Warp bubble geometries with anisotropic fluids: A piecewise analytical approach

We present a comprehensive analytical study of spherically symmetric warp bubble configurations 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.