Chapter 31: Tensors & the Geometry of Physics

Tensors & the Geometry of Physics directional complexity necessitates using nine polarizability coefficients (alpha i j) to define the relationship between the components of P and E. The text emphasizes that these tensor components must transform predictably when the coordinate system is rotated. A critical principle derived from energy considerations establishes that the polarization tensor is generally symmetric, meaning the work done per unit volume to polarize the crystal can be visualized as an energy ellipsoid. When a material is isotropic, this energy ellipsoid simplifies to a sphere. The concept of tensors is then generalized to mechanical systems, specifically introducing the tensor of inertia (I i j) that relates a rigid body’s angular momentum to its angular velocity. Furthermore, the chapter details the stress tensor (S i j), which describes the internal forces (stress) within a solid, defining the force component per unit area across different planes, and noting that this tensor is also generally symmetric. For elastic materials, the highly complex relationship between stress and resulting strain must be governed by the fourth-rank tensor of elasticity (T i j k l), illustrating how tensors of higher rank are required for comprehensive descriptions of complex material properties, reducing the number of independent coefficients from 81 to 21 based on symmetry considerations. The chapter concludes by applying tensors to special relativity with the four-dimensional electromagnetic stress-energy tensor. This four-tensor unifies spatial stress components, energy density, and the flow of energy and momentum (including the Poynting vector) into a single entity, demonstrating how the tensor concept extends naturally to relativistic contexts.