Killing Operator for the Kerr Metric

Abstract

When $\mathcal{D}:E\to F$ 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 $\Phi :J_q(E)\to F=F_0$ 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 ${\mathcal{D}}_1:F_0\to F_1$. When $\mathcal{D}$ is involutive, that is when the corresponding system $R_q=ker(\Phi )$ is involutive, this procedure provides successive first order involutive operators ${\mathcal{D}}_1,...,{\mathcal{D}}_n$ . Though ${\mathcal{D}}_1\circ \mathcal{D}=0$ implies $ad(\mathcal{D})\circ ad({\mathcal{D}}_1)=0$ by taking the respective adjoint operators, then $ad(\mathcal{D})$ may not generate the CC of $ad({\mathcal{D}}_1)$ 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 $R_{q+r}^{(s)}$ 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 $R_1^{(3)}\subset R_1^{(2)}\subset R_1^{(1)}=R_1\subset J_1(T(X))$ for the respective Killing operators.

arXiv commentThis 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

References in corpus (9)

• New identities for linearized gravity on the Kerr spacetime
• The mathematical foundations of general relativity revisited
• All local gauge invariants for perturbations of the Kerr spacetime
• The Conformal Group Revisited
• Compatibility complex for black hole spacetimes
• Minimum Parametrization of the Cauchy Stress Operator
• A Mathematical Comparison of the Schwarzschild and Kerr Metrics
• Minkowski, Schwarzschild and Kerr Metrics Revisited
• Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules