Link: springer.com/article

Negative energy scenarios are the most widely studied for the warp metric. In fact, the prevailing view in the community so far has been that the warp metric necessarily has negative energies. In this work it is shown that the issue of negative energy densities associated with the Alcubierre warp metric with a general form function and similar metrics can be addressed when the whole non-vacuum Einstein equations of the system are examined. To this end, we have considered matter content in the form of anisotropic fluids.We have succeeded in writing the Einstein equations in such a way that some general constraints on the material content become evident. Thismeans that, in rectangular coordinates, the energy density depends necessarily on the tangential pressures of the fluid. For matter such as dust or isotropic fluids we find that that density and other related quantities become identically zero. This makes the negative energy problem spurious. It is also revealed that constructingAlcubierre-based metrics using cylindrical and spherical coordinates results in a system of equations that are amenable to more systematic analysis. The field equations constrain the dependence of the form function and how this impacts the matter content. In all cases we determine that energy density is not mandatory negative, despite the recurrent claims in the literature. This result prompts a reevaluation of the negative energy requirements and underscore the importance of cylindrical and spherical type-warps to demonstrate that negative energy density is not an intrinsic unavoidable feature of warp drives.

Link: springer.com/article

In this work we introduce the Alcubierre warp metric using spherical symmetry. In this way we write the Einstein equations for a perfect fluid and for an anisotropic fluid with cosmological constant. Analysing the energy conditions for both cases, we find that these cases are flexible enough to allow them to be satisfied. We also find that in the time-independent case of the warp bubble, the metric admits a timelike Killing vector and all the energy conditions are satisfied except for the strong energy condition. Moreover, in the time-independent case a barotropic equation of state known from cosmological models naturally arises.

Link: iopscience.iop.org/article

In the present work, we study the consequences of including the lapse func-tion as an additional degree of freedom for a general spherical warp-based geometry. By allowing a non-uniform lapse function to evolve, we find that it is possible to accommodate a fluid that includes heat flow. This broadens the range of fluid types that have been studied in these systems and is consistent with the spherical warp metric. Having added the lapse function, we solved the system of equations using an anisotropic fluid with heat flow. In this way, we can examine the different characteristics of the variables of the system. Next, we study the energy conditions and establish how these are modified by including heat flux for an appropriate generic observer in a locally flat space- time. Finally, we explore all energy conditions using the numerical solutions and verify the regions where they are satisfied.

Link: sciencedirect.com/article

We present a comprehensive analytical study of spherically symmetric warp bubble configura-tions in the framework of classical general relativity. We use a simplified ADM-type metric with a trivial lapse and a non-trivial radial shift function, which resembles a Painlevé–Gullstrand type metric. Employing this metric, our approach leads naturally to an anisotropic energy –momentum tensor characterized by an equation of state that emerges naturally from the equations. To reconcile the strict boundary conditions with the requirement of a localized matter distribution, we adopt a piecewise—defined model for the energy density. This construction allows us to confine possible violations of the dominant energy condition to finite and controlled shells, while ensuring that the weak and null energy conditions are globally satisfied. We illustrate our method with two representative examples: a one-shell exponential decay profile and a double-shell profile incorporating an additional power-law factor. Our results demonstrate that, by properly tuning the model parameters, it is possible to design warp bubble geometries that are not only mathematically consistent, but also physically more feasible, providing a promising stepping stone towards the development of realistic warp bubble models.

Link: sciencedirect.com/article

This article develops static, spherically symmetric “warp-bubble–like” spacetimes sourced by classical electrostatic configurations, using a Painlevé–Gullstrand–type metric with unit lapse in which Einstein’s equations enforce an anisotropic-fluid equation of state \(p_r = -\rho\) and fix \(p_\perp(r)\) once an energy-density profile \(\rho(r)\) is chosen.  By prescribing piecewise-defined electromagnetic-inspired sources—Coulomb, Yukawa-screened, dielectric-layer, and Hulthén-type fields—the authors construct models with flat interiors, curved exteriors, and controlled matching via Israel junction conditions that reveal purely tangential thin shells whenever the field profile is discontinuous, while smooth dielectric profiles avoid such shells altogether.  A systematic analysis of energy conditions shows that pure Coulomb fields satisfy all classical conditions, screened fields remain regular but violate the dominant energy condition due to enhanced tangential stresses, and the dielectric layer satisfies only the dominant condition in its material region, highlighting how realistic, non-exotic electromagnetic matter can generate highly anisotropic yet analytically tractable spacetime geometries.