2021

[1,2]Gabriel Abellán

[1,2,3]Nelson Bolivar

1]Astrum Drive Technologies, Dallas Pkwy Unit 120 B, Frisco, TX., 75034, USA

2]Departamento de Física, Facultad de Ciencias, Universidad Central de
Venezuela, Av. Los Ilustres, Caracas, 1041-A, Venezuela

3]Instituto Balseiro, Centro Atómico Bariloche, Comisión Nacional de Energía
Atómica (CNEA)–Universidad Nacional de Cuyo (UNCUYO), Bariloche, 8400,
Argentina

# Warp drive solutions in spherical coordinates with anisotropic matter
configurations

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Ivaylo Vasilev<EMAIL_ADDRESS>[  [  [

###### Abstract

In this work we study the influence of isotropic and anisotropic fluids on the
spherically symmetric warp metric. We evaluate the energy conditions and the
influence of including a cosmological constant type term. We find that,
considering this term, there is a trade–off between the weak and strong energy
conditions. The obtained solutions are numerical and we solve the system for
both the stationary and the full regime. The influence of imposing the zero
expansion condition has been explored. We find a wide diversity of behaviours
for the solutions. In general there are regions of spacetime where the energy
conditions can be at least partially satisfied. Finally, we calculate the
value of the total mass using the density found in the numerical simulations,
finding examples where it remains positive during the entire evolution of the
system.

###### keywords:

warpdrive, energy conditions, spherical symmetric warp, anisotropic solutions

##  1 Introduction

It is known that in general relativity particles can travel globally at
superluminal velocities Morris:1988cz ; Visser:1989kh ; Krasnikov:1995ad ;
Everett:1997hb . Alcubierre explored this idea Alcubierre:1994tu  and propose
a way to drive matter at velocities higher than the speed of light. The
mechanism proposed in his work creates a distortion of space-time, called a
warp bubble, resulting in a portion of spacetime contracting in front of the
bubble and expanding behind of it as the bubble moves through a geodesic. The
line element proposed by Alcubierre was,

$ds^{2}=-dt^{2}+\left(dx-f(r_{s})v_{s}dt\right)^{2}+dy^{2}+dz^{2}\;,$ (1)

with $r_{s}=\sqrt{(x-x_{s})^{2}+y^{2}+z^{2}}$ and $v_{s}=\frac{dx_{s}}{dt}$.
This corresponds to an ADM-like decomposition of the line element
Arnowitt:1962hi ; Wald1984ucp ; Alcub2008itnr . Studying this metric,
Alcubierre already note that the proposed metric implied the violation of
energy conditions, since it seemed that a negative energy density would be
necessary to sustain such a bubble.

Since Alcubierre’s seminal work, there have been numerous contributions
pointing to improve and a better understanding of the physics behind the warp
metric he proposed. A key interesting aspect has been to apprehend the
properties of spacetimes that would allow superluminal velocities to be
achieved Krasnikov:1995ad ; Everett:1997hb . Among the most relevant
contributions are the modifications to the original metric, allowing for a
significant decrease in the energy involved VanDenBroeck:1999sn . Other
explorations implicated the imposition of constraints on the system, either on
the spacetime or on the matter fluid, as can be seen in Natario’s work
Natario:2001tk  where he proposed a warp drive with zero expansion. Lobo and
Visser Lobo:2004an  also discuss several characteristics of energy-matter of
the warp bubble, determining some interesting properties, for instance, it
must be massless at the centre. It is worth to remark that in these studies
the amount of energy needed is significantly reduced. One aspect that has
received much attention from the very beginning has been the study of energy
conditions as a means of validating the physical feasibility of the warp drive
Olum:1998mu ; Pfenning:1998ua ; Low:1998uy ; VanDenBroeck:1999xs ;
Barcelo:2000zf ; Lobo:2002zf ; Barcelo:2002bv ; Lobo:2004wq ; McMonigal:2012ey
; Alcubierre:2017kqf ; Alcubierre:2017pqm ; Lentz:2020euv ; Santiago:2021aup ;
Schuster:2022ati . The occurrence of horizons and closed time curves as well
as the development of instabilities due to the presence of quantum matter has
also been studied Hiscock:1997ya ; Clark:1999zn ; Gonzalez-Diaz:1999vtr ;
Eroshenko:2022cyx ; Everett:1995nn ; Barcelo:2022vru . This has been the
subject of extensive debate in the community and although much progress has
been made, there is still no definitive consensus.

In the works mentioned so far, the starting point is the metric and from there
the properties of the matter that supports this spacetime are deduced.
However, it is possible to proceed in the reverse way, that is to say, by
fixing some properties of matter and studying how the corresponding spacetime
must be. In a series of papers Santos-Pereira:2020puq ; Santos-Pereira:2021mqp
; Santos-Pereira:2021rsr ; Santos-Pereira:2021mrr ; Santos-Pereira:2021sdm ,
Santos et al. propose to study the warp problem from this last point of view.
They examined how the elements of the warp metric should be constrained if
some kind of matter configuration is imposed, for instance dust or a perfect
fluid. In this way they were able to obtain some relations for the deformation
function given by the Einstein equations. The present work adheres to this
approach. In a previous publication we have proposed an alternative metric to
study the issues associated with warp drive abellan:2023 . This metric
exploits the advantages of having spherical symmetry in the description of
spacetime.
In this work we analyse in depth the consequences on spacetime when
considering different types of fluids.

In section 2 we discuss the most important results related to the warp metric
in spherical coordinates. We also present the energy-momentum tensor that we
will use to describe the type of fluids we will consider. On the other hand we
will give a description of the system of equations obtained in terms of
traveling patterns. This allows us to study the stationary behavior of the
system under study. Section 3 is devoted to evaluate the energy conditions
associated with the problem in question. This will be done from both the fluid
and metric points of view. We will then look at the consequences to the system
after imposing the zero expansion condition. Subsequently, in sections 4 and 5
we will go on to solve numerically the system of equations obtained. Basically
we will study solutions for isotropic fluids and then for anisotropic fluids.
The solutions studied will be both in the steady state and in the full regime.
We also include a discussion of the effect of adding a cosmological constant
type term to the system. In section 6 we will use the results of the
simulations to determine the behavior of the mass as the system evolves.
Finally we make some concluding remarks and considerations for future work.

##  2 Warp drive in spherical coordinates

In this section we present the ingredients of the model we intend to build. We
want to write Einstein’s equations for the metric proposed. That is

$\displaystyle G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi
T_{\mu\nu}\;.$ (2)

Here $R_{\mu\nu}$ is the Ricci tensor, $R$ the Ricci scalar, $g_{\mu\nu}$ the
metric and $T_{\mu\nu}$ the energy–momentum tensor. Also we are working in
geometric units where $G=c=1$.

###  2.1 Metric and Einstein Tensor

In a recent work abellan:2023 , motivated by Alcubierre’s work we propose a
warp line element using spherical coordinates

$\displaystyle ds^{2}=-dt^{2}+(dr-\beta dt)^{2}+r^{2}d\Omega^{2}\;,$ (3)

with $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ and $\beta$ the form
function. Note the non–diagonal element and the fact that $\beta=0$ returns a
Minkowski flat spacetime.

Using this metric we calculate the elements of the Einstein tensor, which are

$\displaystyle G_{00}$ $\displaystyle=$
$\displaystyle\frac{\beta}{r^{2}}\Bigg{[}\left(1-\beta^{2}\right)\left(\beta+2r\frac{\partial\beta}{\partial
r}\right)-2r\beta\frac{\partial\beta}{\partial t}\Bigg{]},$ (4) $\displaystyle
G_{01}$ $\displaystyle=$
$\displaystyle\frac{\beta}{r^{2}}\left(\beta^{2}+2r\beta\frac{\partial\beta}{\partial
r}+2r\frac{\partial\beta}{\partial t}\right),$ (5) $\displaystyle G_{11}$
$\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}\left(\beta^{2}+2r\beta\frac{\partial\beta}{\partial
r}+2r\frac{\partial\beta}{\partial t}\right),$ (6) $\displaystyle G_{22}$
$\displaystyle=$
$\displaystyle-r\Bigg{\\{}\beta\left(2\frac{\partial\beta}{\partial
r}+r\frac{\partial^{2}\beta}{\partial
r^{2}}\right)+\frac{\partial\beta}{\partial
t}+\;r\left[\left(\frac{\partial\beta}{\partial
r}\right)^{2}+\frac{\partial^{2}\beta}{\partial t\partial
r}\right]\Bigg{\\}},$ (7) $\displaystyle G_{33}$ $\displaystyle=$
$\displaystyle-r\sin^{2}{\theta}\Bigg{\\{}\beta\left(2\frac{\partial\beta}{\partial
r}+r\frac{\partial^{2}\beta}{\partial
r^{2}}\right)+\frac{\partial\beta}{\partial
t}+r\left[\left(\frac{\partial\beta}{\partial
r}\right)^{2}+\frac{\partial^{2}\beta}{\partial t\partial
r}\right]\Bigg{\\}}.$ (8)

In general, we consider the form function as a quantity dependent of both time
and radial coordinates. Note also that we only have five non–zero components.
This represents a remarkable simplification with respect to the Alcubierre’s
metric written in Cartesian coordinates.

###  2.2 Energy–Momentum tensor

The next step in writing the Einstein equations consists in fixing the
material content of the system under study. For this we consider an
anisotropic fluid given by

$T_{\mu\nu}^{(a)}=(\rho+p_{\perp})u_{\mu}u_{\nu}+p_{\perp}g_{\mu\nu}+(p_{r}-p_{\perp})s_{\mu}s_{\nu}\;,$
(9)

where we have used the following four–vectors

$u_{\mu}=(-1,0,0,0)\;,\hskip 19.91684pts_{\mu}=(-\beta,1,0,0)\;,$ (10)

these vectors are timelike and spacelike respectively and satisfy the
relations $u^{\mu}u_{\mu}=-1$, $s^{\mu}s_{\mu}=1$ and $u^{\mu}s_{\mu}=0$. In
(9) $\rho$ is the relativistic energy density, $p_{r}$ the radial pressure and
$p_{\perp}$ the tangential pressure.

We are interested in considering the effects of including a cosmological
constant type term. To facilitate the subsequent numerical solution of the
system of equations, we will include this term in the energy–momentum tensor,
namely

$T_{\mu\nu}^{(\Lambda)}=-\frac{\Lambda}{8\pi}g_{\mu\nu}=-\rho_{\Lambda}g_{\mu\nu}\;,$
(11)

with $\Lambda$ (and $\rho_{\Lambda}$) a constant.

So, the total energy–momentum tensor is
$T_{\mu\nu}\\!=\\!T_{\mu\nu}^{(a)}+T_{\mu\nu}^{(\Lambda)}$. Using equations
(9) and (11) we can write in matrix form

$T_{\mu\nu}=\left[\begin{array}[]{cccc}\rho+\beta^{2}p_{r}+(1-\beta)\rho_{\Lambda}&-\beta(p_{r}-\rho_{\Lambda})&0&0\\\
-\beta(p_{r}-\rho_{\Lambda})&p_{r}-\rho_{\Lambda}&0&0\\\
0&0&r^{2}(p_{\perp}-\rho_{\Lambda})&0\\\
0&0&0&r^{2}\sin^{2}\\!\theta\,(p_{\perp}-\rho_{\Lambda})\end{array}\right].$
(12)

It is interesting to compare with the tensor proposed in Santos-
Pereira:2021sdm . We can easily check that the system admits anisotropic
configurations.

###  2.3 Einstein’s equations

Using the Einstein tensor components (4)–(8) and the energy–momentum tensor
(12) we find the Einstein’s equations for this system

$\displaystyle\beta\left(\beta+2r{\frac{\partial\beta}{\partial r}}\right)$
$\displaystyle=$ $\displaystyle 8\pi r^{2}\rho^{(e)}\,,$ (13)
$\displaystyle\beta\left(\beta+2r{\frac{\partial\beta}{\partial
r}}\right)+2r\frac{\partial\beta}{\partial t}$ $\displaystyle=$
$\displaystyle-8\pi r^{2}p_{r}^{(e)},\;\;\;\;\;$ (14)
$\displaystyle\beta^{2}+r{\frac{\partial\beta}{\partial
t}}-r^{2}\left[{\frac{\partial}{\partial
r}}\left(\beta{\frac{\partial\beta}{\partial
r}}\right)+\frac{\partial^{2}\beta}{\partial t\partial r}\right]$
$\displaystyle=$ $\displaystyle 8\pi r^{2}\Delta\,,$ (15)

with the effective quantities $\rho^{(e)}=\rho+\rho_{\Lambda}$,
$p_{r}^{(e)}=p_{r}-\rho_{\Lambda}$ and $\Delta=p_{\perp}-p_{r}$ the anisotropy
factor. From Einstein tensor (5) and (6) the equation (14) arise. The same is
true for the components (7) and (8) which produce the same equation for
$p_{\perp}^{(e)}=p_{\perp}-\rho_{\Lambda}$. That is, we have just three
independent equations to solve. The system (13)–(15) has four degrees of
freedom: $\beta$, $\rho$, $p_{r}$ and $\Delta$.

###  2.4 Stationary, traveling–type solutions

We are interested in the possibility that the system of equations produces a
traveling wave type solution. For this we use the parametrization

$\beta(t,r)=f(r_{s})v_{s}(t)\;.$ (16)

Here $r_{s}=\|r-v_{s}(t)t\|$, with $v_{s}(t)$ a time–dependent spatial
velocity function. In the following we consider this velocity a constant, so
the displacement is uniform in radial direction.

Using this parametrization we find that the equations to be solved for
$r-v_{s}t\geq 0$ are

$\displaystyle
f^{2}+2(r_{s}+v_{s}t)ff^{\prime}=\frac{8\pi}{v_{s}^{2}}(r_{s}+v_{s}t)^{2}\rho^{(e)}\;,$
(17) $\displaystyle
f^{2}+2(r_{s}+v_{s}t)(f-1)f^{\prime}=-\frac{8\pi}{v_{s}^{2}}(r_{s}+v_{s}t)^{2}p_{r}^{(e)}\;,$
(18) $\displaystyle
f^{2}-(r_{s}+v_{s}t)f^{\prime}-(r_{s}+v_{s}t)^{2}[(f^{\prime})^{2}+(f-1)f^{\prime\prime}]=\frac{8\pi}{v_{s}^{2}}(r_{s}+v_{s}t)^{2}\Delta\,,$
(19)

where primes denote derivatives with respect to $r_{s}$. Moreover, for
$r-v_{s}t<0$ the system vanish and we obtain the traveling wave behavior.

The systems of equations (13)-(15) and (17)-(19) describe the complete and
stationary system respectively. To solve it is necessary to provide boundary
conditions. In both cases we will use

$\lim_{r\to 0}\beta=\lim_{r\to\infty}\beta=0\;.$ (20)

An important point to note is that only the complete system requires initial
conditions. This is one of the advantages of working with the equations that
explore travelling patterns.

##  3 Energy conditions and expansion

In this section we study energy conditions, which are constraints imposed on
the energy-momentum tensor so that we can evaluate the viability of the model
and control non-physical aspects of the system Wald1984ucp ; Visser1995lwet ;
Poisson2004rtmb ; Carroll2004sgig ; Curiel:2014zba ; Kontou:2020bta .

###  3.1 Weak energy condition

Weak energy condition (WEC) requires that $u^{\mu}$,
$T_{\mu\nu}u^{\mu}u^{\nu}\geq 0$. That is

$T_{\mu\nu}u^{\mu}u^{\nu}=\rho+\rho_{\Lambda}\geq 0\;.$ (21)

Here $u_{\mu}$ is given by (10) and is a timelike vector. In fact, this vector
defines an Eulerian observer for the system. So, we find that weak energy
condition is satisfied if $\rho+\rho_{\Lambda}\geq 0$. Using Einstein’s
equations, the equivalent condition on the $G_{\mu\nu}$ requires that

$\beta\left(\beta+2r{\frac{\partial\beta}{\partial r}}\right)\geq 0\;.$ (22)

For this result we have used $r^{2}>0$, which simplifies the final expression.

###  3.2 Dominant energy condition

The dominant energy condition (DEC) is equivalent to the WEC, with the
additional requirement that $T^{\mu}_{\nu}u^{\nu}$ is a future-pointing causal
vector. Thus, the weak energy condition and $F^{\mu}F_{\mu}\leq 0$, with
$F^{\mu}=T^{\mu\nu}u_{\nu}$, must be satisfied.

After some straightforward calculations, we find that

$F^{\mu}F_{\mu}=-(\rho+\rho_{\Lambda})^{2}\leq 0\;,$ (23)

Note that, if the conservation of the $T_{\mu\nu}$ is also considered, both
conditions (21) and (23) guarantee the causal structure in local matter
configurations. In terms of the metric components, the dominant condition is
equivalent to

$-\left(\beta+2r\frac{\partial\beta}{\partial r}\right)^{2}\leq 0\;.$ (24)

It is clear that this condition is always satisfied.

###  3.3 Strong energy condition

The strong energy condition (SEC) imposes a bound on a more complicated
expression in $4$ dimensions

$\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}\geq 0\,.$ (25)

Using the energy–momentum tensor (12) and the definition for the anisotropy
factor $\Delta$ we obtain

$\rho-2\rho_{\Lambda}+3p_{r}+2\Delta\geq 0\,.$ (26)

This is a proper generalization of results in Santos-Pereira:2021sdm . The
equivalent form written in terms of $\beta$ is

$-\beta\left(2\frac{\partial\beta}{\partial
r}+r\frac{\partial^{2}\beta}{\partial
r^{2}}\right)+2\frac{\partial\beta}{\partial
t}+r\left[\left(\frac{\partial\beta}{\partial
r}\right)^{2}+\frac{\partial^{2}\beta}{\partial t\partial r}\right]\geq 0\;.$
(27)

This expression is actually rather complex in relation to the others already
discussed.

###  3.4 Null energy condition

Finally, the null energy condition (NEC) is analogous to WEC, with the
timelike vector $u_{\mu}$ replaced by a null vector $k_{\mu}$. That is

$T_{\mu\nu}k^{\mu}k^{\nu}\geq 0\,.$ (28)

Assuming a vector $k^{\mu}=\\{a_{\pm},b,0,0\\}$, with

$a_{\pm}=\frac{b}{\beta\pm 1}\;,$ (29)

which is a lightlike vector $k^{\mu}k_{\mu}=0$. We find that NEC is

$\displaystyle\rho+p_{r}\geq 0\;.$ (30)

Note that the contributions from $\rho_{\Lambda}$ cancel each other. Writing
this condition in terms of $\beta$ we find

$\frac{\partial\beta}{\partial t}\leq 0\;.$ (31)

That is, as time evolves, $\beta$ decreases.

Finally a comment about the cosmological constant type term. It is important
to note that the addition of $\rho_{\Lambda}$ in the model gives additional
freedom, allowing a shift to be made in case of finding conditions that are
not satisfied by themselves. We consider this to be important because it can
help to establish more general procedures to deal with the problems that warp
drive has traditionally suffered with respect to energy conditions. However,
we can see that there is an interplay between the weak and strong conditions.
That is, we can manipulate the term and enforce one of them but necessarily
make the other one worse. We will see examples of this when studying the
solutions.

###  3.5 Expansion

As can be seen in (3), the geometry corresponding to 3-space is flat and
therefore the extrinsic curvature tensor $K_{ij}$ allows encoding information
about the curvature. The extrinsic curvature tensor serves to describe how the
3-dimensional hypersurfaces are embedded in the 4-dimensional space. It is
given by the expression

$K_{ij}=\frac{1}{2\alpha}\Big{(}D_{i}\beta_{j}+D_{j}\beta_{i}-\frac{\partial
g_{ij}}{\partial t}\Big{)}\;,$ (32)

where $D_{i}$ is the covariant derivative relative to the 3-space metric, and
$\alpha$ correspond to lapse function which we assume $\alpha=1$ as usual.
This choice means that timelike curves normal to the hypersurfaces in 3-space
are geodesics, namely Eulerian observers are free falling. According to these
observers, the expansion $\Theta$ of the volume elements is written in terms
of $K_{ij}$ and is given by

$\displaystyle\Theta$ $\displaystyle=$ $\displaystyle-\mathrm{Tr}[K_{ij}]$
(33) $\displaystyle=$ $\displaystyle
2\frac{\beta}{r}+\frac{\partial\beta}{\partial r}\;.$

In fig. 1 is possible to observe the behaviour of expansion for a typical
$\beta$ solution (see next sections). Analogously to Alcubierre’s findings,
there is a deformation that lives in a compact domain. The volume elements are
expanding behind and contracting in a sort of wavefront. We also note the
existence of a flat region inside the bubble. All this will have effects on
the dynamical equations and energy conditions as we will see next.

Figure 1: The plot depicts the expansion of the volume elements $\Theta$ of
the spherical warp bubble in the non-stationary case with $\rho_{0}$ being a
form of parametrization to account for the coordinates “perpendicular to $r$”.
The full expansion is shown in the upper plot. A transverse cut is shown in
the lower plot where it can be noted the flat sitting region and the expansion
and contraction.

It is possible to use equation (33) and impose a zero expansion condition on
the Einstein equations. This restricts the possibilities of fluid evolution
and hence the shape of spacetime. With $\Theta=0$, we find that

$\displaystyle\beta^{2}$ $\displaystyle=$
$\displaystyle-\frac{8\pi}{3}r^{2}\rho\;,$
$\displaystyle\frac{2}{r}\frac{\partial\beta}{\partial
t}-\frac{3}{r^{2}}\beta^{2}$ $\displaystyle=$ $\displaystyle-8\pi p_{r}\;,$
(34) $\displaystyle\frac{3}{r}\frac{\partial\beta}{\partial
t}-\frac{9}{r^{2}}\beta^{2}-\frac{\beta}{r^{2}}\frac{dr}{dt}$ $\displaystyle=$
$\displaystyle 8\pi\Delta\;.$ (35)

These equations are interesting because it is evident that in order to have a
$\beta$ real, $\rho$ must be negative. So we have a close relation between the
expansion factor and the energy density as was already pointed out in
Natario:2001tk .

To conclude this section we find it is worthwhile to evaluate the energy
conditions using the null expansion condition. Using the constraint $\Theta=0$
over expression (33) we find in terms of $\beta$ that

WEC: $\displaystyle\hskip 17.07182pt\beta^{2}\leq 0\;,$ (36) DEC:
$\displaystyle\hskip 17.07182pt\beta^{2}\geq 0\;,$ (37) SEC:
$\displaystyle\hskip 17.07182pt\beta^{2}\geq 0\;,$ (38) NEC:
$\displaystyle\hskip 17.07182pt\displaystyle\frac{\partial\beta}{\partial
t}\leq 0\;.$ (39)

It is clear that the weak condition cannot be satisfied under the zero
expansion regime. However, the dominant and strong conditions are always
satisfied. The null condition does not undergo any modification using the
constraint.

##  4 Case 1: isotropic solutions

Since the system of equations is highly non-linear, we will solve it
numerically. In order to do so, we consider several cases separately.

###  4.1 Non–stationary solutions ($\beta(t,r)$)

First we solve for system (13)–(15) without imposing any constraint. In this
case we set $\Delta=0$ so we reduce de degrees of freedom to three and the
system closes. Also we set $\rho_{\Lambda}=0$ and later we consider the
effects when this is included. In order to solve it is necessary to give an
initial condition for $\beta$. We will use for convenience a Gaussian given by

$\beta(0,r)=\exp{\Big{[}-\frac{(r-0.5r_{e})^{2}}{0.25r_{e}}\Big{]}}\;.$ (40)

Here $r_{e}$ represents the spatial maximum value for the numerical
integration domain.

In figure 2 we have the form function $\beta$, radial pressure $p_{r}$ and the
empirical equation of state $p_{r}(\rho)$ for several times. We can notice
that $\beta$ is decreasing in amplitude as it becomes wider. We also see that
the radial pressure supporting this warp bubble has regions where it is
positive and others where it is negative. This is interesting because it tells
us that this distribution of matter generates non-zero pressure gradients that
could affect the motion of the system. On the other hand, empirical equation
of state shows a completely non-linear behaviour and is multivalued.

Figure 2: Non–stationary Case 1. Form function $\beta$, radial pressure
$p_{r}$ and empirical equation of state for $t=1$ (blue), $3$ (orange), $5$
(green) and $8$ (red).

Below in the graph 3 we have the energy conditions. In general we can observe
that for all of them there are regions where they are fulfilled and others
where they are not. However, we notice that for all the cases they are bounded
and therefore they do not have singular behaviour. This is relevant since it
allows us to use the cosmological constant term to enforce some of them.
However, we see the interplay between the strong and weak conditions as we
have already mentioned. We also find that the null condition is violated in
some regions as has been suggested in Santiago:2021aup .

Figure 3: Non–stationary case 1. From the top we show NEC, SEC and WEC for
$t=1$ (blue), $t=3$ (orange), $t=5$ (green) and $t=8$ (red).

###  4.2 Stationary solutions ($v_{s}f(r_{s})$)

The traveling wave type solutions are interesting because cover the purpose of
what was originally proposed by Alcubierre. They also represent a sort of
stationary behavior for the system under study. In fig. 4 and fig. 5 we show
the stationary solutions of $\beta$, pressure $p_{r}$ and the empirical
equation of state $p_{r}(\rho)$. These are shown for different velocities
$v_{s}$, ranging from $v_{s}=0.3$ (subluminal) to $v_{s}=1.2$ (superluminal).
In fig. 4 velocities $v_{s}=0.3$ and $v_{s}=0.8$ are depicted. As expected the
amplitude of $\beta$ decreases with time, although the lower velocity shows
the largest initial amplitudes. This behavior is also seen in the superluminal
case, fig. 5. The form function $\beta$ displaces its origin with the
velocity, the bubble shape shrinks with larger $v_{s}$.

The radial pressure as in the non–stationary case has positive and negative
regions constrained to the size of $\beta$ for each time and velocity.
Nevertheless, the distribution of the pressures differs from the
non–stationary case in what we think is an interesting feature. For instance,
the boundary values of the pressure that sustains the bubble are finite,
negative at lower $r$ (the origin point of the $\beta$ form), and positive by
the external boundary. This is in contrast with the zero boundary in non-
stationary scenarios. It is worth noting then that the pressure distribution
suggests, for larger subluminal velocities a pressure gradient along the
bubble.

Looking at the empirical equations of state, one can notice the appearance of
regions where $P_{r}$ can be linearly approximated. Nevertheless, we note, as
with the non-stationary solutions, the fact that the relationship between
$P_{r}$ and $\rho$ is multivalued.
We can divide the empirical plots into quadrants, and then observe that the
system possesses regions with different combinations for signs. We want to
remark the upper right quadrant which shows positive signs for $\rho$ and
$P_{r}$. These positive regions has larger contributions for times chosen when
$v_{s}=0.3$, but as the velocity increases the contributions for larger times
are smaller, as can be seen in fig. 4 and even more pronounced in fig. 5 where
only shorter times has both positive contributions.

Figure 4: Stationary case 1 for $v_{s}=0.3$ (left column) and $v_{s}=0.8$
(right column). Each curve corresponds to times $t=15$ (orange), $t=20$ (blue)
and $t=25$ (green)

Figure 5: Stationary case 1 for $v_{s}=1.0$ (left column) and $v_{s}=1,2$
(right column). Each curve corresponds to times $t=15$ (orange), $t=20$ (blue)
and $t=25$ (green) Figure 6: Stationary case 1. From the top down we show the
NEC, SEC and WEC for $t=15$ (orange), $t=20$ (blue) and $t=25$ (green). Left
column $v_{s}=0.3$ and right column $v_{s}=0.8$. Figure 7: Stationary case 1.
From the top down we show the NEC, SEC and WEC for $t=15$ (orange), $t=20$
(blue) and $t=25$ (green). Left column $v_{s}=1.0$ and right column
$v_{s}=1.2$.

##  5 Case 2: anisotropic solutions

In this case we consider an arbitrary anisotropy factor $\Delta$, so it is
necessary to give an additional condition to close the system.

First we will give an equation of state in the extended polytrope form

$p_{r}=K\rho^{\gamma}+\epsilon\rho\;,$ (41)

with $K$, $\gamma$ and $\epsilon$ parameters that need to be provided.

###  5.1 Non–stationary solutions ($\beta(t,r)$)

Solving for this system we show the results in figs. 8 and 9. For the chosen
set of parameters, we can observe in fig. 8 that the form function $\beta$ is
positive and evolves towards a shock wave type behaviour as time progresses.
On the other hand, the pressure has regions where it is positive and negative
and becomes narrower as time progresses. The empirical equation of state shows
a practically linear behaviour for $K=1$ (left column); however, for $K=50$
(right column) the system moves further away from linearity, especially when
the density acquires negative values.

In the fig. 9 we have the energy conditions for the anisotropic case. We can
observe again that for all of them there are regions where they are fulfilled
and others where they are not. Moreover, for all cases we notice that energy
conditions are bounded and therefore do not have a singular or divergent
behaviour. As in the isotropic case, it is possible to use the cosmological
constant term to enforce either the weak condition or the strong condition,
but not both as mentioned above. We also find that the null condition is
violated in some regions.

Figure 8: Non-Stationary case 2. Each curve corresponds to times $t=1$
(orange), $t=3$ (blue) and $t=5$ (green). The polytrope parameters are $K=1$
(left column), $K=50$ (right column). The solutions are numerically stable
around $\epsilon=-0.5$, $\gamma=2$.  Figure 9: Non-Stationary energy
conditions case 2. From the top down we show the NEC, SEC and WEC for $t=1$
(orange), $t=3$ (blue) and $t=5$ (green). The polytrope parameters are $K=1$
(left column), $K=50$ (right column). The solutions are numerically stable
around $\epsilon=-0.5$, $\gamma=2$.

###  5.2 Stationary solutions ($v_{s}f(r_{s})$)

Now we study the traveling wave behaviour of the anisotropic system. In figs.
10 – 15 we show the stationary solutions of $\beta$, pressure $p_{r}$ and the
empirical equation of state $p_{r}(\rho)$. These are shown for different
velocities, from $v_{s}=0.3$ (subluminal) to $v_{s}=1.2$ (superluminal).
Different sets of parameters have been chosen for the equation of state so
that the behaviour of the system in different regimes of the fluid under
consideration can be observed.

In general, the form function $\beta$ displaces its origin with the velocity.
For $\epsilon=-0.5$ in (41), it can be seen in figs. 10 and 11 that $\beta$ is
decreasing as time elapses for $v_{s}=0.3$. However, for $v_{s}=0.8,1.0,1.2$,
we observe that beta is negative and its value for $t=15$ is null while for
intermediate times an increase in the amplitude of the form function is
obtained. It is worth mentioning that as vs get larger, beta gets narrower,
which affects the shape of the bubble as can be deduced from (33). In the case
of radial pressure $p_{r}$, we find that there are regions where it is
positive and others where it is negative. We can also see that as $v_{s}$
increases, the pressure becomes narrower in accordance with $\beta$. Next, we
see the equation of state which produces essentially a linear relationship
between $p_{r}$ and $\rho$ except for $v_{s}=0.3$ where some non-linearity can
be observed.

The analysis for figs. 12–15 is similar to the previous paragraph. However, it
is worth noting that for $\epsilon=-10^{-17}$ you have a completely non–linear
equation of state because both terms in (41) have comparable order of
magnitude.

Figure 10: Stationary case 2 for $v_{s}=0.3$ (left column) and $v_{s}=0.8$
(right column). The polytrope parameters are $K=100$, $\gamma=2$ and
$\epsilon=-0.5$. Each curve corresponds to times $t=11$ (orange), $t=12$
(blue) and $t=15$ (green).  Figure 11: Stationary case 2 for $v_{s}=1.0$ (left
column) and $v_{s}=1.2$ (right column). The polytrope parameters are $K=100$,
$\gamma=2$ and $\epsilon=-0.5$. Each curve corresponds to times $t=11$
(orange), $t=12$ (blue) and $t=15$ (green).  Figure 12: Stationary case 2 for
$v_{s}=0.3$ (left column) and $v_{s}=0.8$ (right column). The polytrope
parameters are $K=100$, $\gamma=2$ and $\epsilon=-50$. Each curve corresponds
to times $t=11$ (orange), $t=12$ (blue) and $t=15$ (green).  Figure 13:
Stationary case 2 for $v_{s}=1.0$ (left column) and $v_{s}=1.2$ (right
column). The polytrope parameters are $K=100$, $\gamma=2$ and $\epsilon=-50$.
Each curve corresponds to times $t=11$ (orange), $t=12$ (blue) and $t=15$
(green).  Figure 14: Stationary case 2 for $v_{s}=0.3$ (left column) and
$v_{s}=0.8$ (right column). The polytrope parameters are $K=100$, $\gamma=2$
and $\epsilon=-10^{-17}$. Each curve corresponds to times $t=11$ (orange),
$t=12$ (blue) and $t=15$ (green).  Figure 15: Stationary case 2 for
$v_{s}=1.0$ (left column) and $v_{s}=1.2$ (right column). The polytrope
parameters are $K=100$, $\gamma=2$ and $\epsilon=-10^{-17}$. Each curve
corresponds to times $t=11$ (orange), $t=12$ (blue) and $t=15$ (green).

As in case 1, it is observed in figs. 16 and 17 that the energy conditions
have regions where it is positive and others where it is negative. Again, it
is possible to use the cosmological constant term to fix either WEC or SEC and
the interplay between them is observed where if one is fulfilled, the other is
necessarily worsened.

One aspect worth noting is what happens with the weak energy condition which
corresponds to the energy density. We can observe that the energy density
takes positive or negative values in various regions of the domain. However,
we notice that in some cases there are time values where the density is
completely positive. This behaviour is then partially lost, which may suggest
some kind of instability.

Figure 16: Stationary case 2. From the top down we show the NEC, SEC and WEC
for $t=11$ (orange), $t=12$ (blue) and $t=15$ (green). Left column $v_{s}=0.3$
and right column $v_{s}=0.8$. The polytrope parameters are $K=100$, $\gamma=2$
and $\epsilon=-0.5$.  Figure 17: Stationary case 2. From the top down we show
the NEC, SEC and WEC for $t=11$ (orange), $t=12$ (blue) and $t=15$ (green).
Left column $v_{s}=1.0$ and right column $v_{s}=1.2$. The polytrope parameters
are $K=100$, $\gamma=2$ and $\epsilon=-0.5$.  Figure 18: Stationary case 2.
From the top down we show the NEC, SEC and WEC for $t=11$ (orange), $t=12$
(blue) and $t=15$ (green). Left column $v_{s}=0.3$ and right column
$v_{s}=0.8$. The polytrope parameters are $K=100$, $\gamma=2$ and
$\epsilon=-50$.  Figure 19: Stationary case 2. From the top down we show the
NEC, SEC and WEC for $t=11$ (orange), $t=12$ (blue) and $t=15$ (green). Left
column $v_{s}=1.0$ and right column $v_{s}=1.2$. The polytrope parameters are
$K=100$, $\gamma=2$ and $\epsilon=-50$.  Figure 20: Stationary case 2. From
the top down we show the NEC, SEC and WEC for $t=11$ (orange), $t=12$ (blue)
and $t=15$ (green). Left column $v_{s}=0.3$ and right column $v_{s}=0.8$. The
polytrope parameters are $K=100$, $\gamma=2$ and $\epsilon=-10^{-17}$.  Figure
21: Stationary case 2. From the top down we show the NEC, SEC and WEC for
$t=11$ (orange), $t=12$ (blue) and $t=15$ (green). Left column $v_{s}=1.0$ and
right column $v_{s}=1.2$. The polytrope parameters are $K=100$, $\gamma=2$ and
$\epsilon=-10^{-17}$.

##  6 Warp density energy requirements

Using the results obtained, it is natural to investigate how the mass is
described by the configurations of matter used. We are interested in determine
to what extend the system violates or not the energy conditions, specifically
the local weak energy condition. To this end we begin from the relation,

$T^{0}_{\;\;0}=\frac{G_{\;\;0}^{0}}{8\pi}\;,$ (42)

which gives an expression for the density

$\rho_{\text{warp}}\equiv\rho=\frac{\beta}{8\pi
r^{2}}\left(\beta+2r{\frac{\partial\beta}{\partial
r}}\right)-\rho_{\Lambda}\;.$ (43)

Next, we can obtain a numerical form of (43) from the numerical solutions
obtained for $\beta$. We use the “volume integral quantifier” proposed by
Visser in sayan:2004 , which amounts to calculating a definite integral for
the relevant coordinate domains.

$\displaystyle
M_{\text{warp}}=\int_{\mathcal{V}}\rho_{\text{warp}}=4\pi\int\\!\\!dr\;r^{2}T^{0}_{\;\;0}=\frac{1}{2}\int
dr\;r^{2}G^{0}_{\;\;0}\;.$ (44)

In this way we can quantify the amount of violation or not of the energy
density to the extent in which these integrals can become positive or negative
for several defined times.

Integrating in the appropriate numerical domain $\mathcal{D}$ given in our
solutions $\beta$ with $\rho_{\Lambda}=0$, we obtain the behaviour for the
analysed cases using several values of $t$ which can be seen in Figs. 22–25.
We note first of all that for both examples taken from case 1 we obtain
positive mass values throughout the whole time progression. The same is true
for the non-stationary case 2. On the other hand, for the stationary case 2 we
find some negative mass values. However, most of the points are positive. This
result is remarkable and contrasts with the widely debated original
Alcubierre’s warp. Apart from observing a large sample of positive values for
the required masses of the warp, we also notice that they are of a rather low
order of magnitude compared to what has been reported in previous work.

For instance if we pick the mass point of case 1, fig. 23 associated with time
$t=2.0$ which has order of magnitude $10^{-11}$ it corresponds to masses of
order $10^{-16}$ kg. Compared to some familiar Newtonian masses, it is
equivalent to $10^{-7}$ the mass of the moon, or $57$ times the mass of mount
Everest.

Figure 22: Numerical estimation of $M_{\text{warp}}$ using a “Volume integral
quantifier” integration for several times. The estimation is for stationary
case 1 with $v_{s}=0.3$. Figure 23: Numerical estimation of $M_{\text{warp}}$
using a “Volume integral quantifier” integration for several times. The
estimation is for non–stationary case 1. Figure 24: Numerical estimation of
$M_{\text{warp}}$ using a “Volume integral quantifier” integration for several
times. The estimation is for stationary case 2 with $v_{s}=1.0$, $K=0.6$ and
$\epsilon=-0.5$. Figure 25: Numerical estimation of $M_{\text{warp}}$ using a
“Volume integral quantifier” integration for several times. The estimation is
for non–stationary case 2 with $K=0.6$ and $\epsilon=-0.5$.

It is expected that since the null energy conditions demands in equation (31)
the $\beta$ function to diminish in time, the values of $M_{\text{warp}}$ will
tend to zero. An interesting aspect of this solutions is the oscillating
behavior shown, so the quantity $M_{\text{warp}}$ seems oscillates and damps
as times evolves. A remarkable difference occurs in fig. 24, where although
the warp mass stays positive it seems to increase in time. We believe that
because the method for finding these solutions is unstable for long times, we
were not able to find the expected decaying regions.

##  7 Final Remarks

In this paper we have explored various solutions to a spherically symmetric
metric describing a warp drive abellan:2023 . For them we have considered as a
source of matter a generic fluid which admits anisotropy and cosmological
constant. Once the Einstein equations for the system under study have been
established, we realise that the anisotropic fluid is the most general that we
can describe with the proposed metric. The metric used here has obvious
advantages over the traditional metric originally proposed by Alcubierre. This
is due to the incorporation of spherical symmetry in describing the residual
flat space that remains in any warp metric. Doing so allows a much more neat
and ready to be solved set of equations.

We have rewritten the set of equations found to accommodate travelling wave
solutions. We found this useful for several reasons. Firstly, it is this type
of solution that was originally proposed by Alcubierre when he gave an
explicit form of the metric, so we wanted to get as close as possible to this
approach in order to be able to compare. Secondly, the set of differential
equations becomes dependent on a single variable and this greatly simplifies
their solution. And thirdly, as a consequence of the previous point, we avoid
the use of initial conditions, which can add a certain degree of arbitrariness
to the solutions obtained. The solutions described in this way can be
considered as the stationary regime to the problem posed.

As we have already described, the problem of energy conditions has dominated
the discussion regarding the feasibility of warp solutions. For this reason,
we include a detailed analysis of the energy conditions from the point of view
of both the energy-momentum tensor and the metric coefficients. Trivially the
dominant energy condition is always satisfied. On the other hand the null
energy condition imposes that the $\beta$ form function must decrease in time
or at least remain unchanged. An interesting aspect occurs when including the
cosmological constant term as a material source. We find that there is a
trade-off between weak and strong energy conditions. We could use the
cosmological constant to fix one of them but then we would necessarily make
the other worse. This relationship is very interesting and worth further
exploration. Another aspect worked on is the incorporation of the zero
expansion condition in the system. This produces a set of equations with
little room for experimentation. However, we believe it is worth taking into
account for future work.

The equations found were solved using numerical methods. For this purpose, we
considered both stationary solutions using the above-mentioned description of
travelling patterns, and complete solutions without imposing any restrictions.
Both types of solutions were applied to isotropic and anisotropic fluids. In
general, it is observed that for certain time instants there are spatial
regions where the energy conditions are satisfied. In the isotropic case it is
interesting to observe the type of empirical equation that appears, reflecting
a multivalued behaviour which may suggest some type of fluid that allows the
occurrence of phase transitions. The characterisation of these fluids is a
very interesting aspect but clearly beyond the scope of the present work.
Regarding the energy conditions, we can observe that all of them show plots
with bounded values. This absence of divergence is a good sign and motivates
us to continue exploring mechanisms to satisfy the conditions in a general
way. In this work we explore the effect of adding a cosmological constant-type
term to the momentum energy tensor. This allows us to correct the behaviour of
either the weak or the strong energy condition but not both. This trade-off
between the two conditions seems to be a general property. We can say that
exploring different kinds of fluids opens up new possibilities and may provide
a way to deal with the constraints imposed by the energy conditions.

With respect to the $\beta$ form function, in general it is observed that its
amplitude is decreasing. However, there are some of the simulated cases in
which the opposite behaviour appears and the amplitude increases. This seems
to suggest a certain instability in the solutions. Something similar can be
observed for the radial pressure, which in most cases is at least partially
positive and only in a few cases completely negative.

Finally, we calculate the value of the total warp mass using the results found
in the numerical simulations. This is useful because it acts as an average
indicator of the amount of total energy needed to sustain the warp structure.
We were able to find values in the parameters that make the integrated energy
density positive at all instants of the system’s evolution. In most cases we
noticed that these integrated values become smaller and smaller with time.
This may support the argument of some instability in the solutions, although
we believe that more tests are needed to have a conclusive result. However,
the fact of finding solutions that produce positive mass values is remarkable
and we believe that further study of this system will allow us to understand
the fundamental characteristics of warp drives and also to understand why
solutions like Alcubierre’s behave as they do.

In summary, we believe that research related to the warp spherical metric
should be continued as it presents an efficient way to study analytically and
numerically the problems associated with the warp drive phenomenon.
Furthermore, it is crucial to continue studying the material content that
supports the warp geometry, distinct and more complex matter configurations,
different metric realizations that could include dissipation, heat flux and
electromagnetic fields can be studied, in this way to pointing to tailor
feasible space-times with these configurations, at least at a theoretical
level.

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