Is an impedance operator necessary causal, and is this an issue of complexity?

Abstract

An impedance operator describes the mapping of a velocity field across a part of a boundary surface, to the traction field across the same part. Understood to represent the solution of a “direct” problem, i.e., the velocity field describes the problem forcing and the traction field part of the solution, the impedance operator is necessary causal. On the other hand, understood to represent the general solution of an “inverse” problem, i.e., the velocity field is part of the observed solution with the traction field representing the problem forcing, the operator need not be causal. Continuing, a uniqueness theorem that applies to the direct problem assures that the impedance operator thusly defined is unique. The lack of a corresponding theorem for the inverse problem suggests that the impedance operator thusly defined need not be unique. This further suggests requiring causality selects from multiple impedance operators, representing multiple solutions to the inverse problem, the one that is unique. This raises two questions. Is the causality that makes the operator unique a requirement of the governing physics? What impact does this have on the concept of impedance as a tool for addressing complexity in dynamical systems?