The Elastodynamics of the Spacetime Continuum as a Framework for Strained Spacetime

We derive the elastodynamics of the spacetime continuum by applying continuum mechanical results to strained spacetime. Based on this model, a stress-strain relation is derived for the spacetime continuum. From the kinematic relations and the equilibrium dynamic equation of the spacetime continuum, we derive a series of wave equations: the displacement, dilatational, rotational and strain wave equations. Hence energy propagates in the spacetime continuum as wave-like deformations which can be decomposed into dilatations and distortions. Dilatations involve an invariant change in volume of the spacetime continuum which is the source of the associated rest-mass energy density of the deformation, while distortions correspond to a change of shape of the spacetime continuum without a change in volume and are thus massless. The deformations propagate in the continuum by longitudinal and transverse wave displacements. This is somewhat reminiscent of wave-particle duality, with the transverse mode corresponding to the wave aspects and the longitudinal mode corresponding to the particle aspects. A continuity equation for deformations of the spacetime continuum is derived, where the gradient of the massive volume dilatation acts as a source term. The nature of the spacetime continuum volume force and the inhomogeneous wave equations need further investigation.