TY - JOUR

T1 - Kaluza–Klein dimensional reduction from elasticity theory of crumpled paper

AU - Adda-Bedia, Mokhtar

AU - Katzav, Eytan

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/3

Y1 - 2023/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85149571229&partnerID=8YFLogxK

U2 - 10.1140/epjp/s13360-023-03828-2

DO - 10.1140/epjp/s13360-023-03828-2

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AN - SCOPUS:85149571229

SN - 2190-5444

VL - 138

JO - European Physical Journal Plus

JF - European Physical Journal Plus

IS - 3

M1 - 198

ER -