On spacetime structure, spacetime transformations and material frame-indifference in solid mechanics

In this work, we provide several insights which are related to the physical origins and the mathematical formulation of the principle of material frame-indifference in solid mechanics. The salient feature of the present approach relies crucially on the concrete understanding that when one models the ambient space as a rigid Euclidean space—one with a constant metric—misses important geometrical information; this information is related to the spacetime structure, the possible spacetime transformations and the spacetime–matter interaction. Upon abandoning this point of view and considering the ambient space as a flat manifold, we can see that material frame-indifference is a spacetime property which follows naturally from a physical postulate, namely that of Leibniz equivalence. We also analyze the connection which exists between material frame-indifference and relativity principles; this analysis vindicates Noll’s initial intuition to call the principle “principle of isotropy of space”. Finally, upon considering the spatial metric as a dynamical object which incorporates the deformation field, we present a formal introduction to the concept of general covariance in a constitutive theory.