Killing Operator for the Kerr Metric
Abstract
When ${\cal{D}}: E \rightarrow F$ is a linear differential operator of order $q$ between the sections of vector bundles over a manifold $X$ of dimension $n$, it is defined by a bundle map $\Phi: J_q(E) \rightarrow F=F_0$ that may depend, explicitly or implicitly, on constant parameters $a, b, c, ...$. A "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1: F_0 \rightarrow F_1$. When ${\cal{D}}$ is involutive, that is when the corresponding system $R_q=ker(\Phi)$ is involutive, this procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ . Though ${\cal{D}}_1 \circ {\cal{D}}=0 $ implies $ad({\cal{D}}) \circ ad({\cal{D}}_1)=0$ by taking the respective adjoint operators, then $ad({\cal{D}})$ may not generate the CC of $ad({\cal{D}}_1)$ and measuring such "gaps" led to introduce extension modules in differential homological algebra. They may also depend on the parameters. When $R_q$ is not involutive, a standard {\it prolongation/projection} (PP) procedure allows in general to find integers $r,s$ such that the image $R^{(s)}_{q+r}$ of the projection at order $q+r$ of the prolongation ${\rho}_{r+s}(R_q) = J_{r+s}(R_q) \cap J_{q+r+s}(E)\subset J_{r+s}(J_q(E)) $ is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr $(m, a)$, Schwarzschild $(m, 0)$ and Minkowski $(0, 0)$ parameters while computing the dimensions of the inclusions $R^{(3)}_1\subset R^{(2)}_1 \subset R^{(1)}_1 =R_1 \subset J_1(T(X))$ for the respective Killing operators.
 Publication:

Journal of Modern Physics
 Pub Date:
 2023
 DOI:
 10.4236/jmp.2023.141003
 arXiv:
 arXiv:2211.00064
 Bibcode:
 2023JMPh...14...31P
 Keywords:

 Physics  General Physics;
 18G15;
 35N10;
 35Q75;
 83C35;
 83C50
 EPrint:
 This paper is largely improving the previous arXiv:2203.11694 (now published in DOI:10.4236/jmp.2022.134036 ) by using new intrinsic homological techniques that have never been introduced in General Relativity The study of contact structures is also revisited along the same lines