Minimum Parametrization of the Cauchy Stress Operator

TL;DR

This paper investigates the minimal parametrization of the Cauchy stress operator across arbitrary dimensions, clarifying historical misconceptions and emphasizing the role of the Einstein operator over the Ricci operator.

Contribution

It proves that previous works on stress operator parametrization used the Einstein operator and not the Ricci operator, clarifying a long-standing confusion.

Findings

All previous works used the Einstein operator, not the Ricci operator.

Clarification of the distinction between the div operator and the Cauchy operator.

Highlights the historical ambiguity regarding Einstein's awareness of prior research.

Abstract

When ${\cal{D}}:\xi \rightarrow \eta$ is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that ${\cal{D}}\xi=\eta$ implies ${\cal{D}}_1\eta=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ when the ground manifold has dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a more difficult " inverse problem " is to look for an operator ${\cal{D}}: \xi \rightarrow \eta$ having the generating CC ${\cal{D}}_1\eta=0$. If this is possible, that is when the differential module defined by ${\cal{D}}_1$ is torsion-free, one shall say that the operator ${\cal{D}}_1$ is parametrized by ${\cal{D}}$ and there is no relation in general between ${\cal{D}}$ and ${\cal{D}}_2$. The parametrization is said to be " minimum " if the differential module defined by ${\cal{D}}$ has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchy stress operator in arbitrary dimension $n$ has attracted many famous scientists (G.B. Airy in 1863 for $n=2$, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for $n=3$, A. Einstein in 1915 for $n=4$) . This paper proves that all these works are using the Einstein operator and not the Ricci operator. As a byproduct, they are all based on a confusion between the so-called $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator which is the formal adjoint of the Killing operator ${\cal{D}}$ parametrizing the Riemann operator ${\cal{D}}_1$ for an arbitrary $n$. Like the Michelson and Morley experiment, it is an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment.