Minkowski, Schwarzschild and Kerr Metrics Revisited

Abstract

In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (10, order 1), Riemann (20, order 2), Bianchi (20, order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. However, even if they exhibited a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars, Killing-Yano tensors or the 2+2-formalism separating the two variables (t,r) from the two other angular variables of space-time. Using the formal theory of systems of partial differential equations and Lie pseudogroups, the purpose of this difficult computational paper is to provide new intrinsic differential homological methods involving the Spencer operator in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. Most of these new tools are now available as computer algebra packages.There are many different ways to look at such a system. The first natural one is to say that the only solution is y = 0 when u = v = 0. The second one is to look for the CC that must be satisfyed by u, v and we may adopt two possible presentations:• Substitute y and obtain the 2 fourth order CC:which are not differentially independent because one can easily check:• However, we also have:that is a second order CC. Finally, we obtain:and we discover that the CC of D = (P, Q) are generated by (A, B) but also by C alone, though any student will hesitate between the two possibilities !. Refering to differential homological algebra as in ([25]) while indicating the order of an operator below its arrow, the same trivial differential module M = 0 (care) defined by D has therefore two split resolutions:

Generating Compatibility Conditions and General Relativity

“…3 ) < dim(R 3 ) < dim(R 4 ) with 11 < 12 < 15.However, we have the general Theorem 2.A.7 in ( [20]) providing the useful prolongation/ projection (PP) procedure, namely that we have ρ r (R (1) q ) = R (1) q+r , ∀r ≥ 0 whenever the symbol g q of R q is 2-acyclic. In the present case, we have indeed ρ r (R…”

“…However, in order to prove that d 2 ξ 0,3 − d 3 ξ 0,2 = 0 or equivalently that d 2 W 3 − d 3 W 2 = 0, the previous procedure cannot work but we must never forget that U, V 2 , V 3 , W 2 , W 3 both belong to j 2 (Ω). Introducing the formal Lie derivative R = L(ξ 1 )ρ, we recall that: we have proved in ( [20]) that the third order CC d 2 W 3 − d 3 W 2 = 0 is not a generating one because it is just a differential consequence of the second order CC R 01,23 = 0.…”

Compatibility Complex for Black Hole Spacetimes

“…However, it can be shown that, on Kerr, there is no second order operator L with symbol σ p (ϑ 2 ) such that LK 0 = 0, and hence K 0 fails to be involutive. It was rightly noted in [29,30] that constructing a full compatibility operator for an involutive version of K 0 is much easier. However, we point out that our K 0 is tied to the fixed notion of gauge symmetry and gauge invariance in linearized gravity, hence we are not free to replace it with its involutive prolongation.…”

“…Nevertheless, this paper is the first to fully demonstrate completeness for a set of gauge invariants for the Kerr spacetime. See [29,30] for work on related problems.…”

“…However, its use requires all differential operators to be explicitly expressed in coordinates and in components, which unfortunately highly obfuscates any geometric structure in K 1 and, more often than not, results in extremely long expressions of doubtful utility. A semi-algorithmic version of this approach has been pursued in the recent papers [29][30][31] and has yet to arrive at a full expression for a compatibility operator K 1 , let alone one as compactly expressed as in [2]. Another drawback of this approach is that the completeness of K 1 is proved by virtue of its algorithmic construction.…”