}|^{2}}{2}-\frac{\partial\phi}{\partial{t}},$
(35)

and doing $f=-g,G=-gz,p=constant$, we find

$G-\frac{p}{\rho}-\frac{|\vec{v}|^{2}}{2}-\frac{\partial\phi}{\partial{t}}=constant.$
(36)

###  II.4 Potential flow

A potential flow is defined according to the following relation of fluid
velocity with a scalar potential

$\vec{v}=\vec{\nabla}\phi,$ (37)

if we extract the divergence of 37

$\vec{\nabla}\cdot\vec{v}=\vec{\nabla}\cdot\vec{\nabla}\phi=\nabla^{2}\phi=0,$
(38)

we find the _Laplace equation_

$\nabla^{2}\phi=0.$ (39)

This type of flow is said to be incompressible. Which implies that
${d\rho}/{dt}=0$, by the continuity equation and considering the definition of
the current 3, we have that

$\vec{\nabla}\cdot\vec{J}=\vec{\nabla}\cdot\rho\vec{v},$ (40)

using the derivative properties, we get

$\vec{\nabla}\cdot\vec{J}=\vec{\nabla}\rho\cdot\vec{v}+\rho\vec{\nabla}\cdot\vec{\nabla}\phi,$
(41)

in this case, by the equation 39 the second term vanishes, what makes the
fluid incompressible is the density gradient $\vec{\nabla}\rho=0$. In this
case, the _Euler equation_ comes from 22

$\frac{\partial\vec{v}}{\partial{t}}=-\frac{1}{\rho}\vec{\nabla}P.$ (42)

So the pressure generates acceleration of the fluid element in the flow
direction.

##  III Electrodynamics as a theory of fluid

Electromagnetism conceived current as a fluid of charge carriers, which
propagated through conductors. The very idea of fields also had at its core
the concept of fluid, in the case of the electric field being a fluid, the
flux was calculated by Gauss’ law

$\vec{\nabla}\cdot{\vec{E}}=\frac{\rho}{\epsilon_{0}}$ (43)

and charge

$q=\int_{V}\rho{dV}.$ (44)

The vortices associated with this fluid are described by Faraday’s law

$\vec{\nabla}\times{\vec{E}}=-\partial_{t}\vec{B}$ (45)

where the magnetic field is associated with the fluid circulation property.
Furthermore, the divergence of the magnetic field is zero

$\vec{\nabla}\cdot{\vec{B}}=0,$ (46)

that shows the inexistence of magnetic sources (monopoles). Completing the
symmetry between the fields, we have the Ampere-Maxwell law

$\vec{\nabla}\times\vec{B}=\mu_{0}\vec{J}+\mu_{0}\epsilon_{0}\partial_{t}\vec{E},$
(47)

where $\mu_{0}\epsilon_{0}\partial_{t}\vec{E}$ is the displacement current
discovered by Maxwell. Here $\vec{J}=q\vec{v}$ is the charge carrier current.
Applying the divergence to 47, we have

$\vec{\nabla}\cdot\vec{\nabla}\times\vec{B}=\mu_{0}\vec{\nabla}\cdot\vec{J}+\mu_{0}\epsilon_{0}\partial_{t}\left(\vec{\nabla}\cdot\vec{E}\right).$
(48)

Substituting 43 and considering the properties of the divergence, we get

$\vec{\nabla}\cdot\vec{J}+\partial_{t}\rho=0,$ (49)

which is the _continuity equation_.

In the absence of sources $q=0,\vec{j}=0$, we have

$\displaystyle\vec{\nabla}\cdot{\vec{E}}=0,$ (50)
$\displaystyle\vec{\nabla}\times{\vec{E}}=-\partial_{t}\vec{B},$ (51)
$\displaystyle\vec{\nabla}\cdot{\vec{B}}=0,$ (52)
$\displaystyle\vec{\nabla}\times\vec{B}=\mu_{0}\epsilon_{0}\partial_{t}\vec{E}.$
(53)

Therefore, the vortex of an electric field implies the temporal variation of
the magnetic field, just as the magnetic vortex implies the temporal variation
of the electric field. Applying the curl (53), we have:

$\nabla^{2}\vec{B}-\frac{1}{c^{2}}\partial^{2}_{t}\vec{B}=0,$ (54)

so we have the wave equation for the magnetic field. We can likewise deduce
for the electric field, by applying the curl to the equation (51)

$\nabla^{2}\vec{E}-\frac{1}{c^{2}}\partial^{2}_{t}\vec{E}=0.$ (55)

Although the fields are similar to fluids, the electromagnetic wave does not
propagate in any medium, this was one of the greatest revolutions in science,
for it put an end to the idea of the luminiferous ether, which the Mickelson-
Morley experiment later showed did not exist.

##  IV Covariance of the Wave Equation

We now are going to check whether the Galileo transformations keep the wave
equation invariant.

$\nabla^{2}\phi-\frac{1}{c^{2}}\partial_{t}\phi=0,$ (56)

where $\phi$ is a scalar field.

Performing the Galileo transformation

$\frac{\partial}{\partial{x}}=\frac{\partial{x^{\prime}}}{\partial{x}}\frac{\partial}{\partial{x^{\prime}}}+\frac{\partial{y^{\prime}}}{\partial{x}}\frac{\partial}{\partial{x}}+\frac{\partial{z^{\prime}}}{\partial{x}}\frac{\partial}{\partial{z}}+\frac{\partial{t^{\prime}}}{\partial{x}}\frac{\partial}{\partial{t}}$
(57)

where

$\frac{\partial{x^{\prime}}}{\partial{x}}=1;\;\frac{\partial{y^{\prime}}}{\partial{x}}=\frac{\partial{z^{\prime}}}{\partial{x}}=\frac{\partial{t^{\prime}}}{\partial{x}}=0\Rightarrow\frac{\partial}{\partial{x}}=\frac{\partial}{\partial{x^{\prime}}},$
(58)

similarly

$\frac{\partial}{\partial{y}}=\frac{\partial}{\partial{y^{\prime}}},\frac{\partial}{\partial{z}}=\frac{\partial}{\partial{z^{\prime}}}.$
(59)

In consequence

$\nabla^{2}=\nabla^{\prime 2}$ (60)

Temporal component:

$\displaystyle\frac{\partial}{\partial{t}}$
$\displaystyle=\frac{\partial{t^{\prime}}}{\partial{t}}\frac{\partial}{\partial{t^{\prime}}}+\frac{\partial{x^{\prime}}}{\partial{t}}\frac{\partial}{\partial{x^{\prime}}}+\frac{\partial{y^{\prime}}}{\partial{t}}\frac{\partial}{\partial{y}}+\frac{\partial{z^{\prime}}}{\partial{t}}\frac{\partial}{\partial{z^{\prime}}}$
$\displaystyle=\frac{\partial}{\partial{t^{\prime}}}-V\frac{\partial}{\partial{x^{\prime}}}=\frac{\partial}{\partial{t^{\prime}}}-\vec{V}\cdot{\vec{\nabla^{\prime}}}.$
(61)

The second time derivative

$\frac{\partial^{2}}{\partial{t}^{2}}=\left(\frac{\partial}{\partial{t^{\prime}}}-\vec{V}\cdot{\vec{\nabla^{\prime}}}\right)\left(\frac{\partial}{\partial{t^{\prime}}}-\vec{V}\cdot{\vec{\nabla^{\prime}}}\right).$
(62)

The transformation to the second derivative:

$\frac{\partial^{2}}{\partial{t^{2}}}=\frac{\partial^{2}}{\partial{t^{\prime
2}}}-2\vec{V}\cdot\nabla^{\prime}\frac{\partial}{\partial{t^{\prime}}}+(\vec{V}).$
(63)

Substituting 63 in 56, we have

$\nabla^{\prime
2}\phi-\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial{t^{\prime
2}}}+\frac{2}{c^{2}}\vec{V}\cdot\nabla^{\prime}\frac{\partial\phi}{\partial{t^{\prime}}}-\frac{1}{c^{2}}\vec{V}\cdot(\vec{\nabla}^{\prime}\vec{V})\cdot\nabla^{\prime}\phi=0.$
(64)

We can see that the wave equation is not covariant against Galileo
transformations, in the case of sound waves, this non-covariance is associated
with the presence of a medium, so the extra term represents motion in relation
to the medium, however electromagnetic waves do not depend on a medium to
propagate, as demonstrated by the Michelson-Morlei experiment.

Having studied the Galileo transformation, we will now deduce a set of
transformations, which are compatible with the propagation of the
electromagnetic wave. Let us imagine that in a given reference frame $S$ the
propagation of a spherical wavefront is

$x^{2}+y^{2}+z^{2}-(ct)^{2}=0,$ (65)

and in another reference frame $S^{\prime}$, the wavefront is described by

$x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-c^{2}t^{\prime 2}=0.$ (66)

For simplicity, we can define that this signal is sent when the origin of both
references coincides with $O\equiv{O}^{\prime}$, in addition the reference
frame$S^{\prime}$ propagates with speed $V$ with respect to the reference
frame $S$. Let us assume that the transformations between the references are
linear. soon

$x^{\prime}=A(x-Vt)$ (67)

and

$t^{\prime}=Bt+Cx.$ (68)

These transformations must meet the following requirements

  1. (i)

A uniform rectilinear movement with respect to $S$ must also be rectilinear
and uniform with respect to $S^{\prime}$;

  2. (ii)

For $V=0$, ($\vec{V}\equiv{V}\hat{x}$), the transformation reduces to
identity; and,

  3. (iii)

When a light signal is sent from $O\equiv{O^{\prime}}$ in $t=t^{\prime}=0$ its
wavefront must propagate with $c$ in both reference frames.

With a simple inspection we verify that 67 and 68 obey those requirements, we
see that the linearity of the transformation guarantees that uniform movements
in a given reference frame are mapped in uniform movements in another
reference frame. We also notice that when $V=0$, we have $x=x^{\prime}$. This
would imply that $C\propto{V}$ and, for simplicity, we are considering that
the referential $S^{\prime}$ propagates in the $x$ direction, also implying
$y=y^{\prime}$, $z=z^{\prime}$.

Replacing 67 in 65

$x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-c^{2}t^{\prime
2}=A^{2}\left(x-Vt\right)^{2}+y^{2}+z^{2}-c^{2}\left(Bt+Cx\right)=0,$ (69)

where $y^{2}+z^{2}=(ct)^{2}-x^{2}$. We rewrite 69, highlighting $x^{2},t$ and
$t^{2}$,

$\left(A^{2}V^{2}-c^{2}B^{2}+c^{2}\right)x^{2}-2\left(A^{2}V+c^{2}BC\right)xt+\left(A^{2}V^{2}-c^{2}B^{2}+c^{2}\right)t^{2}=0.$
(70)

As $x,t$ are linearly independent quantities, we have that 70 only vanishes if
the coefficients vanish. Then we have the following system of three equations
and three variables

$\displaystyle A^{2}V+c^{2}BC$ $\displaystyle=0,$ $\displaystyle
A^{2}-c^{2}C^{2}$ $\displaystyle=1,$ $\displaystyle
B^{2}-\frac{V^{2}}{c^{2}}A^{2}$ $\displaystyle=1.$

Wich we can rewrite

$\displaystyle A^{2}$ $\displaystyle=-\frac{c^{2}}{V}BC,$
$\displaystyle-c^{2}\frac{C}{V}(B+VC)$ $\displaystyle=1,$ $\displaystyle
B^{2}+VBC$ $\displaystyle=B(B+VC)=1,$

and therefore that $A^{2}=B^{2}$. Going back in the system of equations we
have

$A^{2}\left(1-\frac{V^{2}}{c^{2}}\right)=1,$ (71)

where

$A=\pm{B}=\pm\gamma,$ (72)

and

$\gamma=\frac{1}{\sqrt{1-\frac{V^{2}}{c^{2}}}}.$ (73)

We still have

$C=-\frac{V}{c^{2}}\gamma,$ (74)

when arrive the Lorentz transformations

$\displaystyle x^{\prime}$ $\displaystyle=\gamma\left(x-Vt\right),$ (75)
$\displaystyle t^{\prime}$
$\displaystyle=\gamma\left(t-\frac{V}{c^{2}}x\right),$ (76) $\displaystyle
y^{\prime}$ $\displaystyle=y,$ (77) $\displaystyle z^{\prime}$
$\displaystyle=z.$ (78)

The inverse transforms

$\displaystyle x$ $\displaystyle=\gamma\left(x^{\prime}-Vt^{\prime}\right),$
(79) $\displaystyle t$
$\displaystyle=\gamma\left(t^{\prime}+\frac{V}{c^{2}}x^{\prime}\right),$ (80)

in the limit where $V<<c$, we reestablish the Galileo transformations. We can
now generalize the Lorentz transformation to any direction and study the
transverse effects of the Lorentz transformation,

$\hat{V}\equiv\frac{\vec{V}}{V},$ (81)

so in the direction parallel to the motion:

$\displaystyle\vec{x}_{\parallel}$
$\displaystyle=(\vec{x}\cdot\hat{V})\hat{V}=\frac{(\vec{x}\cdot\vec{V})\vec{V}}{V^{2}}$
(82) $\displaystyle\vec{x}_{\perp}$
$\displaystyle=\vec{x}-\vec{x}_{\parallel}=\vec{x}-(\vec{x}\cdot\hat{V})\hat{V},$
(83)

and similarly

$\displaystyle\vec{x^{\prime}}_{\parallel}$
$\displaystyle=\gamma(\vec{x}_{\parallel}-\vec{V}t)$ (84) $\displaystyle
x^{\prime}_{\perp}$ $\displaystyle=\vec{x}_{\perp}$ (85) $\displaystyle
t^{\prime}$
$\displaystyle=\gamma\left(t-\frac{V}{c^{2}}\vec{x}\cdot\hat{V}\right)=\gamma\left(t-\frac{\vec{V}\cdot\vec{x}}{c^{2}}\right).$
(86)

We take the opportunity to demonstrate the covariance of the wave equation
under the Lorentz transformations

$\displaystyle\frac{\partial}{\partial{x}}$
$\displaystyle=\frac{\partial{x^{\prime}}}{\partial{x}}\frac{\partial}{\partial{x^{\prime}}}+\frac{\partial{t^{\prime}}}{\partial{x}}\frac{\partial}{\partial{t^{\prime}}}$
(87) $\displaystyle\frac{\partial^{2}}{\partial{x^{2}}}$
$\displaystyle=\frac{\partial^{2}{x^{\prime}}}{\partial{x^{2}}}\frac{\partial}{\partial{x^{\prime}}}+\frac{\partial{x^{\prime}}}{\partial{x}}\frac{\partial^{2}}{\partial{x^{\prime
2}}}+\frac{\partial^{2}{t^{\prime}}}{\partial{x^{2}}}\frac{\partial}{\partial{t^{\prime}}}+\frac{\partial{t^{\prime}}}{\partial{x}}\frac{\partial^{2}}{\partial{x}\partial{t}},$
(88)

using the transformations 75 and 76

$\displaystyle\frac{\partial{x^{\prime}}}{\partial{x}}$
$\displaystyle=\gamma,$ (89)
$\displaystyle\frac{\partial{t}}{\partial{x^{\prime}}}$
$\displaystyle=\frac{V}{c^{2}}\gamma$ (90)
$\displaystyle\frac{\partial^{2}x^{\prime}}{\partial{x^{2}}}$
$\displaystyle=0=\frac{\partial^{2}t^{\prime}}{\partial{x^{2}}},$ (91)

we rewrite 88

$\frac{\partial^{2}}{\partial{x^{2}}}=\gamma\frac{\partial^{2}}{\partial{x^{\prime
2}}}-\frac{V\gamma}{c^{2}}\frac{\partial^{2}}{\partial{x^{\prime}}\partial{t}}.$
(92)

Following the same reasoning for the time derivative

$\frac{\partial}{\partial{t}}=\frac{\partial{t^{\prime}}}{\partial{t}}\frac{\partial}{\partial{t^{\prime}}}+\frac{\partial{x^{\prime}}}{\partial{t}}\frac{\partial}{\partial{x^{\prime}}},$
(93)

the second derivative:

$\frac{\partial^{2}}{\partial{t^{2}}}=\frac{\partial^{2}t^{\prime}}{\partial{t^{2}}}\frac{\partial}{\partial{t^{\prime}}}+\frac{\partial{t^{\prime}}}{\partial{t}}\frac{\partial^{2}}{\partial{t^{\prime
2}}}+\frac{\partial^{2}x^{\prime}}{\partial{t}^{2}}\frac{\partial}{\partial{x^{\prime}}}+\frac{\partial{x^{\prime}}}{\partial{t}}\frac{\partial^{2}}{\partial{t}\partial{x^{\prime}}}.$
(94)

Using once again the transformations 75 and 76 we have

$\frac{\partial^{2}}{\partial{t^{2}}}=\gamma\frac{\partial^{2}}{\partial{t^{\prime
2}}}-\gamma{V}\frac{\partial^{2}}{\partial{t}\partial{x^{\prime}}},$ (95)

subtracting 92 and 95, we get the wave equation

$\frac{\partial^{2}}{\partial{x^{2}}}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial{t^{2}}}=0,$
(96)

and we thus demonstrate the covariance of the D’Lambertian concerning to the
Lorentz transforms.

##  V The relativistic law for the composition of velocities

Let’s now calculate the instantaneous velocity of the particle, considering
$x^{\prime}=x^{\prime}(t)$, $y^{\prime}=y^{\prime}(t)$,
$z^{\prime}=z^{\prime}(t)$, we can write the velocity $v^{\prime}(t)$ in the
reference $S^{\prime}$.

$v^{\prime}_{x}=\frac{dx^{\prime}}{dt^{\prime}},v^{\prime}_{y}=\frac{dy^{\prime}}{dt^{\prime}},v^{\prime}_{z}(t)=\frac{dz}{dt}$
(97)

the velocity in the $S$ reference, we have:

$v_{x}=\frac{dx}{dt},v_{y}=\frac{dy}{dt},v_{z}=\frac{dz}{dt}$ (98)

where $x^{\prime}(t)$, $y^{\prime}(t)$, $z^{\prime}(t)$ are related to $x(t)$,
$y(t)$, $z(t)$, making the velocity reference frame parallel to the $x$ axis,
or $\vec{x}\parallel\vec{V}$, we write

$dx^{\prime}=\gamma(dx-
Vdt),dt^{\prime}=\gamma(dt-\frac{V}{c^{2}}dx),dy^{\prime}=dy,dz^{\prime}=dz.$
(99)

Thus, we deduce the velocity transformations for $x$, $y$ and $z$ components,
respectively:

$\displaystyle v^{\prime}_{x}$
$\displaystyle=\frac{dx^{\prime}}{dt^{\prime}}=\frac{dt\left(\frac{dx}{dt}-V\right)}{dt\left(1-\frac{V}{c^{2}}\frac{dx}{dt}\right)}=\frac{v_{x}-V}{1-\frac{v_{x}V}{c^{2}}},$
(100) $\displaystyle v^{\prime}_{y}$
$\displaystyle=\frac{dy^{\prime}}{dt^{\prime}}=\frac{dy}{\gamma{dt}\left(1-\frac{V}{c^{2}}\frac{dx}{dt}\right)}=\frac{\sqrt{1-\frac{V^{2}}{c^{2}}}v_{y}}{1-\frac{v_{x}V}{c^{2}}},$
(101) $\displaystyle v^{\prime}_{z}$
$\displaystyle=\frac{dz^{\prime}}{dt^{\prime}}=\frac{dz}{\gamma{dt}\left(1-\frac{V}{c^{2}}\frac{dx}{dt}\right)}.$
(102)

##  VI Space-time interval

In relativity the spatial intervals are replaced by space-temporal intervals,
because time becomes an axis, a coordinate as well as space. Therefore to
describe an event in space-time, we need coordinates. To describe a point in
space, thinking in two dimensions, we can build a space-time interval, which
is an invariant quantity of relativity

$\delta{s}^{2}=(s_{1}-s_{0})^{2}=(x_{1}-x_{0})^{2}-c^{2}(t_{1}-t_{0})^{2},$
(103)

being the same in any inertial frame of reference being

$\delta{s}^{2}<0,$ (104)

wich we call time-type interval. This interval is related to events which are
connected by a causal relationship, the signals that link these events are
slower than light. We call it time interval. If we have

$\delta{s^{2}}=-c^{2}\delta\tau^{2}.$ (105)

We may have a light-type interval,

$\delta{s}^{2}=0,$ (106)

where the signal connecting the two events is a light signal. this is the
boundary of the causal region.

Finally, if we have the interval

$\delta{s}^{2}>0,$ (107)

this type of interval is no longer a causal interval, so the events placed by
these signals cannot be related as cause and effect, space-like region.

These properties define what we call the cone of light (see the Fig. 1), which
is the pictorial representation of the causal structure of relativity.

Figure 1: The Cone of Light represents the causal structure of Special
Relativity. The inner region represents the events, which have causal
connection (time-like region). The signals are at the surface region of the
cone, they propagate with the speed of light, therefore the limit of the
causal zone (light-like region). In the outer zone (space-like region) the
events have no causal connection, as they would be connected by supraluminal
signals.

.

###  VI.1 Inertial References in Special Relativity

It is already known that, from the relativistic view, space and time are
relativistic and interrelated quantities; therefore, the variation in one of
them (eg time) in a given referential $S^{\prime}$ – which moves with respect
to another referential $S$ – implies the variation of the other (eg space).

Based on the Minkowski formulation, although neither time nor space is fixed
(absolute), there is a large invariant which is the “interval four-dimensional
space-time $ds$”. This geometric formulation of relativity helps us to realize
that, in fact, there is a kind of ’balancing’ between categories of space and
time such that $ds$ remains invariant in any inertial frame of reference, even
though space and time transform separately, with the change of reference .
This invariant element $ds$ is called a four-dimensional line element in
Minkowki space and is written as:

  * •

in the reference frame $S$

$ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2},$ (108)

  * •

and in the reference frame $S^{\prime}$

$ds^{\prime 2}=c^{2}dt^{\prime 2}-dx^{\prime 2}-dy^{\prime 2}-dz^{\prime 2}.$
(109)

The frame $S^{\prime}$ moves with velocity $v$ relative to the frame $S$, and
the time and space measurements change from $S^{\prime}$ to $S$ and vice
versa; however it comes that

$ds^{2}=ds^{\prime 2},$ (110)

because this space-temporal interval is kept invariant by referential changes.

To better exemplify the power of synthesis of this elegant Minkowki geometric
formulation, consider an observer in the frame $S$ (in rest) and another one
at the origin $O^{\prime}$ of the moving frame $S^{\prime}$, i.e.
$dx^{\prime}=dy^{\prime}=dz^{\prime}=0$. So, let’s write the following:

$ds^{\prime 2}=c^{2}d\tau^{2}.$ (111)

In the reference frame $S^{\prime}$ the observer considers himself to be at
rest, as he does not perceive his velocity in relation to the reference $S$.
While $dt$ represents an improper time interval measured in $S$, where
$dx^{2}+dy^{2}+dz^{2}=dr^{2}=v^{2}dt^{2}$. In fact, we have $dt\geq{d\tau}$.
As $v$ approaches $c$, we can show that $dt>>d\tau$, which represents an
intense time dilation. Comparing the equations 108 and 110 based on 109, we
get that:

$c^{2}d\tau^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2},$ (112)

or yet

$c^{2}dt^{2}=c^{2}d\tau^{2}+dx^{2}+dy^{2}+dz^{2}.$ (113)

Dividing by $dt^{2}$, we have

$c^{2}\left(\frac{d\tau}{dt}\right)^{2}+\left(\frac{dr}{dt}\right)^{2}=c^{2}.$
(114)

The relation 112 needs a closer analysis. From it we get

$\left(\frac{d\tau}{dt}\right)^{2}=1-\frac{\left(\displaystyle{\frac{dr}{dt}}\right)^{2}}{c^{2}}=\left(1-\frac{v^{2}}{c^{2}}\right)$
(115)

then

$dt=\frac{d\tau}{\sqrt{1-\frac{v^{2}}{c^{2}}}},$ (116)

which represents time dilation. So, if $v$ approaches $c$, it follows that
$dt>>d\tau$, which means that proper time (in the frame $S$), passes much
slower than the improper time $dt$ measured in a frame $S$ “stopped”.

Through the equation 112 above, we will draw an important conclusion about the
compensation (balancing), which exists between certain spatial and temporal
quantities in such a way that, an invariance will be found. Let’s look at the
following terms in the equation 112:

  1. (i)

$\displaystyle{\left(\frac{dr}{dt}\right)^{2}=v^{2}}$, represents the square
of the three-dimensional velocity of the particle;

  2. (ii)

$c^{2}\left(\displaystyle{\frac{d\tau}{dt}}\right)^{2}=c^{2}\left(1-\displaystyle{\frac{v^{2}}{c^{2}}}\right)=c^{2}-v^{2}$,
this term has an abstract connotation. Interpreted as a temporal velocity,
given in the fourth dimension – time; that is, it is a speed of time, or even
a spatial analog for the degree of slowness or speed with which time flows for
a moving particle. Note that $c^{2}\left({d\tau}/{dt}\right)^{2}$ has the
dimension of a squared space velocity itself.

Now, we can easily observe that when we add the two terms, that is,
$(i)+(ii)=v^{2}+(c^{2}-v^{2})=c^{2}$, then we always get a constant or
invariant value, which is $c^{2}$ by actually checking the relation 112, that
is, we have $v^{2}_{t}+v^{2}_{l}=c^{2}$, where the time velocity
$v_{t}=c\sqrt{1-\frac{v^{2}}{c^{2}}}$ and $v_{l}$ is the space velocity. With
this we can see more clearly that the sum of the squares of the space-time
velocity is always constant $c^{2}$ although each of them may vary from one
frame of reference to another. This invariance is represented by $c$, where

$c=\sqrt{v^{2}_{t}+v^{2}_{l}}=\sqrt{c^{2}\left(\frac{d\tau}{dt}\right)^{2}+v^{2}}.$
(117)

The equation 117 means that all objects or particles in any frame of reference
always have a space-time velocity $c$. In this sense, we can also think as the
particles in our universe are on a kind of equipotential surface
$\Phi_{u}\equiv{c^{2}}$ with energy $E=m\Phi_{u}\equiv{mc^{2}}$, or yet using
112, we can write

$E=mc^{2}\left(\frac{d\tau}{dt}\right)^{2}+m\left(\frac{dr}{dt}\right)^{2}\equiv{mc^{2}}.$
(118)

##  VII Fluctuations in barotropic fluids $p=p(\rho)$ under Lorentz symmetry

Matt Visser’s work (Visser, 1993), detecting that Lorentz symmetry could be
extended to barotropic fluids, was seminal, his conclusions gave rise to the
field of study of gravity analog systems, see Unruh (1981), and Visser (1993).
Novello, Visser and Volovik even applied this analogy to black holes Novello
_et al._ (2002), where they even modeled ergo-regions. Let’s now follow Unruh
(1981), Visser (1993), Novello  _et al._ (2002) to review the concept of
acoustic geometry.

The speed with which an oscillation travels through a given spatial interval
$ds$ is related to the time interval spent $dt$ as follows.

$c(\vec{x})=\frac{ds}{dt},$ (119)

rewriting for the space interval

$ds=c(\vec{x})dt,$ (120)

where the pulse speed is given according to a property of the medium in which
it propagates

$c(\vec{x})=\frac{c_{0}}{n(\vec{x})}.$ (121)

The quantity $n(\vec{x})$ represents the refractive index of the medium. Thus,
the time spent traveling through the pulse $\gamma$, is given by

$T_{\gamma}=\int\frac{ds}{c(\vec{x})}$ (122)

replacing 119 in 122 we have

$T_{\gamma}=\int\frac{ds}{c_{0}}n(\vec{x})$ (123)

by Fermat’s principle, the states of the pulse that crosses the medium
$n(\vec{x})$ are given by the minimization of the following integral , which
we call the optical path

$L_{\gamma}=\int{dl}=\int{ds}n(\vec{x})$ (124)

starting from this minimization we can then define the trajectory of a
particle in the fluid, this trajectory we will call geodesic, for that, we
will define a tensor $3\times 3$ called metric

$g_{ij}=n^{2}(\vec{x})\delta_{ij}$ (125)

where $\delta_{ij}$ is a Kronecker delta $i,j=1,2,3$. We have redefined 124,
now we write the length differential associated with the metric

$dl^{2}=g_{ij}(\vec{x})dx^{i}dx^{j}$ (126)

replacing 125 in 126 we have:

$dl^{2}=n^{2}(\vec{x}(s))\delta_{ij}dx^{i}dx^{j}=n^{2}(\vec{x}(s))ds^{2}$
(127)

we then rewrite 124

$L_{\gamma}=\int{dl}=\int{ds}\sqrt{g_{ij}(\vec{x}(s))\frac{dx^{i}}{ds}\frac{dx^{j}}{ds}}$
(128)

minimizing 128, we have the geodesic, which is the path that a free particle
follows in the fluid, or even an expression of Newton’s second law when the
resulting forces are zero.

$\frac{d}{ds}\left[\frac{g_{ij}(\vec{x}(s))\frac{dx^{i}}{ds}}{\sqrt{g_{kl}\frac{dx^{k}}{ds}\frac{dx^{l}}{ds}}}\right]=\frac{\partial{g}_{kl}(\vec{x})}{\partial{x^{i}}}\frac{dx^{k}}{ds}\frac{dx^{l}}{ds}\left(g_{mn}(\vec{x})\frac{dx^{m}}{ds}\frac{dx^{n}}{ds}\right)^{-\frac{1}{2}}$
(129)

###  VII.1 The causal structure

Considering the velocity measured by an observer in the laboratory Novello
_et al._ (2002), we have

$\frac{d\vec{x}}{dt}=c\vec{n}+\vec{v},$ (130)

taking $\vec{n}$ as a unit vector and making the squared modulus of 130, we
write

$\frac{d\vec{x}^{2}}{dt^{2}}=c^{2}+2cv+v^{2},$ (131)

rearranging the terms

$d\vec{x}\cdot{d\vec{x}}-\left(c^{2}+2cv+v^{2}\right)dt^{2}=0,$ (132)

or

$\left(d\vec{x}-\vec{v}dt\right)^{2}-c^{2}dt^{2}=0.$ (133)

This is the geometry where the phonons propagate with velocity $c$, the fluid
flows with velocity $\vec{v}=\vec{\nabla}\theta$, a geometry indistinguishable
from the Lorentz geometry, where we write the element of line

$ds^{2}_{ac}=\frac{\rho}{c}\left[-c^{2}dt^{2}+||d\vec{x}-\vec{v}dt||^{2}\right]$
(134)

from where we can deduce the acoustic metric

$\mathcal{G}^{\mu\nu}=\frac{\rho}{c}\left(\begin{array}[]{cc}c^{2}-v^{2}&-\vec{v}\\\
-\vec{v}&\mathcal{I}\end{array}\right).$ (135)

The metric 134 corresponds to the solution of the Klein Gordon equation

$\Delta\psi=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\psi\right)=0.$
(136)

A curious property of the acoustic metric 134 is that the term
$||{d\vec{x}-\vec{v}dt}||$ is a module, so $d\vec{x}-\vec{v}dt>0$, implying
that ${d\vec{x}}/{dt}>\vec{v}$. If we have $\vec{v}=0$ the sound propagating
in a static fluid follows

$ds^{2}=\frac{\rho}{c}-c^{2}dt^{2}+dx^{2},$ (137)

then recovering the Lorentz metric, this would be known as standing waves. In
another situation, if $v=c$, we have that $ds^{2}=||d\vec{x}||^{2}$, this is
precisely the limit where the fluid does not allow sound propagation, the
metric then coincides with the trivector modulus, being a crystallized fluid,
actually a solid. On the other hand, the case in which $c<v$, we perceive the
fluid flow dragging the sound waves, this would be the case that we would call
_ergoregion_ Novello  _et al._ (2002); Baines  _et al._ (2021), where rest is
impossible, not there would be, in this case, standing waves.

Phonon rays propagate according to the following geodesic:

$\left(c^{2}-v^{2}\right)\left(\frac{dt}{ds}\right)^{2}-2v_{i}\left(\frac{dx^{i}}{ds}\right)\left(\frac{dt}{ds}\right)+1=0$
(138)

solving the quadratic equation 138, we have:

$\frac{dt}{ds}=\frac{-v_{i}\displaystyle{\frac{dx^{i}}{ds}}+\sqrt{c^{2}-v^{2}+||v_{i}\partial{dx^{i}}{ds}||^{2}}}{c^{2}-v^{2}}$
(139)

The equation 139 is the effective geodesic of a phonon, which propagates in
the fluid. Therefore, causality properties of Lorentzian geometry can be found
in material media, thus generating an interdiscipline point of contact, which
can help to understand the propagation of signals in media where the
gravitational field is strong, that is, it can help to better understand the
gravitational collapse.

Considering the metric 134, we write

$ds^{2}=\frac{\rho}{c^{2}}\left[-\left(c^{2}+v^{2}\right)dt^{2}+dx^{2}-vdxdt\right]$
(140)

doing in 140

$\phi=-\frac{v^{2}(r)}{2}=\frac{2GM}{r},$ (141)

which corresponds to a vortex in a fluid, and we will have

$ds^{2}=\rho\left[\left(1-\frac{2GM}{rc^{2}}\right)dt^{2}+dx^{2}-\sqrt{\frac{2MG}{rc^{2}}}dxdt\right]$
(142)

The equation 142 is a Paileve-Gusland metric Volovik (1999), a Lense-Thirring
spacetime variant Baines  _et al._ (2021); Visser (2007).

$ds^{2}_{LH}=-dt^{2}+\left[dr+\sqrt{\frac{2mG}{r}}dt\right]^{2}+r^{2}\left[d\theta^{2}+\sin^{2}(\theta)\left(d\phi-\frac{2J}{r^{3}}dt\right)^{2}\right],$
(143)

for the case $j=0$, being a model for the surroundings of stars and planets.
The metric 142 represents a black hole mimicker in the case of velocity
$v^{2}=c^{2}{r_{h}}/{r}$, we have that $r_{h}$ is the Schwarszchild radius
where the event horizon is formed. This type of system is a good toy model for
simulating quantum effects in the vicinity of a black hole. Being a
contribution of hydrodynamics to the advancement of a future theory that is
able to unify quantum and gravitational phenomena.

##  VIII The Maxwell Tensor

We can define auxiliary fields, called electromagnetic potentials $\Phi$ and
$\vec{A}$. Then we write the electric and magnetic fields in terms of
potentials,

$\vec{E}=-\vec{\nabla}\Phi-\frac{1}{c}\partial_{t}\vec{A}$ (144)
$\vec{B}=\vec{\nabla}\times\vec{A}.$ (145)

These potentials play an important role for example in the Aharonov-Bohm
effect. We can unify the fields and build the tensor formalism. We then define
the potential quadrivector

$A^{\mu}=(A^{0},A^{1},A^{2},A^{3})$ (146)

where $A^{0}=\Phi$, and $A^{1},A^{2},A^{3}$, are the components of the
potential vector, also unifying the derivatives
$\partial_{\mu}=(c\partial_{t},\vec{\nabla})$, or $\vec{\nabla}=\partial^{i}$
where $i=1,2,3$.

We can define the Maxwell tensor

$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.$ (147)

Explicitly we write

$F^{0i}=E^{i}=-\partial^{0}A^{i}-\partial^{i}A^{0}$ (148)
$F^{ij}=\partial^{i}A^{j}-\partial^{j}A^{i}=B_{k}\epsilon^{ijk}.$ (149)

We finally write the matrix associated with the Maxwell tensor

$F^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}0&E^{1}&E^{2}&E^{3}\\\
-E^{1}&0&B^{3}&B^{2}\\\ -E^{2}&-B^{3}&0&B^{1}\\\ -E^{3}&-B^{2}&-B^{1}&0\\\
\end{array}\right].$ (150)

Knowing the continuity equation

$\vec{\nabla}\cdot\vec{j}+c\partial_{t}\rho=0$ (151)

where $\vec{j},\rho$ are the vector current density and the charge, we can
generalize this current (151) using the continuity equation 151

$\partial_{\mu}J^{\mu}=0,$ (152)

then we have the quadridivergence of the quadri-current
$J^{\mu}=(\rho,\vec{j})$. If we act with the derivative operator on the
Maxwell tensor

$\partial_{\mu}F^{\mu\nu}=\mu_{0}J^{\nu}$ (153)

more explicitly if we act
$\partial_{i}F^{0i}=\partial_{i}E^{i}=\nabla{A}^{0}=\frac{\rho}{\epsilon_{0}}$,
and yet, $\partial_{i}F^{ij}=\mu_{0}j^{j}$ where $J^{j}=\vec{j}$.

##  IX Momentum-energy tensor: relativistic fluid

Let’s define the action following Santos  _et al._ (2020); Landau and Lifshitz
(2013)

$S=\int\xi\left(q,\frac{\partial{q}}{\partial{x^{i}}}\right)dVdt=\frac{1}{c}\int\xi{d\Omega}$
(154)

where, $q_{,i}\equiv{\partial{q}}/{\partial{x_{i}}}$ are generalized
variables.

Minimizing the action (154)

$\displaystyle\delta{S}$
$\displaystyle=\frac{1}{c}\int\left(\frac{\partial\xi}{\partial{q}}\delta{q}+\frac{\partial\xi}{\partial{q_{,i}}}\delta{q_{,i}}\right)d\Omega$
$\displaystyle=\frac{1}{c}\int\left[\frac{\partial\xi}{\partial{q}}\delta{q}\frac{\partial}{\partial{x^{i}}}\left(\frac{\partial\xi}{\partial{q_{,i}}}\right)-(\delta{q})\frac{\partial}{\partial{x^{i}}}\frac{\partial\xi}{\partial{q_{,i}}}\right]d\Omega=0,$
(155)

the second term cancels out on the integration in the whole space. We can then
write the equation of motion

$\frac{\partial}{\partial{x^{i}}}\frac{\partial\xi}{\partial{q_{i}}}-\frac{\partial\xi}{\partial{q}}=0,$
(156)

here we assume the sum over repeated indices. Now following a procedure
similar to the one used to verify the conservation of energy

$\frac{\partial\xi}{\partial{x^{i}}}=\frac{\partial\xi}{\partial{q}}\frac{\partial{q}}{\partial{x^{i}}}+\frac{\partial\xi}{\partial{q_{,k}}}\frac{\partial{q_{,k}}}{\partial{x^{i}}},$
(157)

substituting in the equation of motion, considering that $q_{,k,i}=q_{,i,k}$,
we have:

$\frac{\partial\xi}{\partial{x^{i}}}=\frac{\partial}{\partial{x^{k}}}\left(\frac{\partial\xi}{\partial{q_{,k}}}\right)q_{,i}+\frac{\partial\xi}{\partial{q_{,k}}}\frac{\partial{q_{,i}}}{\partial{x^{k}}}=\frac{\partial}{\partial{x^{k}}}\left(q_{,i}\frac{\partial\xi}{\partial{q_{,k}}}\right)$
(158)

using the following property

$\frac{\partial\xi}{\partial{x^{i}}}=\delta^{k}_{i}\frac{\partial\xi}{\partial{x^{k}}}.$
(159)

The momentum-energy tensor in terms of the canonical variables is

$T^{k}_{i}=q_{,i}\frac{\partial\xi}{\partial{q_{,i}}}-\delta^{k}_{i}\xi.$
(160)

Following this form of the momentum-energy tensor, we can write the simplest
form, by Landau and Lifshitz (2013); Carroll (2004),

$T^{\mu\nu}=U^{\mu}P^{\nu},$ (161)

where

$U^{\mu}=c\sqrt{1-\frac{v^{2}}{c^{2}}}\left[1,\frac{v^{\alpha}}{c}\right]$
(162)

is the four-speed of an observer moving along with the fluid and
$P^{\mu}=m_{0}cU^{\mu}$ is the quadri-momentum. We usually specify the
components of the momentum-energy tensor and also define them nominally. We
have the normalization

$g_{\mu\nu}U^{\mu}U^{\nu}=1,$ (163)

the energy density

$T^{00}=\rho,$ (164)

the energy flow

$T^{i0}=cP^{i},$ (165)

and the tension tensor

$T^{ij}=u^{i}p^{i},$ (166)

here Latin indices limit if $i,j=1,2,3$, Greek indices to $\mu,\nu=0,1,2,3$.
The momentum-energy tensor is a symmetric matrix, that is,

$T^{\mu\nu}=T^{\nu\mu}.$ (167)

Relating the momentum-energy tensor energy momentum to the kinematic structure
of special relativity, we have

$T^{\mu\nu}=m_{0}c\sqrt{1-\frac{v^{2}}{c^{2}}}U^{\mu}U^{\nu}.$ (168)

The conservation of the momentum tensor energy has a fundamental consequence,

$\partial_{\mu}T^{\mu\nu}=m_{0}c^{2}(\partial_{\mu}U^{\mu})U^{\nu}=0.$ (169)

We can define the four-force as

$f^{\mu}=\partial_{t}(mU^{\mu}).$ (170)

The Lorentz force is also a four-vector

$f^{\nu}=\frac{q}{c}U_{\mu}F^{\mu\nu},$ (171)

by Newton’s law we arrive that

$m_{0}c\partial_{t}U^{\mu}=\frac{q}{c}U_{\mu}F^{\mu\nu},$ (172)

replacing 172 in 169, we have

$\partial_{\mu}T^{\mu\nu}=\frac{q}{c}U_{\mu}F^{\mu\nu},$ (173)

the four-current can be written as $J^{\mu}=qU^{\mu}$, and we have then

$\partial_{\mu}T^{\mu\nu}=\frac{1}{c}J_{\mu}F^{\mu\nu}.$ (174)

By the equation 153 we arrive that

$\partial_{\mu}T^{\mu\nu}=\frac{\mu_{0}}{c}(\partial^{\alpha}F_{\alpha\mu})F^{\mu\nu},$
(175)

using product derivative properties

$\displaystyle\partial^{\alpha}(F_{\alpha\mu}F^{\mu\nu})$
$\displaystyle=(\partial^{\alpha}F_{\alpha\mu})F^{\mu\nu}+F_{\alpha\mu}\partial^{\alpha}F^{\mu\nu}$
(176) $\displaystyle\partial^{\alpha}(F_{\alpha\mu}F^{\mu\nu})$
$\displaystyle-
F_{\alpha\mu}\partial^{\alpha}F^{\mu\nu}=(\partial^{\alpha}F_{\alpha\mu})F^{\mu\nu},$
(177)

and evoking the symmetry property

$\displaystyle(\partial^{\alpha}F_{\alpha\mu})F^{\mu\nu}$
$\displaystyle=\left[\partial^{\alpha}(F^{\mu\nu}F_{\alpha\mu})-F_{\alpha\mu}\partial^{\alpha}F^{\mu\nu}\right]$
$\displaystyle F_{\alpha\mu}\partial^{\alpha}F^{\mu\nu}$
$\displaystyle=\frac{1}{2}(F_{\alpha\mu}\partial^{\alpha}F^{\mu\nu}+F_{\mu\alpha}\partial^{\mu}F\nu\alpha)$
$\displaystyle=\frac{1}{2}F_{\alpha\mu}(\partial^{\alpha}F^{\alpha\mu}+\partial^{\mu}F^{\alpha\mu})=-\frac{1}{2}F_{\alpha\mu}\partial^{\alpha}F^{\mu\alpha}$
$\displaystyle=-\frac{1}{4}\partial^{\nu}(F_{\alpha\mu}F^{\mu\alpha}),$

we have

$\frac{1}{c}F^{\mu\nu}J_{\mu}=\frac{1}{4}\left[\partial(F^{\alpha\mu}F_{\alpha\mu})-\frac{1}{4}\partial^{\nu}(F_{\mu\alpha}F^{\mu\alpha})\right],$
(178)

then

$\partial_{\mu}T^{\mu\nu}=\frac{1}{4\pi}\partial_{\alpha}\left(F^{\nu\mu}F^{\alpha}_{\mu}-\frac{1}{4}g^{\nu\alpha}F_{\mu\rho}F^{\mu\rho}\right).$
(179)

Finally the energy-momentum tensor for the electromagnetic field is

$T^{\nu\alpha}_{em}=\frac{1}{4\pi}\left(-F^{\nu\mu}F^{\alpha}_{\mu}-\frac{1}{4}g^{\nu\alpha}F_{\mu\rho}F^{\mu\rho}\right).$
(180)

Operating with $g_{\nu\alpha}$, we easily verify that the trace of this
electromagnetic tensor is null. In a matrix, we write the energy-momentum
tensor as

$T^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}T^{00}&T^{01}&T^{02}&T^{03}\\\
T^{10}&T^{11}&T^{12}&T^{13}\\\ T^{20}&T^{21}&T^{22}&T^{23}\\\
T^{30}&T^{31}&T^{32}&T^{33}\\\ \end{array}\right].$ (181)

Knowing the antisymmetry property of Maxwell’s tensor
$F^{\mu\nu}=-F^{\nu\mu}$, easily verifiable in 150, applying this same
property in 180 we notice that $T^{\mu\nu}=T^{\nu\mu}$. The components
$T^{ij}$ with $i,j=1,2,3$ form the so-called tension tensor. The components
$T^{0i}$

$T^{0i}_{em}=-\frac{1}{4\pi}F^{0\mu}F^{i}_{\mu},$ (182)

is. for the Poyting vector $\vec{S}=\vec{E}\times\vec{B}$, the directional
density of energy propagation.

##  X Isotropic fluid

Inspired by 160 we define the momentum-energy tensor of an ideal fluid as
general as possible (without anisotropy) Carroll (2001) as

$T^{\mu\nu}=(\rho+p)U^{\mu}U^{\nu}+pg^{\mu\nu}$ (183)
$T^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}\rho&0&0&0\\\ 0&-p&0&0\\\
0&0&-p&0\\\ 0&0&0&-p\\\ \end{array}\right].$ (184)

The momentum-energy tensor trace is

$T=\rho-3p,$ (185)

and the energy density source is given by

$\rho=T^{\mu\nu}U_{\mu}U_{\nu}.$ (186)

We can set the projector $P_{\mu\nu}=g_{\mu\nu}-U_{\mu}U_{\nu}$, so we find
the pressure

$-\frac{1}{3}P_{\mu\nu}T^{\mu\nu}=p,$ (187)

and the momentum-energy tensor can be rewritten in projector terms

$T^{\mu\nu}={\rho}U^{\mu}U^{\nu}+pP^{\mu\nu}.$ (188)

We also have cases of a momentum-energy tensor

$T^{\mu\nu}=\rho{U^{\mu}U^{\nu}},$ (189)

which is called a dust momentum-energy, in this case we have pressure $p=0$. A
radiation fluid is described by

$T^{\mu\nu}=p(U^{\mu}U^{\nu}+g^{\mu\nu}),$ (190)

and we have an equation of state $p=3\rho$. We then define the barotropic
factor

$\omega=\frac{p}{\rho},$ (191)

where this constant factor assumes values according to the type of matter
Carroll (2004), see Table 1.

Table 1: Barotropic factor. matter | $\omega$
---|---
ordinary | $0$
radiation | $\frac{1}{3}$
curvature | $-\frac{1}{3}$
vacuum | $-1$

No imposition on the momentum-energy tensor is made _a priori_ , Santos  _et
al._ (2020). Some fluids are repulsive, others may violate causality or are
ultra-relativistic. These energy conditions classify fluids according to these
criteria that we will address next. Let a given time like vector be

$\displaystyle t^{\mu}$ $\displaystyle=\gamma(1,a,b,c),$ (192)
$\displaystyle\gamma$ $\displaystyle=\frac{1}{1-a^{2}-b^{2}-c^{2}},$ (193)

with $g_{\alpha\beta}t^{\alpha}t^{\beta}$ so that $a^{2}+b^{2}+c^{2}<1$, we
have

$T_{\mu\nu}U^{\mu}U^{\nu}\geq 0$ (194)

and for a null vector $l^{\mu}=(1,a^{\prime},b^{\prime},c^{\prime})$
,$1=a^{\prime 2}+b^{\prime 2}+c^{\prime 2}$, we have

$T_{\mu\nu}l^{\mu}l^{\nu}\geq 0.$ (195)

We deduce from this that

$\rho=T^{\mu\nu}U_{\mu}U_{\nu},T_{\mu\nu}l^{\mu}l^{\nu}=(\rho+p)(U_{\mu}l^{\mu})^{2}.$
(196)

This implies that $\rho\geq 0$ and $(\rho+p)\geq 0$. Then arises the so-called
_Weak Energy Condition_ that we will discuss next.

###  X.1 Weak Energy Condition

Let’s analyze the weak energy condition

$T_{\mu\nu}t^{\mu}t^{\nu}=(\rho+a^{2}p+b^{2}p+c^{2}p)\geq 0,$ (197)

if we do $a=b=c=0$ we have $\rho\geq 0$, alternatively we do two of the null
constants, for example $b=c=0$ and $a=1$ we have $(\rho+p)\geq 0$. We write
then

$\rho\geq 0;(\rho+p)\geq 0.$ (198)

We can still do

$\frac{\rho+p}{\rho}=1+\frac{p}{\rho},$ (199)

barotropic fluids are known from the equation of state $p=\omega\rho$, so we
rewrite the weak energy condition

$\omega\geq-1.$ (200)

This condition is associated with the causality of fluid flow. Therefore, the
fluid flows respecting the Carroll (2001) light cone. There is a weaker
version of this energy condition, which we will now address.

###  X.2 No energy condition

Proceeding in the same way as for the weak energy condition Carroll (2001),
but using light-like vectors, like this

$T_{\mu\nu}l^{\mu}l^{\nu}\geq 0,$ (201)

where

$T_{\mu\nu}l^{\mu}l^{\nu}=\rho+a^{\prime 2}p+b^{\prime 2}p+c^{\prime 2}p\geq
0,$ (202)

if we do $a^{\prime 2}+b^{\prime 2}+c^{\prime 2}=1$, then

$\rho+p\geq 0.$ (203)

This condition admits negative density $\rho<0$, in cases of ultra-
relativistic fluids, so $l^{\mu}l_{\mu}=0$, we use light-like vectors.

###  X.3 Dominant energy condition

An observer with four-velocity $U^{\mu}$, will see a four-velocity
$-T^{\mu}_{\nu}U^{\nu}$, that is, we have that $-T^{\mu}_{\nu}U^{\nu}$ cannot
be space-like, which is equivalent to saying
$T_{\mu\nu}T^{\nu}_{\lambda}t^{\mu}t^{\lambda}\leq 0$ then we have

$T_{\mu\nu}t^{\mu}t^{\nu}\geq
0;T_{\mu\nu}T^{\nu}_{\lambda}t^{\mu}t^{\lambda}\leq 0.$ (204)

From $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$, we get $\rho\geq 0$. The current quad
is not a space-like vector, which implies that
$\gamma^{2}\left(-\rho^{2}+(a^{2}+b^{2}+c^{2})p^{2}\right)$, if we make
$b=c=0$, we are left with $\rho^{2}{\geq}a^{2}p^{2}$ and $a<1$, this implies
that $\rho\geq|p|$. We finally write

$\rho\geq 0;\rho\geq|p|.$ (205)

The energy conditions above are not the only ones, we are going to state two
more energy conditions, which are associated with the presence of
gravitational field. We will do this after some considerations about the
momentum-energy tensor in curved spaces.

##  XI The Riemman tensor and the covariant derivative

The Riemmann tensor is the mathematical entity that measures the curvature in
a Riemanian or pseudo-Riemanian manifold in the case of manifolds used in
general relativity, Carroll (2004). Let any 4-$v^{\alpha}$ be, in the absence
of torsion

$[\nabla_{\mu},\nabla_{\nu}]v^{\alpha}=R^{\alpha}_{\mu\nu\beta}v^{\beta},$
(206)

where $[\nabla_{\mu},\nabla_{\nu}]$ is the covariant derivative commutator. In
a curved space-time it is not possible to transport a 4-vector parallel to
itself, because the “lines” are generalized and are called geodesics, so at
each point the curvature of space causes the vector to be modified, this
modification then needs to be computed when we transport a 4-vector in
parallel, the dot product is conserved. Therefore we can think of a dot
product between a generic 4-vector $U^{\mu}$ and a base $e_{\mu}$,

$U=\mathcal{U}^{\alpha}e_{\alpha}.$ (207)

Studying the covariant derivative
$\partial_{\nu}U=\partial_{\nu}\mathcal{U}^{\alpha}_{r}e_{\alpha}+\mathcal{U}^{\alpha}_{r}\partial_{\nu}e_{\alpha}$,
we can write the base variation
$\partial_{\nu}e_{\mu}=\Gamma^{\lambda}_{\mu\nu}e_{\lambda}$. This is how we
write the covariant derivative

$\nabla_{\mu}U_{\nu}=\partial_{\mu}U_{\nu}-\Gamma^{\alpha}_{\mu\nu}U_{\alpha}.$
(208)

Now, doing

$\nabla_{\beta}\nabla_{\mu}U_{\nu}=\nabla_{\beta}\partial_{\mu}U_{\nu}+(\nabla_{\beta}\Gamma^{\alpha}_{\mu\nu})U_{\alpha}+\Gamma^{\alpha}_{\mu\nu}\nabla_{\beta}U_{\alpha},$
(209)

and permuting the indices, we get

$\nabla_{\mu}\nabla_{\beta}U_{\nu}=\nabla_{\mu}\partial_{\beta}U_{\nu}+(\nabla_{\mu}\Gamma^{\alpha}_{\beta\nu})U_{\alpha}+\Gamma^{\alpha}_{\beta\nu}\nabla_{\mu}U_{\alpha}.$
(210)

Subtracting 210 from 209 and considering
$\Gamma^{\alpha}_{\mu\nu}=\Gamma^{\alpha}_{\nu\mu}$, which implies the absence
of torsion, then we have the Riemann tensor

$R^{\lambda}_{\mu\nu\kappa}=\partial_{\nu}\Gamma^{\lambda}_{\mu\kappa}-\partial_{\kappa}\Gamma^{\lambda}_{\mu\nu}-\Gamma^{\lambda}_{\sigma\kappa}\Gamma^{\sigma}_{\mu\nu}+\Gamma^{\lambda}_{\sigma\nu}\Gamma^{\sigma}_{\mu\kappa}.$
(211)

The quantities $\Gamma^{\alpha}_{\beta\kappa}$, are known as connections,
representing exactly the curvature correction when we move the vector parallel
to itself. Let us then consider the metric tensor $g_{\mu\nu}$ and apply the
covariant derivative to it. We therefore have

$\nabla_{\alpha}g_{\mu\nu}=\partial_{\alpha}g_{\mu\nu}+\Gamma^{\beta}_{\mu\nu}g_{\beta\alpha}+\Gamma^{\beta}_{\alpha\mu}g_{\beta\nu}$
(212)
$\nabla_{\mu}g_{\nu\alpha}=\partial_{\mu}g_{\nu\alpha}+\Gamma^{\beta}_{\nu\alpha}g_{\beta\mu}+\Gamma^{\beta}_{\mu\nu}g_{\beta\alpha}$
(213)
$\nabla_{\nu}g_{\alpha\mu}=\partial_{\nu}g_{\alpha\mu}+\Gamma^{\beta}_{\alpha\mu}g_{\beta\nu}+\Gamma^{\beta}_{\nu\alpha}g_{\beta\mu}$
(214)

adding 212 and 213 and subtracting 214, remembering that in a pseudoriemanian
manifold $\nabla_{x}g_{yz}~{}=~{}0$, we get the connection written in terms of
the derivatives of metric

$\Gamma^{\beta}_{\alpha\mu}=\frac{1}{2}g^{\beta\nu}\left(\partial_{\alpha}g_{\mu\nu}+\partial_{\mu}g_{\nu\alpha}-\partial_{\nu}g_{\alpha\mu}\right).$
(215)

##  XII Momentum-energy tensor as source for gravitational field

The action of gravitational fields in general relativity is associated with
the curvature of spacetime, these spacetimes are associated with riemanine
varieties Carroll (2004). The curvature of these manifolds is measured by the
Riemaann 211 that we rewrite next

$R^{\lambda}_{\mu\nu\kappa}=\partial_{\nu}\Gamma^{\lambda}_{\mu\kappa}-\partial_{\kappa}\Gamma^{\lambda}_{\mu\nu}-\Gamma^{\lambda}_{\sigma\kappa}\Gamma^{\sigma}_{\mu\nu}+\Gamma^{\lambda}_{\sigma\nu}\Gamma^{\sigma}_{\mu\kappa}.$
(216)

We have the connection defined in the manifold under which we are working, in
this case, the Christoffel symbols, which we deduce in the equation 215

$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\nu\sigma}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right).$
(217)

The Ricci tensor is found by doing the contraction $\lambda=\nu$, i.e.

$R_{\mu\kappa}=R^{\nu}_{\mu\nu\kappa}.$ (218)

The Ricci scalar is then

$R=g^{\mu\kappa}R_{\mu\kappa},$ (219)

these objects characterize the curvature of space-time. We can now define the
Einstein-Hilbert action Santos  _et al._ (2020),Carroll (2004)

$S_{H}=\int{R}\sqrt{-g}d^{4}x$ (220)

using minimal coupling with a $S_{M}$ matter action, we have

$\delta{S}=\delta{S_{H}}+8\pi\delta{S_{M}}=0,$ (221)

assuming $G=c=1$ ie

$\delta{S}=\int\left(\frac{\delta(\sqrt{-g}R)}{\delta{g^{\mu\nu}}}+\frac{8\pi}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{M})}{\delta{g^{\mu\nu}}}+\frac{8\pi}{\sqrt{-g}}\frac{\partial(\sqrt{g}L_{M})}{\partial(\partial_{k}g^{k\nu})}\delta(\partial_{k}g^{k\mu}))\right)\delta{g}^{\mu\nu}d^{4}x=0,$
(222)

where $\mathcal{L}_{M}=\sqrt{-g}L_{M}$ is the Lagrangian density of matter.
Rearranging the terms, we have

$\delta{S}=\int\left(\frac{\delta\sqrt{-g}}{\delta{g^{\mu\nu}}}\frac{R}{2\sqrt{-g}}-\frac{\delta{R}}{\delta{g^{\mu\nu}}}+\frac{8\pi}{\sqrt{-g}}\frac{\delta(\mathcal{L}_{M})}{\delta{g^{\mu\nu}}})\right)\sqrt{-g}\delta{g}^{\mu\nu}d^{4}x,$
(223)

We call the momentum-energy tensor

$T^{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta(\mathcal{L}_{M})}{\delta{g^{\mu\nu}}},$
(224)

where

$R_{\mu\nu}=\frac{\delta{R}}{\delta{g^{\mu\nu}}},$ (225)

we then arrive at Einstein’s equation,

$R^{\mu\nu}-\frac{R}{2}g^{\mu\nu}-8\pi{T^{\mu\nu}}=0.$ (226)

Writing the Einstein tensor

$G^{\mu\nu}=R^{\mu\nu}-\frac{R}{2}g^{\mu\nu},$ (227)

we have

$G^{\mu\nu}=8\pi{T^{\mu\nu}}.$ (228)

By Bianchi’s identity we have, Carroll (2004),

$g^{\nu\sigma}g^{\mu\lambda}\left(\nabla_{\lambda}R_{\rho\sigma\mu\nu}+\nabla_{\rho}R_{\sigma\lambda\mu\nu}+\nabla{\sigma}R_{\lambda\rho\mu\nu}\right)=0,$
(229)

if we contract this expression, we get

$\nabla^{\mu}R_{\rho\mu}-\nabla_{\rho}R+\nabla^{\mu}R_{\rho\mu}=0,$ (230)

ie

$\nabla^{\mu}R_{\rho\mu}=\frac{1}{2}\nabla_{\rho}R.$ (231)

Then we arrive at the conservation of the momentum-energy tensor

$\nabla_{\mu}G^{\mu\nu}=\nabla_{\mu}\left[R^{\mu\nu}-\frac{R}{2}g^{\mu\nu}\right]=8\pi\nabla_{\mu}{T^{\mu\nu}}=0.$
(232)

The conservation of the momentum-energy tensor generates the so-called
Tolemam-Opennhaimer-Volkov equation.

We can now study two energy conditions, associated with gravitational regimes.

###  XII.1 Dominant null energy condition

Let us now consider two energy conditions

$T_{\mu\nu}l^{\mu}l^{\nu}\geq
0;T_{\mu\nu}T^{\nu}_{\lambda}l^{\mu}l^{\lambda}\leq 0,$ (233)

the first $T_{\mu\nu}l^{\mu}l^{\nu}$, has already been mentioned and implies
$\rho+p\geq 0$, in turn
$T_{\mu\nu}T^{\nu}_{\lambda}l^{\mu}l^{\lambda}=\rho^{2}+(a^{\prime
2}+b^{\prime 2}+c^{\prime 2})p$ . We know that $a^{\prime 2}+b^{\prime
2}+c^{\prime 2}=1$, so $\rho+p\leq 0$. Logically the only remaining condition
is $p=-\rho$. This energy condition excludes all sources, excluded by the
Dominant Energy Condition, except vacuum energy, Carroll (2001)

$p=-\rho.$ (234)

From the equation 161, we see that $T^{\mu\nu}=pg^{\mu\nu}$ The zero energy
condition is associated with the cosmological constant. The solution of the
Einstein equation for this type of energy-momentum tensor, implies a maximal
spacetime $R_{\mu\nu}\propto{g_{\mu\nu}}$ Santos  _et al._ (2020),Carroll
(2001),Carroll (2004).

###  XII.2 Strong energy condition

This condition is linked to attractive gravity, using the Einstein equation
$R_{\mu\nu}=\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)$, we have

$R_{\mu\nu}t^{\mu}t^{\nu}=\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)t^{\mu}t^{\nu}\geq
0,$ (235)

$T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$ implies $\rho+p\geq 0$, whereas

$\frac{1}{2}Tg_{\mu\nu}t^{\mu}t^{\nu}=-\frac{1}{2}(\rho-3p)$ (236)

by the equation 185.

So,

$R_{\mu\nu}t^{\mu}t^{\nu}=\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)t^{\mu}t^{\nu}=\rho+p+\frac{1}{2}(\rho-3p)\geq
0,$ (237)

as $\rho+p\geq 0$, we can only write that $\rho-3p\geq 0$.

We finally write

$\rho+p\geq 0;\rho-3p\geq 0.$ (238)

Violation of this condition generates repulsive gravity.

Figure 2: Energy conditions are applied to perfect fluids. Representing
possible values of energy density and pressure, by Carroll (2001).

.

###  XII.3 Isotropic TOV demonstration: ideal fluid in a generic metric

We start from the equation (161) and derive $\nabla_{\mu}T^{\mu\nu}$, then

$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}\left(p+\rho\right)U^{\mu}U^{\nu}+(\rho+p)\left((\nabla_{\mu}U^{\mu})U^{\nu}+U^{\nu}\nabla_{\mu}U^{\mu}\right)+\nabla_{\mu}pg^{\mu\nu}.$
(239)

The equation of continuity, or conservation of mass is

$\nabla_{\mu}(\rho{U^{\mu}})=U^{\mu}\nabla_{\mu}\rho+(\rho+p)\nabla_{\mu}U^{\mu}=0,$
(240)

then we see that

$\nabla_{\mu}T^{\mu\nu}=(\nabla_{\mu}p)U^{\mu}U^{\nu}+(\rho+p)U^{\nu}\nabla_{\mu}U^{\mu}+(\nabla_{\mu}p)g^{\mu\nu}.$
(241)

We know that
$\nabla_{\mu}U^{\nu}=\partial_{\mu}U^{\nu}+\Gamma^{\nu}_{\alpha\mu}U^{\alpha}$,
and that we have four-speed $U^{t}=\sqrt{g^{00}},U^{1}=0$. So the only non-
zero component of the 4-speed derivative is

$\nabla_{r}U^{t}=\partial_{r}(\sqrt{g^{tt}})+\Gamma^{t}_{tr}\sqrt{g^{tt}},$
(242)

and we can write

$(\nabla_{\mu}p)g^{00}+(\rho+p)\sqrt{g^{00}}\nabla_{\mu}\sqrt{g^{00}}+(\nabla_{\mu}p)g^{00}=0.$
(243)

Contracting with the metric

$2\nabla_{\mu}p+(\rho+p)\sqrt{g^{00}}\Gamma^{\nu}_{00}g^{tt}g_{\mu\nu}g^{00}=0,$
(244)

we finally get

$\nabla_{\mu}p=-(\rho+p)\sqrt{g^{00}}\Gamma^{\nu}_{00}g^{00}g_{\mu\nu}(g^{00})^{-1}.$
(245)

We then arrive at the equation that represents the conservation of mass-energy

$\nabla_{\mu}p=-(\rho+p)\sqrt{g^{00}}\Gamma^{\nu}_{00}g_{\mu\nu}.$ (246)

###  XII.4 Demonstration of Isotropic TOV: metric of a spherical object

Let’s study the Tolman-Oppenheimer-Volkoff(TOV) equation for a given spherical
mass distribution. TOV is the equation that corresponds to the hydrostatic
equilibrium of a relativistic fluid. We can also interpret TOV as the
conservation of the momentum-energy tensor Glendenning (1997).

The metric of a spherical distribution of mass $m$

$ds^{2}=-e^{\Phi(r)}dt^{2}+\left(1-\frac{2m}{r}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin(\theta)d\phi^{2}\right),$
(247)

being

$g^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}-e^{\Phi(r)}&0&0&0\\\
0&\left(1-\frac{2m}{r}\right)^{-1}&0&0\\\ 0&0&r^{2}&0\\\
0&0&0&r^{2}\sin(\theta)\\\ \end{array}\right].$ (248)

We can then calculate the Christoffel symbols associated with the metric of a
spherical distribution

$\Gamma^{t}_{tr}=\Phi^{\prime},$ (249)
$\Gamma^{r}_{tt}=\Phi^{\prime}e^{2\Phi}\left(1-\frac{2m}{r}\right),$ (250)
$\Gamma^{r}_{rr}=\frac{rm^{\prime}-m}{r^{2}-2mr},$ (251)
$\Gamma^{\theta}_{r\theta}=\Gamma^{\phi}_{r\phi}=\frac{1}{r},$ (252)
$\Gamma^{\phi}_{\theta\phi}=-\sin(\theta)\cos(\theta),$ (253)
$\Gamma^{r}_{\theta\theta}=\Gamma^{\theta}_{\phi\phi}=\csc^{2}(\theta),$ (254)
$\Gamma^{r}_{\phi\phi}=2m-r.$ (255)

Here it must be said that we are using a system of units where mass and radius
have the same unit. The Ricci tenso,r associated with these connections, has
the following components

$R_{tt}=e^{2\Phi}\left[\left(\Phi^{\prime\prime}+\Phi^{\prime
2}\right)\left(1-\frac{2m}{r}\right)+\Phi^{\prime}\left(\frac{2r-3m-rm^{\prime}}{r^{2}}\right)\right],$
(256)
$R_{rr}=\left(1-\frac{2m}{r}\right)^{-1}\left[\frac{(rm^{\prime}-m)(2+rm^{\prime})}{r^{3}}\right]-\Phi^{\prime\prime}-\Phi^{\prime
2},$ (257)
$R_{\theta\theta}=\csc^{2}(\theta)R_{\phi\phi}=(2m-r)\Phi^{\prime}+m^{\prime}+\frac{m}{r}.$
(258)

The Ricci’s scalar,

$R=g^{\mu\nu}R_{\mu\nu}=2\left[\frac{2m^{\prime}}{r}+\Phi^{\prime}(3m-2r+rm^{\prime})-\left(1-\frac{2m}{r}\right)\left(\Phi^{\prime\prime}+\Phi^{\prime
2}\right)\right].$ (259)

Using the $rr$ component of the Einstein equation,

$G_{rr}=\frac{2}{r}\left(\Phi^{\prime}-\frac{m}{1-\frac{2m}{r}}\right)=\frac{8\pi{p}}{1-\frac{2m}{r}},$
(260)

we then find

$\Phi^{\prime}=\frac{m+4{\pi}r^{3}p}{r(r-2m)}.$ (261)

The $tt$ component is

$G_{tt}=\frac{2m^{\prime}e^{2\Phi}}{r^{2}}=8\pi\rho{e^{2\Phi}}$ (262)

where $m^{\prime}=\frac{dm}{dr}=4\pi\rho{r^{2}}$, is the continuity equation.

We can write the momentum-energy tensor

$T^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}\rho{e}^{-2\Phi}&0&0&0\\\
0&(1-\frac{2m}{r})p&0&0\\\ 0&0&\frac{p}{r^{2}}&0\\\
0&0&0&p\frac{\csc^{2}(\theta)}{r^{2}}\\\ \end{array}\right].$ (263)

Since the functions are dependent on the $r$ coordinate, we have the covariant
derivative of the momentum-energy tensor,

$\displaystyle\nabla_{r}T^{r\nu}$
$\displaystyle=\partial_{r}T^{r\nu}+T^{\sigma\mu}\Gamma^{r}_{\sigma\mu}+T^{r\sigma}\Gamma^{\nu}_{\sigma\nu}$
(264)
$\displaystyle=\partial_{r}T^{rr}+T^{rr}(\Gamma^{\nu}_{r\nu}+\Gamma^{r}_{rr})+T^{\theta\theta}\Gamma^{r}_{\theta\theta}+T^{\phi\phi}\Gamma^{r}_{\phi\phi}$
$\displaystyle=\left(1-\frac{2m}{r}\right)\left[\frac{dp}{dr}+(\rho+p)\Phi^{\prime}\right]=0$

so rises the TOV equation

$\frac{dp}{dr}=-(\rho+p)\frac{m+4{\pi}pr^{3}}{r(r-2m)},$ (265)

together with the continuity equation,

$\frac{dm}{dr}=4\pi\rho{r^{2}},$ (266)

form the so-called structure equations for a Glendenning (1997) star.

##  XIII Anisotropic fluid

We define now the momentum-energy tensor of an anisotropic ideal gas, as seen
in Santos  _et al._ (2020); Carroll (2001, 2004); Chan  _et al._ (2009),

$T^{\mu\nu}=(\rho+p_{t})U^{\mu}U^{\nu}+pg^{\mu\nu}+(p_{t}-p)s^{\mu}s^{\nu},$
(267)

where $s^{\mu}$ is perpendicular to the four-velocity fluid flow
$U^{\mu}=(1,0,0,0)$, $s^{\mu}U_{\mu}=0$.

In matrix terms

$T^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}\rho&0&0&0\\\ 0&p&0&0\\\
0&0&p_{t}&0\\\ 0&0&0&p_{t}\\\ \end{array}\right].$ (268)

The element

$\Delta=p_{t}-p,$ (269)

which is the so-called _anisotropic factor_ , $\Delta>0,p_{t}>p$, that is, a
repulsive factor – or $\Delta<0,p_{t}<p$, in this case the anisotropy
collaborates with the gravitational action. The trace of the momentum-energy
tensor is

$T=\rho-p-2p_{t}.$ (270)

We will now study the limitations known as energy conditions Santos  _et al._
(2020); Chan  _et al._ (2009). We will considerate energy conditions similar
to the conditions for the isotropic case, similar in the sense that time-like
and light-like vectors will be chosen to suit the momentum-energy tensor with
different components.

###  XIII.1 Low energy condition

As in the previous sections the weak energy condition is

$T_{\mu\nu}t^{\mu}t^{\nu}\geq 0,$ (271)

being $t^{\mu}$ time-like vector, we establish that $t^{\mu}=(1,a,b,c)$, then
$T_{\mu\nu}t^{\mu}t^{\nu}=\rho+ap+bp_{t}+cp_{t}$. Choosing $a=b=c=1$ we arrive
at the positivity of the trace

$\rho+p+2p_{t}\geq 0.$ (272)

Here the barotropic factor is modified, $1+\omega+2\frac{p_{t}}{\rho}\geq 0$

$\omega\geq-1-2\frac{p_{t}}{\rho},$ (273)

the causality is then modified, taking into account the tangential pressure.

###  XIII.2 null energy condition

Using the null vectors $l^{\mu}=(1,a^{\prime},b^{\prime},c^{\prime})$ the null
energy condition is

$T_{\mu\nu}l^{\mu}l^{\nu}\geq 0.$ (274)

We verify that, $T_{\mu\nu}l^{\mu}l^{\nu}=\rho+a^{\prime 2}p+(b^{\prime
2}+c^{\prime 2})p_{t}$, choosing $b^{\prime}=c^{\prime}=0$, obligatorily we
have $a^{\prime}=1$, and we get $\rho+p\geq 0$. Alternatively we make
$a^{\prime}=0$ and we are left with $\rho+p_{t}\geq 0$, therefore

$\rho+p\geq 0,\rho+p_{t}\geq 0.$ (275)

We obtained two similar inequalities, which represent conditions for ultra-
relativistic fluids.

###  XIII.3 Dominant energy condition

The dominant energy condition is

$T_{\mu\nu}t^{\mu}t^{\nu}\geq
0,T_{\mu\nu}T^{\nu}_{\lambda}t^{\mu}t^{\lambda}\leq 0.$ (276)

From $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$, we get $\rho\geq 0$. The current quad
not being a space-like vector implies that
$\gamma^{2}\left(-\rho^{2}+(a^{2}p^{2}+(b^{2}+c^{2})p_{t})\right)$, if we make
$b=c=0$, we are left with $\rho^{2}{\geq}a^{2}p^{2}$ and $a<1$, implies that
$\rho\geq|p|$. Following a similar reasoning $a=0$, we arrive at
$\rho\geq|p_{t}|$. We finally write

$\rho\geq 0;\rho\geq|p|;\rho\geq|p_{t}|.$ (277)

###  XIII.4 Dominant null energy condition

The dominant null energy condition is expressed

$T_{\mu\nu}l^{\mu}l^{\nu}\geq 0;T_{\mu\nu}T^{\nu}_{\lambda}l^{\mu}l^{\nu}\leq
0,$ (278)

where $T_{\mu\nu}l^{\mu}l^{\nu}$, has already been calculated and implies
$\rho+p\geq 0,\rho+p_{t}\geq 0$, in turn
$T_{\mu\nu}T^{\nu}_{\lambda}l^{\mu}l^{\nu}=\rho^{2}+a^{\prime 2}p+(b^{\prime
2}+c^{\prime 2})p_{t}$. We know that $a^{\prime 2}+b^{\prime 2}+c^{\prime
2}=1$, making $b^{\prime}=c^{\prime}=0$, so $\rho+p\leq 0$. Logically the only
remaining condition is $p=-\rho$. We can still make $a^{\prime}=0$, implying
that $\rho+p_{t}\leq 0$. So we also have $p_{t}=-\rho$, so the vacuum energy
is not anisotropic. In summary

$p_{t}=p=-\rho$ (279)

expresses the results.

###  XIII.5 Strong energy condition

This condition is linked to the attractive gravity, using the Einstein
equation we have

$\displaystyle R_{\mu\nu}$
$\displaystyle=\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)$ $\displaystyle
R_{\mu\nu}t^{\mu}t^{\nu}$
$\displaystyle=\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)t^{\mu}t^{\nu}\geq
0,$ (280)

again $T_{\mu\nu}t^{\mu}t^{\nu}\geq 0$, implies $\rho-p-2p_{t}\geq 0$, already
$\frac{1}{2}Tg_{\mu\nu}t^{\mu}t^{\nu}=-\frac{1}{2}(\rho-p-2p_{t})$ by the
equation 270. So
$R_{\mu\nu}t^{\mu}t^{\nu}=\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)t^{\mu}t^{\nu}=\rho+p+2p_{t}+\frac{1}{2}(\rho-p-2p_{t})\geq
0$, as $\rho+p+2p_{t}\geq 0$ we can only write that $\rho-p-2p_{t}\geq 0$. We
finally write

$\rho+p+2p_{t}\geq 0;\rho-p-2p_{t}\geq 0.$ (281)

The violation of this condition generates repulsive gravity.

###  XIII.6 Demonstration of anisotropic TOV

As we did in the isotropic case, we write the metric here in terms of the
functions $\nu(r),\lambda(r)$.

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}(\theta)d\phi^{2}),$
(282)

we can now use the Einstein equation 232. The $tt$ component is

$e^{-\lambda(r)}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}=8\pi\rho,$
(283)

and the $rr$ component

$e^{-\lambda(r)}\left(\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right)-\frac{1}{r^{2}}=8\pi{p}.$
(284)

for angular components $\theta\theta$, it generates

$\frac{1}{2}\left(\frac{1}{2}(\nu^{\prime})^{2}+\nu^{\prime\prime}-\frac{1}{2}\lambda^{\prime}\nu^{\prime}+\frac{1}{r}(\nu^{\prime}-\lambda^{\prime})\right)=8\pi{p_{t}}.$
(285)

The equation for $p_{t}$ represents the anisotropic pressure, here the
difference with the isotropic case is manifested. This is the usual model for
anisotropic strange stars.

Let us consider the hydrostatic equilibrium Santos  _et al._ (2020);
Glendenning (1997)

$F_{g}+F_{hydro}+F_{ani}=0,$ (286)

where $F_{g}=\frac{1}{2}(\rho+p)d_{r}\nu(r)$ is the gravitational force,
$F_{hydro}=-\frac{dp}{dr}$ the hydrostatic force and the anisotropic force
$F_{ani}=2\frac{(p_{t}-p)}{r}$.

We can then write the TOV in its anisotropic version as

$\frac{dp}{dr}=-m\frac{\rho+p}{r^{2}}\frac{d\nu(r)}{dr}+\frac{2}{r}(p_{t}-P).$
(287)

The first installment of the TOV is identical to the isotropic case, as the
equations 260 are very similar to the equation 284,
$\nu^{\prime}(r)=\Phi^{\prime}(r)$. We have finally written the TOV,

$\frac{dp}{dr}=-(\rho+p)\frac{m+4{\pi}pr^{3}}{r(r-2m)}+\frac{2}{r}(p_{t}-p),$
(288)

which is the isotropic TOV.

##  XIV The hydrostatic stability in a De-Sitter geometry

It is also possible to construct a momentum-energy tensor taking into account
the interaction of matter with the cosmological constant. In this case the
Einstein equation is modified to

$R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}+\Lambda{g}_{\mu\nu}=8\pi{T_{\mu\nu}},$ (289)

the homogeneous version of this equation would be the Einstein equation for
vacuum, known as the De Sitter equation Santos  _et al._ (2020); Carroll
(2001, 2004),

$R_{\mu\nu}-\frac{R}{2}g_{\mu\nu}+\Lambda{g}_{\mu\nu}=0,$ (290)

which has as a solution

$R_{\mu\nu}=\Lambda{g}_{\mu\nu},$ (291)

a metric-proportional Ricci tensor. We could alternatively think about a
metric-proportional energy-momentum tensor $T_{\mu\nu}=\Lambda{g}_{\mu\nu}$,
this condition is associated with the De Sitter metric

$ds^{2}=-\left(1-\frac{\Lambda{r^{2}}}{3}\right)dt^{2}+\frac{1}{1-\frac{\Lambda{r^{2}}}{3}}dl^{2}+r^{2}(d\theta^{2}+\sin(\theta)d\phi^{2}),$
(292)

and the momentum-energy tensor proportional to the metric. That implies an
equation of state 161

$p=-\rho,$ (293)

see Carroll (2001),Carroll (2004). Having a barometric factor $\omega=-1$, an
equation of state of this type obeys the NDEC energy condition.

To effectively calculate the TOV we need a metric Santos  _et al._ (2020).

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}(\theta)d\phi^{2}\right),$
(294)

explaining the components of the metric
$g_{tt}=e^{\nu},g_{rr}=-e^{\lambda},g_{\theta\theta}=-r^{2},g_{\phi\phi}=-r^{2}\sin^{2}(\theta)$

$g^{\mu\nu}g_{\nu\lambda}=\delta^{\mu}_{\lambda}.$ (295)

We then write the metric in matrix form

$g^{\mu\nu}=\left[\begin{array}[]{rrcccccccc}e^{-\nu}&0&0&0\\\
0&-e^{\lambda}&0&0\\\ 0&0&-\frac{1}{r^{2}}&0\\\
0&0&0&\frac{1}{(r\sin{\theta})^{2}}\\\ \end{array}\right],$ (296)

here we used the metric functions $\nu=\nu(r)$ and $\lambda=\lambda(r)$. We
calculate the Ricci tensor,
$R_{\mu\nu}=\partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}-\partial_{\nu}\Gamma^{\alpha}_{\mu\alpha}+\Gamma^{\alpha}_{\beta\alpha}\Gamma^{\beta}_{\mu\nu}-\Gamma^{\alpha}_{\beta\mu}\Gamma^{\beta}_{\alpha\nu}$,
always remembering that $R^{\alpha}_{\beta}=g^{\alpha\mu}R_{\mu\beta}$. We can
write the components of the Einstein equation:

$tt$ component,

$\displaystyle G^{t}_{t}$ $\displaystyle=-8\pi{T^{t}}_{t},$ $\displaystyle
8\pi\rho$
$\displaystyle=e^{-\lambda}\left(\frac{\lambda^{\prime}}{r}-\frac{1}{r^{2}}\right)+\frac{1}{r^{2}}-\Lambda;$
(297)

$rr$ component,

$\displaystyle G^{r}_{r}$ $\displaystyle=8{\pi}T^{r}_{r},$ $\displaystyle
8\pi{p}$
$\displaystyle=e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}+\Lambda.$
(298)

For a static star we consider

$\frac{d\rho}{dt}=\frac{dp}{dt}=0.$ (299)

Thus, the covariant derivative of the momentum-energy tensor induces the
following result

$(\rho+p)(\partial_{\mu}u_{\sigma})u^{\mu}+\partial_{\sigma}p+\partial_{\mu}pu^{\mu}u_{\sigma}=0.$
(300)

We remember the $rr$ component of the metric $g_{rr}=-e^{\lambda}$, so

$(\rho+p)(\partial_{\mu}u_{t})u^{\mu}-\partial_{r}p=0,$ (301)

like this

$\partial_{r}p=-\frac{(\rho+p)}{2}\frac{d\nu}{dr}.$ (302)

We now define the mass for a spherical shell of radius $r$ as

$m(r)=4\pi\int^{r}_{0}\rho(r^{\prime})r^{\prime 2}dr^{\prime},$ (303)

differentiating the mass and substituting in the temporal component of the
Einstein equation 297 we obtain

$\frac{dm}{dr}=-\frac{1}{2}\frac{d}{dr}\left[e^{-\lambda}r-r+\frac{\Lambda{r^{3}}}{3}\right],$
(304)

integrating we have

$2\int^{r}_{0}\frac{dm}{dr}dr=-\int^{r}_{0}\frac{1}{2}\frac{d}{dr}\left[e^{-\lambda}r-r+\frac{\Lambda{r^{3}}}{3}\right]dr,$
(305)

so we finally arrive at the $rr$ component of the metric

$e^{-\lambda(r)}=1-\frac{2m(r)}{r}-\frac{\Lambda{r^{3}}}{3},$ (306)

adding the radial and temporal components of the Einstein equation,
respectively the equations 297 and 298, we have,

$8\pi(\rho+p)=\frac{\lambda^{\prime}e^{-\lambda}}{r}+\frac{e^{-\lambda}\nu^{\prime}}{r}.$
(307)

Deriving the equation 306 with respect to the radial coordinate we have

$\lambda^{\prime}e^{-\lambda}=\frac{2m^{\prime}}{r}-\frac{2m}{r^{2}}+\frac{2\Lambda{r}}{3},$
(308)

we now substitute the equations 308 and 306 in the equation 307, and we get

$8\pi(\rho+p)=\left(\frac{2m^{\prime}}{r}-\frac{2m}{r^{3}}+\frac{2\Lambda}{3}\right)+\frac{1}{r}\frac{d\nu}{dr}\left(1-\frac{2m}{r}-\frac{\Lambda{r^{2}}}{3}\right).$
(309)

Substituting 302 in the equation 309, we finally arrive at TOV with
cosmological constant,

$\frac{dp}{dr}=-\rho\left(1+\frac{p}{\rho}\right)\frac{m+4\pi{p}r^{3}-\frac{\Lambda{r^{3}}}{3}}{r^{2}(1-\frac{2m}{r}-\frac{\Lambda{r^{2}}}{3})}.$
(310)

The introduction of the cosmological constant generates an interesting effect,
if we make $m=0$, we have an effect associated with vacuum pressure

$\frac{dp}{dr}=-\rho\left(1+\frac{p}{\rho}\right)\frac{4\pi{p}r^{3}-\frac{\Lambda{r^{3}}}{3}}{r^{2}(1-\frac{\Lambda{r^{2}}}{3})}.$
(311)

We usually consider that the vacuum equation of state is $p=-\rho$ , obeying
the energy condition 233, would induce the above equation to be 311
identically null, so the pressure of the vacuum was a constant. However, if we
consider the effect of the cosmological constant term as an extra pressure,
similarly to what we do in the presence of tangential pressure, we could
consider the part associated with the cosmological constant ($\Lambda>0$) to
be an anisotropy. An extra negative pressure, which attempts to compensate for
the effects of radial pressure.

##  XV Hydrodynamics with quantum potential

###  XV.1  Mandelung formalism

The interpretation of Bohn-De’s bloglie Quantum Mechanics bears a strong
resemblance to a theory of fluids. Being the wave function a current density
Chiarelli (2019) is notable the presence of a continuity equation. One way to
introduce quantum effects in a relativistic scenario is through the Mandelung
formalism Chiarelli (2019). Where precisely we have a quantum potential
introduced.

Madelung’s formalism can be a good working reference for the mesoscale
Chiarelli (2019) classical/quantum description describing the interaction
between classical and quantum phenomena.

Following Chiareli Tsekov (2012), we now write Klein Gordon’s equation for a
function $\psi_{m}$, which is a mass field,

$\partial_{\mu}\partial^{\mu}\psi_{m}=-\frac{m^{2}c^{2}}{h^{2}}\psi_{m}.$
(312)

The function $\psi_{m}$, is associated with a scalar field $s_{m}$, as follows

$\psi_{m}=|\psi^{*}_{m}|\exp\left[\frac{i}{h}s_{m}\right],$ (313)

in this hydrodynamics the momentum is defined as

$p^{m}_{\mu}=\partial_{\mu}s_{m}.$ (314)

We return the Klein-Gordon equation 312 and write

$\partial_{\mu}s_{m}\partial^{\mu}s_{m}-h^{2}\frac{\partial_{\mu}\partial^{\mu}|\psi_{m}|}{|\psi_{m}|}-m^{2}c^{2}=0.$
(315)

Since 312 and 315 are a system of Klein Gordon equations in Minkowski space,
coupled to conserved current,

$\frac{\partial}{\partial{q}_{\mu}}\left(|\psi_{m}|^{2}\frac{\partial{S}}{\partial{q}_{\mu}}\right)=m\frac{\partial{J}_{\mu}}{\partial{q}_{\mu}}=0.$
(316)

Starting from 313 we get $s_{m}$ in terms of the massive field $\psi_{m}$

$s_{m}=\frac{h}{2i}\ln\left[\frac{\psi_{m}}{\psi^{*}_{m}}\right].$ (317)

We write the current

$J_{\mu}=\left(c\rho,-j_{i}\right)=\frac{ih}{2m}\left(\psi^{*}_{m}\frac{\partial\psi_{m}}{\partial{q_{\mu}}}-\psi_{m}\frac{\partial\psi^{*}_{m}}{\partial{q}_{\mu}}\right),$
(318)

which characterizes an incompressible fluid.

We write the 4-momentum in analogy with hydrodynamics

$p^{m}_{\mu}=\left(\frac{E}{c},-p_{i}\right)=-\frac{\partial{s_{\mu}}}{\partial{q^{\mu}}},$
(319)

and relate the current to the momentum 315

$J_{\mu}=-|\psi_{m}|^{2}\frac{p_{\mu}}{m},$ (320)

making clear the analogy with hydrodynamics, where the fluid density takes the
place of the probability density

$\rho=\frac{|\psi|}{mc^{2}}\frac{\partial{s_{m}}}{\partial{t}}.$ (321)

Considering the equations 313 and 319 we write below

$\frac{\partial{s}_{m}}{\partial{q}^{\mu}}\frac{\partial{s}_{\mu}}{\partial{q}_{\mu}}=p^{m}_{\mu}p^{\mu}_{m}=\left(\frac{E^{2}}{c^{2}}-p^{2}_{m}\right)=m^{2}c^{2}\left(1-\frac{V_{qu}}{mc^{2}}\right),$
(322)

where we have that the quantum potential is linked to the massive field
$\psi_{m}$ being given by

$V_{qu}=-\frac{h}{m}\frac{\partial_{\mu}\partial^{\mu}|\psi_{m}|}{\psi_{m}}.$
(323)

The scatter relation expressed in 322 is very similar to the scatter relation
found in the reference Cruz and da Silva (2018) . However, the scalar
potential $\phi$ defined by Cruz and da Silva (2018), has a kinematic
interpretation associated with the energy-time uncertainty principle, being
intimately associated with a clock hypothesis and a deformed metric, while the
quantum potential $V_{qu}$323, has an interpretation associated with matter
waves. Anyway, the equation 323 represents a De-Broglie Bohm type potential.
Other authors Falciano  _et al._ (2010) and Zloshchastiev (2020a, b) interpret
the potential 323 in different ways, but in both cases, the interpretations
are associated with properties of a conformal geometry, which would act as an
effective geometry capable of introducing quantum effects.

The cosmological constant is written by Chiarelli (2019) in terms of the
matter fields $\psi_{m}$ and the scalar field $V_{qu}$,

$\Lambda=-\frac{mc^{2}}{\gamma}|\psi_{m}|^{2}\left(1-\sqrt{1-\frac{V_{qu}}{mc^{2}}}\right),$
(324)

this is yet another point of similarity between the works of Nassif da Cruz
(2019)and Tsekov (2012), since the scalar field equation in Cruz and da Silva
(2018) has a functional form very similar to the form of 324, and we have
already demonstrated in Cruz and da Silva (2018) that the field $\phi$
associated with the Symmetric Special Relativity (SSR) kinematics has
inflationary properties. It is notable that two researchers independently
arrived at similar results for the same problem. In the work of Chiarelli
(2019), the large $\gamma$ has a relation with the line element

$\gamma=\frac{c}{\sqrt{g_{\mu\nu}v^{\mu}v^{\nu}}},$ (325)

where $v^{\mu},v^{\nu}$, are canonical 4-speeds.

###  XV.2 Schutz formalism

The Schutz formalism is a generalization of the Clebsch formalism Alvarenga
_et al._ (2017) where velocity could be written in terms of three potentials
$\vec{v}=\vec{\nabla}\phi+\alpha\vec{\nabla}\beta$, so we deal with three
scalar potentials $\phi,\alpha,\beta$. The reason for defining this type of
vector is that the curl of a gradient is null, but a linear combination of
gradients of scalar fields is not null
$\vec{\nabla}\times\vec{v}=\vec{\nabla}\alpha\times\vec{\nabla}\beta$, thus
eliminating an apparent contradiction in the quantization of and in the
definition of vorticity, which we will see in the following sections.

In the Schutz formalism the 4-speed can be written in terms of six scalar
fields, thus consisting of a generalization of the Clebsch formalism,

$u_{\mu}=k^{-1}\left(\partial_{\mu}\phi+\alpha\partial_{\nu}\beta+\theta\partial_{\mu}s\right).$
(326)

Scalar potentials can be interpreted as thermodynamic potentials, where

$k=\frac{\rho+p}{\rho_{0}},$ (327)

is the enthalpy. Over the enthalpy the Hawking-Ellis energy conditions 205,
198, 203 and the 233 condition impose a propagation of signals at infinite
speed, this condition being linked to the cosmological constant, so this
formalism in a cosmological constant space would imply an infinite enthalpy.
We also define the scalar potential $s$ that corresponds to entropy per
particle, where the orthogonality relations hold, which links the potentials
to each other

$u^{\mu}\partial_{\mu}\phi=-k,$ (328)

where two conservation laws appear

$\displaystyle u^{\mu}\partial_{\mu}\alpha$ $\displaystyle=0$ (329)
$\displaystyle u^{\mu}\partial_{\mu}\beta$ $\displaystyle=0.$ (330)

We set the temperature

$u^{\mu}\partial_{\mu}\theta=T,$ (331)

then the orthogonality with the entropy gradient, thus defining the isentropic
trajectories, in the space of scalar fields

$u^{\mu}\partial_{\mu}s=0.$ (332)

Another link to be established between the potentials derives from the
orthogonality property $u^{\mu}u^{\nu}g_{\mu\nu}=-1$,

$m^{2}=-g^{\mu\nu}\left(\partial_{\mu}\phi+\alpha\partial_{\mu}\beta+\theta\partial_{\mu}s\right)\left(\partial_{\nu}\phi+\alpha\partial_{\nu}\beta+\theta\partial_{\nu}s\right),$
(333)

implying, therefore, that the enthalpy $k$ is the normalizing element. A
second order equation, can also be defined considering 326, we have

$\rho_{0}\nabla^{\mu}u_{\mu}=0.$ (334)

The set of equations 328, 329, 330, 331 and 332 implies the termination of the
following action:

$I=\int\sqrt{-g}pd^{4}x,$ (335)

here we have $p=p(k,s)$ from which we deduce the first law of thermodynamics
as a conserved current.

$\delta{P}=\rho_{0}\delta{m}-\rho_{0}T\delta{s}$ (336)

##  XVI The Momentum Energy Tensor

A Momentum Energy tensor for an ideal fluid in the Schutz formalism.

$T_{\mu\nu}=\rho{U}_{\mu}U_{\nu}+P_{\mu\nu}$ (337)

here $P_{\mu\nu}=g_{\mu\nu}+U_{\mu}U_{\nu}$ where we will consider for
$g_{\mu\nu}$ the acoustic metric 134

$\mathcal{G}^{\mu\nu}=\frac{\rho}{c}\left(\begin{array}[]{cc}c^{2}-v^{2}&-\vec{v}\\\
-\vec{v}&\mathcal{I}\end{array}\right)$ (338)

Considering an incompressible fluid $v_{\mu}=\nabla_{\mu}\phi$ where $\phi$ is
a scalar field, we then have

$T_{\mu\nu}=(\rho+p)\nabla_{\mu}\phi\nabla_{\nu}\phi-p\mathcal{G}_{\mu\nu},$
(339)

so we have a momentum energy tensor for an incompressible fluid. Starting from
the Einstein-Hilbert action 220 ${\int}d^{4}x\sqrt{-g}\left(R+16\pi{p}\right)$
and following Alvarenga  _et al._ (2017), we can consider the construction of
the Einstein field equation for fluids governed by the tensor energy momentum
339.

$\delta\left(\sqrt{-g}R\right)=\left(R_{\sigma\mu}-\frac{1}{2}g_{\sigma\mu}R\right)\sqrt{-g}\delta{g}^{\sigma\nu}$
(340)

and

$\frac{\delta{\sqrt{-g}p}}{\delta{g}_{\alpha\beta}}=\frac{1}{2}pg_{\alpha\beta}\sqrt{-g}+\frac{\delta{p}}{\delta\mu}\frac{\delta\mu}{\delta{g}^{\alpha\beta}}\sqrt{-g}.$
(341)

We know from 328 that

$\displaystyle\frac{\partial{k}}{\partial{g}^{\sigma\mu}}$
$\displaystyle=-\frac{k}{2}U_{\sigma}U_{\ mu}$
$\displaystyle\frac{\partial{p}}{\partial\mu}$
$\displaystyle=\frac{\rho+p}{k}.$

Therefore

$\frac{\delta{\sqrt{-g}p}}{\delta{g}^{\sigma\mu}}=\frac{\sqrt{-g}}{2}\left[pg_{\sigma\nu}+\left(\rho+p\right){k}U_{\sigma}U_{\nu}\right].$
(342)

We put together the equations 340 and 341, and then arrive at the Einstein
field equation for the energy-momentum tensor

$R_{\sigma\nu}-\frac{R}{2}g_{\sigma\nu}=\frac{\sqrt{-g}}{2}\left[pg_{\sigma\nu}+\left(\rho+p\right){k}U_{\sigma}U_{\nu}\right],$
(343)

hence a field equation for the scalar fields associated with thermodynamic
potentials. We therefore find a link between general relativity and
thermodynamics.

##  XVII Superfluids a possible model for antigravity

Many authors Volovik (2001); Visser (1993); Zloshchastiev (2010); Santos and
Almeida (2022), have been looking for a model to express what would be
antigravity, the cosmological constant Khoury  _et al._ (2018) and cosmic
inflation Cruz and da Silva (2018) are two theoretical hypotheses that find a
phenomenon similar to superfluidity. The properties of superfluids such as
energy propagation towards the capillary, the system boundary, could serve as
a toy model for a possible antigravity. Models analogous to gravity,
especially with superfluids, have received great attention from the scientific
community, phenomena of superfluids on the astrophysical scale as in neutron
stars Martin and Urban (2016) and models of ultracompact objects Cruz  _et
al._ (2020), also found in superfluidity a phenomenology to be compared.

Let’s consider the two-fluid Khalatnikov model Huang (2016), where we have two
fluids with different densities $\rho_{s},\rho_{n}$ and different currents
$\vec{j}_{s}$ and $\vec{j}_{n}$ which are orthogonal to each other and both
respect the continuity equation 14. Then we define the currents

$\vec{j}_{s}=\rho_{s}\vec{v}_{s},$ (344)

and the perpendicular current

$,\vec{j}_{n}=\rho_{n}\vec{v}_{n}$ (345)

where 345 is perpendicular to the flowline Huang (2016) and $\vec{j}_{s}$
corresponds to a ring around the flowline.

###  XVII.1 Landau’s criteria of superfluidity

Following Volovik (2001); Huang (2016) we will establish the Landau criterion
for the existence of a superfluid. Landau’s theory of superfluids is invariant
by Galilean transformations between reference frames. Therefore, considering
two reference frames $S$ and $S^{\prime}$ we have that the momentum is
transformed as follows

$\vec{P^{\prime}}=\vec{P}+M\vec{v}.$ (346)

In this case, we have $M$ as the total mass of the fluid. We can now write the
transformation for the kinetic energy

$E^{\prime}=E+\vec{P}\cdot\vec{v}+\frac{Mv^{2}}{2}.$ (347)

Therefore we have a fluid motion with constant velocity $\vec{v}$ with respect
to the capillary wall. In case any momentum excitation appears, we need to
describe it in relation to the referential $S$, which is in the fluid,
therefore a moving referential. We have that the total energy of the fluid is
given by

$E+\epsilon(p),$ (348)

so we have that $E$ is the energy of the ground state and $\epsilon(p)$ is the
energy of the excited state, the appearance of $\epsilon(p)$, configures the
destruction of the superfluid. In the reference frame $S$ the capillary wall
moves with velocity $-\vec{v}$. The appearance of the excited state modifies
347 as follows

$E^{\prime}=E_{0}+\epsilon(p)+\vec{p}\cdot{v}+\frac{Mv^{2}}{2}.$ (349)

In 348 excitations appear under favorable energy conditions, namely

$\epsilon(p)+\vec{p}\cdot\vec{v}<0.$ (350)

The 350 condition implies energy dissipation to the capillary wall, then to
the system boundary, thus destroying the superfluid. This condition therefore
occurs when we establish a minimum velocity $v_{c}$ for the appearance of
excitations,

$v_{c}=min\frac{\epsilon(\vec{p})}{p}.$ (351)

The equation 351 defines the critical Landau velocity $v_{c}$, the superfluid
flow velocity, associated with the first excited degree that we call the roton
Huang (2016). When dealing with relativistic fluids, the dispersion relation
is $\epsilon(p)=\sqrt{p^{2}+M^{2}}$, in this case

$\lim_{p\rightarrow 0}\frac{d\epsilon(p)}{dp}=0,$ (352)

for minimum values of $p$, then we get that 351 is null, however, when dealing
with linear scatter relations $\epsilon(p)\propto{p}$, we have that

$\lim_{p\rightarrow 0}\frac{d\epsilon(p)}{dp}=V,$ (353)

a constant value of $V$, so we have that the excited states have a minimum
velocity. As a good approximation, we can handle small values of momentum in
the scattering relation $\epsilon(p)\propto{p^{2}}$, reducing this scattering
relation to a linear relation $\epsilon(p)\propto{p}$, implying the appearance
of a minimum velocity for the excitations. An important classification for the
excitations by 348 is as follows: phonos have a linear dispersion relation
$\epsilon(p)\propto{p}$, whereas the roton is the first minimum, being
characterized by the dispersion relation
$\epsilon(p)=\Delta+\frac{(p-p_{0})^{2}}{2m}$, in turn, the maxon is the
opposite of the roton, being a maximum
$\epsilon(p)=\Delta^{\prime}-\frac{(p-p_{0})^{2}}{2m}$. It is possible to
interpret the superfluid excitations as another fluid, orthogonal to the first
one, so we can use the formulation of Huang (2016); Volovik (2001), which
makes this work much clearer. In this case, we then write the total density of
both fluids as $\rho=\rho_{s}+\rho_{n}$ and the current as
$\vec{j}=\vec{j}_{s}+\vec{j}_{n}$, so for both fluids the number of
quasiparticles per volume is $n=\sqrt{\rho}$, $\rho$ is the magnitude of the
order factor $\psi=\rho e^{\left[i\theta\right]}$.

###  XVII.2 Building collective variables for the components $\vec{j}_{s}$ and
$\vec{j}_{n}$

Let us now study our system in terms of the collective variables
$n(\textbf{p})$ which is the density of quasi-particles and $\vec{v}$ the
velocity of the fluid Volovik (2001), we define, according to Volovik citevolo
, the distribution of quasiparticles in thermal equilibrium at a temperature
$T$

$f_{\tau}(p,v)=\left(\exp\left[\frac{\tilde{E}-\vec{p}\cdot\left(\vec{v}_{n}-\vec{v}_{s}\right)}{T}\right]\pm
1\right)^{-1},$ (354)

where $+$ represents fermion and $-$ represents Boson. The scatter relation
for $\tilde{E}$ is the same as the 349 relation. What makes us able to write,
starting from 354,

$\vec{w}=\vec{v}_{s}-\vec{v}_{n}.$ (355)

We refer to $\vec{w}$ as counterflow current

$\vec{j}_{q}=n_{nik}\vec{w}.$ (356)

We can now calculate the quasi-particle density, which is given according to
Volovik (2001),

$n_{ni\kappa}=-\Sigma_{p}\frac{p_{i}p_{\kappa}}{m}\frac{\partial{f}_{\tau}}{\partial{E}},$
(357)

where $n_{ik}=n\delta_{ik}-n_{nik}$.

###  XVII.3 The quantization of a superfluid

A superfluid respects the Gross-Pitaevski equation Huang (2016)

$\frac{-\hbar^{2}}{2m}\nabla^{2}\psi+\left(g\psi^{*}\psi-\mu_{0}\right)\psi=i\hbar\frac{\partial\psi}{\partial{t}},$
(358)

where $g=4\pi\frac{a\hbar}{m}$. Although we have discussed in the previous
sections an elaboration where a fluid would respect a Klein Gordon equation
312 with an exponential type solution 313. We had not commented on the
possibility of this object representing a superfluid Chiarelli (2019), we will
leave a more in-depth debate on this topic for another opportunity. In this
case, we are associating the Schutz formalism with a superfluid since the two
formalisms, Schutz and Mandelung, are the most used to treat quantum
potentials. The Mandelung formalism has a dispersion relation 322. Schutz’s
formalism, on the other hand, allows an immediate connection with
thermodynamics, which can be interesting in the treatment of superfluids. In
this sense, the solution of the equation 358 is an exponential
$\psi=\rho{e^{\sigma}}$, so we have, as in the Mandelung and Schutz formalism
326, the velocity being described like the gradient of a scalar field

$\vec{v}:=\vec{\nabla}\sigma.$ (359)

The quantization of the velocity, however, occur when we do the line integral

$\int\vec{v}\cdot\textbf{ds}.$ (360)

Usually, for a gradient vector this integral would be zero, but here we write

$\int\vec{v}\cdot\textbf{ds}=2n\pi\hbar,$ (361)

where $n=0,1,2,3,...$ Implying that velocity is associated with a vortex and

$v=\frac{2\pi\hbar{n}}{mr},$ (362)

by Visser (1993), $\vec{v}$ obeys the equations 425, 426, being able to be a
model for the dark matter Harko and Madarassy (2022). But the issue is really
the quantization of velocity, it is usually expected that
$\int\vec{\nabla}\sigma\cdot\textbf{ds}=0$, the fact that 361 is not null have
enigmatic consequences, which are still the subject of intense study Huang
(2016). One of the open discussions is the relation with the simple
connectedness of the space Xiong  _et al._ (2014); Stone (2000). So the vector
359 cannot be written solely as a gradient. Having thus a component that
allows us to write $\vec{\nabla}\times\vec{v}\neq 0$, following Huang (2016)
we write

$\vec{v}=\vec{\nabla}\phi+\vec{v}_{ind},$ (363)

it offers us

$\vec{\nabla}\times\vec{v}=\vec{\nabla}\times\vec{\nabla}\phi+\vec{\nabla}\times\vec{b}.$
(364)

We define the vorticity

$\vec{k}=\vec{\nabla}\times\vec{v}_{ind}.$ (365)

Considering a vector tangent to the flowline of the vortex $\vec{s}(\xi,t)$,
where $\xi$ is a parameter that runs on the flowline itself and $t$o time. In
this sense we can write the vorticity in terms of this vector

$\vec{k}(\vec{r},t)=k_{0}\int{ds}\delta\left(\vec{r}-\vec{s}(\xi,t)\right),$
(366)

the enormous similarity with electromagnetism, suggests that the velocity
$\vec{b}$, must be written by the Biort-Savat law, being $\vec{b}$ associated
with the rotation velocity of the vortex around the line of flow.

$\vec{v}_{int}:=\frac{k}{4\pi}\int{d\xi^{\prime}}\frac{(\vec{s}(\xi^{\prime},t)-\vec{s}(\xi,t))}{|\vec{s}(\xi^{\prime},t)-\vec{s}(\xi,t)|^{3}},$
(367)

where $\vec{s}_{1}$ is a particular point on the flowline and extends to all
points on the flowline

$d\vec{s}(\xi,t)=\frac{\partial\vec{s}}{\partial\xi}d{\xi}+\frac{\partial\vec{s}}{\partial{t}}dt,$
(368)

where

$\displaystyle\vec{s}^{\prime}$
$\displaystyle=\frac{\partial\vec{s}}{\partial\xi}$
$\displaystyle\dot{\vec{s}}$
$\displaystyle=\frac{\partial\vec{s}}{\partial{t}}.$

We can relate $\vec{b}$,$\dot{\vec{s}},\vec{s}^{\prime}$,

$\dot{\vec{s}}=\vec{v}_{ind}(\vec{s})+\vec{v}+\frac{\alpha}{|\vec{s}^{\prime}|}\
vec{s}^{\prime}\times\left(\vec{v}_{ns}-\vec{v}_{ind}\right)-\alpha^{\prime}\frac{\vec{s}^{\prime\prime}}{|\
vec{s}^{\prime}|}\times\left[\frac{\vec{s}^{\prime}}{|\vec{s}^{\prime}|}\times\left(\vec{v}_{ns}-\vec{v}_{ind}\right)\right],$
(369)

with the module $|\vec{s}^{\prime\prime}|=\frac{1}{R}$, where $R$ is the
radius of the circulation ring, so there is a link between the vector
$\vec{s}^{\prime\prime}$ is the speed modulus given by 362. Knowing that the
vectors
$\vec{s}^{\prime},\vec{s}^{\prime\prime},\vec{s}^{\prime}\times\vec{s^{\prime\prime}}$
generate a moving base like Frenet-Serrat Bini  _et al._ (1999), then we also
obtain a relation of the circulation velocity with this basis.

Returning to 371 and making $\dot{\vec{s}}(\xi,t)\equiv\vec{V}_{L}(\xi,t)$, as
well as $\vec{v}_{s}(\vec{s}(\xi,t),t)\equiv\vec{V}_{sl}(\xi,t)$ we write the
Magnus force, associated with the friction between the layers of the fluid,

$\frac{f_{m}}{\rho_{s}k_{0}}=\frac{\vec{s}}{|\vec{s}^{\prime}|}\times\left(\vec{v}_{L}-\vec{v}_{sl}\right).$
(370)

The magnus force 370, indicates the importance of Galilean symmetry in this
fluid.

###  XVII.4 Iordenski force

After establishing the Galinean symmetry for the excitations of superfluids
346, we will study the appearance of a force between these excited states,
which is known as the Iordenski force Huang (2016); Stone (2000),

$\vec{F}_{io}=-D\left[e_{z}\times\left(\vec{u}_{e}-V_{L}\right)\right]+D^{\prime}\left[e_{z}\times\left(\vec{u}_{e}-\vec{V}_{L}\right)\right].$
(371)

Here we have that $e_{z}$ is defined by the direction of a cyon type system
Stone (2000), where $\vec{\Phi}=\phi{e_{z}}$, the velocity $V_{L}$ is the
velocity of the vortex. Therefore, we consider the particles to be vortices,
so they would be better identified as quasiparticles, thus having an internal
structure, $u_{e}$ is the propagation velocity of the particle. We note that
there is a Galilean symmetry $\vec{u}_{e}-\vec{V}_{L}$, so this force is, in a
good sense, a force dependent on a Galilean transformation. The Iordenski
force is largely similar to the Lorentz force and the field $\vec{\Phi}$
carries an Aharonov-Bohm Stone (2000) structure at its origin. However, for
developments in the next sections, our interest is in the relation between the
iordenski force and Galilean symmetry. The Iordeski force brings a novelty,
which is a force associated with the conservation of Galilean symmetry, we
have a different perspective of the Galilean systems, where the usual Galilean
symmetry is valid.

###  XVII.5 Relativistic two fluids Hydrodynamycs – An application to the
Schutz formalism

Following Xiong  _et al._ (2014) we define an ideal fluid based on the
superfluids defined above 355,

$\Pi^{jk}=\rho_{s}v^{j}_{s}v^{k}_{s}+\rho_{n}v^{j}_{n}v^{k}_{n}+p\delta_{jk},$
(372)

and its conservation

$\frac{\partial{j}^{k}}{\partial{t}}+\partial_{j}\Pi^{jk}=0.$ (373)

We introduce a gauge transformation

$\displaystyle\frac{\partial}{\partial{t}}$
$\displaystyle\rightarrow\nabla-i\phi,$ (374) $\displaystyle\vec{\nabla}$
$\displaystyle\rightarrow\vec{\nabla}-i\vec{A},$ (375)

we then write a Lagrangian that respects the nonlinear Schroedinger equation

$\mathcal{L}=\mathcal{L}_{0}+\rho\phi-\vec{j}\cdot\vec{A}+\frac{\rho}{2}A^{2}.$
(376)

In this Lagrangian we have the validity of Galileo’s symmetry.

$\displaystyle\vec{A}$
$\displaystyle=\alpha\left(\vec{v}_{s}-\vec{v}_{n}\right),$ (377)
$\displaystyle\phi$ $\displaystyle=\vec{v}_{n}\cdot\vec{A}.$ (378)

Constituting itself as a fundamental symmetry for the very emergence of the
excitations of the superfluid 354, according to the criterion of Landau 351.
Such symmetries configure a potential

$U=-\frac{1}{2}\left(\vec{v}_{n}-\vec{v}_{s}\right)\frac{\partial\rho_{n}}{\partial\rho}-\frac{1}{2\rho}\vec{\nabla}\left[\rho_{n}\left(\vec{v}_{n}-\vec{v}_{s}\right)\right],$
(379)

which appears in the nonlinear Schroedinger equation, also known as the Gross
Pitaevski equation 358

$i\partial_{t}\psi=-\frac{1}{2}\nabla^{2}\psi+\left(|\psi|^{2}-1+U\right)\psi.$
(380)

We will now see a perturbation in Galileo’s symmetry, so we start to study
superfluids no longer described by the nonlinear Schroedinger equation.

###  XVII.6 NLKG-Mandelung’s formalism and superfluids

For a nonlinear Klein Gordon equation, we define a complex scalar field as we
did in the Mandelung formalism 314

$\phi(\vec{x},t)=F(\vec{x},t)e^{[i\sigma(\vec{x},t)]}.$ (381)

The flow in the tensor formalism, for example 326

$v^{\mu}_{s}=\partial^{\mu}\sigma,$ (382)

the associated trivector

$\vec{v}_{s}=\frac{\vec{\nabla}\sigma}{\partial^{0}\sigma},$ (383)

we have mandatory that $\frac{|\vec{v}_{s}|}{c}<1$.

We write the action for a scalar field

$\mathcal{L}=\int{d^{4}x}\sqrt{-g}\left(g^{\mu\nu}\partial_{\mu}\phi^{*}\partial_{\nu}\phi+V\right),$
(384)

the conserved current generated by 384 is

$\left(\square-V^{\prime}\right)\phi=0$ (385)

where

$\displaystyle V^{\prime}$
$\displaystyle=\frac{d\mathcal{L}}{d(\phi^{*}\phi)}$ (386)
$\displaystyle\square\phi$
$\displaystyle\equiv\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi\right).$
(387)

We then write the hydrodynamic equations for 381

$\displaystyle\left(\square-V^{\prime}\right)F+F\nabla^{\mu}\sigma\nabla_{\mu}\sigma$
$\displaystyle=0,$ (388) $\displaystyle
2\nabla^{\mu}F\nabla_{\mu}\sigma+F\nabla^{\mu}\nabla_{\mu}\sigma$
$\displaystyle=0,$ (389)

so we define the current associated with 381

$j^{\mu}=\frac{1}{2i}\left(\phi^{*}\partial^{\mu}\phi-\phi\partial^{\mu}\phi^{*}\right)=F^{2}\partial^{\mu}\sigma.$
(390)

The similarity of the action 384 with the action of a k-essence model Babichev
_et al._ (2008) is remarkable. As an example of 326 we now define

$w^{\mu}:=\partial^{\mu}\alpha+\xi\partial^{\mu}\beta,$ (391)

here $\alpha,\xi,\beta$ are independent variables that respect the Schutz
formalism 328, 329, 330, 331, 332 with these variables we can write some
Lorentzian invariants and it still solves the problem of the contradiction
between the definition of the velocity vector as a gradient 359 and the
velocity to Ampere’s law 362. For this case $\partial_{[\mu}w_{\nu]}\neq 0$,

$\displaystyle I_{1}$ $\displaystyle=\frac{F^{2}}{2}v^{\mu}v_{\nu},$ (392)
$\displaystyle I_{2}$ $\displaystyle=F^{2}v^{\mu}w_{\mu},$ (393)
$\displaystyle I_{3}$ $\displaystyle=\frac{F^{2}}{2}w^{\mu}w_{\mu}.$ (394)

We write the extended Lagrangian with respect to 384

$\mathcal{L}=\mathcal{L}_{0}+\mathcal{F}(I_{1},I_{2},I_{3}).$ (395)

We explicitly write the Schutz formalism

$\left[\square-V^{\prime}-(1+f^{\prime}_{1})v_{\mu}v^{\mu}-2f^{\prime}_{2}w_{\mu}v^{\mu}-f^{\prime}_{3}w_{\mu}w^{\mu}\right]F=0$
(396)

with $f^{\prime}=\frac{\partial\mathcal{F}}{\partial{I}_{n}}$. We still have
other conserved currents

$\displaystyle\partial_{\mu}j^{\mu}$ $\displaystyle=0,$ (397)
$\displaystyle\partial_{\mu}s^{\mu}$ $\displaystyle=0,$ (398) $\displaystyle
s^{\mu}\partial_{\mu}\xi$ $\displaystyle=0,$ (399) $\displaystyle
s^{\mu}\partial_{\mu}\beta$ $\displaystyle=0,$ (400)

fitting the same thermodynamic interpretation of the equations 328, 329, 330,
331 and 332.

Here we return to the equation 382

$v^{\mu}=(\tilde{\mu},\vec{\nabla}\sigma)=\tilde{\mu}\left(1,\vec{v}_{s}\right)$
(401)

and a perpendicular fluid

$v^{\mu}_{n}=\tilde{s}\left(1,\vec{v}_{n}\right)$ (402)

we establish normalization

$\tilde{\mu}=\gamma_{s}\mu,$ (403)

implying a Lorentz factor

$\gamma_{s}=\frac{1}{\sqrt{1-\frac{v^{2}_{s}}{c^{2}}}}$ (404)

the same for $\tilde{s}$

$\tilde{s}=\gamma_{n}s$ (405)
$\gamma_{n}=\frac{1}{\sqrt{1-\frac{v^{2}_{n}}{c^{2}}}}.$ (406)

The dot product between 401 and 402 is

$v^{\mu}_{s}v^{n}_{\mu}=-c^{2}y=-c^{2}\tilde{\mu}\tilde{s}\left(1-\frac{\vec{v}_{s}\cdot\vec{v}_{n}}{c^{2}}\right)=-c^{2}\mu{s}\gamma_{n}\gamma_{s}\left(1-\frac{\vec{v}_{s}\cdot\vec{v}_{n}}{c^{2}}\right),$
(407)

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

$\vec{v}_{ns}=\frac{\vec{v}_{n}-\vec{v}_{s}}{1-\frac{\vec{v}_{s}\cdot\vec{v}_{n}}{c^{2}}}$
(408)

so the squared modulus $||\vec{v}_{ns}||=||\vec{v}_{s}-\vec{v}_{n}||^{2}$ is
calculated as

$\frac{v^{2}_{ns}}{c^{2}}=1-\frac{\mu^{2}s^{2}}{y^{4}}.$ (409)

It is established here a relation of the product of the modulus of the
difference between the perpendicular and circulation velocities
$||\vec{v}_{s}-\vec{v}_{n}||^{2}$ with the entropy of the system.

##  XVIII Superfluid Vacuum theory – a brief review

In this section, we will study a formalism that combines superfluids with
effective metrics, see ,Volovik (2001), we consider the linear phonon spectrum
in the momentum

$E(p,n)\rightarrow{c}(n)|\vec{p}|$ (410)

where $c(n)$ is given in 121. Phonons represent the collective quanta of the
excited modes of a superfluid, whose velocity is given by

$c^{2}(n)=\frac{n}{m}\left(\frac{d^{2}\epsilon}{dn^{2}}\right).$ (411)

Following the concept of acoustic metric 134, we can associate the geometric
structure with hydrodynamic quantities,

$\displaystyle g^{00}$ $\displaystyle=-\frac{1}{mnc},$ (412) $\displaystyle
g^{0i}$ $\displaystyle=-\frac{v^{i}_{s}}{mn^{2}},$ (413) $\displaystyle
g^{ij}$ $\displaystyle=\frac{c^{2}\delta^{ij}-v^{i}_{s}v^{j}_{s}}{mnc},$ (414)
$\displaystyle g_{00}$
$\displaystyle=-\frac{mn}{c}\left(c^{2}-v^{2}_{s}\right),$ (415)
$\displaystyle g_{0i}$ $\displaystyle=\frac{mn}{c}v_{si},$ (416)
$\displaystyle\sqrt{-g}$ $\displaystyle=\frac{m^{2}n^{2}}{c^{2}}.$ (417)

Then the components of the metric $g_{\mu\nu}$ are associated with the density
of states $n$, the mass $m$, the modulus of the fluid velocity $v$, the
modulus of the propagation of signals $c$ and to the velocity vector of the
flow line $v_{si}$. The “gravitons” of this acoustic geometry respect the
geodesic

$g^{\mu\nu}p_{\mu}p_{\nu}=0,$ (418)

and the dispersion relation

$\left(\tilde{E}-\vec{p}\cdot\vec{v}_{s}\right)^{2}=(cp)^{2}.$ (419)

###  XVIII.1 Mandelung formalism applied to dark matter

We use 313 to describe quantum vacuum fluctuations, interpreting 313 as a
wavefunction $\Psi(\textbf{r},t)$, in three-dimensional Euclidean space.
Building then the normalization condition Zloshchastiev (2010)

$<\psi|\psi>=\int_{\nu}\rho{d\nu}=\mathcal{M},$ (420)

where $\mathcal{M}$ is the total mass and $\nu$ is the volume of a fluid, so
we write the mass density of the fluid $\rho=|\psi|^{2}$. The function of 313
is governed by the Schroedinger equation

$\left[-i\hbar\partial_{t}-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{ext}(\vec{r},t)+F\left(|\psi|^{2}\right)\right]\psi=0,$
(421)

where $m$ is the mass of the constituent particle, $V_{ext}(\vec{r},t)$ is an
external potential and $F(\rho)$ is a properly chosen function. The wave
function can be formally derived as a minimizing condition of a functional
action that follows the Lagrangian

$\mathcal{L}=\frac{i\hbar}{2}\left(\psi\partial_{t}\psi^{*}-\psi^{*}\partial_{t}\psi\right)+\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}+V_{ext}(\vec{r},t)|\psi|^{2}+V(|\
psi|^{2}).$ (422)

The excitations governed by 422, can be understood as a fluid analogous to
gravity, thus having an induced geometry as defined in 134, which we rewrite
below, considering the velocity as given in 314

$\mathcal{G}^{\mu\nu}=\frac{\rho}{c}\left(\begin{array}[]{cc}-[c^{2}_{s}-\eta^{2}(\nabla{S})^{2}]&-\eta\vec{\nabla}s\\\
-\eta\vec{\nabla}s&\mathcal{I}\end{array}\right),$ (423)

the factor being $\eta=\frac{\hbar}{m}$ and
$s=s(\vec{r},t)=-i\ln\left(\frac{\psi(\vec{r},t)}{|\psi(\vec{r},t)|}\right)$,
we still need to establish that $c_{s}$ is the speed of sound. In Mandelung’s
representation $\psi=\sqrt{\rho}e^{is}$ is a unitary matrix in three
dimensions. We also established that $c_{s}>\eta|\nabla{s}|$, we are talking
here, therefore, about small oscillations of small amplitude of a condensate,
moving along the induced geodesic, as in 139. In this example, authors like
Zloshchastiev (2020a) consider that waves propagate with the speed of sound in
a material medium, therefore $c_{s}=\sqrt{\frac{dp}{d\rho}}$, thus
establishing , that space-time behaves as a material medium under certain
conditions, we refer here to what Zloshchastiev (2020a) calls fluid/gravity,
in this formulation it is not possible to remove the backgroud, and the metric
itself corresponds to matter distribution, this fact produces a slightly
different interpretation of the Einstein field equation 232 deduced from the
Einstein-Hilbert action gains a new interpretation here. For both the left
side corresponds to matter and the right side to a geometry induced by matter

$T^{\mu\nu}_{induces}{\equiv}R^{\mu\nu}-\frac{R}{2}g^{\mu\nu}.$ (424)

So the difference between 232 and 424 is that 232 is understood as a
differential equation for unknown terms of the metric. On the other hand 424
corresponds to a coupling of the test particles, where the condition of small
oscillations is not violated. Another difference is that 424 is seen by
R-Observer. In this formalism called fluid-gravity we define two types of
observers: the R-observer that is able to observe small measurements, small
moments, small deviations and an F-observer that is in the Euclidean space
Zloshchastiev (2020b). This approximation can be applied to dark matter Harko
and Madarassy (2022). This entire formulation respects Eulerian relations,
which are crystallized in the continuity equation

$\frac{dn}{dt}+\vec{\nabla}\cdot\vec{v}=0,$ (425)

and in Euller’s equation

$m\frac{d\vec{v}}{dt}=m\left[\partial_{t}\vec{v}+(\vec{v}\cdot\vec{\nabla})\vec{v}\right]=-\vec{\nabla}\left[U(\vec{r})+U_{rot}(\vec{r})+u_{0}n-\frac{\hbar^{2}}{2m}\frac{\Delta\sqrt{n}}{\sqrt{n}}\right],$
(426)

and the Thomas-Fermi approximation

$\nabla\left[U(\vec{r})+U_{rot}(\vec{r})+u_{0}n\right]=0.$ (427)

In 427 we have that $n=\psi(\vec{r},t)$

$U_{rot}(\vec{r})=-w^{2}r^{2},$ (428)

where $w$ is the rotation speed of the dark matter Galactic Halo Harko and
Madarassy (2022). The equilibrium of the superfluid associated with dark
matter is given by

$\Delta\rho(\vec{r})+k^{2}\left[\rho(\vec{r})+\rho_{b}(\vec{r})+\rho_{disc}(\vec{r})-\frac{w^{2}}{2G\pi}\right]=0,$
(429)

here $k^{2}=\frac{Gm^{3}}{a\hbar^{3}}=\frac{4{\pi}Gm^{2}}{u_{0}}$,
$\rho=mn(\vec{r},t)$, $a$ is the scattering length in the equation 427, we
have $U(\vec{r})$ being the gravitational potential, which is usually worked
out as having three parts, associated with the gravitational potential of
baryonic matter, dark matter and the system disorder Harko and Madarassy
(2022).

##  XIX Conclusions

The study of the momentum-energy tensor is a fundamental topic in relativity,
the review of classical conditions of energy is a necessity given its
relevance to several lines of research in astrophysics and cosmology. The
emergence of dark and exotic relativistic fluids, brings the need to review
their energy conditions especially in terms of causality. The anisotropy of
the momentum-energy tensor is a topic that still raises many doubts among
students and the cosmological constant is increasingly being associated with
ultra-dense objects. We review these themes by calculating the conservation of
the momentum-energy tensor, we seek to contribute to the discussion about
energy conditions, both in the isotropic and anisotropic case, identifying
exactly which energy condition is associated with vacuum energy.

We built the superfluid structure via Landau formalism, introduced Galilean
and Lorentzian symmetries and some of its aspects applied to superfluids were
discussed.

In the future, we intend to extend this study to a review of the SVT theory,
also to study relativistic fluids associated with nuclear astrophysics such as
Bjorken and Gupsen fluids and review properties of relativistic fluids with
viscosity, in addition to investigating further the causal structure,
especially in dark and tachionic fluids. Exploring fluid-gravity
correspondence is also one of our goals.

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