Fluid dynamics in the warp drive spacetime geometry

Abstract

The Alcubierre warp drive metric is a spacetime geometry featuring a spacetime distortion, called a warp bubble, where a massive particle inside it acquires global superluminal velocities, or warp speeds. This work presents solutions of the Einstein equations for the Alcubierre metric having fluid matter as gravity source. The energy–momentum tensor considered has two fluid contents, the perfect fluid and the parametrized perfect fluid (PPF), a tentative more flexible model whose aim is to explore the possibilities of warp drive solutions with positive matter density content. Santos-Pereira et al. (Eur Phys J C 80:786, 2020) already showed that the Alcubierre metric having dust as source connects this geometry to the Burgers equation, which describes shock waves moving through an inviscid fluid, but led the solutions back to vacuum. The same happened for two out of four solutions subcases for the perfect fluid. Other solutions for the perfect fluid indicate the possibility of warp drive with positive matter density, but at the cost of a complex solution for the warp drive regulating function. Regarding the PPF, solutions were also obtained indicating that warp speeds could be created with positive matter density. Weak, dominant, strong and null energy conditions were calculated for all studied subcases, being satisfied for the perfect fluid and creating constraints in the PPF quantities such that a positive matter density is also possible for creating a warp bubble. Summing up all results, energy–momentum tensors describing more complex forms of matter or field distributions generate solutions for the Einstein equations with the warp drive metric where a negative matter density might not be a strict precondition for attaining warp speeds.

1. Introduction

It is well known that in general relativity particles can travel globally at superluminal speeds, whereas locally they cannot surpass the light speed. The warp drive spacetime geometry advanced by Alcubierre uses these physical properties to propel material particles at superluminal speeds. It creates a limited spacetime distortion, called warp bubble, such that the spacetime is contracted in front of it and expanded behind the bubble as it moves along a geodesic. This warp drive metric is such that a particle trapped inside this bubble would locally move at subluminal speeds, whereas the bubble with the particle inside acquires global superluminal velocities, or warp speeds. In a seminal paper, Alcubierre also concluded that the warp metric would imply the violation of the energy conditions since it appeared that a negative energy density would be required for the creation of the warp bubble.

Since this original work many authors have contributed to our understanding of the theoretical details of the Alcubierre warp drive metric and the feasibility of matter particles acquiring warp speeds. Ford and Roman applied quantum inequalities to calculate the amount of negative energy required to transport particles at superluminal speeds. They concluded that such energy requirements would be huge, so the amount of negative energy density necessary for the practical construction of a warp bubble would be impossible to achieve. Pfenning and Ford also used quantum inequalities to calculate the necessary bubble parameters and energy for the warp drive viability, reaching an enormous amount of energy, ten orders of magnitude greater than the mass-energy of the entire visible universe, also being negative. Hiscock computed the vacuum energy–momentum tensor (EMT) of a reduced two-dimensional quantized scalar field of the warp drive spacetime. He showed that in this reduced context that the EMT diverges if the apparent velocity of the bubble is greater than the speed of light. Such a divergence is connected to the construction of an horizon in this two-dimensional spacetime. Due to the semiclassical effects, the superluminal travel via warp drive might be unfeasible. For example, to an observer within the warp drive bubble, the backward and forward walls look like the horizon of a white hole and of a black hole, respectively, resulting from Hawking radiation.

The issue of superluminal speeds of massive particles traveling faster than photons has also been studied by Krasnikov, who argued that this would not be possible if some conjectures for globally hyperbolic spacetimes are made. He described some spacetime topologies and their respective need of the tachyon existence for the occurrence of travel at warp speeds. This author also advanced a peculiar spacetime where superluminal travel would be possible without tachyons, named the Krasnikov tube by Everett and Roman, who generalized the metric designed by Krasnikov by proposing a tube in the direction of the particle's path providing a connection between Earth and a distant star. Inside the tube the spacetime is flat and the lightcones are opened out in order to allow for one direction superluminal travel. For the Krasnikov tube to work they showed that huge quantities of negative energy density would also be necessary. Since the tube does not possess closed timelike curves, it would be theoretically possible to design a two way non-overlapping tube system such that it would work as a time machine. In addition, the EMT is positive in some regions. Both the metric and the obtained EMT were thoroughly analyzed in later work.

A relevant contribution to warp drive theory was made by van de Broeck, who demonstrated that a small modification of the original Alcubierre geometry greatly diminishes to a few solar masses the total negative energy necessary for the creation of the warp bubble distortion, a result that led him to hypothesize that other geometrical modifications of this type could further reduce in a dramatic fashion the amount of energy necessary to create a warp drive bubble. Natario designed a new warp drive with zero expansion by using spherical coordinates and the X-axis as the polar one. Lobo and Visser discussed that the center of the warp bubble, as proposed by Alcubierre, needs to be massless. A linearized model for both approaches was introduced and it was demonstrated that for small speeds the amassed negative energy inside the warp field is a robust fraction of the particle's mass inside the center of the warp bubble. Lee and Cleaver have looked at how external radiation might affect the Alcubierre warp bubble, possibly making it energetically unsustainable, and how a proposed warp field interferometer could not detect spacetime distortions. Mattingly et al. discussed curvature invariants in the Natario and Alcubierre warp drives.

In a previous paper we have considered some of these issues, but from a different angle. The Alcubierre metric was not advanced as a solution of the Einstein equations, as it was originally proposed simply as an ad hoc geometry designed to create a spacetime distortion such that a massive particle inside it travels at warp speeds, whereas locally it never exceeds the light speed. The basic question was then which possible types of matter-energy sources are capable of creating a warp bubble. To answer this question the Einstein equations have to be solved with some form of EMT as a source. The simplest one to start with is incoherent matter. Following this line of investigation, we showed that the dust solutions of the Einstein equations for the warp drive metric implied in vacuum, that is, such a distribution is incapable of creating a warp bubble; nevertheless, the Burgers equation appeared as part of the solution of the Einstein equations. In addition, since the Burgers equation describes shock waves moving in an inviscid fluid, it was also found that these shock waves behave as plane waves.

In this paper we generalize the results obtained in that reference by following the next logical step, that is, investigating the perfect fluid as EMT source for the Alcubierre metric. We also propose a slightly generalized perfect fluid EMT, called here the parametrized perfect fluid (PPF), in order to produce a tentatively more flexible model such that the pressure may have different parameter values. The aim is to see if more flexible EMT distributions could relax the original requirement that warp speeds could only be achieved by means of a negative matter density.

For the perfect fluid EMT solutions we found that two out of four subcases turn out to be the dust solution of the previous reference where both the matter density and the pressure vanish, but the Burgers equation also appears as a result of the solutions of the Einstein equations. Two other subcases, however, indicate that warp speeds are possible with positive matter density, but at the cost of a complex solution for the warp metric regulating function. Weak, dominant, strong and null energy conditions were calculated for both EMTs and all perfect fluid solutions satisfy them. In the case of the PPF, two out of four solutions give rise to a nonlinear equation of state linking various pressures to the matter density. Other solutions produced results where a nonvanishing pressure occurs with a vanishing matter density, a condition considered unphysical and then dismissed. The solutions also produced parameters and equations of state related to pressure and inequalities that satisfy all the energy conditions. These results indicate that energy–momentum tensors describing more complex forms of matter distributions generate solutions for the Einstein equations with the warp drive metric where negative matter density might not be a strict precondition.

The plan for the paper is as follows. In Sect. 2 we briefly review the Alcubierre warp drive theory and present the relevant equations and all nonzero components of the Einstein tensor for the warp drive metric. In Sect. 3 the Einstein equations are then solved and solutions presented for the warp drive metric having a perfect fluid gravity source. In Sect. 4 the nonzero components of the Einstein tensor in the warp drive geometry are written in terms of the PPF EMT. Solutions for this more flexible EMT are also obtained and studied in all subcases. Section 5 presents the EMT divergence of both the perfect fluid and the PPF, whereas Sect. 6 discusses the energy conditions for the two types of EMTs. Section 7 provides further discussions on the results presented in the previous sections, and Sect. 8 presents our conclusions.

2. Einstein equations

We shall start this section by briefly reviewing the Alcubierre warp drive metric. Subsequently, the nonzero components of the Einstein tensor of this metric will also be explicitly shown. The expressions presented in this section form the basic set of equations required for the next sections.

2.1 The Alcubierre warp drive geometry

The geometry advanced in Alcubierre's paper may be written as follows:

ds² = − ( α² − β_i βⁱ ) dt² + 2 β_i dxⁱ dt + γ_ij dxⁱ dx^j (2.1)

where dτ is the proper time lapse, α is the lapse function, βⁱ is the spacelike shift vector and γ_ij is the spatial metric for the hypersurfaces. (Throughout this paper Greek indices will range from 0 to 3, whereas the Latin ones indicate the spacelike hypersurfaces and will range from 1 to 3.) The lapse function α and the shift vector βⁱ are functions to be determined, whereas γ_ij is a positive-definite metric on each of the spacelike hypersurfaces, for all values of time, a feature that makes the spacetime globally hyperbolic.

Alcubierre assumed the following particular parameter choices for Eq. (2.1):

α = 1 (2.2)

β¹ = − v_s(t) f [ r_s(t) ] (2.3)

β² = β³ = 0 (2.4)

γ_ij = δ_ij (2.5)

Hence, the Alcubierre warp drive metric follows as Eq. (2.6): a line element whose time-time coefficient is minus the quantity one minus the square of v_s(t) f(r_s), which carries a cross term in dx dt proportional to v_s(t) f(r_s), and whose spatial part is the flat sum dx² + dy² + dz².

Here v_s(t) is the velocity of the center of the bubble moving along the curve x_s(t). This is given by the following expression:

v_s(t) = dx_s(t)/dt (2.7)

The function f(r_s) is the warp drive regulating function. It describes the shape of the warp bubble, which is given by the following expression:

f(r_s) = [ tanh( σ (r_s + R) ) − tanh( σ (r_s − R) ) ] / [ 2 tanh(σ R) ] (2.8)

where σ and R are parameters to be determined. The variable r_s(t) defines the distance from the center of the bubble at coordinates x_s(t), 0, 0 to a generic point x, y, z on the surface of the bubble, given by the following equation:

r_s(t) = √( [ x − x_s(t) ]² + y² + z² ) (2.9)

From the above one can see that the motion is one-dimensional, since the x-coordinate is the only one perturbed by the function x_s(t).

Let us now adopt Alcubierre's original notation by assuming

β = − β¹ = v_s(t) f(r_s) (2.10)

in Eq. (2.3).

2.2 Einstein tensor components

The nonzero components of the Einstein tensor for the metric (2.6) are then given by Eqs. (2.11) to (2.20). They are built entirely from the single function β and its derivatives: the energy-density component G00 collects the squared transverse gradients of β along y and z with a coefficient set by β itself; the momentum components G01, G02 and G03 collect the mixed second derivatives of β; G11 is proportional to the same squared transverse gradients; G12, G13 and G23 collect products of first derivatives with mixed second derivatives along the transverse directions and time; and G22 and G33 collect the second derivatives of β along the direction of travel and in time, together with the squared gradients. Every component vanishes when β is uniform, which is the statement that a flat shift profile carries no source.

3. Perfect fluid

Besides incoherent matter, or dust, already studied in the previous paper, the simplest matter-energy distribution to be considered as a gravity source for the possible creation of a warp bubble, and then warp speeds, is the perfect fluid. Hence, this section will discuss matter content solutions of the Einstein equations considering a perfect fluid matter source EMT for the Alcubierre metric.

3.1 Perfect fluid content solutions

The EMT for a perfect fluid may be written as follows:

T_αβ = (μ + p) u_α u_β + p g_αβ (3.1)

where μ is the matter density, p is the fluid pressure, g_αβ is the metric tensor and u_α is the four-velocity of an observer inside the fluid. Perfect fluids have no shear stress, rotation, heat conduction or viscosity, nevertheless this ideal fluid provides a more complex matter content than simple dust, allowing us to study if a warp bubble can be created with this gravity source and how the respective gravity field equations solutions can be understood.

For the metric (2.6) the perfect fluid EMT assumes the matrix form of Eq. (3.2), whose leading entry is μ + β² p, whose time-space entries are −βp, and whose remaining diagonal entries are the single pressure p.

Solving the Einstein equations component by component with this source produces an equation of state for the Alcubierre metric having a perfect fluid gravity source, in which the pressure is three times the matter density. The component G23 is zero since T23 is zero, which leads to either the derivative of β with respect to y, or the derivative of β with respect to z, or both, being equal to zero — and this fork defines Case 1 and Case 2, each of which splits again into subcases a and b.

(The case-by-case solution of the Einstein equations for the perfect fluid, and the summary of it in Table 1, are omitted for length; the complete text is at the source.)

4. Parametrized perfect fluid

Let us propose a generalization of the perfect fluid EMT having seven quantities, namely the mass-energy density μ, the β function and five different pressures A, B, C, D and p. The last quantity D is a momentum density parameter. In the perfect fluid of Eq. (3.2) the pressure denoted by p is all the same in the EMT, a constraint that has been relaxed here. Let us call the perfect fluid generalization with the quantities above the parametrized perfect fluid (PPF). Its respective EMT may be written as the matrix of Eq. (4.1), whose leading entry is μ + β² p, whose time-space entries are −βD, and whose remaining diagonal entries are the three distinct pressures A, B and C.

The quantities A, B, C, D and p will not be assumed to be constants, but rather functions of the spacetime coordinates t, x, y and z. This is clearly a more flexible EMT than the perfect fluid, and it is being proposed here as a tentative model in order to explore the consequences of more complex EMTs in terms of generating possible positive matter solutions of the Einstein equations.

(The solution of the Einstein equations for the PPF in Sect. 4, its discussion in Sect. 4.1, the resulting equations of state in Sect. 4.1.1, the divergence conditions of Sect. 5 and the summary in Table 2 are omitted for length; the complete text is at the source. The outcome is stated in Sects. 7 and 8 below.)

6. Energy conditions

The energy conditions are the well-known inequalities applied to the matter content in physical systems as boundary conditions and to test if the energy of such systems follows positive constraint values.

This section aims at obtaining these conditions for the Alcubierre warp drive geometry considering the previously discussed EMTs for both the perfect fluid and the PPF. Our focus will be on the main classical inequalities, namely, the weak, dominant, strong and null energy conditions. Our analysis starts with the PPF EMT defined in Eq. (4.1) in order to constrain its quantities so that the inequalities are satisfied for each of the four just named conditions. Then the same analysis for the perfect fluid is performed by reducing the results to the perfect fluid choice, but considering only the physically plausible cases.

6.1 Weak energy condition (WEC)

The WEC requires that at each point of the spacetime the EMT contracted twice with the observer four-velocity be greater than or equal to zero (6.1)

This is valid for any timelike vector u, whose norm is negative, as well as for any null vector k, whose norm is zero. For an observer with a unit tangent vector v at some point of the spacetime, the local energy density measured by any observer is non-negative. Considering the Eulerian, or normal, observers from Alcubierre's paper with four-velocity given by the expressions

u^α = (1, β, 0, 0), and u_α = (−1, 0, 0, 0) (6.2)

and the EMT given by Eq. (4.1), the WEC expression is computed from the sum of the T00 term, twice the T01 term and the T11 term, each weighted by the corresponding components of the four-velocity.

(The full evaluation of the weak, dominant, strong and null energy conditions for every subcase, and their summary in Table 3, are omitted for length; the complete text is at the source. The results are that the perfect fluid satisfies all four conditions — the null condition requiring only that the sum of matter density and pressure be non-negative, which, given that the pressure is three times the density, is fulfilled whenever the matter density is not negative — and that the PPF satisfies them under explicit constraints, chiefly that the pressure A be greater than or equal to zero. The null condition further requires that the shift β, which equals v_s(t) f(r_s), cannot take the exact value of the speed of light, nor its negative, while values above the speed of light remain available.)

7. Further discussions

This section discusses some points, and raises others, all related to the physics of the warp drive as suggested by the results presented in the previous sections. It aims at offering some thoughts that may be important in fostering further understanding on how a superluminal travel can be achieved.

7.1 Regulating function and the Burgers equation

The regulating function (2.8) describes the shape of the warp bubble, but it is not uniquely determined. However, the integration of the Einstein equations in both the perfect fluid and the PPF led to the appearance of generic functions in the dynamic equations which may end up connected to the regulating function, a situation that adds to its nonuniqueness. This means that physically feasible superluminal speeds will require the specification of these generic functions, possibly by boundary conditions. We shall show below an example of this situation using the dust solution.

Considering Eqs. (2.3) and (2.10), the partial derivatives of β with respect to time and to x are given in Eqs. (7.1) and (7.2) in terms of the derivative of the regulating function with respect to r_s and the derivative of r_s with respect to the corresponding coordinate, with the simplified notation f = f[r_s(t)] and v_s = v_s(t). Substituting these into the Burgers equation (3.19) yields Eq. (7.3), in which the second derivative of x_s with respect to time, the term carried by v_s and the term carried by v_s squared times f sum to an arbitrary function h(t).

The partial derivative of Eq. (2.8) may be written as follows: the derivative of f with respect to r_s equals σ divided by twice the hyperbolic tangent of σR, multiplied by the difference of the squared hyperbolic secants of σ(r_s + R) and σ(r_s − R) (7.4)

The partial derivatives of r_s yield

∂r_s/∂t = ± v_s(t) (7.5)

∂r_s/∂x = ±1 (7.6)

and remembering that in the dust case β is a function of x and t, then r_s(t) is given, from Eq. (2.9), by the absolute value of x minus x_s(t) (7.7)

Considering the expressions above, Eq. (7.3) may be rewritten in the form of Eq. (7.8), in which the second derivative of x_s with respect to time, weighted by f, together with two terms in v_s squared carrying the function F over twice the hyperbolic tangent of σR, sum to h(t), where

F(r_s) ≡ sech²[ σ (r_s + R) ] − sech²[ σ (r_s − R) ] (7.9)

Remembering that the regulating function f[r_s(t)], as defined by Eq. (2.8), can be approximated by a top hat function when σ is large compared with R, in this limit Eq. (7.8) takes the following form inside the bubble:

d²x_s/dt² = h(t) (7.10)

where f = 1. Outside the bubble f = 0 and then h(t) = 0, which means plane shock waves described by the inviscid Burgers equation.

Hence, the nonuniqueness of the shift vector β = v_s(r_s) f[r_s(t)] arises not only from the fact that the function h(t) is arbitrary, but also because the regulating function f[r_s(t)] only requires a top hat behavior with null values outside the bubble. So, any well-behaved function that respects such constraints may be part of a solution of the Burgers equation. Moreover, from the energy conditions calculated for the PPF, as summarized in Table 3, one can see that β plays a fundamental role in Cases 1a and 2a for the null and dominant energy conditions to be satisfied, which adds further physical constraints to its behavior. So, it is clear that the nonlinearity of the Einstein equations imply that the generic functions appearing in their integration become entangled with the regulating function in a non-trivial manner.

To extend this analysis to the perfect fluid and PPF contents requires further constraints on the shift vector, something which at this stage would be done in an entirely arbitrary manner. Perhaps in the future this can be done under more physically plausible reasoning.

7.2 Anisotropic fluids

The PPF proposed in Sect. 4 aimed at offering an alternative EMT for solving the Einstein equations endowed with the warp drive metric. As we shall see below, the PPF can actually be seen as an anisotropic fluid with heat flux.

In general relativity the energy–momentum tensor T_μν represents the source of energy and momentum, where T00 is the flow of energy across a surface of constant time (energy density), T0i is the energy flux across a surface in the i direction (constant xⁱ), Ti0 is the momentum density and Tij is the momentum flux. If we choose a comoving frame of reference that moves with the same velocity as the fluid this means that particles in this fluid will have zero velocity and the flux of energy will be only through the flux of heat, and the momentum flux will be via some sort of dissipative phenomena such as viscosity, thermal radiation or even some sort of electromagnetic type of radiation.

The general stress–energy tensor of a relativistic fluid can be written in the form

T^αβ = μ u^α u^β + p h^αβ + u^α q^β + u^β q^α + π^αβ (7.11)

where

h_αβ = g_αβ + u_a u_b (7.12)

projects tensors onto hypersurfaces orthogonal to u^α, μ is the matter density, p is the fluid static pressure, q^α is the heat flux vector and π^αβ is the viscous shear tensor. The world lines of the fluid elements are the integral curves of the four-velocity vector u^α. The heat flux vector and viscous shear tensor are transverse to the world lines, that is,

q_a u^a = 0, and π_ab u^b = 0 (7.13)

In terms of coordinates we can write the energy–momentum tensor for a general fluid as the block matrix of Eq. (7.14), with the scalar function ε in the time-time slot, the three-vector heat flux q_a in the time-space slots, and the three-by-three viscous stress tensor π_ab in the space-space block, which is symmetric and traceless. Both q_a and π_ab have, respectively, three and five linearly independent components. Together with the density μ and the static pressure p, this makes a total of ten linearly independent components, which is the number of linearly independent components in a four-dimensional symmetric rank two tensor. We noticed that the Einstein tensor components are highly nonlinear for the warp drive metric and the off-diagonal terms require those free parameters for a non-overdetermined solution.

For a non-curved metric, i.e., the Minkowski metric η_αβ, the energy–momentum tensor for a perfect fluid with anisotropic pressures can be written as the diagonal matrix of Eq. (7.15), carrying μ, then the three pressures p_x, p_y and p_z. Isotropic static pressure means that the three pressures are equal to one another. The perfect fluid has no heat flux or dissipative phenomena, so the heat flux vector and the viscous shear tensor both vanish. This special case with dust content is the well-known EMT, that is,

T^αβ = μ u^α u^β + p h^αβ = (μ + p) u^α u^β + p g^αβ (7.16)

The PPF proposed in Sect. 4 has the matrix form given by Eq. (4.1), which may be broken down as the sum of a perfect fluid in a warp drive background as given by Eq. (3.2), and a dissipative fluid with the heat flux four-vector given by q_α = −½ (q0, q1, 0, 0). Hence the symmetrized product of heat flux and four-velocity, Eq. (7.17), has q0 and q1 in the first row and column and zeros elsewhere, since the four-velocity for the moving frame is u^α = (−1, 0, 0, 0). The isotropic term is the diagonal matrix of Eq. (7.18) carrying π00, π01, π02 and π03, and the density-plus-pressure combination transforms as

μ + β² p → (μ + π00 + q0) + β² p (7.19)

So, the four parameters of the PPF are as follows:

A = π01 + p (7.20)

B = π02 + p (7.21)

C = π03 + p (7.22)

D = p − q1/β (7.23)

The tensor π_αβ must be traceless, giving us one more equation to solve for the free parameters (anisotropic pressures)

π00 + π01 + π02 + π03 = 0 (7.24)

From the above one can see that the warp drive metric endowed with the PPF allows one to make a study of fluid anisotropy coupled with possible dissipative effects that could lead to a warp drive bubble. Perfect fluids are well known to be part of solutions for the Einstein equations, this being the case of the standard FLRW cosmological model that accounts for the expansion, isotropy and spatial homogeneity of the universe. On the other hand, anisotropic imperfect fluids offer a more complex source of gravitational effects, presenting dissipative processes, shear and bulk viscous pressures, interaction between fluids, radiation processes, electromagnetic interaction and even collision between particles, charged or not. These fluids are even known to avoid the big bang singularity in cosmological models. MacCallum discussed various ways of generating anisotropy such as the presence of electromagnetic fields, the presence of viscous terms and the anisotropic stresses due to the anisotropic expansion of a cloud of collisionless particles. Another way to account for viscosity, heat and energy flux is the interaction of two or more fluids.

7.3 Other aspects of warp drive physics

The points presented above regarding the physical feasibility of superluminal travel with the Alcubierre spacetime geometry by no means exhaust this discussion. Several issues remain open, with each of them deserving separate studies that are beyond the scope of this paper, since in here we focused on the basic properties of the solutions of the Einstein equations of the warp drive metric with fluid content. With respect to the open issues, one can point out the amount of mass-energy density, exotic or not, necessary for the feasibility for the warp drive in the context of both the perfect fluid and the PPF, and also quantum effects and the question of stability or instability in our solutions. These issues deserve further investigations and are the subject of ongoing research.

8. Conclusions

In this work we have analyzed the Einstein equations for the Alcubierre warp drive metric having as gravity source two types of energy–momentum tensors (EMT) for a fluid, namely the perfect fluid and the parametrized perfect fluid (PPF). The latter is defined by allowing the EMT pressure components of the perfect fluid to be different from one another and dependent on all coordinates.

After obtaining the components of the Einstein tensor for the warp drive metric we calculated the dynamic equations for both fluids by solving the respective Einstein equations. Solutions were found in the form of various equations of state, and, by further imposing the null divergence for the EMTs, new constraints were also found for the various variables. The weak, dominant, strong and null energy conditions were also calculated, which implied further constraints upon the free quantities for the EMTs. For the perfect fluid these were on the matter density μ and pressure p. For the PPF that occurred these were on the pressure components p, A, B, C, the matter density for the fluid μ and the momentum component D.

We found two main groups of solution subcases possessing different conditions for each EMT. For the perfect fluid, one solution may be interpreted as requiring that the warp bubble can only be viable with negative matter density. The alternative interpretation is that the warp bubble is possible with positive matter density, but in this case the regulating function f(r_s), which shapes the warp bubble, becomes a complex function. This comes from results allowing the possibility that the function β = v_s(t) f(r_s) may have complex solutions once the matter density is positive. Other results in both fluids reinforce our earlier finding that the warp bubble necessary for generating superluminal velocities, or warp speeds, can be interpreted as a shock wave from classical fluid dynamics theory.

We mentioned four cases arising from the solutions of the Einstein equations: 1a, 1b, 2a and 2b. However, Cases 1a and 2a are very similar, or equal, to each other, and we have the same for Cases 1b and 2b. For this reason they were grouped together in the tables that summarize all results.

Specifically, Cases 1b and 2b for the perfect fluid reduced the solutions to the ones found for dust content already studied in the previous paper, that is, a vacuum solution unable to create a warp bubble, but which connects the warp metric to the inviscid Burgers equation, also yielding β as a function of t and x coordinates. The null EMT divergence is satisfied and a continuity equation was also found.

Cases 1b and 2b for the PPF resulted in an equation of state of the form μ = −β² p, the coordinate dependency of the β function became β = β(x, t) and a non-homogeneous Burgers equation emerged. However, the pressures were constrained in such a way that this PPF EMT solution for the warp drive was dismissed as unphysical.

Cases 1a and 2a for the perfect fluid produced an equation of state relating pressure and matter density given by p = 3μ. The β function dependencies became β = β(y, t) in the former case and β = β(z, t) in the latter case, and both produced a differential equation for β that either requires a negative density for β to be a real valued function, or a positive matter density which then leads to a complex solution for β, which in turn leads to a complex regulating function f(r_s) whose possible real part would then be related to a physically viable warp bubble.

Cases 1a and 2a for the PPF resulted in an equation of state relating almost all quantities, in the form μ = β²(2D − A − p) + A/3, which is valid for both cases. The coordinate dependency on β that occurred, respectively, resulted in β = β(y, t) and β = β(z, t). The function β is also governed by the first order differential equations (4.73) and (4.74), respectively.

The null EMT divergence was calculated, producing further sets of very nonlinear differential equations constraining all quantities in the PPF which could be used, in principle, to determine all pressures and matter density in this fluid. For the perfect fluid, Cases 1a and 2a are reduced to a continuity equation including the function β, which is interpreted as playing the role of the flow velocity vector field. Cases 1b and 2b reduced the PPF to either the trivial condition of Minkowski flat spacetime with no warp drive, or an EMT with all components being zero, that is, a vacuum case.

It has already been seen in the previous paper that the Burgers equation appears to be connected to the dust solution of the warp drive metric, which is in fact a vacuum solution. Cases 1b and 2b of the perfect fluid became reduced to the dust solution, and a non-homogeneous form of the Burgers equation appears in these respective cases for the PPF, although the whole solutions in these cases were dismissed as unphysical. The solutions that presented themselves as the most plausible ones for creating warp speeds, 1a and 2a for both the perfect fluid and the PPF, do not generate a Burgers equation.

The weak, dominant, strong and null energy conditions were also studied in the context of the perfect fluid and PPF energy–momentum tensors for a warp drive metric. The resulting expressions were found to satisfy all conditions in the perfect fluid EMT. Regarding the PPF, specific expressions constraining its EMT quantities were obtained in order to satisfy these energy conditions, but they do not necessarily lead to the conclusion that negative matter density is always necessary for viable warp speeds, particularly because in the PPF the pressure A must assume values equal to or greater than zero.

Summing up, the results of this paper indicate that warp speeds might be physically viable in the context of positive matter density as some solutions of the Einstein equations for both fluids keep open this possibility. Nevertheless, such a situation creates the additional issue in the perfect fluid context concerning the meaning of a possible complex regulating function in the warp metric, a result that may be interpreted as a caveat, or major stumbling block. Such a difficulty does not appear to occur in the PPF scenario, although this fluid was considered here mainly as a hypothetical model whose aim was to investigate whether or not new possibilities arise in the solutions of the Einstein field equations considering more complex energy–momentum tensors having the Alcubierre warp drive metric. On this front it seems then that the initial conclusions about the unphysical nature of the warp drive, or the impossibility of generating warp speeds, may not be not as stringent as initially thought, or, perhaps, are not valid at all.

Acknowledgements

We are grateful to the referees for useful comments. E.M.C.A. thanks CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazilian scientific support federal agency, for partial financial support, Grants numbers 406894/2018-3 and 302155/2015-5.

Data Availability Statement: This manuscript has no associated data or the data will not be deposited. Authors' comment: This is a theoretical work. There is no data available and, hence, no data will be deposited.

(The 42-item reference list is omitted for length; the complete text is at the source.)