Emergence of the Hooke Tensor from Spacetime Permeability Kernels

Elastic wave propagation is conventionally described by local constitutive relations and differential operators, with the Hooke tensor and mass density introduced as fundamental material parameters. In this formulation, locality and instantaneous response are assumed a priori. In contrast, experimentally accessible information on elastic media is encoded in spacetime response functions relating applied forces to observable displacements. In this work we develop a response-based formulation of elasticity in which the primary dynamical object is a tensorial spacetime permeability kernel. No assumption of local constitutive behavior is imposed at the outset. We show that, under minimal and physically verifiable conditions, the inverse response kernel admits a controlled low-frequency and longwavelength expansion. The quadratic sector of this expansion defines an effective local elastic operator. Within this framework, the mass density and the Hooke tensor arise as second-order derivatives of the inverse spacetime permeability with respect to frequency and wavevector. For isotropic media, this structure reduces uniquely to the Lamé form, providing explicit expressions for the elastic moduli as emergent quantities. The domain of validity of the local elastic description is determined by the breakdown of the quadratic approximation, thereby quantifying the onset of non-local elastic behavior. This formulation places classical elasticity on the same conceptual footing as electromagnetic constitutive theory, where permeability is understood as a response function rather than a fundamental constant, and establishes a systematic basis for extending elastic descriptions to non-local media.