The purpose of this self-contained but difficult paper is to revisit general relativity (GR) and gauge theory (GT), but also Elasticity (EL) and electromagnetism (EM) in view of the latest mathematical developments existing today in group theory, system theory and module theory, namely:

Systems: The orders of the successive operators appearing in the conformal Killing resolution highly depend on the dimension n , a result confirmed in 2016 by A. Quadrat (INRIA) while using computer algebra [1]. They are respectively 3 → 1 5 → 3 5 → 1 3 → 0 when n = 3 , 4 → 1 9 → 2 10 → 2 9 → 1 4 → 0 when n = 4 and 5 → 1 14 → 2 35 → 1 35 → 2 14 → 1 5 → 0 when n = 5 . This result leads to revisit conformal geometry.Groups: The Ricci and the Maxwell tensors have only to do with the second order jets of the conformal group, called elations by Cartan (1922) [2]-[5]. This result questions the mathematical foundations of both general relativity (GR) and gauge theory (GT).Modules: Contrary to the Maxwell equations, the Einstein equations cannot be parametrized by a potential. Such a result is not coherent with the use of “extension modules” in homological algebra and the Cauchy stress equations must not be thus confused with the divergence-type condition for the Einstein tensor obtained by contracting the Bianchi identities [6]-[12]. This result questions the origin and existence of both gravitational waves and black holes [12][13].

We briefly recall the historical framework leading to these new results.

The concept of “group” has been introduced in mathematics for the first time by E. Galois (1830) and slowly passed from algebra to geometry with the work of S. Lie on Lie groups (1880) and Liepseudogroups (1890) of transformations. The concept of a finite length differential sequence, now called Janet sequence, has been described for the first time as a footnote by M. Janet (1920). Then, the work of D. C. Spencer (1970) has been the first attempt to use the formal theory of systems of partial differential equations in order to study the formal theory of Lie pseudogroups. However, the linear and nonlinear Spencer sequences for Lie pseudogroups, though never used in physics, largely supersede the “Cartan structure equations” (1905) and are quite different from the “Vessiotstructure equations” (1903), introduced for the same purpose but still totally unknown today because they have never been acknowledged by E. Cartan or successors [14]-[16]:

Meanwhile, mixing differential geometry with homological algebra, M. Kashiwara (1970) [[6] has created “differential homological algebra”, in order to study differential modules by means of double duality and the corresponding extension modules but his work has only been accessible in 1995 and a similar work of U. Oberst in 1990 [9] is only restricted to systems with constant coefficients (See [17] for more references and Zbl 1079.93001).

By chance, unexpected arguments have been introduced by the brothers E. and F. Cosserat (1909) in order to revisit elasticity and by H. Weyl (1918) in order to revisit electromagnetism through a unique differential sequence only depending on the structure of the conformal group. However, while the Cosserat brothers were only using (translations + rotations), Weyl has only been dealing with (dilatation + elations) as we shall explain [18]-[20].

The initial 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 visiting student of D.C. Spencer in the close-by 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?

After recalling the negative answer we already provided in 1995 [7], the main purpose of this paper is to use the new techniques of differential double duality in order to revisit the mathematical foundations of general relativity and gauge theory that are leading to gravitational waves [12][13]. We point out the fact that all the formulas presented could be obtained by computer algebra while using the packages developed by my former PhD student A. Quadrat and collaborators.

The origin of this striking but difficult paper is a series of lectures given at the Albert Einstein Institute (AEI, Potsdam, October, 23-27, 2017) “General Relativity and Gauge Theory: Beyond the Mirror” (hal-01632085, 03-11-2017) and a more applied presentation under the title “From Elasticity to Electromagnetism: Beyond the Mirror” (arXiv:1802.02430) published in [3]. In this approach, we advise the reader to have a look to the photo-elastic beam experiment (photo taken by the author) showing the link that may exist between Hooke constitutive laws in elasticity (EL) and Minkowski constitutive laws in electromagnetism (EM). We also point out the way to use high level mathematical tools and their phenomenological byproducts done by J.C. Maxwell, explaining why the interference pattern is made by hyperbolas, a result highly not evident at first sight!

We finally say that this paper is a kind of wink to the English writer Lewis Carroll who first wrote his famous book “Alice in Wonderland” in 1865 but added a second part six years later under the title “Beyond the Mirror” in which Alice, sitting in front of a mirror, is falling asleep and dreams that she is passing through this mirror, discovering the strange things that happen on the other side. After spending a life-time on general relativity (GR), it has been quite a surprise to discover the close link existing between differential homological algebra and the mathematical foundations of GR, not foundduringone century (up to our knowledge) that we now point out.

Starting with the well known linear map C : S 2 T * → S 2 T * : R i j → E i j = R i j − 1 2 ω i j t r ( R ) between symmetric covariant tensors, where ω is a metric with d e t ( ω ) ≠ 0 and t r ( R ) = ω r s R r s , we may introduce the linear second order operators R i c c i : Ω → R and E i n s t e i n : Ω → E obtained by linearization over ω and we have the relation E i n s t e i n = C ∘ R i c c i where C does not depend on any conformal factor and is only invertible when n ≥ 3 because t r ( E ) = ( 2 − n ) 2 t r ( R ) [21]. We recall the method used in any textbook for studying gravitational waves, which “surprisingly “ brings the same map C : Ω → Ω ¯ = Ω − 1 2 ω   t r ( Ω ) in order to introduce the key composite operator X : Ω ¯ → Ω → E which is such that E i n s t e i n = X ∘ C .

The first goal will be to prove that only the use of differential homological algebra, a mixture of differential geometry (differential sequences, formal adjoint) and homological algebra (module theory, double duality, extension modules) totally unknown by physicists, is able to explain why the Einstein operator (with 6 terms) defined above is useless as it can be replaced by the Ricci operator (with 4 terms only) in the search for gravitational waves equations. Indeed, the Einstein operator is self-adjoint as we shall see later on when ω is the Minkowski metric, contrary to the Ricci operator [6][8][9][12][13]. Taking the respective (formal) adjoint operators (that is multiplying by convenient test functions and integrating by parts), we get:

Meanwhile, the Riemann operator can be considered as a second order operator describing the compatibility conditions (CC) for the Killing operator ξ ∈ T → ℒ ( ξ ) ω = Ω ∈ S 2 T * with standard notations where ℒ is the Lie derivative [16][22]. In this new framework, we shall prove that we no longer need to use the Bianchi operator as the first order CC for the Riemann operator. Also, we shall prove that the relative parametrization with div-type differential constraints needed in order to keep only the Dalembert operator in the wave equations has nothing to do with any gauge transformation in the corresponding adjoint differential sequence, but has only to do with the search for a minimal parametrization, exactly like Maxwell in 1870 for elasticity [13][23].

Example 2.1: When n = 2 , the stress equations become d 1 σ 11 + d 2 σ 12 = 0 , d 1 σ 21 + d 2 σ 22 = 0 . Their second order parametrization σ 11 = d 22 ϕ , σ 12 = σ 21 = − d 12 ϕ , σ 22 = d 11 ϕ has been provided by George Biddell Airy (1801-1892) in 1863. It can be simply recovered as follows:

When constructing a long prismatic dam with concrete, we may transform a problem of 3-dimensional elasticity into a problem of 2-dimensional elasticity by supposing that the axis x 3 of the dam is perpendicular to the river with Ω i j ( x 1 , x 2 ) , ∀ i , j = 1 , 2 and Ω 33 = 0 because the rocky banks of the river are supposed to be fixed and we have d 22 Ω 11 + d 11 Ω 22 − 2 d 12 Ω 12 = 0 . Introducing the two Lamé constants ( λ , μ ) in order to describe the constitutive relations of an homogeneous isotropic medium, we may restrict them from the standard case n = 3 to the case n = 2 by setting:

even though σ 33 = 1 2 λ ( Ω 11 + Ω 22 ) = 1 2 λ t r ( Ω ) . Let us consider the right square of the diagram below with locally exact rows, where any vector bundle is simply denoted by its fiber dimension:

Taking into account the linearization of the only component of the Riemann tensor over the Euclidean metric ω when n = 2 and substituting the Airy parametrization, we obtain as in [13]:

where the linearized scalar curvature t r ( 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. The Airy potential ϕ has nothing to do with the perturbation Ω of the metric ω and the Airy operator is nothing else but the adjoint of the Riemann operator, that is A i r y = a d ( R i e m a n n ) .

Example 2.2: When n = 3 , we may now use the left square of the following diagram with locally exact rows:

where the self-adjoint operator B e l t r a m i = a d ( R i e m a n n ) has been introduced by E. Beltrami in 1892. We may substitute the 3-dimensional constitutive relations with Lamé constants ( λ , μ ) in the Cauchy stress equations and get, when f → ~ g → (gravity) is now the right member:

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 f → = ρ ∂ 2 ξ → / ∂ t 2 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 ∇ → ∧ ( ∇ → ∧ ξ → ) = ∇ → ( ∇ → ⋅ ξ → ) − Δ ξ → , we discover the existence of two types of elastic waves A → exp i ( k → ⋅ x → − ω t ) with wave vector k → , period T , pulsation ω = 2 π / T along standard notations. We obtain thus the longitudinal and transversal waves with different speeds v T < v L , which are really existing because they are responsible for earthquakes:

These comments pushed the author to a systematic use of the formal adjoint of an operator as explained and illustrated in the platform “ideXlab” on the net.

Let us explain the origin of the definition of extension modules in homological algebra by means of an elementary example. With d 22 ξ = η 2 , d 12 ξ = η 1 for D , we get d 1 η 2 − d 2 η 1 = ζ for the CC D 1 . Then a d ( D 1 ) is defined by μ 2 = − d 1 λ , μ 1 = d 2 λ while a d ( D ) is defined by ν = d 12 μ 1 + d 22 μ 2 but the CC of a d ( D 1 ) are simply generated by ν ' = d 1 μ 1 + d 2 μ 2 with ν = d 1 ν ′ . Using operators, we have the three differential sequences:

where D 1 generates the CC of D in the upper sequence but a d ( D ) does not generate the CC of a d ( D 1 ) in the lower sequence, even though D 1 ∘ D = 0 ⇒ a d ( D ) ∘ a d ( D 1 ) = 0 , contrary to what happens in the Poincaré sequence for the exterior derivative used in electromagnetism for exhibiting the Maxwell equations when n = 4 . We shall see that this “gap” brings the need to introduce the first extension module e x t 1 ( M ) of the differential module M determined by D .

More generally, using the same notation for a vector bundle and its set of local sections, when E → D F is a given operator, its formal adjoint is ∧ n T * ⊗ E * ← a d ( D ) ∧ n T * ⊗ F * where E * and F * are respectively obtained from E and F by inverting the transition matrices, like T and T * . The stress is thus not a tensor but a 2-contravariant tensor density, that is σ ∈ ∧ n T * ⊗ S 2 T [12][13][17].

Before going ahead, let us prove that there may be mainly two types of differential sequences, the Janet sequence introduced by M. Janet in 1920 [24], having to do with the tools we have studied, and a different sequence called Spencer sequence introduced by D. C. Spencer in 1970 with totally different operators ([14] does not contain explicit examples while the examples in the Introduction of [15] have no relation with the core of the book). For this, if E is a vector bundle over the base X , we introduce the q -jet bundle J q ( E ) with sections ξ q : ( x ) → ( ξ k ( x ) , ξ i k ( x ) , ξ i j k ( x ) , ⋯ ) transforming like the sections j q ( ξ ) : ( x ) → ( ξ k ( x ) , ∂ i ξ k ( x ) , ∂ i j ξ k ( x ) , ⋯ ) up to order q . The Spencer operator d : J q + 1 ( E ) → T * ⊗ J q ( T ) allows to compare these sections by considering the differences ( ∂ i ξ k ( x ) − ξ i k ( x ) , ∂ i ξ j k ( x ) − ξ i j k ( x ) , ⋯ ) and so on. It can be extended to an operator:

with standard multi-index notation for exterior forms and one can easily check that d ∘ d = 0 . This is the reason for which we have used the same notation for the Spencer operator and the exterior derivative that are thus “interlaced” in the above formula. Such a notation also avoids any confusion with the ring D of differential operators that will be introduced in the next section. Its restriction to the symbol g q + 1 = R q + 1 ∩ S q + 1 T * ⊗ E ⊂ J q + 1 ( E ) is minus the Spencer map:

with δ 2 = δ ∘ δ = 0 because ξ μ + 1 i + 1 j , I k d x i ∧ d x j = 0 with lengths | ν | = q + 1 , | μ | = q . A symbol g q is said to be involutive if all the δ -sequences are exact at all the ∧ s T ∗ ⊗ g q + r and finite type if there exists r large enough such that g q + r = 0 . It is known that if g q is involutive and finite type, then necessarily g q = 0 [8][22][25]. A vector field ξ i ( x ) ∂ i may be written as ξ ∈ T but, in the module framework, we must choose ξ i ∈ K when ξ i d i ∈ D , using the formal notation d i ξ k for operators but ( ∂ i ξ k ( x ) − f i k ( x ) ) for the Spencer operator when dealing with sections of J 1 ( T ) .

When ω is a non-degenerate metric with Christoffel symbols γ and Levi-Civita isomorphism j 1 ( ω ) ≃ ( ω , γ ) while T = T ( X ) is the tangent bundle to X , we consider the second order involutive system R 2 ⊂ J 2 ( T ) defined by considering the first order Killing system ℒ ( ξ ) ω = 0 , adding its first prolongation ℒ ( ξ ) γ = 0 and using ξ 2 instead of j 2 ( ξ ) . Looking for the first order generating compatibility conditions (CC) D 1 of the corresponding second order operator D just described, we may then look for the generating CC D 2 of D 1 and so on. We may proceed similarly for the injective operator T → j 2 C 0 ( T ) = J 2 ( T ) , finding successively C 0 ( T ) → D 1 C 1 ( T ) and C 1 ( T ) → D 2 C 2 ( T ) induced by d . When n = 2 and ω is the Euclidean metric, we have a Lie group of isometries with the 3 infinitesimal generators { ∂ 1 , ∂ 2 , x 1 ∂ 2 − x 2 ∂ 1 } . If we now consider the Weyl group defined by ℒ ( ξ ) ω = 2 A ω with A = c s t and ℒ ( ξ ) γ = 0 , we have to add the only dilatation x 1 ∂ 1 + x 2 ∂ 2 and get the strict inclusions R 2 ⊂ R ˜ 2 ⊂ J 2 ( T ) . As for the conformal system R ^ 3 ⊂ J 3 ( T ) , according to [4], we have to add the two elations θ 1 = 1 2 ( ( x 1 ) 2 + ( x 2 ) 2 ) ∂ 1 + x 1 x 2 ∂ 2 and θ 2 obtained by exchanging x 1 with x 2 . Both systems have vanishing third symbol and we have the strict inclusions R 3 ⊂ R ˜ 3 ⊂ R ^ 3 ⊂ J 3 ( T ) with respective dimensions 3 < 4 < 6 .

For a later use, it is important to notice that, if we define the infinitesimal conformal transformations by ℒ ( ξ ) ω = 2 A ( x ) ω , ( ℒ ( ξ ) γ ) i j k = δ i k A i ( x ) + δ j k A i ( x ) − ω i j ω k r A r ( x ) while considering the couple ( ω , γ ) as a geometric object, we let the reader check that ∂ i A − A i = 0 ⇒ ∂ i A j − ∂ j A i = 0 a reason for using the factor 2 and explaining the link existing between the work of Weyl in [20] and the Spencer operator.

Collecting the results and exhibiting the induced kernel upper differential sequence, we get the following commutative fundamental diagram I where the upper down arrows are monomorphisms while the lower down arrows are epimorphisms Φ 0 , Φ 1 , Φ 2 [3][22][25]:

It follows that “Spencer and Janet play at see-saw”, the dimension of each Janet bundle being decreased by the same amount as the dimension of the corresponding Spencer bundle is increased. The Poincaré sequence for the exterior derivative d is ∧ 0 T * → d ∧ 1 T * → d ∧ 2 T * → 0 but it is only at the end of the paper that we shall understand the link with Maxwell equations when n = 4 .

Let A be a unitary ring, that is 1 , a , b ∈ A ⇒ a + b , a b ∈ A , 1 a = a 1 = a and even an integral domain ( a b = 0 ⇒ a = 0 or b = 0 ) with field of fractions K = Q ( A ) . However, we shall not always assume that A is commutative, that is a b may be different from b a in general for a , b ∈ A . We say that M = A M is a left module over A if x , y ∈ M ⇒ a x , x + y ∈ M , ∀ a ∈ A or a right module M B over B if the operation of B on M is ( x , b ) → x b , ∀ b ∈ B . If M is a left module over A and a right module over B with ( a x ) b = a ( x b ) , ∀ a ∈ A , ∀ b ∈ B , ∀ x ∈ M , then we shall say that M = A M B is a bimodule. Of course, A = A A A is a bimodule over itself. We define the torsion submodule t ( M ) = { x ∈ M | ∃ 0 ≠ a ∈ A , a x = 0 } ⊆ M and M is a torsion module if t ( M ) = M or a torsion-free module if t ( M ) = 0 . We denote by h o m A ( M , N ) the set of morphisms f : M → N such that f ( a x ) = a f ( x ) . We finally recall that a sequence of modules and maps is exact if the kernel of any map is equal to the image of the map preceding it. When A is commutative, h o m ( M , N ) is again an A -module for the law ( b f ) ( x ) = f ( b x ) as we have ( b f ) ( a x ) = f ( b a x ) = f ( a b x ) = a f ( b x ) = a ( b f ) ( x ) . In the non-commutative case, things are more complicate and, given A M and A N B , then h o m A ( M , N ) becomes a right module over B for the law ( f b ) ( x ) = f ( x ) b (See [2][8][17] for more details or [12][26][27] for homological algebra).

Definition 3.1: A module F is said to be free if it is isomorphic to a (finite) power of A called the rank of F over A and denoted by r k A ( F ) while the rank r k A ( M ) of a module M is the rank of a maximum free submodule F ⊂ M . It follows from this definition that M / F is a torsion module. In the sequel we shall only consider finitely presented modules, namely finitely generated modules defined by exact sequences of the type F 1 → d 1 F 0 → p M → 0 where F 0 and F 1 are free modules of finite ranks m 0 and m 1 often denoted by m and p in examples. A module P is called projective if there exists a free module F and another (projective) module Q such that P ⊕ Q ≃ F .

Proposition 3.2: For any short exact sequence 0 → M ′ → f M → g M ″ → 0 , we have the important relation r k A ( M ) = r k A ( M ′ ) + r k A ( M ″ ) , even in the non-commutative case. As a byproduct, if M admits a finite length free resolution ⋯ → d 2 F 1 → d 1 F 0 → p M → 0 , we may define the Euler-Poincarécharacteristic χ A ( M ) = ∑ r ( − 1 ) r r k A ( F r ) = r k A ( M ) .

The following classical proposition is quite useful:

Proposition 3.3: A short exact sequence splits if one of the following three conditions holds:

There exists a monomorphism v : M ″ → M called lift of g and such that g ∘ v = i d M ″ .There exists an epimorphism u : M → M ′ called lift of f and such that u ∘ f = i d M ′ .There exist isomorphisms φ = ( u , g ) : M → M ′ ⊕ M ″ and ψ = f + v : M ′ ⊕ M ″ → M that provide an isomorphism M ≃ M ′ ⊕ M ″ with f ∘ u + v ∘ g = i d M and thus k e r ( u ) = i m ( v ) .

These conditions are automatically satisfied if M ″ is free or projective.

Using the notation M * = h o m A ( M , A ) , for any morphism f : M → N , we shall denote by f * : N * → M * the morphism which is defined by f * ( h ) = h ∘ f , ∀ h ∈ h o m A ( N , A ) and satisfies r k A ( f ) = r k A ( i m ( f ) ) = r k A ( f * ) , ∀ f ∈ h o m A ( M , N ) . We may take out M in order to obtain the deleted sequence ⋯ → d 2 F 1 → d 1 F 0 → 0 and apply h o m A ( • , A ) in order to get the sequence ⋯ ← d 2 * F 1 * ← d 1 * F 0 * ← 0 [8][12][17][26][27].

Proposition 3.4: If we define the extension modules e x t A 0 ( M ) = k e r ( d 1 * ) = h o m A ( M , A ) = M * and e x t i ( M ) = e x t A i ( M ) = k e r ( d i + 1 * ) / i m ( d i * ) , ∀ i ≥ 1 , they do not depend on the resolution chosen and are torsion modules for i ≥ 1 .

We now study modules over the ring D = K [ d 1 , ⋯ , d n ] = K [ d ] of differential operators with coefficients in a differential field K with n commuting derivations ( ∂ 1 , ⋯ , ∂ n ) , also called D -modules. Any operator is of the form P = ∑ a μ d μ and its (formal) adjoint is a d ( P ) = ∑ ( − 1 ) | μ | d μ a μ but we may also use the rule a d ( a ) = a , a d ( d i ) = − d i , a d ( P Q ) = a d ( Q ) a d ( P ) with d i a = a d i + ∂ i a .

In a more general setting, if a differential operator ξ → D η is given, a direct problem is to find generating compatibility conditions (CC) as an operator η → D 1 ζ such that D ξ = η ⇒ D 1 η = 0 . Conversely, given η → D 1 ζ , the inverse problem will be to look for ξ → D η such that D 1 generates the CC of D and we shall say that D 1 is parametrized by D if such an operator D is existing. Of course the main problem will be to solve these two problems for all the physical explicit or physical operators that we shall meet.

Theorem 3.5: If M is the differential module defined by a differential operator D and N is the differential operator similarly defined by a d ( D ) , we have the crucial formula t ( M ) = e x t 1 ( N ) and D can be parametrized if and only if t ( M ) = 0 . As a d ( a d ( D ) ) = D , we have also t ( N ) = e x t 1 ( M ) .

Introducing the morphism ϵ : M → M ** such that ϵ ( m ) ( f ) = f ( m ) , ∀ m ∈ M , ∀ f ∈ M * and defining the differential modules M and N from D 1 instead of D (care), we obtain:

Corollary 3.6: (Reflexivity test) Checking whether M is reflexive or not, that is to find out a parametrization if e x t 1 ( N ) = t ( M ) = 0 which can be again parametrized amounts to prove that ϵ is an isomorphism of D -modules in the following long exact sequence:

The 5 steps are described as follows, where a d ( D ) generates the CC of a d ( D 1 ) , D ′ 1 generates the CC of D = a d ( a d ( D ) ) , a d ( D − 1 ) generates the CC of a d ( D ) and D ′ generates the CC of D − 1 :

Corollary 3.7: In the differential module framework, if F 1 → D 1 F 0 → p M → 0 is a finite free presentation of M = c o k e r ( D 1 ) with t ( M ) = 0 , then we may obtain an exact sequence F 1 → D 1 F 0 → D E of free differential modules where D is the parametrizing operator. However, there may exist other parametrizations F 1 → D 1 F 0 → D ′ E ′ called minimal parametrizations such that c o k e r ( D ′ ) is a torsion module and we have thus r k D ( M ) = r k D ( E ′ ) .

Example 3.8: When n = 3 , the div operator can be parametrized by the curl operator which can be itself parametrized by the grad operator. However, the new minimal parametrization − d 3 ξ 2 = η 1 , d 3 ξ 1 = η 2 , d 1 ξ 2 − d 2 ξ 1 = η 3 ⇒ d 1 η 1 + d 2 η 2 + d 3 η 3 = 0 cannot be again parametrized.

The differential module M defined by the first set of Maxwell equations ∧ 2 T * → d   ∧ 3 T * is reflexive because we have the adjoint differential sequence 0 ← ∧ 4 T * ← d   ∧ 3 T * ← d   ∧ 2 T * ← d   ∧ 1 T * . Similarly, the differential module defined by the second set of Maxwell equations is also reflexive because it is defined by the adjoint operator ∧ 3 T * ← d   ∧ 2 T * which is the adjoint of the parametrization ∧ 1 T * → d   ∧ 2 T * defined by A → d A = F .

Similarly, it is even less evident that the differential module defined by the Cauchy operator is reflexive as we shall see when n = 3 but only torsion-free as we saw when n = 2 .

Theorem 3.9: The Einstein operator, namely the linearization of the Einstein tensor over the locally constant Minkowski metric ω , is self-adjoint and Cauchy = ad(Killing) is parametrized by ad(Ricci).

Proof: First of all, the linearizations of the Christoffel symbols γ i j k and the Riemann tensor ρ l , i j k are:

Setting t r ( Ω ) = ω r s Ω r s , we deduce Γ r i r = 1 2 d i t r ( Ω ) and get:

Setting E i j = R i j − 1 2 ω i j t r ( R ) with t r ( R ) = ω r s R r s , we obtain the linear Einstein operator (6 terms):

It is essential to notice that the Ricci operator is not self-adjoint because we have for example:

and a d provides a term appearing in − ω i j t r ( R ) but not in 2 R i j .

After two integrations by parts, we obtain successively for ad(Ricci):

Setting t r ( λ ) = ω i j λ i j , we may change the indices in order to factor out Ω i j and finally get the adjoint of the Einstein operator:

the 6 terms being exchanged between themselves with ( 1 , 2 , 3 , 4 , 5 , 6 ) → ( 1 , 6 , 3 , 4 , 5 , 2 ) .

Example 3.10: When n = 3 and the Euclidean metric, we obtain:

We let the reader check that eight to twelve terms are disappearing each time, a reason for which nobody saw that the Einstein equations had been written exactly (up to sign) by E. Beltrami in 1892 in order to parametrize the Cauchy stress equations while using 6 stress functions ϕ i j = ϕ j i in place of Ω i j = Ω j i , 25 years before Einstein [ 3] [12][13]. The comparison needs no comment!

Remark 3.11: When n = 3 , we have already noticed in the first page of [2] that one has to take into account the factors “2” in the duality summation with Lagrange multipliers ( λ i j = λ j i ) which is used in order to exhibit the operator a d ( E i n s t e i n ) , namely (Compare to [11]):

In the present situation, this is sufficient in order to obtain a self-adjoint operator as follows: