Coordinate-Time and Proper-Time Brachistochrones in Tolman--Oppenheimer--Volkoff Stellar Interiors.

We formulate the relativistic brachistochrone problem inside static, spherically symmetric stellar interiors generated by the Tolman--Oppenheimer--Volkoff equations for a prescribed equation ofstate. The stellar geometry is determined by the pressure,energy-density, mass and lapse profiles obtained from the TOV system.For a massive test particle released from rest at the stellar surface, we derive two distinct variational problems: the minimization of the coordinate travel time t measured by an observer at infinity, and theminimization of the proper time τ measured along the particle worldline. Both problems reduce to Fermat-type variational principles on the spatial geometry of the static slice, with two different effective refractive indices. The cyclic angular coordinate yields generalized Snell laws and integral formulae for the opening angle, the turning radius and the travel times. A universal weak-field expansion in the compactnessparameter μ=rₛ/R shows that the difference between the coordinate-time and proper-time turning radii, q⁽ᵗ⁾-q(τ), is determined at O(μ) for any equation of state by two one dimensional quadratures and one derivative, without solving the full TOV system. In the constant-density limit this reduces to the closed form μ\,q₀(1-q₀²)²/4. These results provide a framework for comparing relativistic brachistochrone curves in constant-density interiors, polytropic stars and realistic neutron-star equations of state.