_{1}

^{*}

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 (DOI:10.4236/jmp.2018.910125) 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
→ M when
*m*
→ 0 and K
→
S when
*a*
→ 0, the CC of S do not provide the CC of M when
*m*
→ 0 while the CC of K do not provide the CC of S when a
→ 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
*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 algebra 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 if each Spencer sequence is isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by the corresponding Lie algebra.

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 [

DEFINITION 1.1: A system of order q on E is an open vector subbundle

The most difficult but also the most important theorem has been discovered by M. Janet in 1920 [

THEOREM 1.2: If

COROLLARY 1.3: (PP procedure) If a system

The paper will be organized as follows:

· First of all, starting with an arbitrary system

though they are of course fully determined by the first order CC of the final involutive system

· The same procedure will be applied to the two first order systems of infinitesimal Lie equations allowing to define the Killing operator for the S-metric and the K-metric while comparing the respective results obtained. We may say that the case of the S-metric has already been treated in the publication quoted in the abstract but that it took us two years just for daring to engage in dealing similarly with the K-metric as anybody can understand by looking at the components of the Riemann tensor in the literature. It has been a surprising “miracle” to discover in the proof of Theorem 4.2 that there was a unique but tricky way to bring this problem to a purely mathematical and relatively simple computation on Lie equations and their prolongations.

· In the case of the S-metric, starting with the system

· Then, the case of the K-metric seems to be similar as it is also leading to the strict inclusions

· Finally, we are able to relate these results to the computation of certain extension modules in differential homological algebra, showing why the mathematical foundations of conformal geometry in arbitrary dimension and general relativity must be entirely revisited in the light of these results.

MOTIVATING EXAMPLE 1.4: With

First of all we have to look for the symbol

Since that moment, we have to consider the two possibilities:

·

We have thus the Janet sequence:

or, equivalently, the exact sequence of differential modules over

where p is the canonical projection onto the residual differential module.

·

As

Finally, we have

It remains to find out the CC for

It follows that we have only one second order and one third order CC:

but, surprisingly, we are left with the only generating second order CC

We let the reader prove as an exercise (see [

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

We may consider successively the following systems of decreasing dimensions

The last system is involutive with the following Janet tabular:

The generic solution is of the form

satisfying the only first order CC:

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 [

We let the reader prove that the space of solutions has dimension 12 over

satisfying the only fourth order CC

It follows that we have the unexpected differential resolution:

with, from left to right,

and the long δ-sequence:

in which

However,

Accordingly, R_{3} is thus involutive and the only CC

MOTIVATING EXAMPLE 1.7: With

We obtain at once through crossed derivatives

satisfying

is not formally exact because

is indeed formally exact because

but not strictly exact because

It follows from these examples and the many others presented in [

However, 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 Killing operators we want to study, the problem becoming only a “mathematical” one but surely not a “physical” one.

IMPORTANT REMARK 1.8: The intrinsic properties of a system with constant coefficients may drastically depend on these coefficients, even if the systems do not appear to be quite different at first sight. Using jet notations, let us consider the second order system

and the adjoint sequence:

though the CC sequence that must be used with

On the contrary, if

and the CC sequence does coincide with the adjoint sequence:

It is thus essential to notice that

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 [

With canonical projection

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

We shall use this result, first found exactly 40 years ago [

EXAMPLE 1.10: The Janet tabular in Example 1.4 with

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

In the Boyer-Lindquist (BL) coordinates

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

Though this system

We obtain in particular, modulo

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

and notice that the

However, as we are dealing with sections,

and a new prolongation only brings the single equation

Knowing that

and we have replaced by “×” the only “dot” (non-multiplicative variable) that cannot provide vanishing crossed derivatives and thus involution of the symbol

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

Denoting by

In this diagram, not depending any longer on m, we have now

We notice the vanishing of the Euler-Poincaré characteristics:

We point out that, whatever is the sequence used or the way to describe

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

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

where we have set

as a well known way to recover the Schwarschild metric. We notice that t or

with Euler-Poincaré characteristic

Using now cartesian space coordinates

and the fundamental diagram

The involutive system produced by the PP procedure does not depend on

THE ONLY IMPORTANT OBJECT IS THE GROUP, NOT THE METRIC

Let us now introduce the Riemann tensor

that can be considered as an infinitesimal variation. As for the Ricci tensor

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 [

However, introducing

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

Summing, we see that all terms in

Setting

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 ( [

a result showing that certain third order CC may be differential consequences of the Bianchi identities (see [

and, comparing to the previous computation for

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

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

LEMMA 4.1:

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

with

that is, after division by

Finally, after eliminating the last term, we get:

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

in a coherent way with the result

for the S metric when

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

coefficient of

With

Similarly, multiplying

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

Substituting, we obtain:

a situation leading to modify

and with

Finally, multiplying

Using the rational coefficients belonging to the differential field

One has the classical orthonormal decomposition:

and defining:

in which the coefficient of

^{1} and dX^{2} are respectively proportional to

We may obtain simpler formulas in the corresponding basis, in particular the 6 components with only two 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 (

We have successively:

It must be noticed that we have been able to factorize the six components with only two different indices by

After tedious computations, we obtain:

which is indeed vanishing when

Introducing the formal Lie derivative

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

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

It follows that

The first row proves that

Accordingly, we only need to take into account

Similarly, we also obtain

where we have to set

Hence, taking into account

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

and thus

In addition, we have

We have also:

The following invariants are obtained successively in a coherent way:

However, as

These results are leading to

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

Using the previous identity for

Taking into account the fact that

A similar procedure could have been followed by using

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

Keeping in mind the study of the S-metric and the fact that

Then, taking into account the fact that

The leading determinant does not vanish when

In the case of the K-metric, we may use the relations already framed in order to keep only the four parametric jets

if we use the fact that

As a byproduct, we are now left with the two (complicated) equations

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

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

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

· We also discover the summation

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

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

Hence, the rank of the system with respect to the 4 parametric jets

In the case of the K-metric with

With

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

Q.E.D.

Finally, we know from [

with the algebraic bracket bilinearly defined by

It follows that

and we may choose only the 2 parametric jets

The system is not involutive because it is finite type with

It remains to make one more prolongation in order to study

Surprisingly and contrary to the situation found for the S metric, we have now a trivially involutive first order system with only solutions

a result that cannot be even imagined from [

members the values obtained from

while replacing

Using one more prolongation, all the sections (care again) vanish but

Like in the case of the S metric,

REMARK 4.4: We have in general ( [

that is, in our case

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

THEOREM 4.5: The operator

Proof: We provide successively the explicit corresponding parametrizations:

·

Cauchy operator

Airy operator

It is clear that the test function f has nothing to do with the metric ω ( [

·

which does not seem to be self-adjoint but is such that

which is indeed self-adjoint. Keeping

which is minimum because

· _{2} generates the CC of D_{1}, then

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 [

EXAMPLE 4.7: (Weyl tensor for

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

The true reason is that the symbol

Of course, these operators can be obtained by using computer algebra like in ([

The starting point is the

Substracting the fourth row from the first row and multiplying the fourth row by

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

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

The combination of this example with the results announced in [

First of all, comparing the M-metric, the S-metric and the K-metric by using the corresponding systems of first order infinitesimal Lie equations, we may summarize the results previously obtained by repeating that, when E = T, the smaller is the background Lie 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, which are torsion 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 like Lie algebroids or differential extension modules and their purely mathematical consequences. Summarizing this paper in a few words, we do really believe that “God used group theory rather than computer algebra when He created the World”!

The author declares no conflicts of interest regarding the publication of this paper.

Pommaret, J.-F. (2020) A Mathematical Comparison of the Schwarzschild and Kerr Metrics. Journal of Modern Physics, 11, 1672-1710. https://doi.org/10.4236/jmp.2020.1110104