Exact Traversable Wormhole Solutions in Finsler--Randers Gravity: Geometry, Physical Viability, and Equilibrium

Abstract We construct a static, spherically symmetric traversable wormhole in Finsler--Randers (FR) gravity using the osculating-Riemannian formulation, in which the spacetime anisotropy is encoded by a weak time-like vector field $\zeta_a(r)$ rather than introduced by hand at the matter level. Starting from the general FR field equations for a static spherically symmetric line element, we specialize to a zero-tidal-force Morris--Thorne wormhole and derive closed-form expressions for the energy density, radial pressure, and tangential pressure sourcing a fixed shape function $b(r)=r_0^2/r$. Adopting a decaying weak-field ansatz for the anisotropy field, $\zeta_0(r)=\zeta_c\,r_0/r$, we show that the amount of null-energy-condition (NEC) violating matter required to hold the throat open decreases monotonically with $\zeta_c$, and that the radial extent of the NEC-violating region shrinks from the entire spacetime (the general-relativistic limit $\zeta_c\to0$) to a compact neighbourhood of the throat. We further derive the generalized hydrostatic-equilibrium (TOV-type) equation for this geometry and identify an explicit geometric force $F_\zeta$, sourced by the gradient of $\zeta_0(r)$, that supplements the usual hydrostatic and anisotropic forces. The model reduces smoothly to the general-relativistic Morris--Thorne wormhole as $\zeta_0\to0$, and all reported effects remain within the weak-anisotropy regime $|\zeta_0|\ll1$ under which the field equations were derived. We further examine tidal forces (finding the radial component vanishes identically for every observer), sound speeds and the Herrera cracking criterion, a corrected convergent exotic-matter measure, a Herrera-type complexity factor adapted to the wormhole geometry, an analytic classification of the critical anisotropy strength across the shape-function and anisotropy-decay parameter space, a thin-shell realization with reduced surface exotic density, the absence of a photon sphere with the corresponding light-deflection angle, and a linearized generalization to a nonzero redshift function.

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