In the Boyer-Lindquist (BL) coordinates, the Schwarzschild metric is and, let us introduce with the 4 formal derivatives,

. With speed of light and where m is a constant, the metric can be written in the diagonal form:

with a surprisingly simple determinant.

Using the notations of differential modules or jet theory, we may consider the infinitesimal Killing equations:

where we have introduced the Christoffel symbols through the standard Levi-Civita isomorphism while setting in the differential field K of coefficients [15]. 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 where L is the formal Lie derivative:

Though this system 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 defined by the 40 equations in the initial coordinates. From the symmetry, it is clear that such a system has at least 4 solutions, namely the time translation and, using cartesian coordinates, the 3 space rotations.

We obtain in particular, modulo:

We may also write the Schwarzschild metric in cartesian coordinates as:

and notice that the 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 . Such a result also amounts to say that the spatial projection of any Killing vector on the radial spatial unit vector vanishes beause r must stay invariant.

However, as we are dealing with sections, implies, , … but NOT (care), these later condition being only brought by one additional prolongation and we have the strict inclusions that we rename as. 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.

Knowing that, , , we have thus obtained the 15 equations defining with 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 has the two parametric jets and is not 2-acyclic. Adding, we finally achieve the PP procedure with the 16 equations defining the system with, namely:

and we have replaced by “×” the only “dot” (non-multiplicative variable) that cannot provide vanishing crossed derivatives and thus involution of the symbol with the only parametric jets. It is easy to check that, having minimum dimension equal to 4, is formally integrable, though not involutive as it is finite type with with parametric jet and to exhibit 4 solutions linearly independent over the constants. We let the reader prove as an exercise that the dimension of the Spencer -cohomology at is but we have proved in [15] that its restriction to is of dimension 1 only. We obtain:

THIS SYSTEM IS NOT INVOLUTIVE BUT DOES NOT DEPEND ON m ANY LONGER

Denoting by with the prolongation of, 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 and is of order 2 like 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, then is parametrizing the Cauchy operator 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 of potentials and the number of fields (compare to [16] ).

We now write the Kerr metric in Boyer-Lindquist coordinates:

where we have set as usual and we check that:

as a well known way to recover the Schwarschild metric. We notice that t or do not appear in the coefficients of the metric. 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, we shall obtain the only two possible infinitesimal generators. We shall prove that the new first order system is involutive, contrary to the case of the S metric. Accordingly, we have the fundamental diagram I with fiber dimensions:

with Euler-Poincaré characteristic. 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, produced by the non-zero components of the Weyl tensor and thus, at best, as these zero order equations will be obtained after only two prolongations. They depend on and we should obtain therefore eventually CC of order 2 without any way to know about the desired third order CC.

Using now cartesian space coordinates with, , we have only to study the following first order involutive system for with coefficients no longer depending on, providing the only generator:

and the fundamental diagram

The involutive system produced by the PP procedure does not depend on any longer. Accordingly, this final result definitively proves that, as far as differential sequences are concerned:

THE ONLY IMPORTANT OBJECT IS THE GROUP, NOT THE METRIC

Let us now introduce the Riemann tensor and use the metric in order to raise or lower the indices in order to obtain the purely covariant tensor. Then, using r as an implicit summation index, we may consider the formal Lie derivative on sections:

that can be considered as an infinitesimal variation. As for the Ricci tensor, we notice that though we have only:

The 6 non-zero components of the Riemann tensor are known to be:

First of all, we notice that:

We obtain therefore:

Similarly, we also get:

We also 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 [15]. Using the Spencer operator and the fact that, we first obtain the 3 third order CC:

However, introducing in the right member as in the motivating examples, we have 3 PD equations for, namely:

Using two prolongations and eliminating the third order jets, we obtain successively:

Summing, we see that all terms in and disappear and that we are only left with terms in, including in particular the second order jets, namely:

Setting, , , , with , we obtain the additional strikingly unusual third order CC for:

Nevertheless, 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 ( [15], Lemma 3.B.3). Finally, we have:

a result showing that certain third order CC may be differential consequences of the Bianchi identities (see [15] for details). Finally, we notice that:

and, comparing to the previous computation for, 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.

Though we shall provide explicitly all the details of the computations involved, we shall change the coordinate system in order to confirm these results by only using computer algebra as less as possible. The idea is to use the so-called “rational polynomial” coefficients while setting anew:

in order to obtain over the differential field :

with now and . For a later use, it is also possible to set

.

As this result will be crucially used later on, we have:

LEMMA 4.1:.

Proof: As an elementary result on matrices, we have:

with because and is thus equal to:

that is, after division by and:

Finally, after eliminating the last term, we get:

that is (Compare to [ ] and [ ]):

in a coherent way with the result obtained

for the S metric when. For a later use, we have obtained .

Q.E.D.

Contrary to the S-metric, the main “trick” for studying the K-metric is to take into account that the partition between the zero and nonzero terms will not change if we use convenient coordinates, even if the nonzero terms may change. Meanwhile, we notice that the most important property of the K-metric is the

existence of the off-diagonal term, that is the

coefficient of in the metric which is indeed. We may obtain therefore successively the Killing equations for the Kerr type metric, using sections of jet bundles and writing simply while framing the principal derivative of:

With, multiplying by, by and adding, we notice that:

Similarly, multiplying by (care to the factor 2), we get:

Substracting, we obtain therefore the tricky formula (see the previous Lemma):

Substituting, we obtain:

a situation leading to modify, and, similar to the one found in the Minkowski case with, , when. We also obtain with and:

and with and:

Finally, multiplying by, by and adding, we finally obtain (see the Lemma again)

Using the rational coefficients belonging to the differential field , the nonzero components of the corresponding Riemann tensor can be found in textbooks.

One has the classical orthonormal decomposition:

and defining:

in which the coefficient of is while the coefficient of is indeed. We have

and make thus the Minkowski metric appearing in a purely algebraic way. We now use the new coordinates, and it follows that the conditions, are invariant under such a change of basis because dX¹ and dX² are respectively proportional to and. Indeed, as and thus, the new symbol of while as mixed tensors.

We may obtain simpler formulas in the corresponding basis, in particular the 6 components with only two different indices are proportional to while the 3 components with all four different indices are proportional to.

In the original rational coordinate system, the main nonzero components of the Riemann tensor can only be obtained by means of computer algebra. For helping the reader to handle the literature, for example the book “Computations in Riemann Geometry” written by Kenneth R. Koehler that can be found on the net with a free access, we refer to the seventh chapter on “Black Holes”. We notice that ω→−ω, that is to say changing the sign of the metric, does not change the Christoffel symbols () and the Riemann tensor () but changes the sign of (). For this reason, we have adopted the sign convention of this reference for the explicit computation of these later components as the products and quotients used in the sequel will not be changed.

We have successively:

It must be noticed that we have been able to factorize the six components with only two different indices by and the three components with four different indices by, a result not evident at first sight but coherent with the orthogonal decomposition.

After tedious computations, we obtain:

which is indeed vanishing when for the S metric, both with:

Introducing the formal Lie derivative and using the fact that is a tensor, the system contains the new equations:

Taking into account the original first order Killing equations, we obtain successively:

and we must add:

These linear equations are not linearly independent because:

Also, linearizing while using the Kronecker symbol, we get:

Thus, introducing the Ricci tensor and linearizing, we get:

It follows that and we have in particular:

The first row proves that is a linear combination of and. Then, if we want to solve the three other equations with respect to, and, the corresponding determinant is, up to sign:

Accordingly, we only need to take into account.

Similarly, we also obtain:

where we have to set.

Hence, taking into account, we just need to use and.

However, using the previous lemma, we obtain the formal Lie derivative:

and thus with.

In addition, we have and thus .

We have also:

The following invariants are obtained successively in a coherent way:

However, as, then and can be both divided by a and we get the new invariant:

These results are leading to, , thus to, and after substitution. In the case of the S-metric, only the first invariant can be used in order to find.

Taking into account the previous result, we obtain the two equations:

Using the fact that we have now:

we may multiply the first equation by, the second by and sum in order to obtain:

Using the previous identity for, we obtain therefore:

Taking into account the fact that and substituting, we finally obtain:

A similar procedure could have been followed by using and.

Now, we must distinguish among the 20 components of the Riemann tensor along with the following tabular where we have to take into account the identity:

In this tabular, the vanishing components obtained by computer algebra are put in a box, the nonzero components of the left column do not vanish when and the other components vanish when. Also, the 11 (care) lower components can be known from the 10 upper ones.

Keeping in mind the study of the S-metric and the fact that, , , while framing the leading terms not vanishing when, we get:

Then, taking into account the fact that, we obtain similarly:

The leading determinant does not vanish when because, in this case, all terms are vanishing and we are left with the two linearly independent framed terms, a result amounting to and in the case of the S-metric in [15].

In the case of the K-metric, we may use the relations already framed in order to keep only the four parametric jets on the right side. We may also rewrite them as follows:

if we use the fact that in the inverse metric.

As a byproduct, we are now left with the two (complicated) equations and where the dots mean linear combinations of with coefficients in K and the study of the Killing operator is quite more difficult in the case of the K-metric. Of course, it becomes clear that the use of the formal theory is absolutely necessary as an intrinsic approach could not be achieved if one uses solutions instead of sections. Indeed the strict inclusion cannot be even imagined if one does believe that, brings and. The computation could have been done with and because and.

The next hard step will be to prove that the other linearized components of the Riemann tensor do not produce any new different first order equation. The main idea will be to revisit the new linearized tabular with:

Putting the leading terms into a box, we have the identity that must be combined with the following formulas:

and so on, allowing to compute the 11 (care) lower terms from the 2 + 4 + 4 = 10 upper ones.

We have thus the following successive eleven logical inter-relations:

Keeping in mind the four additional equations and their consequences that have been already framed, both with the vanishing components of the Riemann tensor, namely:

we get successively:

As we have already exhibited an isomorphism, we may use only the later right set of parametric jet components. Using the previous logical relations while framing the leading terms not vanishing a priori when, there is only one possibility to choose four components of the linearized Riemann tensor, namely:

In order to understand the difficulty of the computations involved, we propose to the reader, as an exercise, to prove “directly” that the two following relations:

are only linear combinations of the previous ones.

We are facing two technical problems “spoilting”, in our opinion, the use of the K metric:

· With in place of, we have and the leading term of becomes proportional to with a wrong sign that cannot allow using. A similar comment is valid for the four successive leading terms.

· We also discover the summation in with a wrong sign that cannot allow introducing 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:

THEOREM 4.2: The rank of the previous system with respect to the four jet coordinates is equal to 2, for both the S and K-metrics. We obtain in particular the two striking identities:

R 03 , 13 + a ( 1 − c 2 ) R 01 , 03 = 0 ,             R 02 , 03 + a ( r 2 + a 2 ) R 03 , 23 = 0

Proof: In the case of the S-metric with, only the framed terms may not vanish and, denoting by “~” a linear proportionality, we have already obtained:

Hence, the rank of the system with respect to the 4 parametric jets 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, the study is much more delicate.

With, the coefficients of the metric of the previous system on the basis of the above parametric jets are proportional to the symmetric matrix:

Indeed, we have successively for the common factor:

and similarly for the common factor:

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.

Proof: According to the previous theorem, we have as we can choose the 4 parametric jets and. Using the introductory diagram with and thus , we obtain at once in a purely intrinsic way. We may thus start afresh with the new first order system obtained from after 2 prolongations. This result is thus obtained totally independently of any specific GR technical object like the Teukolski scalars, the Killing-Yano tensors or even the Penrose spinors introduced in [8] [9] [10] [11] [16].

Q.E.D.

Finally, we know from [2] [4] [12] [15] [17] [18] [19] that if is a system of infinitesimal Lie equations, then we have the algebroid bracket defined on sections by the following formula not depending on the lift of:

with the algebraic bracket bilinearly defined by and such that:

It follows that is such that with because we have obtained a total of 6 new different first order equations. We have on sections (care again) the 16 (linear) equations of as follows:

and we may choose only the 2 parametric jets among to which we must add in any case as they are not appearing in the Killing equations and their prolongations.

The system is not involutive because it is finite type with and cannot be thus involutive.

It remains to make one more prolongation in order to study with strict inclusions in order to study the third order CC for already described for the Schwarzschild metric in [15].

Surprisingly and contrary to the situation found for the S metric, we have now a trivially involutive first order system with only solutions, , ,. 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, or, equivalently, , and thus 4 third order CC coming from the 4 following components of the Spencer operator:

a result that cannot be even imagined from [8] [9] [10] [11] [16]. Of course, proceeding like in the motivating examples, we must substitute in the right

members the values obtained from and set for example

while replacing and 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 and, a result leading to in a coherent way with the only nonzero Killing vectors. We have indeed:

Like in the case of the S metric, is not involutive but is involutive. However, contrary to the S metric with, now for the K metric and is trivially involutive with a full Janet tabular having 16 rows of first order jets and 2 rows of zero order jets.

REMARK 4.4: We have in general ( [2] [5] p 339, 345):

that is, in our case. However, we have indeed the equality even if the conditions of Theorem 1.1 are not satisfied because is not 2-acyclic. Indeed, the Spencer map is not injective and we let the reader check as an exercise that its kernel is generated by and the Spencer δ-cohomology is such that because the cocycles are defined by the equations. 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 is not involutive with for the S-metric while is involutive with 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 [8] [9] [10] [11].

The following result even questions the usefulness of the whole previous approach:

THEOREM 4.5: The operator admits a minimum parametrization by the operator with 1 potential when, found in 1863. It admits a canonical self-adjoint parametrization by the operator with 6 potentials when, found in 1892 and modified to a mimimum parametrization by the operator Maxwell with 3 potentials, found in 1870 or Morera found in 1892. More generally, it admits a canonical parametrization by the operator with potentials that can be modified to a relative parametrization by with potentials which is nevertheless not minimum when, 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:

· : Multiplying the linearized Riemann operator by a test function and integrating by parts, we obtain (care to the factor 2 involved):

Cauchy operator

Airy operator

It is clear that the test function f has nothing to do with the metric ω ( [5], Introduction).

· We now present the original Beltrami parametrization:

which does not seem to be self-adjoint but is such that. 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 with, we obtain the Maxwell parametrization:

which is minimum because. 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 [20] for the technical details).

· This is far more complicated 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 [4] [12] [17] [21] or papers [12] [13] [14] [15] [22], the linear Spencer sequence is (locally) isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by a Lie algebra with equal to the dimension of the largest group of invariance of the metric involved. When, 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, the relations,. It follows that, if D₂ generates the CC of D₁, then is parametrizing, a fact not evident at all, even when for the Cosserat couple-stress equations exactly described by [18]. Passing to the differential modules point of view with the ring (even an integral domain) of differential operators with coefficients in a differential field K, this result amounts to say that. As it is known that such a result does not depend on the differential resolution used or, equivalently, on the differential sequence used, if generates the CC of in the Janet sequence, then is parametrizing and this result is still true even if 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 (second order for the M-metric, a mixture of second and third order for the S-metric and K-metric), then is, in any case, parametrizing the Cauchy operator for any. 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 is of order 3 when as we shall see below [6] [13] [14] [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 the torsion elements (the so-called Lichnerowicz waves in [22] ) for the equations because the Einstein equations cannot be parametrized and the extension modules are torsion modules [5] [7] [13] [19]. Such a result is only depending on the group structure of the conformal group of space-time that brings the canonical splitting without any reference to a background metric as it is usually done [4] [15] [19] [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 introducing the so-called Einstein parametrizing operator [19] [22]. A minimum parametrization of the Cauchy operator when with 6 potentials can be found by keeping only the Lagrange multipliers with used in [13] while setting like Morera when.

EXAMPLE 4.7: (Weyl tensor for and euclidean metric) We proved in ( [21], p 156-158) and more recently in [14] [22] [23] that, for, the natural “geometric object” corresponding to the Weyl tensor is no longer providing a second order differential operator but by a third order Weyl operator with first order CC in the differential sequence:

corresponding to the differential sequence of D-modules where p is the canonical residual projection:

The true reason is that the symbol of is finite type with second prolongation while its first prolongation is not 2-acyclic. It is important to notice that the operators are acting on the left on column vectors in the upper sequence but on the right on row vectors in the lower sequence though we have in any case the identities and.

Of course, these operators can be obtained by using computer algebra like in ([21], Appendix 2) but one may check at once that and are completely different operators while the operator is far from being self-adjoint even though it is described by a operator matrix. Our purpose is to prove that it can be nevertheless transformed in a very tricky way to a self-adjoint operator, exactly like the curl operator in 3-dimensional classical geometry because. It does not seem that these results are known today.

The starting point is the first order operator matrix defining the conformal Killing operator, namely:

Substracting the fourth row from the first row and multiplying the fourth row by, we obtain the operator matrix:

Adding the fourth row to the first, we obtain the operator matrix:

Adding the first row to the fourth row and dividing by 2, we obtain the operator matrix:

Multiplying the second, fourth and fifth row by −1, then multiplying the central column of the matrix thus obtained by −1, we finally obtain the operator matrix:

We now care about transforming given in ( [21], p 158) by the 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:

and check that like in the Poincaré sequence for where. As the new corresponding operator is homogeneous and of order 3 (care), we obtain locally, a result not evident at first sight (compare to [21], p 157).

The combination of this example with the results announced in [14] [23] 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.