Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

Killing Operator for the Kerr Metric

“…The case of the Killing operator for the Minkowski metric is well known but the study of the Killing operator for the Schwarzschild and Kerr metrics is rather recent and striking as it proves that both the numbers of generating second order CC and the numbers of generating third order CC may change J.-F. Pommaret DOI: 10.4236/jmp.2023.141003 33 Journal of Modern Physics [12] [13] [14] [15]. We provide a few reasons for which we totally disagree with these publications, in particular with ( [13]), as we already explained in the Introduction of ( [11]).…”

“…The coefficients of the linear equations lin involved depend on the Riemann tensor as in ( [11]). Accordingly, we may choose only the 2 parametric jets ( )…”

“…in such a way that the CC of ξ η =  is of the form 1 0 η =  . As shown in many books ([1]- [7]) and papers ([8] [9] [10] [11]), such a problem may be quite difficult because the order of the generating CC may be quite high. Proceeding in this way, we may construct the CC 2 1…”

“…3) Our third doubts came from an elementary example (See [11], end of Introduction) that we shall revisit in the following motivating examples. Indeed, a (difficult) theorem of homological algebra is saying that the only intrinsic concept that can be attached to a module is made by the extension modules that do not depend on the resolution used for their computation.…”

“…A key step in the procedure for constructing differential sequences will be to use the following (difficult) theorems and corollary (For Spencer cohomology and acyclicity or involutivity (See [1] [3] [5] for details and compare to [20] [21]) (See [7] [8] [11] for more details):…”