Killing Operator for the Kerr Metric

TL;DR

This paper investigates the differential operators defining symmetries of Kerr, Schwarzschild, and Minkowski spacetimes, focusing on their compatibility conditions and the effects of parameters on these structures.

Contribution

It applies advanced differential homological algebra techniques to analyze the Killing operators in specific spacetime metrics, revealing parameter dependencies and involutivity properties.

Findings

Dimensions of the inclusions of the prolongation systems are computed.

Parameter dependencies of the compatibility conditions are characterized.

The structure of the Killing operators varies with spacetime parameters.

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.