Generating Compatibility Conditions and General Relativity

The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.


Introduction
If X is a manifold of dimension n with local coordinates ( )

T T X =
, the q-symmetric tensor bundle * q S T and the bundle * r T ∧ of r-forms.In General Relativity, there may be different solutions of Einstein equations in vacuum like the Minkowski, the Schwarzschild and the Kerr metrics for example.
For fixing the notations and with more details, if is a nondegenerate metric, that is ( ) 0 det ω ≠ , and if q j denotes all the derivatives of an object up to order q, we may construct the Christoffel symbols γ through the Levi-Civita isomorphism ( ) ( ) , j ω γ ω  and, using the language of jet bundles, ( ) , ω γ is a section of ( ) J S T that will be simply written ( ) ( ) self-adjoint with 6 terms though the Ricci operator is not with only 4 terms.Recently, many physicists (See [5] [6] [7] [8] [9]) have tried to construct the compatibility conditions (CC) of the Killing operator for various types of background metrics, in particular the three ones already quoted, namely an operator order 1, 2,3, till we stop because of noetherian arguments ( [10]).
• These CC only depend on the Lie algebra structure (dimension of the solution space and commutation relations) of the corresponding Killing operator, which, even though it is finite dimensional with dimension ( ) that is 10 obtained for the Minkowski metric, may have dimension 4 for the Schwarzschild metric and dimension 2 for the Kerr metric.• The only two canonical sequences that can be constructed from an operator or a system, namely the Janet and Spencer sequences, are structurally quite different.Indeed, the Janet bundles 0 , , n F F  appearing in the Janet sequence are concerned wit geometric objects like , , ω γ ρ , while the Spencer bundles 0 , , n C C  are far from being related with geometric objects, the simplest example being ( ) In the case of Lie equations considered, the central concept is not the system but rather the group as it can be seen at once from the construction of the Vessiot structure equations ([3] [11] [12] [13]).
The authors who have studied these questions had in mind that the total number of generating CC could be considered as a kind of "differential transcendence degree", also called "degree of generality" by A. Einstein in his letters to E. Cartan of 1930 on absolute parallelism ( [14]), the modern definition being that of the "differential rank" ([10] [12] [15] [16]).We must say that Cartan, being unable to explain to Einstein his theory of exterior systems, just copied the work of Janet published in 1920 ( [17]) in his letters to Einstein, published later on as the only paper he wrote on the PD approach, but without ever quoting Janet who suffered a lot from this behavior and had to turn to mechanics.
Such a result will be obtained in the framework of differential modules as its explanation in the framework of differential systems is much more delicate and technical ( [10] [12] [18]).
First of all, with our previous assumptions,

[ ]
main and we can restrict our study to finitely generated differential modules which are therefore finitely presented (See [14] for more details).Let thus M be defined by a finite free presentation giving rise to the long exact sequence: where the differential operator  is acting on the right by composition with action law ( ) , p is the canonical residual projection and is called the differential module of equations and is thus finitely generated because D is a noetherian differential domain.The following useful proposition proves the additivity property of the differential rank and is used in the next two corollaries ( [3] [10] [12]): PROPOSITION 1.2:If we have a short exact sequence 0 0 COROLLARY 1.3:If  is a linear partial differential operator with coeffi- cients in a differential field K and ( ) ad  is the formal adjoint that can be obtained formally or through an integration by parts, then In actual practice, working in the system framework, starting with a system ( ) R J E ⊂ of order q on E and introducing the canonical projection ( ) ( ) B + is defined by more generating PD equations than the ones defining r B both with its prolongations, and start to get equality when r is large enough in the projective limit . The striking result is that there may be gaps in the procedure, that is we shall even provide a tricky example where one can have a single generating CC of order 3, then no new generating CC of order 4 and 5, but suddenly a new generating CC of order 6 ending the procedure.We do not believe that such situations were even known to exist.

Motivating Examples
We provide below three examples, pointing out that it is quite difficult to exhibit such examples.
and K =  while keeping an upper index for any unknown, let us consider the following system ( ) , , par ξ ξ ξ = and corresponding Janet tabular: It is easy to check that all the second order jets vanish and that the general solution { } As the non-multiplicative variable written with the sign ×cannot be used, the symbol 1 g is not involutive because it is finite type with 2 0 g = .This system is trivially FI because it is made by homogeneous PD equations.We have the following commutative diagrams:  The next result points out the importance of the Spencer δ -cohomology.
Indeed, we shall prove that the last symbol diagram is commutative and exact.In particular, the lower left map δ is surjective and thus the upper right induced map is also surjective while these two maps have isomorphic kernels.
For this, we notice that the 3 components of that is to say by two linearly independent equations.Accordingly, in the left column we have: An unusual snake-type diagonal chase left to the reader as an exercise proves that the induced map is surjective with a kernel isomorphic to ( ) H g .This is indeed a crucial result because it also proves that the additional CC has only to do with the single second order component of the Riemann tensor in dimension 2, a striking result that could not even be imagined by standard methods.Moreover, we know that if a system ( ) when it is homogeneous like in this case, and its symbol that s is the smallest integer such that q s g + becomes 2-acyclic (or involutive), then the generating CC are of order at most 1 s + ([3] [10] [12]).Collecting the above results, we find the 3 first order differentially independent generating CC coming from the Janet tabular and the additional single second order generating CC describing the 2-dimensional Riemann operator, that is the linearized Riemann tensor in the space ( ) An elementary computation provides the second order CC:  , , where p is the canonical (residual) projection.We check indeed that 1 4 5 2 0 − + − = but this sequence is quite far from being even strictly exact.Of course, as 2 R is involutive, we may set ⊗ and obtain the corresponding canonical second Spencer sequence which is induced by the Spencer operator: Proceeding inductively as we did for finding the second order CC, we may obtain by combinatorics the following formally exact sequence: and we have ( ) , that is three generating first order CC which are differentially independent, plus their 9 prolongations, plus one second order CC which is nevertheless not differentially independent.Hence we have a total number of 3 1 4 + = generating CC but this number has nothing to do with any differential transcendence degree because , , Ψ Ψ Ψ .
We finally compute the corresponding (canonical) Janet sequence by quotient.
For this, we must use the trivial second Spencer sequence: with 2 20 40 30 8 0 − + − + =.The (canonical) Janet sequence is thus: and dimensions: us consider the following linear inhomogeneous system: may not be involutive or the coordinate system may not be δ -regular.However, changing linearly the local coordinates with → + → , we obtain the Ja- net tabular for 2 g : We let the reader check as an exercise that 2 g is not 2-acyclic by counting the dimensions in the long sequence: and that 3 g is involutive, thus 2-acyclic, with characters ( ) We obtain from the main theorem . It is easy to check that ( ) . We may thus consider the new second order system ( ) with a strict inclusion and We may start again with 2 R′ and study its symbol 2 g′ defined by the 3 li- near equations with the following Janet tabular obtained after doing the same change of local coordinates as before: This symbol is neither 2-acyclic nor involutive but its prolongation 3 g′ , de- fined by the 8 equations: is involutive with characters ( ) 0, 0, 2 and we may consider again the system: , , u v w , an elementary but tedious computation, we shall use a trick, knowing in advance that the generating CC must be of order 1 1 2 + = because 2 g′ had to get one prolongation in order to become involutive and thus 2-acyclic. • Step 2: It thus remains to find out the CC for ( ) , u v in the initial inhomogeneous system.As we have used two prolongations in order to exhibit 2 R′ , we have second order formal derivatives of u and v in the right members.Now, from the above argument, we have second order CC for the new right members and could hope therefore for a fourth order generating CC.The trick is to use the three different brackets of operators that can be obtained.
We have in a formal way: brings the third order CC: brings the fourth order CC: ( ) We have indeed the identity = and thus ( ) rentially dependent, that is B is a new generating fourth order CC which is not a consequence of the prolongations of A. Again, the total number of generating CC, that is 1 1 2 + = , has nothing to do with the differential transcendence de- gree of the CC differential module which is ( ) ( ) we now prove that a slight change of the equations may provide quite important changes in the number and order of the CC.Such an example is the only one that we could have found in more than 40 years of computing CC in mathematics and applications.For this, let us consider the new system: Before starting, we first notice that it is a prioiri not evident to discover that R R ∞ = is a finite dimensional vector space over K with ( ) However such a result can be obtained by direct integration (Compare to the Janet example treated in the introduction of [12]).
• Step 1: The symbol 2 g is defined by may not be involutive or the coordinate system may not be δ -regular.However, we obtain the Janet tabular for 2 g : and thus the Janet tabular for 3 g : We let the reader check as an exercise that 2 g is not 2-acyclic by counting the dimensions in the long sequence: and that 3 g is involutive, thus 2-acyclic, with characters ( ) 0, 0, 4 as in the previous example.It follows that ( ) . It is easy to chek that ( ) 2 y u v x u = − − .We may thus consider the new second order system ( ) with a strict inclusion and ( ) We may start again with 2 R′ and study its symbol 2 g′ defined by the 3 li- near equations with the following Janet tabular obtained after doing the same change of local coordinates as before: This symbol is not involutive but its prolongation 3 g′ , defined by the 8 equa- tions: J is involutive with characters ( ) 0, 0, 2 and we may consider again the system: As before, instead of writing out the system 3 R′ and studying its formal inegrability by an elementary but tedious computation, we shall use a trick, knowing in advance that the generating CC must be of order at least 1 1 2 + = because 2 g′ had to get one prolongation in order to become involutive and thus 2-acyclic. • Step 2: It thus remains to find out the CC for ( ) , u v in the initial inhomogeneous system.As we have used two prolongations in order to exhibit 2 R′ , we have second order formal derivatives of u and v in the right members.Now, from the above argument, we have second order CC for the new right members and could hope therefore for a fourth order generating CC.The trick is to use the three different brackets of operators that can be obtained.
We obtain in a formal way: brings the third order CC: brings the new first order equation: ( ) Accordingly, we may start afresh with the new system ( ) and thus a CC of order 5, namely: ( ) We obtain therefore a new sixth order CC: which cannot be a differential consequence of A. After tedious computations, one can find the differential identity: ( ) The corresponding simplest free resolution, written with differential modules, is thus: Again, the total number of generating CC, that is 1 1 2 + = , has nothing to do with the differential transcendence degree of the CC differential module which is still ( ) ( )

Mathematical Tools
Instead of starting with a linear system ( ) R J E ⊂ of order q on E, let us start with a bundle map ( ) Φ and let us consider the linear PD operator ( ) . The general case of the successive prolongations with 0 r ≥ is described by the following commutative and exact diagram: q r q r r r q r q r r r q r r q r r q r q r r r with symbol-map induced in the upper symbol sequence ([19] [20] [21]).
Chasing in this diagram while applying the "snake" lemma ([10] [22] [23]), we obtain the long exact connecting sequence: which is thus connecting in a tricky way FI (lower left) with CC (upper right).
Needless to say that absolutely no classical procedure can produce such a result which is thus totally absent from the GR papers already quoted.
Setting ( ) ( ) , we have equivalently the shorter long exact sequence: ( ) As a possible interpretation, ( ) dim Q is the total number of CC of order 0,1, up to r included.However, the problem to solve is to study the structure of the projective limit of vector bundles made by the induced epimorphisms Of course, as it is mostly realized in the examples, we have to sup- pose that q R is sufficiently regular in such a way that the q r R + are vector bundles 0 r ∀ ≥ and that the ( ) ( ) s q r s q r q r q r s R R π + + + + + + = are also vector bundles, such a situation being in particular always realized when ( ) R J E ⊂ or D are defined over a differential field K.In this case, introducing the filtered noethe- of differential operators with coefficients in K, we may introduce a differential module M with induced filtration in such a way that the system ( ) associated with M with ( ) course automatically FI (care).Following Macaulay in ( [24]), we have already proved in many places ([3] [10]) that R is a differential module for the Spencer operator * : : by the explicit formula: ( ) It is important to notice that such an operator/system is far from being formally integrable because: As can be seen from the examples previously presented, starting with r Ψ for a given r, the main problem is to compare the epimorphism ( ) with the morphism ( ) ( ) ( ) in the following commutative diagram which may not be exact: ( ) r r q r q r r r q r q r r r q r r q r r q r q r r r where the central row is induced from the long exact sequence: and may not be exact.   ([12Proposi- tions 10, p 83) or ([10], Remark 2.9, p 315) that ( ) . We have thus a projective limit of systems, each one being defined by more equations than the preceding one and such a procedure must finish with a FI system that can even be prolonged, as we shall see in the examples, in order to obtain an involutive system that may be used to start a Janet sequence.The decision to stop is provided by the maximum order of the CC obtained, namely of order bounded by 1 r s t + + = if the system ( ) s q r R + is involutive or at least with a 2-acyclic symbol. The idea is to use the composite morphism while chasing in order to prove that any element of ( ) ( ) Let us now deal with the symbol cohomology by chasing in the following commutative diagram: where neither the first nor the second upper columns may be exact and where the left column may not be exact, unless g q is involutive or 2-acyclic.Chasing with the same notations, we obtain: PROPOSITION 3.3: There exists an exact sequence: The upper left arrows are not in general epimorphisms and it may be sometimes useful to consider r h as a kind of symbol in the more abstract diagram: where the rows are now exact.However, understanding the meaning of r h as a kind of new symbol may not be possible unless is a monomorphism, that is when q g is 2-acyclic and r h is 1-acyclic, that is when q g is also 3-acyclic (or involutive).Once more, we understand the crucial importance of 2-acyclicity but we recall that the only symbol known to be 2-acyclic without being involutive is the symbol of the conformal Killing system whenever 4 n ≥ , which is also 3-acyclic whenever We have the Janet tabular for 1 2 3 The two CC are:

S T E S T ⊗ 
and 1

2
F Q  while 1 0 Q = as there is no CC of or- der 1.From the snake lemma and a chase, we obtain the long exact connecting sequence when 0 r = : relating FI (lower left) to CC (upper right).By composing the epimorphism and the long exact sequence: which is nevertheless not a long ker/coker exact sequence by counting the dimensions as we have 6 15 12 1 2 0 − + − = ≠ .
The above diagrams illustrate perfectly the three propositions of Section 2. We have in particular: ( and the formally exact sequence, which is nevertheless not strictly exact though 1 2 1 0 − + =: We remind the reader that, contrary to the situation met with FI systems where the exactness on the jet level is obtained inductively from the exactness on the symbol level, here we discover that we may have the exactness on the jet level without having exactness on the symbol level.First of all, let us compute the dimensions and the parametric jets that will be used in the following diagrams.
, , , , , par par y y y y y


It is not at all evident to study these diagrams.We We have already proved dent at first sight explaining why the only second order additional generating CC is nothing else than the Riemann tensor in dimension equal to 2. We have explained in ( [4]) that such a system has its origin in the study of the integration of the Killing system for the Schwarzschild metric, which is not FI.With more details, let us use the Boyer-Lindquist coordinates ( ) ( ) where m is a constant, the metric can be written in the diagonal form: Using the notations that can be found in the theory of differential modules, let us consider the Killing equations: where we have introduced the Christoffel symbols γ while setting  ).In particular, as the Ricci part is vanishing by assumption, we may identify the Riemann part with the Weyl splitting part as tensors ( [3]) and it is possible to prove (using a tedious direct computation or computer algebra) that there are only 6 non-zero components.It is important to notice that this result, bringing a strong condition on the zero jets because of the Lie derivative of the Weyl tensor and thus on the first jets, involves indeed the first derivative of the Weyl tensor because we have a term in ( ) A ′ ′′ .When 0 Ω = , we obtain after 2 prolongations the additional 5 new first order PD equations: As we are dealing with sections, 1 0 ξ = does imply 1,1 0 ξ = and 0,0 0 ξ = but does not imply 1,0 0 ξ = , these later condition being only brought by one ad- ditional prolongation and we have the strict inclusions Hence, it remains to determine the dimensions of the subsystems ( ) ( ) ( ) R , having minimum dimension equal to 4, is for- mally integrable, though not involutive as it is finite type, and to exhibit 4 solu-tions linearly independent over the constants.Indeed, we must have 0 c ξ = where c is a constant and we may drop the time variable not appearing elsewhere while using the equation 1 0 ξ = .It follows that , , , f g ξ θ φ ξ θ φ = = while , f g are solutions of the first, second and fifth equations of Killing type wih a general solution depending on 3 constants, a result leading to an elementary probem of 2-dimensional elasticity left to the reader as an exercise.The system ( ) R is formally integrable while the system ( )

R
is involutive.Having in mind the PP procedure, it follows that the CC could be of order 2, 3 and even 4.
Equivalently, we may cut the integration of this system into three systems: 1) First of all, we have 1 0 ξ = and thus 1 2) Then, we may consider 0 3) Finally, we arrive to the FI system with the same properties as the ones found for Example 2.1: that is with 3 generating first order CC and 1 additional second order generating CC.
Proceeding like in the motivating examples, we may introduce the inhomogeneous systems: ( ) However, we have the linearization formulas: and obtain therefore the formulas: ( ) with two similar ones for , , , determined by the 15 10 5 4 4 2 = + = + + second order CC that we have exhi- bited.Now, after one prolongation, we get: ( ) and thus 1 0,0 0 0,1 0 d d ξ ξ − = .Similarly, we have: In order to proceed further on, we notice that the generating CC of order 3 already found can be written as: Using crossed derivatives, we get: Finally, as already noticed, the symbol is not involutive and even 2-acyclic because otherwise there should only be first order CC for the right members defining the system ( )

R J T ′ ⊂
. As a byproduct, we have, at least on the symbol level, the second order CC: As shown in ( [4]), the study of the Killing system for the Kerr metric is even more difficult because the space of solutions is reduced from 4 already given to the 2 infinitesimal generators { } , t φ ∂ ∂ only.Accordingly, we discover that the Schwarzschild and the Kerr metrics do behave quite differently and there is thus no hope at all for selecting specific solutions of the Einstein equations in vacuum.
We consider this result as a key challenge when questioning the origin and existence of gravitational waves in general relativity and believe this problem has never been pointed out clearly for the very simple reason that the underlying mathematics are not known by physicists.

EXAMPLE 2.2 REVISITED:
Coming back to the system ( ) with a strict inclusion and second members ( ) , , u v w u v x u = − − , let us exchange 1 x with 2 x in order to have an involutive third order symbol 3 g′ in δ -regular coordinates and con- sider the system 3 R′ with now which is formally exact on the jet level, even if it is not strictly exact because the first operator is not FI, and we check that 1 2 5 7 4 1 0 − + − + − =.We notice that the part between 0 F and 4 F is typically a Janet sequence for 1  .
It follows that we have the following long exact sequence on the level of jets, 5 r ∀ ≥ − : (

called
Einstein equations are described by 0 ij =  or, equivalently, by 0 ij ρ = when 4 n = .Now, if  is a fibered manifold over X with fiber dimension m and local coordinates ( ) we may introduce the tangent bundle ( ) T  over  with local coordinates ( ), , ,x y u v and the vertical bundle ( ) V  with local coordinates ( over  .We shall denote by the capital letters the respective linearizations of , , ω γ ρ which are sections of the respective vertical bundles.Introducing the Lie derivative  of geometric objects, it is therefore possible to introduce the cor- responding first order Killing operator Ricci operator.For example, it is known that − Ω .We haveproved in ([1] [2] [3][4]) that the so-called gravitational waves equations are nothing else than ( ) ad Ricci by introducing the formal adjoint operator.It is important to notice that the Einstein operator 1 2

DEFINITION 1 . 1 :
The differential rank ( ) D rk M over D of a differential module M is the differential rank over D of the maximum free differential submodule F of M and we have the short exact sequence 0

4 :
(Euler-Poincaré characteristic) For any finite free differential resolution of a differential module M, then but not free in general and we may look for a minimum number of generators which may be differentially dependent in general as we shall see in the next examples.It thus remains to provide examples of such computations showing that these two numbers are not related and must therefore be found totally independently in general, apart from the very exceptional situation met when there is only a single generating CC.
 →  →  →  →  → → with Euler-Poincaré characteristic 2 5 13 19 12 3 0 − + − + − = but, as before, there is a matrix 260 280 × at least and we doubt about the use of computer algebra, even on such an elementary example.With used as a middle row of the first diagram with dimensions:  →  →  →  → → so that we have again 2 17 31 21 5 0 − + − + = in a coherent way with the fact that ( ) and thus the Janet tabular for 3 g : Journal of Modern Physics things are changing after that.As such a property is intrinsic, coming back to the original system of coordinates, we have after one more prolongation: Instead of doing the same change of variables, writing out the system 3 R′and study its formal inegrability with corresponding 9 11 20 + = CC for ( ) dim g =  .We obtain from the main theorem

PROPOSITION 3 . 1 :
We have only in general: the Spencer operator by d in place of the standard notation D of the literature that could be confused with the ring D of differential operators, we have the following commutative diagram:

PROPOSITION 3 . 2 :
Ψ .With more details, setting for simplicity ( ) obtain the follow- ing crucial proposition (See[4], Example 2.A.9) through a chase left to the reader as an exercise: There exists a short exact sequence: while the other ones are what we called identity to zero like: obtain the following dia- gram with exact central and lower rows whenever 1 r ≥ .

EXAMPLE 2 .
1 REVISITED: in the differential field K of coefficients.As in the previous Macaulay example and in order to avoid any further confusion between sections and derivatives, we shall use the sectional point of view and rewrite the previous equations in the symbolic form ( ) ξ ω = Ω ∈ where L is the formal Lie derivative: far from being involutive because it is finite type with second symbol 2 0 g = defined by the 40 equations 0 k ij v = in the initial coordinates.From the symmetry, it is clear that such a system has at least 4 solutions, namely the time translation ∂ − ∂ ∂ − ∂ .These results are brought by the formal Lie derivative of the Weyl tensor because the Ricci tensor vanishes by assumption and we have the splitting Riemann Ricci Weyl ⊕  according to the fundamental diagram II that we discovered as early as in 1988 ([25]), still not acknowledged though it can be found in ([1] [2] [3]

4 4
dim R = , we have thus obtained the 15 equations defining 1 R′ with even simplify these equations and get a system not depending on A anymore:

2 j
Ω along the ker/coker exact sequence: containing the leading term 00 d U after substitution.
a gene- rating one because it is just a differential consequence of the second order CC V d V ,producing therefore a third order CC that cannot be reduced by means of any Bianchi identity, that is we finally have 15 generating second order CC and 4 new generating third order CC, in a manner absolutely similar to that of all the motivating examples of this paper.

, let us exchange 1 x with 2 x in order to have an involutive third order symbol 3 g′ in δ -regular coordinates but the system 3 R′
, a result not evident to grasp at first sight because it comes from the lack of formal integrability of 2 R and the strict inclusion