Casimir--like wormholes

We investigate spherically symmetric spacetimes sourced by matter with an energy density scaling as ρ ∝ 1/r 4 and obeying the equation of state p r = ωρ. We derive the most general solution of Einstein's field equations for this class of sources and identify a broad family of geometries, including black holes, naked singularities, and traversable wormholes, parametrized by the integration constants and the equation of state parameter ω. We determine the conditions for traversability and show that only two classes of matter can support traversable wormholes: Casimir-type matter with ω > 0 and phantom matter with ω < -1, whereas quintessence and conventional dark energy are excluded by the flare-out condition at the throat. For the genuine Casimir case ω = 3, we find that the transverse pressure approaches p t → -ρ near the central naked singularity at r = 0, implying that the complete Casimir stress-energy tensor, characterized by ρ ∝ -1/r 4 , p r = 3ρ, and p t = -ρ, is dynamically recovered in the vicinity of the origin. This behavior is not imposed a priori but emerges directly from Einstein's field equations. Adopting a purely classical framework in which the amplitude of the energy density is treated as a free parameter, we obtain traversable wormhole solutions with throat radii ranging from microscopic to astrophysical scales. Within this setting, Garattini's Planck-scale Casimir wormhole is recovered as a particular case when the quantum Casimir energy scale is imposed. We further show that the total amount of exotic matter required is finite and scales linearly with the throat 2 radius. We analyze the traversability constraints, including tidal forces and transit times, and determine the minimum throat radius required for comfortable human traversal. Finally, we present embedding diagrams illustrating three distinct geometric regimes depending on the value of ω.