The 4-dim intrinsic (material) Riemannian metric G of the material 4-D space-time continuum P is utilized as the characteristic of the aging processes developing in the material. Manifested through variation of basic material characteristics such as density, moduli of elasticity, yield stress, strength, and toughness., the aging process is modeled as the evolution of the metric G (most importantly of its time component G00) of the material space-time P embedded into 4-D Newtonian space-time with a Euclidean metric. The evolutional equation for the metric G is derived by the classical variational approach. Construction of the Lagrangian for an aging elastic medium and the derivation of a system of coupled elastostatic and aging equations constitute the central part of the work. The external and internal balance laws associated with symmetries of material and physical space-time geometries are briefly reviewed from a new viewpoint.