A Mathematical Comparison of the Schwarzschild and Kerr Metrics

A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M) , Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper, we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S($m$) and K($m,a$) are depending on constant parameters in such a way that S $\rightarrow $ M when $m \rightarrow 0$ and K $\rightarrow$ S when $ a \rightarrow 0$, the CC of S do not provide the CC of M when $m \rightarrow 0$ while the CC of K do not provide the CC of S when $a\rightarrow 0$. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the {\it prolongation/projection} (PP) procedure, we provide for the first time all the CC for K in an intrinsic way, showing that they only depend on the underlying Killing algebras and that the role played by the Spencer operator is crucial. We get K$<$S$<$M with $2<4<10$ for the Killing algebras and explain why the formal search of the CC for M, S or K are strikingly different, even though each Spencer sequence is isomorphic to the tensor product of the Poincar\'{e} sequence for the exterior derivative by the corresponding Lie algebra.


1) INTRODUCTION
In order to explain the type of problems we want to solve, let us start adding a constant parameter to the example provided by Macaulay in 1916 that we have presented in a previous paper for other reasons ( [14]). We first recall the following key definition: DEFINITION 1.1: A system of order q on E is an open vector subbundle R q ⊆ J q (E) with prolongations ρ r (R q ) = R q+r = J r (R q ) ∩ J q+r (E) ⊆ J r (J q (E)) and symbols ρ r (g q ) = g q+r = S q+r T * ⊗ E ∩ R q+r ⊆ J q+r (E) only depending on g q ⊆ S q T * ⊗ E. For r, s ≥ 0, we denote by R (s) q+r = π q+r+s q+r (R q+r+s ) ⊆ R q+r the projection of R q+r+s on R q+r , which is thus defined by more equations in general. The system R q is said to be formally integrable (FI) if we have R (s) q+r = R q+r , ∀r, s ≥ 0, that is if all the equations of order q + r can be obtained by means of only r prolongations. The system R q is said to be involutive if it is FI with an involutive symbol g q . We shall simply denote by Θ = {f ∈ E | j q (f ) ∈ R q } the "set" of (formal) solutions. It is finally easy to prove that the Spencer operator D : J q+1 (E) → T * ⊗J q (E) restricts to D : R q+1 → T * ⊗R q .
The most difficult but also the most important theorem has been discovered by M. Janet in 1920 ( [7]) and presented by H. Goldschmidt in a modern setting in 1968 ( [5]). However, the first proof with examples is not intrinsic while the second, using the Spencer operator, is very technical and we have given a quite simpler different proof in 1978 ( [7], also [9], [10]) that we shall use later on for studying the Killing equations for the Schwarzschild and Kerr metrics: THEOREM 1.2: If R q ⊂ J q (E) is a system of order q on E such that its first prolongation R q+1 ⊂ J q+1 (E) is a vector bundle while its symbol g q+1 is also a vector bundle, then, if g q is 2-acyclic, we have ρ r (R (1) q ) = R (1) q+r . COROLLARY 1.3: (PP procedure) If a system R q ⊂ J q (E) is defined over a differential fiel K, then one can find integers r, s ≥ 0 such that R (s) q+r is formally integrable or even involutive.
Starting with an arbitrary system R q ⊂ J q (E), the main purpose of the next crucial example is to prove that the generating CC of the operator D = Φ 0 • j q : E jq −→ J q (E) Φ0 −→ J q (E)/R q = F 0 , though they are of course fully determined by the first order CC of the final involutive system R (s) q+r produced by the up/down PP procedure, are in general of order r + s + 1 like the Riemann or Weyl operators, but may be of strictly lower order.
MOTIVATING EXAMPLE 1.4 : With m = 1, n = 3, q = 2, let us consider the second order linear system R 2 ⊂ J 2 (E) with dim(R 2 ) = 8 and parametric jets {y, y 1 , y 2 , y 3 , y 11 , y 12 , y 22 , y 23 }, defined by the two inhomogeneous PD equations where a is a constant parameter: P y ≡ y 33 = u, Qy ≡ y 13 + a y 2 = v First of all we have to look for the symbol g 2 defined by the two linear equations y 33 = 0, y 13 = 0. The coordinate system is not δ-regular and exchanging x 1 with x 2 , we get the Janet board: y 33 = 0 y 23 = 0 1 2 3 1 2 • It follows that g 2 is involutive, thus 2-acyclic and we obtain from the main theorem ρ r (R 2 ) = R (1) 2+r . However, R 2 ⊂ R 2 with a strict inclusion because R Since that moment, we have to consider the two possibilities: • a = 0: The initial system becomes y 33 = u, y 13 = v and has an involutive symbol. It is thus involutive because it is trivially FI as the left members are homogeneous with only one generating first order CC, namely u 3 − v 1 = 0. We have dim(g 2+r ) = 4 + r and the following commutative and exact diagrams: We have thus the Janet sequence: or, equivalently, the exact sequence of differential modules over D = Q[d 1 , d 2 , d 3 ] = Q[d]: where p is the canonical projection onto the residual differential module.
• a = 0: When the coefficients are in a differential field of constants, for example if a ∈ Q is invertible, we may choose a = 1 like Macaulay ( [14]). It follows that g (1) 2 is still involutive but we have the strict inclusion g (1) 2 ⊂ g 1 and thus the strict inclusion R 2 ) = 7 < 8. We may thus continue the PP procedure and obtain the new strict inclusion R = u y 23 = v 3 − u 1 y 22 = v 2 − v 13 + u 11 y 13 + y 2 = v 2 is easily seen to be involutive, we achieve the PP procedure, obtaining the strict intrinsic inclusions and corresponding fiber dimensions: Finally, we have ρ r (R (2) 2 ) = ρ r ((R (1) 2 ) (1) ) = (ρ r (R (1) 2 )) (1) = (R (1) 2+r ) (1) = R (2) 2+r . It remains to find out the CC for (u, v) in the initial inhomogeneous system. As we have used two prolongations in order to exhibit R (2) 2 , we have second order formal derivatives of u and v in the right members. Now, as we have an involutive system, we have first order CC for the new right members and could hope therefore for third order generating CC. However, we have successively the 4 CC: It follows that we have only one second and one third order CC: but, surprisingly, we are left with the only generating second order CC v 33 − u 13 − u 2 = 0 which is coming from the fact that the operator P commutes with the operator Q.
We let the reader prove as an exercise (See [14], [20] for details) that dim(R r+2 ) = 4r + 8, ∀r ≥ 0 and thus dim(R 3 ) = 12, dim(R 4 ) = 16 in the following commutative and exact diagrams where E is the trivial vector bundle with dim(E) = 1 and dim(g r+2 ) = r + 4, ∀r ≥ 0: We have thus the formally exact sequence: or, equivalently, the exact sequence of differential modules over D as before: which is nevertheless not a Janet sequence because R 2 is not involutive.
MOTIVATING EXAMPLE 1.5: We now prove that the case of variable coefficients can lead to strikingly different results, even if we choose them in the differential field K = Q(x 1 , x 2 , x 3 ) of rational functions in the coordinates that we shall meet in the study of the S and K metrics. In order to justify this comment, let us consider the simplest situation met with the second order system R 2 ⊂ J 2 (E): We may consider successively the following systems of decreasing dimensions 8 > 7 > 6 > 4: The last system is involutive with the following Janet tabular: The generic solution is of the form y = b(x 1 ) + cx 3 and it is rather striking that such a system has constant coefficients (This will be exactly the case of the S and K metrics but similar examples can be found in [10]). We could hope for 9 generating CC up to order 4 but tedious computations, left to the reader as a tricky exercise, prove that we have in fact, as before, only 2 generating third order CC described by the following involutive system, namely: satisfying the only first order CC: of differential operators with coefficients in K, we obtain the sequence of D-modules: where the order of an operator is written under its arrow. This example proves that even a slight modification of the parameter can change the corresponding differential resolution.
MOTIVATING EXAMPLE 1.6: We comment a tricky example first provided by M. Janet in 1920, that we have studied with details in ( [9], [11]). Using jet notations with n = 3, m = 1, q = 2, K = Q(x 2 ), let us consider the inhomogeneous second order system: We let the reader prove that the space of solutions has dimension 12 over Q and that we have r = 0, s = 5 in such a way that R 2 is involutive and even finite type with a zero symbol. Accordingly, we have dim(R 2 ) = 12. Passing to the differential module point of view, it follows that dim K (M ) = 12 and rk D (M ) = 0. According to the general results presented, we have thus to use 5 prolongations and could therefore wait for CC up to order ... 6 !!!. In fact, and we repeat that there is no hint at all for predicting this result in any intrinsic way, we have only two generating CC, one of order 3 and ... one of order 6 indeed, namely: satisfying the only fourth order CC: It follows that we have the unexpected differential resolution: with Euler-Poincaré characteristic rk D (M ) = 1 − 2 + 2 − 1 = 0 as expected. In addition, if we introduce a constant parameter a by replacing the coefficient x 2 by ax 2 , we obtain 2ay 112 = v 33 − ax 2 v 11 − u 22 and obtain the same conclusions as before. We point out the fact that, when a = 0, the system y 33 = u, y 22 = v, which is trivially FI because it is homogeneous, has a symbol g 2 which is neither involutive (otherwise it should admit a first order CC), nor even 2-acyclic because we have the parametric jets: par 2 = (y 11 , y 12 , y 13 , y 23 ), par 3 = (y 111 , y 112 , y 113 , y 123 ), par 4 = (y 1111 , y 1112 , y 1113 , y 1123 ) and the long δ-sequence: in which dim(B 2 (g 2 )) = 12 − 4 = 8, dim(Z 2 (g 2 )) = 12 − 3 = 9 ⇒ dim(H 2 (g 2 ) = 9 − 8 = 1 = 0. However, g 3 is involutive with the following Janet tabular for the vertical jets (v ijk ) ∈ S 3 T * : Accordingly, R 3 is thus involutive and the only CC v 33 − u 22 = 0 is of order 2 because we need one prolongation only to reach involution and thus 2-acyclicity.
MOTIVATING EXAMPLE 1.7: With m = 1, n = 2, q = 2, K = Q, let us consider the inhomogeneous second order system: We obtain at once through crossed derivatives y = u 11 − v 12 − v and, by substituting, two fourth order CC for (u, v), namely: satisfying B 12 + B − A 11 = 0 . However, we may also obtain a single CC for (u, v), namely while C is a section of F ′ 1 , the jet prolongation sequence: is not formally exact because 4 − 28 + 30 − 2 = 4 = 0, while the corresponding long sequence: is indeed formally exact because 4 − r 2 +11r+30) 2 + (r 2 + 7r + 12) − (r 2 +3r+2) 2 = 0 but not strictly exact because R 2 is quite far from being FI as we have even R It follows from these examples and the many others presented in ( [20]) that we cannot agree with ( [1][2][3][4]). Indeed, it is clear that one can use successive prolongations in order to look for CC of order 1, 2, 3, ... and so on, selecting each time the new generating ones and knowing that Noetherian arguments will stop such a procedure ... after a while !. However, it is clear that, as long as the numbers r and s are not known, it is not effectively possible to decide in advance about the maximum order that must be reached. Therefore, it becomes clear that exactly the same procedure must be applied when looking for the CC of the various Killing operators we want to study, the problem becoming a " mathematical " one, surely not a " physical " one.
IMPORTANT REMARK 1.8: Taking the adjoint operators, it is essential to notice that ad(D) generates the CC of ad(D 1 ) when a = 0, a result leading to ext 1 (M ) = 0 but this is not true when a = 0, a result leading to ext 1 (M ) = 0 ( [10], [18], [21], [22]). Hence we discover on such an example that the intrinsic properties of a system with constant or even variable coefficients may drastically depend on these coefficients, even though the correspondig systems do not appear to be quite different at first sight. Accordingly, we have: WHEN A SYSTEM IS NOT FI, NOTHING CAN BE SAID " A PRIORI " ABOUT THE CC, THAT IS WITHOUT ANY EXPLICIT COMPUTATION OR COMPUTER ALGEBRA.
Comparing the sequences obtained in the previous examples, we may state: DEFINITION 1.9: A differential sequence is said to be formally exact if it is exact on the jet level composition of the prolongations involved. A formally exact sequence is said to be strictly exact if all the operators/systems involved are FI (See [14] for more details). A strictly exact sequence is called canonical if all the operators/systems are involutive. The only known canonical sequences are the Janet and Spencer sequences that can be defined independently from each other.
With canonical projection Φ 0 = Φ : J q (E) ⇒ J q (E)/R q = F 0 , the various prolongations are described by the following commutative and exact introductory diagram: Chasing along the diagonal of this diagram while applying the standard "snake" lemma, we obtain the useful long exact connecting sequence: which is thus connecting in a tricky way FI (lower left) with CC (upper right).
We finally recall the Fundamental Diagram I that we have presented in many books and papers, relating the (upper) canonical Spencer sequence to the (lower) canonical Janet sequence, that only depends on the left commutative square D = Φ• j q with Φ = Φ 0 when one has an involutive system R q ⊆ J q (E) over E with dim(X) = n and j q : E → J q (E) is the derivative operator up to order q: We shall use this result, first found exactly 40 years ago ( [7]) but never acknowledged, in order to provide a critical study of the comparison between the S and K metrics.
We notice that 6 − 16 + 14 − 4 = 0, 1 − 10 + 20 − 15 + 4 = 0 and 1 − 4 + 4 − 1 = 0. In this diagram, the Janet sequence seems simpler than the Spencer sequence but, sometimes as we shall see, it is the contrary and there is no rule. We invite the reader to treat similarly the cases a = 0 and a = x 3 .
Using the notations of differential modules or jet theory, we may consider the infinitesimal Killing equations: ). As in the Macaulay example just considered and in order to avoid any further confusion between sections and derivatives, we shall use the sectional point of view and rewrite the previous 10 equations in the symbolic form Ω ≡ L(ξ 1 )ω ∈ S 2 T * where L is the formal Lie derivative: Though this system R 1 ⊂ J 1 (T ) has 4 equations of class 3, 3 equations of class 2, 2 equations of class 1 and 1 equation of class 0, it is far from being involutive because it is finite type with second symbol g 2 = 0 defined by the 40 equations v k ij = 0 in the initial coordinates. From the symetry, it is clear that such a system has at least 4 solutions, namely the time translation ∂ t ↔ ξ 0 = 1 ⇔ ξ 0 = A and, using cartesian coordinates (t, x, y, z), the 3 space rotations y∂ z − z∂ y , z∂ x − x∂ z , x∂ y − y∂ x . We obtain in particular, modulo Ω: We may also write the Schwarzschild metric in cartesian coordinates as: )dr 2 − (dx 2 + dy 2 + dz 2 ), rdr = xdx + ydy + zdz and notice that the 3 × 3 matrix of components of the three rotations has rank equal to 2, a result leading surely, before doing any computation, to the existence of one and only one zero order Killing equation rξ r = xξ x + yξ y + zξ z = 0 ⇒ ξ 1 = ξ r = 0. Such a result also amounts to say that the spatial projection of any Killing vector on the radial spatial unit vector (x/r, y/r, z/r) vanishes beause r must stay invariant.
However, as we are dealing with sections, ξ 1 = 0 implies ξ 0 0 = 0, ξ 1 1 = 0, ξ 2 2 = 0 ... but NOT (care) ξ 1 0 = 0, these later condition being only brought by one additional prolongation and we have the strict inclusions R Hence, it remains to determine the dimensions of these subsystems and their symbols, exactly like in the Macaulay example. We shall prove in the next section that two prolongations bring the five new equations: and a new prolongation only brings the single equation with dim(R ′ 1 ) = 20 − 15 = 5 and let the reader draw the corresponding Janet tabular for the 4 equations of class 3, the 4 equations of class 1, the 3 equations of class 0 and the 3 equations of class 2. The symbol g ′ 1 has the two parametric jets (v 3 2 , v 1 0 ) and is not 2-acyclic. Adding ξ 1 0 = 0 ⇔ ξ 0 1 = 0, we finally achieve the PP procedure with the 16 equations defining the system R " 1 = R 1 with dim(R " 1 ) = 20 − 16 = 4, namely: and we have replaced by "×" the only " dot " (non-multiplicative variable) that cannot provide vanishing crossed derivatives and thus involution of the symbol g " 1 with the only parametric jets (v 3 2 , v 1 0 ). It is easy to check that R " 1 , having minimum dimension equal to 4, is formally integrable, though not involutive as it is finite type with dim(g " 1 ) = 16 − 15 = 1 ⇒ g " 1 = 0 with parametric jet v 3 2 and to exhibit 4 solutions linearly independent over the constants. We let the reader prove as an exercise that the dimension of the Spencer δ-cohomology at ∧ 2 T * ⊗ g " 1 is dim((H 2 (g " 1 )) = 3 = 0 but we have proved in ( [19]) that its restriction to (x 2 , x 3 ) is of dimension 1 only. We obtain: , it is the involutive system provided by the prolongation/projection (PP) procedure, we are in position to construct the corresponding canonical/involutive (lower) Janet and (upper) Spencer sequences along the following fundamental diagram I that we recalled in the Introduction. In the present situation, the Spencer sequence is isomorphic to the tensor product of the Poincaré sequence by the underlying 4-dimensional Lie algebra G, namely: In this diagram, not depending any longer on m, we have now C r = ∧ r T * ⊗ R " 2 and D is of order 2 like j 2 while all the other operators are of order 1: We notice the vanishing of the Euler-Poincaré characteristics: We point out that, whatever is the sequence used or the way to describe D 1 , then ad(D 1 ) is parametrizing the Cauchy operator ad(D) for the S metric. However, such an approach does not tell us explicitly what are the second and third order CC involved in the initial situation.
In actual practice, all the preceding computations have been finally used to reduce the Poincaré group to its subgroup made with only one time translation and three space rotations !. On the contrary, we have proved during almost fourty years that one must increase the Poincaré group (10 parameters), first to the Weyl group (11 parameters by adding 1 dilatation) and finally to the conformal group of space-time (15 parameters by adding 4 elations) while only dealing with he Spencer sequence in order to increase the dimensions of the Spencer bundles, thus the number dim(C 0 ) of potentials and the number dim(C 1 ) of fields (Compare to [6]).

b) KERR METRIC
We now write the Kerr metric in Boyer-Lindquist coordinates: where we have set ∆ = r 2 − mr + a 2 , ρ 2 = r 2 + a 2 cos 2 (θ) as usual and we check that: as a well known way to recover the Schwarschild metric. We notice that t or φ do not appear in the coefficients of the metric and thus, as the maximum subgroup of invariance of the Kerr metric must be contained in the maximum subgroup of invariance of the Schwarzschild metric because of the above limit when a → 0, we obtain the only possible 2 infinitesimal generators {∂ t , ∂ φ } and we have the fundamental diagram I with fiber dimensions: with Euler-Poincaré characteristic 4 − 58 + 152 − 168 + 88 − 18 = 0. Comparing the surprisingly high dimensions of the Janet bundles with the surprisingly low dimensions of the Spencer bundles needs no comment on the physical usefulness of the Janet sequence, despite its purely mathematical importance. In addition, using the same notations as in the preceding section, we shall prove that we have now the additional zero order equations ξ r = 0, ξ θ = 0 produced by the non-zero components of the Weyl tensor and thus, at best, dim(R 1 ) = 2 as these zero order equations will be obtained after only two prolongations. They depend on j 2 (Ω) and we should obtain therefore eventually dim(Q 2 ) = 10 + dim(R 3 ) ≥ 12 CC of order 2 without any way way to know abut the desired third order CC.
Using now cartesian space coordinates (x, y, z) with ξ z = 0, xξ x + yξ y = 0, we have only to study the following first order involutive system for ξ x = ξ with coefficients no longer depending on (a, m), providing the only generator x∂ y − y∂ x : The involutive system produced by the PP procedure does not deend on (m, a) any longer. Accordingly, this final result definitively proves that, as far as differential sequences are concerned:

3) SCHWARZSCHILD METRIC REVISITED
Let us now introduce the Riemann tensor (ρ k l,ij ) ∈ ∧ 2 T * ⊗ T * ⊗ T and use the metric in order to raise or lower the indices in order to obtain the purely covariant tensor (ρ kl,ij ) ∈ ∧ 2 T * ⊗ T * ⊗ T * . Then, using r as an implicit summation index, we may consider the formal Lie derivative on sections: First of all, we notice that: We obtain therefore: 2r 2 ξ 1 = 0 Similarly, we also get: and so on. We obtain for example, among the second order CC: and thus, among the first prolongations, the third order CC that cannot be obtained by prolongation of the various second order CC while taking into account the Bianchi identities ([MSK]). Using the Spencer operator and the fact that ξ 1 ∈ j 2 (Ω), we obtain indeed: In addition, introducing ξ 1 ∈ j 2 (Ω) in the right member as in the motivating examples, we have 3 PD equations for (ξ 2 , ξ 3 ), namely: Using two prolongations and eliminating the third order jets, we obtain successively: Summing, we see that all terms in ξ 2 and ξ 3 disappear and that we are only left with terms in ξ 1 , including in particular the second order jets ξ 1 22 , ξ 1 33 , namely: (Ω), we obtain the new strikingly unusual third order CC for Ω: However, in our opinion at least, we do not believe that such a purely "technical " relation could have any "physical " usefulness and let the reader compare it with the CC already found in ( [19], Lemma 3.B.3). In addition and contrary to this situation, we have successively: a result showing that certain third order CC may be differential consequrences of the Bianchi identities (See [19] for details). Finally, we notice that: R 23,23 ≡ 2ρ 23,23 (ξ 2 2 + ξ 3 3 ) + ξ∂ρ 23,23 = 3msin 2 (θ)ξ 1 = 0 and, comparing to the previous computation for (ξ 2 , ξ 3 ), nothing can be said about the generating CC as long as the PP procedure has not been totally achieved with a FI or involutive system.
• In addition, we also discover the summation ρ 01,23 + ρ 03,21 in R 01,03 with a wrong sign indeed that cannot allow to introduce ρ 02,31 as one could hope. A similar comment is valid for the four successive summations.
Nevertheless, we obtain the following unexpected formal linearized result that will be used in a crucial intrinsic way for finding out the generating second order and third order CC: The rank of the previous system with respect to the four jet coodinates (ξ 1 3 , ξ 2 0 , ξ 1 0 , ξ 2 3 ) is equal to 2, for both the S and K metrics. We obtain in particular the two striking identities: Proof: In the case of he S metric with a = 0, only the framed terms may not vanish and, denoting by " ∼ " a linear proportionality, we have already obtained mod(j 2 (Ω)): R 01,03 ∼ ξ 1 3 , R 03,23 ∼ ξ 0 2 , R 02,03 = 0, R 03,13 = 0 Hence, the rank of the system with respect to the 4 parametric jets (ξ 1 3 , ξ 2 0 , ξ 1 0 , ξ 2 3 ) just drops to 2 and this fact confirms the existence of the 5 additional first order equations obtained, as we saw, after two prolongations.
In the case of the K metric with a = 0, the study is much more delicate. With a 0 = 1, the coefficients of the 4 × 4 metric of the previous system on the basis of the above parametric jets are proportional to the symmetric matrix: 1 a a a 2 a 1 a 2 a a a 2 a 2 a 3 a 2 a a 3 a 2     Indeed, we have successively for the common factor −a(1 − c 2 ): Row 3 ξ 2 3 → ρ 03,12 + ρ 02,13 = 3a 3 mc(1−c 2 )(3r 2 −a 2 c 2 ) 2(r 2 +a 2 c 2 ) 3 and similarly for the common factor − a (r 2 +a 2 ) : Row 2 ξ 2 0 → ρ 23,23 − ω 00 ω 22 ρ 03,03 = 3mr(r 2 +a 2 ) 2 (r 2 −3a 2 c 2 ) 2(r 2 +a 2 c 2 ) 3 We do not believe that such a purely computational mathematical result, though striking it may look like, could have any useful physical application and this comment will be strengthened by the next theorem provided at the end of this section. Q.E.D.

COROLLARY 4.3:
The Killing operator for the K metric has 14 generating second order CC.
Taking therefore into account that the metric only depends on (x 1 = r, x 2 = cos(θ)) we obtain after three prolongations the first order system: Surprisingly and contrary to the situation found for the S metric, we have now a trivially involutive first order system with only solutions (ξ 0 = cst, ξ 1 = 0, ξ 2 = 0, ξ 3 = cst). However, the difficulty is to know what second members must be used along the procedure met for all the motivating examples. In particular, we have again identities to zero like d 0 ξ 1 − ξ 1 0 = 0, d 3 ξ 2 − ξ 2 3 = 0 or, equivalently, d 3 ξ 1 − ξ 1 3 = 0, d 0 ξ 2 − ξ 2 0 = 0 and thus 4 third order CC coming from the 4 following components of the Spencer operator: = 0 a result that cannot be even imagined from ( [1][2][3][4][5][6]). Of course, proceeding like in the motivating examples, we must substitute in the right members the values obtained from j 2 (Ω) and set for example ξ 1 1 = − 1 2ω11 ξ∂ω 11 while replacing ξ 1 and ξ 2 by the corresponding linear combinations of the Riemann tensor already obtained for the right members of the two zero order equations.
Using one more prolongation, all the sections (care again) vanish but ξ 0 and ξ 3 , a result leading to dim(R " 1 ) = 2 in a coherent way with the only nonzero Killing vectors {∂ t , ∂ φ }. We have indeed: Like in the case of the S metric, R 3 is not involutive but R 4 is involutive. However, contrary to the S metric with g " 1 = 0, now g " 1 = 0 for the K metric and R " 1 is trivially involutive with a full Janet tabular having 16 rows of first order jets and 2 rows of zero order jets.
that is, in our case R 1 ). However, we have indeed the equality R 1 ) even if the conditions of Theorem 1.1 are not satisfied because g ′ 1 is not 2-acyclic. Indeed, the Spencer map δ : ∧ 2 T * ⊗ g ′ 1 → ∧ 3 T * ⊗ T is not injective and we let the reader check as an exercise that its kernel is generated by {v 0 1,01 , v 3 2,23 } and the Spencer δ-cohomology is such that dim(H 2 1 (g ′ 1 ) = 2 = 0 because the cocycles are defined by the equations v k i,jr + v k j,ri + v k r,ij = 0. Hence, contrary to what could be imagined, the major difference between the S and K metrics is not at all the existence of off-diagonal terms but rather the fact that R " 1 is not involutive with g " 1 = 0 for the S metric while R " 1 is involutive with g " 1 = 0 for the K metric. This is the reason for which one among the four third order CC must be added with two prolongations for the S metric while the four third order CC are obtained in the same way from the Spencer operator for the K metric. Of course no classical approach can explain this fact which is lacking in ( [1][2][3][4]).
The following result even questions the usefulness of the whole previous approach: THEOREM 4.5: The operator Cauchy = ad(Killing) admits a minimum parametrization by the operator Airy = ad(Riemann) with 1 potential when n = 2, found in 1863. It admits a canonical self-adjoint parametrization by the operator Beltrami = ad(Riemann) with 6 potentials when n = 3, found in 1892 and modified to a mimimum parametrization by the operator M axwell with 3 potentials, found in 1870. More generally, it admits a canonical parametrization by the operator ad(Riemann) with n 2 (n 2 − 1)/12 potentials that can be modified to a relative parametrization by ad(Ricci) with n(n + 1)/2 potentials which is nevertheless not minimum when n ≥ 4, found in 2007. In all these cases, the corresponding potentials have nothing to do with the perturbation of the metric. Such a result is also valid for any Lie group of transformations, in particular for the conformal group in arbitrary dimension.
Proof: We provide successively the explicit corresponding parametrizations: • n = 2 : Multiplying the linearized Riemann operator by a test function φ and integrating by parts, we obtain (care to the factor 2 involved): We now present the original Beltrami parametrization: which does not seem to be self-adjoint but is such that d r σ ir = 0. Accordingly, the Beltrami parametrization of the Cauchy operator for the stress is nothing else than the formal adjoint of the Riemann operator. However, modifying slightly the rows, we get the new operator matrix: which is indeed self-adjoint. Keeping (φ 11 = A, φ 22 = B, Φ 33 = C) with (φ 12 = 0, φ 13 = 0, φ 23 = 0), we obtain the M axwell parametrization: which is minimum because n(n − 1)/2 = 3. However, the corresponding operator is FI because it is homogeneous but it is not evident at all to prove that it is also involutive as we must look for δ-regular coordinates (See [15] for the technical details).
• n ≥ 4 This is far more complicate and we do believe that it is not possible to avoid using differential homological algebra, in particular extension modules. As we found it already in many books ( [8], [9], [16], [18]) or papers ( [12], [13], [21][22][23]), the linear Spencer sequence is (locally) isomorphic to the tensor product of a Poincaré type sequence for the exterior derivative by a Lie algebra G with dim(G) ≤ n(n + 1)/2 equal to the dimension of the largest group of invariance of the metric involved. When n = 4, this dimension is 10 for the M-metric, 4 for the S-metric and 2 for the K-metric. As a byproduct, the adjoint sequence roughly just exchanges the exterior derivatives up to sign and one has for example, when n = 3, the relations ad(grad) = −div, ad((div) = −grad. It follows that, if D 2 generates the CC of D 1 , then ad(D 2 ) is parametrizing ad(D 1 ), a fact not evident at all, even when n = 2 for the Cosserat couple-stress equations exactly described by ad(D 1 ) ( [12]). Passing to the differential modules point of view with the ring (even an integral domain) D = K[d 1 , ..., d n ] = K[d] of differential operators with coefficients in a differential field K, this result amounts to say that ext 1 D (M, D) = ext 1 (M ) = 0. As it is known that such a result does not depend on the differential resolution used or, equivalently, on the differential sequence used, if D 1 generates the CC of D in the Janet sequence, then ad(D 1 ) is parametrizing ad(D) and this result is still true even if D is not involutive. In such a situation, which is the one considered in this paper, the Killing operators for the M-metric, the S-metric and the K-metric are such that, whatever are the generating CC D 1 (second order for the M-metric, a mixture of second and third order for the S-metric and K-metric), then ad(D 1 ) is, in any case, parametrizing the Cauchy operator ad(D) for any D : T → S 2 T * : ξ → L(ξ)ω. Once more, the central object is the group, not the metric. The same results are also valid for any Lie group of transformations, in particular for the conformal group in arbitrary dimension, even if the operator D 1 is of order 3 when n = 3 as we shall see below ( [16], [20][21][22][23]). Q.E.D.
REMARK 4.6: Accordingly, the situation met today in GR cannot evolve as long as people will not acknowledge the fact that the components of the Weyl tensor are similarly playing the part of torsion elements (the so-called Lichnerowicz waves in [17]) for the equations Ricci = 0, a result only depending on the group structure of the conformal group of space-time that brings the canonical splitting Riemann = W eyl ⊕ Ricci without any reference to a backgroung metric as it is usually done ( [8], [9], [15], [18], [21][22][23]). It is an open problem to know why one may sometimes find a SELF-ADJOINT OPERATOR. It is such a confusion that led to introduce the so-called Einstein parametrizing operator ( [17]).
We now care about transformingD 2 given in ( [16], p 158) by the 5 × 3 operator matrix: Dividing the first column by 2 and the fourth column by −2, then using the central row as a new top row while using the former top row as new bottom row, we obtain the operator matrix D ′ 2 : and check that ad(D ′ 2 ) = −D ′ like in the Poincaré sequence for n = 3 where ad(div) = −grad. As the new corresponding operatorD ′ 1 is homogeneous and of order 3 (care), we obtain locally ad(D ′ 1 ) =D ′ 1 , a result not evident at first sight (Compare to [16], p 157). The combination of this example with the results announced in ( [22]) brings the need to revisit almost entirely the whole conformal geometry in arbitrary dimension and we notice the essential role performed by the Spencer δ-cohomology in this new framework.

5) CONCLUSION
First of all, we may summarize the results previously obtained by saying that "Janet and Spencer play at see-saw " because we have the formula dim(C r ) + dim(F r ) = dim(C r (E)) and the sum thus only depends on (n, m, q) with n = dim(X), m = dim(E) and q is the order of the involutive operator allowing to construct the sequences, but not on the underlying Lie group or Lie pseudogroup group when E = T . Hence, the smaller is the background group, the smaller are the dimensions of the Spencer bundles and the higher are the dimensions of the Janet bundles. As a byproduct, we claim that the only solution for escaping is to increase the dimension of the Lie group involved, adding successively 1 dilatation and 4 elations in order to deal with the conformal group of space-time while using the Spencer sequence instead of the Janet sequence. In particular, the Ricci tensor only depends on the elations of the conformal group of space-time in the Spencer sequence where the perturbation of the metric tensor does not appear any longer contrary to the Janet sequence. It finally follows that Einstein equations are not mathematically coherent with group theory and formal integrability. In other papers and books, we have also proved that they were also not coherent with differential homological algebra which is providing intrinsic properties as the extension modules do not depend on the sequence used for their definition, a quite beautiful but difficult theorem indeed. The main problem left is thus to find the best sequence and/or the best group that must be considered. Presently, we hope to have convinced the reader that only the Spencer sequence is clearly related to the group background and must be used, on the condition to change the group. As a byproduct, we may thus finally say that the situation will not evolve in GR as long as people will not acknowledge the existence of these new purely mathematical tools and their purely mathematical consequences. Summarizing this paper in a few words, we do believe that " God used group theory rather than computer algebra when He created the World "!.