Generating Compatibility Conditions and General Relativity

Abstract

The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the 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. 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 a kind of differential transcendence degree, also called degree of generality by A. Einstein in his 1930 letters to E. Cartan.The motivating examples that we provide are among the rare ones known in the literature and could be used as testing examples for future applications of computer algebra.

Compatibility Complex for Black Hole Spacetimes

“…However, it can be shown that, on Kerr, there is no second order operator L with symbol σ p (ϑ 2 ) such that LK 0 = 0, and hence K 0 fails to be involutive. It was rightly noted in [29,30] that constructing a full compatibility operator for an involutive version of K 0 is much easier. However, we point out that our K 0 is tied to the fixed notion of gauge symmetry and gauge invariance in linearized gravity, hence we are not free to replace it with its involutive prolongation.…”

“…Nevertheless, this paper is the first to fully demonstrate completeness for a set of gauge invariants for the Kerr spacetime. See [29,30] for work on related problems.…”

“…However, its use requires all differential operators to be explicitly expressed in coordinates and in components, which unfortunately highly obfuscates any geometric structure in K 1 and, more often than not, results in extremely long expressions of doubtful utility. A semi-algorithmic version of this approach has been pursued in the recent papers [29][30][31] and has yet to arrive at a full expression for a compatibility operator K 1 , let alone one as compactly expressed as in [2]. Another drawback of this approach is that the completeness of K 1 is proved by virtue of its algorithmic construction.…”

Homological Solution of the Lanczos Problems in Arbitrary Dimension

“…ad  is parametrizing the operator ( ) However, we shall discover that the dimension 4 n = , which is particularly "fine" for the classical Killing sequence, is particularly "bad" for the conformal Killing sequence, a result not known after one century because it cannot be understood without using the Spencer δ-cohomology in the following commutative diagram which is explaining therefore what we shall call the "Lanczos secret". Following ( [21]) and the fact that the two central vertical δ-sequences are exact, this diagram allows to construct the Bianchi operator 2 1 2 : F F →  as generating CC for the Riemann operator…”

Minimum Parametrization of the Cauchy Stress Operator

“…REMARK 1.1.4: As long as the Prolongation/Projection (PP) procedure has not been achieved in order to get an involutive system, nothing can be said about the CC (fine examples can be found in [6] and the recent [10]). A proof that the second order system defined by Einstein equations is involutive has been given by J. Gasqui in 1982 but this paper cannot be applied to the minimum parametrizations that need specific δ -regular coordinates as we shall see [11].…”