Wormhole solutions with radial equation of state profiles in general relativity

Motivated by the absence of observational evidence for both wormholes and the validity of a constant equation of state for the matter supporting such geometries, we consider a position-dependent relation of the form p r + ω ( r ) ρ = 0 . Within the framework of general relativity and the Morris-Thorne wormhole geometry, two functional forms of the equation-of-state parameter are analyzed, namely an inverse-power model ω ( r ) = ω 0 + ω 1 ( r 0 r ) and a logarithmic model ω ( r ) = ω 0 + ω 1 ln ( r r 0 ) , where ω 0 and ω 1 are constants. The corresponding shape functions are obtained analytically, and the resulting thermodynamic quantities are derived from the gravitational field equations. The geometric requirements for a traversable wormhole, including the throat condition, flare-out condition, and asymptotic flatness, are examined. The energy conditions are analyzed, and the violation of the radial null energy condition near the throat is confirmed, as expected in GR wormhole solutions. A graphical analysis is performed to illustrate the behavior of the shape function, energy density, equation of state parameter, and embedding diagrams. Moreover, the equilibrium of the configuration is investigated using the generalized Tolman–Oppenheimer–Volkoff equation, showing that the hydrostatic and anisotropic forces balance each other. Finally, the amount of exotic matter is quantified through the VIQ, indicating that the required NEC-violating matter can be made arbitrarily small near the throat.

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