Kaluza–Klein dimensional reduction from elasticity theory of crumpled paper

During the last century, two theories using the concept of dimensional reduction have been developed independently. The first, known as Föppl–von Kármán theory, uses Riemannian geometry and continuum mechanics to study the shaping of thin elastic structures which could become as complex as crumpled paper. The second one, known as Kaluza–Klein theory, uses Minkowskian geometry and general relativity to unify fundamental interactions and gravity under the same formalism. Here we draw a parallel between these two theories in an attempt to use concepts from elasticity theory of plates to recover the Einstein–Maxwell equations. We argue that Kaluza–Klein theory belongs to the same conceptual group of theories as three-dimensional elasticity, which upon dimensional reduction leads to the Föppl–von Kármán theory of two-dimensional elastic plates. We exploit this analogy to develop an alternative Kaluza–Klein formalism in the framework of elasticity theory in which the gravitational and electromagnetic fields are, respectively, associated with stretching-like and bending-like deformations. We show that our approach of dimensional reduction allows us to retrieve the Lagrangian densities of both gravitational, electromagnetic and Dirac spinors fields as well as the Lagrangian densities of mass and charge sources.