Abstract
In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the {\it Vessiot structure equations}. In all these cases, they discovered that the {\it compatibility conditions} (CC) for the corresponding Killing operator were involving {\it a mixture of both second order and third order CC} and their idea has been to exhibit only a {\it minimal number of generating ones}. However, even if they exhibited a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. Using the formal theory of systems of partial differential equations and Lie pseudogroups, the purpose of this difficult computational paper is to provide new intrinsic differential and homological methods involving the Spencer operator in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools are now available as computer algebra packages.
Highlights
In many recent technical papers, a few physicists working on General Relativity (GR) are trying to construct high order differential sequences while starting with the Killing operator for a given metric (Minkowski, Schwarzschild, Kerr, ...) ([1] [2] [3] [4] [5])
A few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry
We show how formally integrable (FI) is related to compatibility conditions (CC) by means of an homological procedure known as the “long exact connecting sequence” which is the main byproduct of the so-called snake lemma used when chasing in diagrams
Summary
In many recent technical papers, a few physicists working on General Relativity (GR) are trying to construct high order differential sequences while starting with the Killing operator for a given metric (Minkowski, Schwarzschild, Kerr, ...) ([1] [2] [3] [4] [5]). Vessiot in 1903 [7] [8]), that is 10 when n = 4 (space-time) with the Minkowski metric, and discover that, perhaps for this reason, they have to exhibit an unexpected mixture of generating compatibility conditions (CC) of order 2 and 3 On another side, they are clearly aware of the fact that their results are far from being intrinsic and cannot be adapted to other metrics or dimensions. Janet in 1920 ([26]), describes the way to use a certain prolongation/projection (PP) procedure absolutely needed in order to transform any sufficiently regular system into a formally integrable system and, even an involutive system, that is a situation where we know that the generating CC are described by a first order involutive system and the possibility to construct a canonical Janet or Spencer sequence.
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