Killing Operator for the Kerr Metric

Abstract

When D : E → 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 Φ : J q ( E ) → 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 D 1 : F 0 → F 1 . When D is involutive, that is when the corresponding system R q = ker (Φ) is involutive, this procedure provides successive first order involutive operators D 1 , ..., D n . Though D 1 ◦ D = 0 implies ad ( D ) ◦ ad ( D 1 ) = 0 by taking the respective adjoint operators, then ad ( D ) may not generate the CC of ad ( D 1 ) and measuring such ”gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R q is not involutive, a standard 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 ρ r + s ( R q ) = J r + s ( R q ) ∩ J q + r + s ( E ) ⊂ 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 ⊂ R (2)1 ⊂ R (1)1 = R 1 ⊂ J 1 ( T ( X )) for the respective Killing operators. Other striking motivating examples are also presented.