Generating Compatibility Conditions and General Relativity

J.-F Pommaret

doi:10.4236/jmp.2019.103025

Abstract

The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.

Highlights

The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics
We prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra
Collecting the above results, we find the 3 first order differentially independent generating CC coming from the Janet tabular and the additional single second order generating CC describing the 2-dimensional Riemann operator, ( ) that is the linearized Riemann tensor in the space x1, x2 :

Summary

Introduction

( ) If X is a manifold of dimension n with local coordinates x1, , xn , let us in-. J.-F. The authors who have studied these questions had in mind that the total number of generating CC could be considered as a kind of “differential transcendence degree”, called “degree of generality” by A We obtain rkD ( L) − p + m − rkD ( M ) =0 and it follows from noetherian arguments that the differential module L ⊂ D p is finitely generated but not free in general and we may look for a minimum number of generators which may be differentially dependent in general as we shall see in the examples.

Motivating Examples

Mathematical Tools

Applications

Conclusions