Covariant Dynamical Systems Formulation of the Tolman-Oppenheimer-Volkoff Equations

Abstract

We revisit static, spherically symmetric perfect-fluid stellar models in General Relativity within the framework of the $1+1+2$ semi-tetrad formalism. For locally rotationally symmetric static spacetimes, the Tolman-Oppenheimer-Volkoff system can be expressed as a covariant first-order dynamical system and, after suitable normalization, reformulated as a three-dimensional autonomous flow for a general equation of state (EoS). In the case of a linear EoS, the system reduces further to a planar dynamical system whose finite and asymptotic equilibrium points, together with their stability properties, admit a clear geometrical interpretation in terms of covariant variables. For more general equations of state, such as the polytropic case, the dynamics naturally acquire a genuinely three-dimensional character. Beyond providing a compact, covariant, and physically transparent reformulation of the relativistic stellar problem, the present analysis clarifies how the standard metric description is encoded within a global phase-space structure constructed from geometrically meaningful covariant variables.