Why Gravitational Waves Cannot Exist

The purpose of this short but difficult paper is to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT) in the light of a modern approach to nonlinear systems of ordinary or partial differential equations, using new methods from Differential Geometry (D.C. Spencer, 1970), Differential Algebra (J.F. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). The main idea is to identify the differential indeterminates of Ritt and Kolchin with the jet coordinates of Spencer, in order to study Differential Duality by using only linear differential operators with coefficients in a differential field K. In particular, the linearized second order Einstein operator and the formal adjoint of the Ricci operator are both parametrizing the 4 first order Cauchy stress equations but cannot themselves be parametrized. In the framework of Homological Algebra, this result is not coherent with the vanishing of a certain second extension module and leads to question the proper origin and existence of gravitational waves. As a byproduct, we also prove that gravitation and electromagnetism only depend on the second order jets (called elations by E. Cartan in 1922) of the system of conformal Killing equations because any 1-form with value in the bundle of elations can be decomposed uniquely into the direct sum ( where R is a section of the Ricci bundle of symmetric covariant 2-tensors and the EM field F is a section of the vector bundle of skew-symmetric 2-tensors. No one of these purely mathematical results could have been obtained by any classical approach. Up to the knowledge of the author, it is also the first time that differential algebra in a modern setting is applied to study the specific algebraic feature of most equations to be found in mathematical physics, particularly in GR.

As a byproduct, we also prove that gravitation and electromagnetism only depend on the second order jets (called elations by E. Cartan in 1922) of the system of conformal Killing equations because any 1-form with value in the bundle of elations can be decomposed uniquely into the direct sum ( ) , R F

Introduction
The first motivation for studying the methods used in this paper has been a 1000$ challenge proposed in 1970 by J. Wheeler in the physics department of Princeton University while the author of this paper was a student of D.C.
Spencer in the closeby mathematics department: Is it possible to express the generic solutions of Einstein equations in vacuum by means of the derivatives of a certain number of arbitrary functions like the potentials for Maxwell equations?
During the next 25 years and though surprising it may look like, no progress at all has been made towards any solution, either positive or negative. We now explain the way we found the (negative) solution of this challenge in 1995 [1].
Let us consider a manifold X of dimension n with local coordinates   [7].
In classical elasticity, the stress tensor density σ σ φ = = −∂ , 22 11 σ φ = ∂ has been provided by George Biddell Airy (1801-1892) in 1863 [8]. It can be simply recovered in the following manner: ( ) ( ) 11 We get the linear second order system: Introduction of [11], we may transform a problem of 3-dimensional elasticity into a problem of 2-dimensional elasticity by supposing that the axis 3 x of the dam is perpendicular to the river with ( ) 1 2 , , , 1, 2 ij x x i j Ω ∀ = and 33 0 Ω = because of the rocky banks of the river. We may introduce the two Lamé constants ( ) where the linearized scalar curvature ( ) tr R is allowing to define the Riemann operator in the previous diagram, namely the only compatibility condition (CC) of the Killing operator. It remains to exhibit an arbitrary homogeneous polynomial solution of degree 3 and to determine its 4 coefficients by the boundary pressure conditions on the upstream and downstream walls of the dam. Of course, the Airy potential φ has nothing to do with the perturbation Ω of the metric ω and the Airy parametrization is nothing else but the formal adjoint operator ( ) We discover at once that the origin of elastic waves is shifted by one step backwards, from the right square to the left square of the diagram. Indeed, using inertial forces for a medium with mass ρ per unit volume in the right member of Cauchy stress equations because of Newton law and the vector identity It is this comment that pushed me to use the formal adjoint of an operator, knowing already that an operator and its (formal) adjoint have the same differential rank (See later on). In the case of the conformal Killing operator, the second order CC are generated by the Weyl operator, linearization of the Weyl tensor over ω when 4 n ≥ . The particular situation pour 3 n = will be studied in the last section and its corresponding 5 third order CC are not known after one century [6]. Finally, the Bianchi operator describing the CC of the Riemann operator does not appear in this scheme.
Summarizing what we have just said, the study of elastic waves in continuum mechanics only depends on group theory because it has only to do with one differential sequence and its formal adjoint, combined together by means of constitutive relations. We have proved in many books [4] [5] and in [6] [13] [14] that the situation is similar for Maxwell equations, a result leading therefore to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT), thus also of Electromagnetism (EM).
Knowing already M.P. Malliavin as I gave a seminar on the "Deformation Theory of Algebraic and Geometric Structures" [6] [15], I presented in 1995 a seminar at IHP in Paris, proving the impossibility to parametrize Einstein equations, a result I just found [1]. One of the participants called my attention on a recently published translation from japanese of the 1970 master thesis of M. Kashiwara that he just saw on display in the library of the Institute [16]. This has been the true starting of the story because I discovered that the duality involved in the preceding approach to physics was only a particular example of a much =   , contrary to what happened in the previous diagram. We shall see that this comment brings the need to introduce the first extension module ( ) 1 ext M of the differential module M determined by  .
In a more intrinsic setting, using the same notation for a vector bundle and its set of (local) sections, we shall have: In the meantime, following U. Oberst [22] [23], a few persons were trying to adapt these methods to control theory and, thanks to J.L. Lions, I have been able to advertise about this new approach in a european course, held with succes during 6 years [5] and continued for 5 other years in a slightly different form [24]. By chance I met A. Quadrat, a good PhD student interested by control and computer algebra and we have been staying alone because the specialists of Algebraic Analysis were pure mathematicians, not interested at all by applications.
As a byproduct, it is rather strange to discover that the impossibility to parametrize Einstein equations, that we shall prove in Section 4, has never been acknowledged by physicists but can be found in a book on control because it is now known that a control system is controllable if and only if it is parametrizable [24] [25].
The following example of a double pendulum will prove that this result, still not acknowledged today by engineers, is not evident at all. For this, let us consider two pendula of respective length 1 l and 2 l attached at the ends of a rigid bar sliding horizontally with a reference position ( ) with respect to the vertical, it is easy to prove from the Newton principle that the equations of the movements does not depend on the respective masses 1 m and 2 m of the pendula but only depend on the respective lengths and gravity g along the two formulas: where :  3  2  1  1  3  2  2  1  3  2  3  3  1  1  2   2  1  1  2  2  1  3  3  3  1  2 , , , , If we define the differential rank of an operator by the maximum number of differentially independent second member, this is clearly an involutive differential operator with differential rank equal to 2 because ( ) 1 2 , ξ ξ can be given arbitrarily and thus ( ) 1 2 , η η can be given arbitrarily or, equivalently, because the differential rank of div is of course equal to 1 as div has no CC. Now, the involutive system parametrized by one arbitrary function because both 1 ξ and 2 ξ are autonomous in the sense that they both satisfy to at least one partial differential equation (PDE). Accordingly, we discover that div can be parametrized by the curl through 3 arbitrary functions ( ) which, in turn, cannot be parametrized again. Such a situation is similar to the one met in hunting rifles that may have one, two or more trigger mechanisms that can be used successively. It happens that the possibility to have one parametrization of div is an intrinsic property described by the vanishing of ( ) ext M , and so on, but such a result has no classical interpretation. It follows that certain parametrizations are "better" than others and no student should even imagine the minimal parametrization of div that we have presented above. A similar procedure has been adopted by J.C. Maxwell [28] and G. Morera [29] when they modified the parametrization of the Cauchy stress equations obtained by E. Beltrami in 1892 (see [30] and [31] for more details and references or [32] [33] and [34] [35] for computer algebra calculations).
It is clear from the beginning of this Introduction that an isometry is a solution of a nonlinear system in Lie form [2] [5] [6] and that we have linearized this system over the identity transformation in order to study elastic waves.
However, in general, no explicit solution may be known but most nonlinear systems of OD or PD equations of mathematical physics (constant riemannian curvature is a good example in [36]) are defined by differential polynomials. This is particularly clear for riemannian, conformal, complex, contact, symplectic or unimodular structures on manifolds [6]. Hence, in Section 2 we shall provide the main results that exist in the formal theory of systems of nonlinear PD equations in order to construct a formal linearization. The proof of many results is quite difficult as it involves delicate chases in 3-dimensional diagrams [2] [5] [11]. In physics, the linear system obtained may have coefficients in a certain differential field and we shall need to revisit differential algebra in Section 3 because Spencer and Kolchin never clearly understood that their respective works could be combined. It will follow that the linear systems will have coefficients in a differential field K and we shall have to introduce the of differential operators with coefficients in K, which is even an integral domain. This fact will be particularly useful in order to revisit differential duality in Section 4 before applying it to physics in Section 5 and concluding in the last Section 6. This paper is an extended and improved These purely mathematical results question the origin and existence of gravitational waves.

Differential Geometry
If X is a manifold with local coordinates ( ) We may introduce new coordinates ( ) x y y transforming like: ij q x y y y x y = called jet coordinates and sections ( ) x y A x y B x ϕ = = + we say that  is an affine bundle over X and we define the associated vector bundle  over X by the local coordinates ( ) . Of course, when  is an affine bundle over X with associated vector bundle E over X, we have ( ) . With a slight abuse of language, we shall set For a later use, if  is a fibered manifold over X and f is a section of  , we denote by V  by f as the vector bundle over X obtained when replacing ( ) , , , , x f x v in each chart. A similar construction may also be done for any affine bundle over  . Loking at the transition rules of ( ) q J  , we deduce easily the following results: but we shall not specify the tensor product in general.
any order and a short exact sequence: there is an induced exact sequence: is called the first prolongation of q  and we may define the subsets q r +  . In actual practice, if the system is = the first prolongation is defined by adding the as identities on X or at least over an open subset U X ⊂ . Differentiating the first relation with respect to i x and substracting the second, we finally obtain: and the Spencer operator restricts to we obtain: In general, neither q g nor q r g + are vector bundles over In a purely algebraic setting, one has: PROPOSITION 2.9: There exists a map Proof: Let us introduce the family of s-forms . We obtain at once ( ) Q.E.D.
The kernel of each δ in the first case is equal to the image of the preceding δ but this may no longer be true in the restricted case and we set: involutive if it is n-acyclic and finite type if 0 q r g + = becomes trivially involutive for r large enough. In particular, if q g is involutive and finite type, Having in mind the example of 0 0 x xx xy y xy − = ⇒ = with rank changing at 0 x = , we have: PROPOSITION 2.11: If q g is 2-acyclic and 1 q g + is a vector bundle over q  , then q r g + is a vector bundle over , 1 Proof: We may define the vector bundle 1 F over q  by the following ker/coker exact sequence where we denote by the image of the central map: and we obtain by induction on r the following commutative and exact diagram of vector bundles over  indicates the number of y that can be given arbitrarily.
Using the exactness of the top row in the preceding diagram and a delicate 3-dimensional chase, we have (See [2] and [11], p. 336 for the details):  is a system of order q on  such that 1 q g + is a vector bundle over q  and q g is 2-acyclic, then there is an exact sequence: We finally obtain the following crucial Theorem and its Corollary (Compare to [2], p. 72-74 or [11], p. 340 to [37]): This is all what is needed in order to study systems of algebraic ordinary differential (OD) or partial differential (PD) equations.

Differential Algebra
We now present in an independent manner two OD examples and two PD examples showing the difficulties met when studying differential ideals and ask the reader to revisit them later on while reading the main Theorems. As only a few results will be proved, the interested reader may look at [ and thus a is neither prime nor perfect, that is equal to its radical, but ( ) rad a is perfect as it is the intersection of the prime differential ideal generated by y with the prime differential ideal generated by 2 4 x y y − and 2 xx y − , both containing xxx y . EXAMPLE 3.2: With the notations of the previous Example, let us consider the (proper) differential ideal A ⊂ a generated by the differential polynomial and thus a is neither prime nor perfect but ( ) rad a is a perfect differential ideal and even a prime differential ideal p because we obtain easily from the last section that the resisual differential ring { } [ ] 1 2 11 , , , k y k y y y y p is a differential integral domain. Its quotient field is thus the differential field , , , K Q k y k y y y y = p with the rules: changing slightly the notations for using the letter v only when looking at the symbols. It is at this point that the problem starts because 2  is indeed a fibered manifold with arbitrary parametric jets ( ) 1 2 11 , , , y y y y but ( ) is no longer a fibered manifold because the dimension of its symbol changes when 11 1 y = . We understand therefore that there should be a close link existing between formal integrability and the search for prime differential ideals or differential fields. The solution of this problem has been provided as early as in 1983 for studying the "Differential Galois Theory" but has never been acknowledged and is thus not known today ([3] [5]). The idea is to add the third order PDE 111 0 y = and thus the linearized PDE 111 0 Y = obtaining therefore a third order involutive system well defined over K with symbol 3 0 g = . We invite the reader to treat similarly the two previous examples and to compare.
DEFINITION 3.5: A differential ring is a ring A with a finite number of commuting derivations ( ) We shall say that a differential extension p is a finitely generated differential extension of K and we may define the evaluation epimorphism with kernel p by calling η or y the residue of y modulo p . If we study such a differential extension L K , by analogy with Section 2, we shall say that q R or q g is a vector bundle over q  if one can find a certain number of maximum rank determinant D α that cannot be all zero at a generic solution of q p defined by differential polynomials P τ , that is to say, according to the Hilbert Theorem of Zeros, we may find polynomials The following Lemma will be used in the next important Theorem: LEMMA 3.9: If p is a prime differential ideal of a is a prime differential ideal.
PROPOSITION 3.14: If ζ is differentially algebraic over K η and η is differentially algebraic over K, then ζ is differentially algebraic over K. Setting ξ ζ η = − , it follows that, if L K is a differential extension and L ∈ η ξ , are both differentially algebraic over K, then ξ η + , ξη and i d ξ are differentially algebraic over K.
, we have the two towers K L N ⊂ ⊂ and K M N ⊂ ⊂ of differential extensions and we may therefore define the new . However, if only L K and M K are known and we look for such an N containing both L and M, we may use the universal property of tensor products an deduce the existence of a differential The construction of an abstract composite differential field amounts therefore to look for a prime differential ideal in which is a direct sum of integral domains [3]. DEFINITION 3.15: A differential extension L of a differential field K is said to be differentially algebraic over K if every element of L is differentially algebraic over K. The set of such elements is an intermediate differential field

K L
′ ⊆ , called the differential algebraic closure of K in L. If L K is a differential extension, one can always find a maximal subset S of elements of L that are differentially transcendental over K and such that L is differentially algebraic over K S . Such a set is called a differential transcedence basis and the number of elements of S is called the differential transcendence degree of L K . THEOREM 3.16: The number of elements in a differential basis of L K does not depent on the generators of L K and his value is Moreover, if K L M ⊂ ⊂ are differential fields, then THEOREM 3.17: If L K is a finitely generated differential extension, then any intermediate differential field K′ between K and L is also finitely generated over K. EXAMPLE [3] how to overcome this problem but this is out of the scope of this paper. It is finally important to notice that the fundamental differential isomorphism [3] [43] [44]: is the Hopf dual of the projective limit of the action graph isomorphisms between fibered manifolds: . The corresponding automorphic system 2 1 x y y ω = in Lie form where ω is a geometric object as in the Introduction and its prolongations has been introduced as early as in 1903 by E. Vessiot [7] [45] as a way to study principal homogeneous spaces (PHS) for Lie pseudogroups, This is all what is needed in order to study systems of infinitesimal Lie equations defined, like the classical and conformal Killing systems, over ω  where ω is a geometric object solution of a system of algebraic Vessiot structure equations (constant Riemann curvature, zero Weyl tensor).

Differential Duality
Let A be a unitary ring, that is 1, , , These conditions are automatically satisfied if M ′′ is free or projective. Let A be a differential ring, that is a commutative ring with n commuting We shall use thereafter a differential integral domain A with unit 1 A ∈ whenever we shall need a differential field ( )    The following technical Lemma is crucially used in the next proposition: Now, with operational notations, let us consider the two differential sequences: Now, exactly like we defined the differential module M from  , we may define the differential module N from ( ) ad  . For any other presentation of M with an accent, we have [11] [48]: THEOREM 4.10: The modules N and N′ are projectively equivalent, that is one can find two projective modules P and P′ such that and we obtain therefore ( ) ( ),  . It follows that ( ) ker  is a torsion module and, as we already know that ( )  . Accordingly, a torsion-free (  injective)/reflexive (  bijective) module is described by an operator that admits respectively a single/double step parametrization. Q.E.D.
We now turn to the operator framework; DEFINITION 4.12: If a differential operator ξ η →  is given, a direct problem is to find generating compatibility conditions (CC) as an operator , the inverse problem will be to look for ξ η →  such that 1  generates the CC of  and we shall say that 1  is parametrized by  if such an operator  is existing.
We finally notice that any operator is the adjoint of a certain operator because COROLLARY 4.14: In the differential module framework, if t M = , then we may obtain an exact sequence of free differential modules where  is the parametrizing operator. However, there may exist other parametrizations called minimal parametrizations such that ( ) coker ′  is a torsion module and we have thus REMARK 4.15: The following chains of inclusions and short exact sequences allow to compare the main procedures used in the respective study of differential extensions and differential modules:   2  1  2  1  2  1  1  11  11 , , , , , , y y y y y y .
The linearized system 1 0 Y =  over L is: Multiplying by test functions ( )  1  1  2  2  3  3  3  1  1  2  1 of course but also the additional zero order CC: which provides a torsion element ω satisfying Setting Y y δ = as the standard variational notation used by engineers, we obtain easily 0 ω δω ∧ ≠ and ω cannot therefore admit an integrating factor, a result showing that K is its own differential algebraic closure in L. x φ φ ξ It defines therefore a free differential module M D which is thus reflexive and even projective. Any resolution of this module splits, like the short exact sequence

Applications
We start this section with a general (difficult) result on the actions of Lie groups, covering at the same time the study of the classical and conformal Killing systems.
For this, we notice that the involutive first Spencer operator 1 0 . We obtain therefore the crucial formula: when q is large enough allowing to exhibit an isomorphism between the canonical Spencer sequence and the tensor product of the Poincaré sequence by  when q is large enough in such a way Q.E.D.
We now study what happens when 3 n ≥ because the case 2 n = has already been provided, proving that conformal geometry must be entirely revisited.
The study of the conformal case is much more delicate. As 0 F can be described by trace-free symmetric tensors, we have and it remains to discover the operator that will replace the Riemann operator.
Having in mind the diagram of Proposition 2.11 and the fact that ( ) • NO CC order 2: • OK CC order 3: ( ) Once again, the central third order operator is self-adjoint as can be easily seen by proving that the last 5 3 → operator, obtained in [6] by means of computer algebra, can be chosen to be the transpose of the first 3 5 → conformal Killing operator, just by changing columns.
This result can also be obtained by using the fact that, when an operator/a system is formally integrable, the order of the generating CC is equal to the number of prolongations needed to get a 2-acyclic symbol plus 1 ( [5], p. 120, [6]). In the present case, neither 1 g nor 2 g are 2-acyclic while 3 0 g = is trivially involutive, so that ( ) 3 1 1 3 − + =.
• 4 n = : In the classical case, we may proceed as before for exibiting the 20 components of the second order Riemann operator and the 20 components of the first order Bianchi operator.
The study of the conformal case is much more delicate and still unknown.  avoiding therefore the Lorenz (no "t") gauge condition for the EM potential [55]. Indeed, let us start with the Minkowski constitutive law with electric constant 0  and magnetic constant 0 µ such that 0, 0 y y y y + = + = , that is 1 y and 2 y are separately killed by the second order Laplace operator 11 22 d d ∆ = + .
Collecting the above results, we obtain the striking theorem: without any reference to a gauge transformation in order to obtain a (minimum) relative parametrization (see [31] and [56] for details and explicit examples). When 4 n = we finally obtain the adjoint sequences: This last result even strengthens the doubts we already had about the origin and existence of gravitational waves.

Conclusions
Whenever ( ) q q R J E ⊆ is an involutive system of order q on E, we may define the Janet bundles r F for 0,1, , r n = by the short exact sequences: The mathematical structures of electromagnetism and gravitation only depend on the second order jets.