Gravitational Waves and Pommaret Bases

TL;DR

This paper explores the algebraic and homological structures underlying differential operators in physics, especially gravitational waves, using Pommaret bases, differential modules, and duality, revealing new insights into their parametrization and invariants.

Contribution

It introduces a novel algebraic framework connecting Pommaret bases, differential modules, and homological algebra to analyze gravitational wave equations.

Findings

Differential modules can be torsion-free, enabling parametrization.

The algebraic structures relate to physical invariants like the Weyl tensor.

Gravitational waves are linked to specific algebraic properties of differential operators.

Abstract

The first finite length differential sequence, now called {\it Janet sequence}, has been introduced by Janet in 1920. Thanks to the first book of Pommaret in 1978, this algorithmic approach has been extended by Gerdt, Blinkov, Zharkov, Seiler and others who introduced Janet and Pommaret bases in computer algebra. After 1990, new intrinsic tools have been developed in homological algebra with the definition of {\it extension differential modules} through the systematic use of {\it double differential duality} (Zbl 1079.93001). If an operator ${\cal{D}}_1$ generates the compatibility conditions (CC) of an operator ${\cal{D}}$, then the {\it adjoint operator} $ad( {\cal{D}})$ may not generate the CC of $ad({\cal{D}}_1)$. Equivalently, an operator ${\cal{D}}$ with coefficients in a differential field $K$ can be parametrized by an operator ${\cal{D}}_{-1}$ iff the differential module $M$ defined by ${\cal{D}}$ is torsion-free, that is $t(M)={ext}^1_D(N,D) = 0$ when $N$ is the differential module defined by $ad( {\cal{D}})$ and $D$ is the ring of differential operators with coefficients in $K$. Also $R = hom_K(M,K)$ is a differential module for the Spencer operator $d:R \rightarrow T^* \otimes R$, first introduced by Macaulay in 1916 with {\it inverse systems}. When ${\cal{D}}$ is the self-adjoint Einstein operator, it is not evident that $t(M)\neq 0$ is generated by the Weyl tensor having only to do with the group of conformal transformations. Gravitational waves are not coherent with these results because the stress-functions parametrizing the Cauchy = ad (Killing) operator have nothing to do with the metric, {\it exactly like the Airy or Maxwell functions in elasticity}. Similarly, the Cauchy operator has nothing to do with any contraction of the Bianchi operator.