Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)The elastic properties of solids are described in close analogy with General Relativity, by locally gauging the translational group of space-time. Electron interactions with the crystal lattice are thus generated by enforcing full gauge invariance, with the introduction of a gauge field. Elementary excitations are associated with the local gauge, contrasting to the usual interpretation as Goldstone bosons emerging from global symmetry breaking. In the linear limit of the theory, the gauge field displays elastic waves, that we identify with acoustic phonons, when the field is quantized. Coupling with the electronic part of the system yields the standard electron-phonon interaction. If spin-orbit effects are included, unusual couplings emerge between the strain field and the electronic spin current, leading to novel physics that may be relevant for spintronic applications.473Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)