Minimum Parametrization of the Cauchy Stress Operator
aa r X i v : . [ m a t h . G M ] J a n MINIMUM PARAMETRIZATION OFTHE CAUCHY STRESS OPERATOR
J.-F. POMMARETCERMICS, Ecole des Ponts [email protected]: 0000-0003-0907-2601
ABSTRACT
When D : ξ → η is a linear differential operator, a ”direct problem ” is to find the generatingcompatibility conditions (CC) in the form of an operator D : η → ζ such that D ξ = η implies D η = 0. When D is involutive, the procedure provides successive first order involutive operators D , ..., D n when the ground manifold has dimension n . Conversely, when D is given, a moredifficult ” inverse problem ” is to look for an operator D : ξ → η having the generating CC D η = 0. If this is possible, that is when the differential module defined by D is torsion-free, oneshall say that the operator D is parametrized by D and there is no relation in general between D and D . The parametrization is said to be ” minimum ” if the differential module defined by D has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchystress operator in arbitrary dimension n has attracted many famous scientists (G.B. Airy in 1863for n = 2, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for n = 3, A. Einstein in 1915for n = 4) . The aim of this paper is to prove that all these works are explicitly using the Einsteinoperator (which cannot be parametrized) and not the Ricci operator. As a byproduct, they areall based on a confusion between the so-called div operator induced from the Bianchi operator D and the Cauchy operator which is the formal adjoint of the Killing operator D parametrizing theRiemann operator D for an arbitrary n . We also present the similar situation met in the study ofcontact structures when n = 3. Like the Michelson and Morley experiment, it is an open historicalproblem to know whether Einstein was aware of these previous works or not, as the comparisonneeds no comment. KEY WORDS
Differential operator; Differential sequence; Killing operator; Riemann operator; Bianchi operator;Cauchy operator; Electromagnetism; Elasticity; General relativity; Gravitational waves.1 ) INTRODUCTION
We start recalling the basic tools from the formal theory of systems of partial differential (PD)equations and differential modules needed in order to understand and solve the parametrizationproblem presented in the abstract. Then we provide the example of the system of infinitesimal Lieequations defining contact transformations and conclude the paper with the general parametriza-tion problem existing in continuum mechanics for an arbitrary dimension of the ground manifold.As these new tools are difficult and not so well known, we advise the interested reader to followthem step by step on the explicit motivating examples illustrating this paper.
A) SYSTEM THEORY :If X is a manifold of dimension n with local coordinates ( x ) = ( x , ..., x n ), we denote as usualby T = T ( X ) the tangent bundle of X , by T ∗ = T ∗ ( X ) the cotangent bundle , by ∧ r T ∗ the bundleof r-forms and by S q T ∗ the bundle of q-symmetric tensors . More generally, let E be a vectorbundle over X with local coordinates ( x i , y k ) for i = 1 , ..., n and k = 1 , ..., m simply denoted by( x, y ), projection π : E → X : ( x, y ) → ( x ) and changes of local coordinate ¯ x = ϕ ( x ) , ¯ y = A ( x ) y .We shall denote by E ∗ the vector bundle obtained by inverting the matrix A of the changes ofcoordinates , exactly like T ∗ is obtained from T . We denote by f : X → E : ( x ) → ( x, y = f ( x ))a global section of E , that is a map such that π ◦ f = id X but local sections over an open set U ⊂ X may also be considered when needed. Under a change of coordinates, a section transformslike ¯ f ( ϕ ( x )) = A ( x ) f ( x ) and the changes of the derivatives can also be obtained with more work.We shall denote by J q ( E ) the q-jet bundle of E with local coordinates ( x i , y k , y ki , y kij , ... ) = ( x, y q )called jet coordinates and sections f q : ( x ) → ( x, f k ( x ) , f ki ( x ) , f kij ( x ) , ... ) = ( x, f q ( x )) transforminglike the sections j q ( f ) : ( x ) → ( x, f k ( x ) , ∂ i f k ( x ) , ∂ ij f k ( x ) , ... ) = ( x, j q ( f )( x )) where both f q and j q ( f ) are over the section f of E . For any q ≥ J q ( E ) is a vector bundle over X with projection π q while J q + r ( E ) is a vector bundle over J q ( E ) with projection π q + rq , ∀ r ≥ DEFINITION 1.A.1 : A linear system of order q on E is a vector sub-bundle R q ⊂ J q ( E ) anda solution of R q is a section f of E such that j q ( f ) is a section of R q . With a slight abuse oflanguage, the set of local solutions will be denoted by Θ ⊂ E .Let µ = ( µ , ..., µ n ) be a multi-index with length | µ | = µ + ... + µ n , class i if µ = ... = µ i − =0 , µ i = 0 and µ + 1 i = ( µ , ..., µ i − , µ i + 1 , µ i +1 , ..., µ n ). We set y q = { y kµ | ≤ k ≤ m, ≤ | µ | ≤ q } with y kµ = y k when | µ | = 0. If E is a vector bundle over X and J q ( E ) is the q - jet bundle of E , thenboth sections f q ∈ J q ( E ) and j q ( f ) ∈ J q ( E ) are over the section f ∈ E . There is a natural way todistinguish them by introducing the Spencer operator d : J q +1 ( E ) → T ∗ ⊗ J q ( E ) with components( df q +1 ) kµ,i ( x ) = ∂ i f kµ ( x ) − f kµ +1 i ( x ). The kernel of d consists of sections such that f q +1 = j ( f q ) = j ( f q − ) = ... = j q +1 ( f ). Finally, if R q ⊂ J q ( E ) is a system of order q on E locally defined bylinear equations Φ τ ( x, y q ) ≡ a τµk ( x ) y kµ = 0 and local coordinates ( x, z ) for the parametric jets up toorder q , the r - prolongation R q + r = ρ r ( R q ) = J r ( R q ) ∩ J q + r ( E ) ⊂ J r ( J q ( E )) is locally defined when r = 1 by the linear equations Φ τ ( x, y q ) = 0 , d i Φ τ ( x, y q +1 ) ≡ a τµk ( x ) y kµ +1 i + ∂ i a τµk ( x ) y kµ = 0 and has symbol g q + r = R q + r ∩ S q + r T ∗ ⊗ E ⊂ J q + r ( E ) if one looks at the top order terms . If f q +1 ∈ R q +1 isover f q ∈ R q , differentiating the identity a τµk ( x ) f kµ ( x ) ≡ x i and substracting theidentity a τµk ( x ) f kµ +1 i ( x )+ ∂ i a τµk ( x ) f kµ ( x ) ≡
0, we obtain the identity a τµk ( x )( ∂ i f kµ ( x ) − f kµ +1 i ( x )) ≡ d : R q +1 → T ∗ ⊗ R q . More generally, we have the restriction: d : ∧ s T ∗ ⊗ R q +1 → ∧ s +1 T ∗ ⊗ R q : ( f kµ,i ( x ) dx I ) → (( ∂ i f kµ,i ( x ) − f kµ +1 i ,I ( x )) dx i ∧ dx I )with standard multi-index notation for exterior forms and one can easily check that d ◦ d = 0.The restriction of − d to the symbol is called the Spencer map δ : ∧ s T ∗ ⊗ g q +1 → ∧ s +1 T ∗ ⊗ g q and δ ◦ δ = 0 similarly ([22-25],[28],[41],[49]). DEFINITION 1.A.2 : A system R q is said to be formally integrable when all the equations oforder q + r are obtained by r prolongations only , ∀ r ≥ π q + r + sq + r : R q + r + s → Rq + r are epimorphisms ∀ r, s ≥ Pommaretbasis and where one may have to change linearly the independent variables if necessary , is intrin-sic even though it must be checked in a particular coordinate system called δ - regular ([22],[25],[29]). • Equations of class n : Solve the maximum number β nq of equations with respect to the jets oforder q and class n . Then call ( x , ..., x n ) multiplicative variables . • Equations of class i ≥
1: Solve the maximum number β iq of remaining equations with respectto the jets of order q and class i . Then call ( x , ..., x i ) multiplicative variables and ( x i +1 , ..., x n ) non-multiplicative variables . • Remaining equations equations of order ≤ q −
1: Call ( x , ..., x n ) non-multiplicative variables .In actual practice, we shall use a Janet tabular where the multiplicative ”variables” are in upperleft position while the non-multiplicative variables are represented by dots in lower right position.
DEFINITION 1.A 3 : A system of PD equations is said to be involutive if its first prolongationcan be obtained by prolonging its equations only with respect to the corresponding multiplicativevariables. In that case, we may introduce the characters α iq = m ( q + n − i − q − n − i )! − β iq for i = 1 , ..., n with α q ≥ ... ≥ α nq ≥ dim ( g q ) = α q + ... + α nq while dim ( g q +1 ) = α q + ... + nα nq . REMARK 1.A.4 : As long as the
Prolongation / Projection (PP) procedure has not been achievedin order to get an involutive system, nothing can be said about the CC (Fine examples can befound in [41] and the recent [45]).
REMARK 1.A.5 : A proof that the second order system defined by Einstein equations is in-volutive has been given by J. Gasqui in 1982 but this paper cannot be applied to the minimumparametrizations that need specific δ -regular coordinates as we shall see ([8]).When R q is involutive, the linear differential operator D : E j q → J q ( E ) Φ → J q ( E ) /R q = F oforder q is said to be involutive and its space of solutions is defined by the kernel exact sequence0 → Θ → E −→ F . One has the canonical linear Janet sequence (Introduced in [19]):0 −→ Θ −→ E D −→ F D −→ F D −→ ... D n −→ F n −→ compatibility conditions (CC)of the preceding one. Similarly, introducing the Spencer bundles C r = ∧ r T ∗ ⊗ R q /δ ( ∧ r − T ∗ ⊗ g q +1 )we obtain the canonical linear Spencer sequence induced by the Spencer operator:0 −→ Θ j q −→ C D −→ C D −→ ... D n −→ C n −→ B) MODULE THEORY :Let K be a differential field with n commuting derivations ( ∂ , ..., ∂ n ) and consider the ring D = K [ d , ..., d n ] = K [ d ] of differential operators with coefficients in K with n commuting for-mal derivatives satisfying d i a = ad i + ∂ i a in the operator sense. If P = a µ d µ ∈ D = K [ d ],the highest value of | µ | with a µ = 0 is called the order of the operator P and the ring D withmultiplication ( P, Q ) −→ P ◦ Q = P Q is filtred by the order q of the operators. We have the filtration ⊂ K = D ⊂ D ⊂ ... ⊂ D q ⊂ ... ⊂ D ∞ = D . As an algebra, D is gener-ated by K = D and T = D /D with D = K ⊕ T if we identify an element ξ = ξ i d i ∈ T with the vector field ξ = ξ i ( x ) ∂ i of differential geometry, but with ξ i ∈ K now. It follows that D = D D D is a bimodule over itself, being at the same time a left D -module by the composition P −→ QP and a right D -module by the composition P −→ P Q . We define the adjoint functor ad : D −→ D op : P = a µ d µ −→ ad ( P ) = ( − | µ | d µ a µ and we have ad ( ad ( P )) = P both with ad ( P Q ) = ad ( Q ) ad ( P ) , ∀ P, Q ∈ D . Such a definition can be extended to any matrix of operators3y using the transposed matrix of adjoint operators (See [4],[12],[28],[29],[34],[38],[48] for more de-tails and applications to control theory or mathematical physics).Accordingly, if y = ( y , ..., y m ) are differential indeterminates, then D acts on y k by setting d i y k = y ki −→ d µ y k = y kµ with d i y kµ = y kµ +1 i and y k = y k . We may therefore use the jet coor-dinates in a formal way as in the previous section. Therefore, if a system of OD/PD equationsis written in the form Φ τ ≡ a τµk y kµ = 0 with coefficients a ∈ K , we may introduce the free dif-ferential module Dy = Dy + ... + Dy m ≃ D m and consider the differential module of equations I = D Φ ⊂ Dy , both with the residual differential module M = Dy/D
Φ or D -module and wemay set M = D M if we want to specify the ring of differential operators. We may introducethe formal prolongation with respect to d i by setting d i Φ τ ≡ a τµk y kµ +1 i + ( ∂ i a τµk ) y kµ in order toinduce maps d i : M −→ M : ¯ y kµ −→ ¯ y kµ +1 i by residue with respect to I if we use to denotethe residue Dy −→ M : y k −→ ¯ y k by a bar like in algebraic geometry. However, for simplicity,we shall not write down the bar when the background will indicate clearly if we are in Dy orin M . As a byproduct, the differential modules we shall consider will always be finitely gener-ated ( k = 1 , ..., m < ∞ ) and finitely presented ( τ = 1 , ..., p < ∞ ). Equivalently, introducing the matrix of operators D = ( a τµk d µ ) with m columns and p rows, we may introduce the morphism D p D −→ D m : ( P τ ) −→ ( P τ Φ τ ) over D by acting with D on the left of these row vectors whileacting with D on the right of these row vectors by composition of operators with im ( D ) = I . The presentation of M is defined by the exact cokernel sequence D p D −→ D m −→ M −→
0. We noticethat the presentation only depends on
K, D and Φ or D , that is to say never refers to the conceptof (explicit local or formal) solutions. It follows from its definition that M can be endowed with a quotient filtration obtained from that of D m which is defined by the order of the jet coordinates y q in D q y . We have therefore the inductive limit ⊆ M ⊆ M ⊆ ... ⊆ M q ⊆ ... ⊆ M ∞ = M with d i M q ⊆ M q +1 and M = DM q for q ≫ D r M q ⊆ M q + r , ∀ q, r ≥ DEFINITION 1.B.1 : An exact sequence of morphisms finishing at M is said to be a resolution of M . If the differential modules involved apart from M are free, that is isomorphic to a certainpower of D , we shall say that we have a free resolution of M .Having in mind that K is a left D -module with the action ( D, K ) −→ K : ( d i , a ) −→ ∂ i a andthat D is a bimodule over itself, we have only two possible constructions : DEFINITION 1.B.2 : We may define the right differential module hom D ( M, D ). DEFINITION 1.B.3 : We define the system R = hom K ( M, K ) and set R q = hom K ( M q , K ) asthe system of order q . We have the projective limit R = R ∞ −→ ... −→ R q −→ ... −→ R −→ R .It follows that f q ∈ R q : y kµ −→ f kµ ∈ K with a τµk f kµ = 0 defines a section at order q and we may set f ∞ = f ∈ R for a section of R . For an arbitrary differential field K , such a definition has nothingto do with the concept of a formal power series solution ( care ). PROPOSITION 1.B.4 : When M is a left D -module, then R is also a left D -module. Proof : As D is generated by K and T as we already said, let us define:( af )( m ) = af ( m ) , ∀ a ∈ K, ∀ m ∈ M ( ξf )( m ) = ξf ( m ) − f ( ξm ) , ∀ ξ = a i d i ∈ T, ∀ m ∈ M In the operator sense, it is easy to check that d i a = ad i + ∂ i a and that ξη − ηξ = [ ξ, η ] is thestandard bracket of vector fields. We finally get ( d i f ) kµ = ( d i f )( y kµ ) = ∂ i f kµ − f kµ +1 i and thusrecover exactly the Spencer operator of the previous section though this is not evident at all . Wealso get ( d i d j f ) kµ = ∂ ij f kµ − ∂ i f kµ +1 j − ∂ j f kµ +1 i + f kµ +1 i +1 j = ⇒ d i d j = d j d i , ∀ i, j = 1 , ..., n and thus d i R q +1 ⊆ R q = ⇒ d i R ⊂ R induces a well defined operator R −→ T ∗ ⊗ R : f −→ dx i ⊗ d i f . Thisoperator has been first introduced, up to sign, by F.S. Macaulay as early as in 1916 but this is stillnot ackowledged ([17]). For more details on the Spencer operator and its applications, the readermay look at ([31],[37-39],[42]). 4.E.D. DEFINITION 1.B.5 : With any differential module M we shall associate the graded module G = gr ( M ) over the polynomial ring gr ( D ) ≃ K [ χ ] by setting G = ⊕ ∞ q =0 G q with G q = M q /M q +1 and we get g q = G ∗ q where the symbol g q is defined by the short exact sequences:0 −→ M q − −→ M q −→ G q −→ ⇒ −→ g q −→ R q −→ R q − −→ −→ D q − −→ D q −→ S q T −→ gr q ( D ) ≃ S q T and we may set as usual T ∗ = hom K ( T, K ) in a coherent way with differential geometry.The two following definitions, which are well known in commutative algebra, are also valid(with more work) in the case of differential modules (See [28] for more details or the references[9],[21],[29],[47] for an introduction to homological algebra and diagram chasing).
DEFINITION 1.B.6 : The set of elements t ( M ) = { m ∈ M | ∃ = P ∈ D, P m = 0 } ⊆ M isa differential module called the torsion submodule of M . More generally, a module M is called a torsion module if t ( M ) = M and a torsion-free module if t ( M ) = 0. In the short exact sequence0 → t ( M ) → M → M ′ →
0, the module M ′ is torsion-free. Its defining module of equations I ′ isobtained by adding to I a representative basis of t ( M ) set up to zero and we have thus I ⊆ I ′ . DEFINITION 1.B.7 : A differential module F is said to be free if F ≃ D r for some integer r > define rk D ( F ) = r . If F is the biggest free dfferential module contained in M , then M/F is a torsion differential module and hom D ( M/F, D ) = 0. In that case, we shall define the differential rank of M to be rk D ( M ) = rk D ( F ) = r . PROPOSITION 1.B.8 : If 0 → M ′ → M → M ” → rk D ( M ) = rk D ( M ′ ) + rk D ( M ”).In the general situation, let us consider the sequence M ′ f −→ M g −→ M ” of modules which maynot be exact and define B = im ( f ) ⊆ Z = ker ( g ) ⇒ H = Z/B . LEMMA 1.B.9 : The kernel of the induced epimorphism coker ( f ) → coim ( g ) is isomorphic to H . Proof : It follows from a snake chase in the commutative and exact diagram where coim ( g ) ≃ im ( g ):0 ↓ H ↓ ↓ ↓ → B → M −→ coker ( f ) → ↓ k ↓ → Z → M g −→ coim ( g ) → ↓ ↓ ↓ H ↓ inversion ” of arrows. Indeed,when an operator is injective, that is when we have the exact sequence 0 → E D −→ F with dim ( E ) = m, dim ( F ) = p , like in the case of the operator 0 → E j q −→ J q ( E ), on the contrary, usingdifferenial modules, we have the epimorphism D p D −→ D m →
0. The case of a formally surjectiveoperator, like the div operator, described by the exact sequence E D −→ F → → D p D −→ D m → M → D has no CC.
2) PARAMETRIZATION PROBLEM
In this section, we shall set up and solve the minimum parametrization problem by comparingthe differential geometric approach and the differential algebraic approach. In fact, both sides areessential because certain concepts, like ” torsion ”, are simpler in the module approach while oth-ers, like ” involution ” are simpler in the opertor approach. However, the reader must never forgetthat the ” extension modules ” or the ” side changing functor ” are pure product of differentialhomological algebra with no system counterpart. Also, the close link existing between ” differentialduality ” and ” adjoint operator ” may not be evident at all, even for people quite familiar withmathematical physics ([4],[28],[38]).Let us start with a given linear differential operator η D −→ ζ between the sections of two givenvector bundles F and F of respective fiber dimension m and p . Multiplying the equations D η = ζ by p test functions λ considered as a section of the adjoint vector bundle ad ( F ) = ∧ n T ∗ ⊗ F ∗ andintegrating by parts, we may introduce the adjoint vector bundle ad ( F ) = ∧ n T ∗ ⊗ F ∗ with sections µ in order to obtain the adjoint operator µ ad ( D ) ←− λ , writing on purpose the arrow backwards, thatis from right to left. As any operator is the adjoint of another operator because ad ( ad ( D )) = D ,we may decide to denote by ξ ad ( D ) ←− µ the generating CC of ad ( D ) by introducing a vector bundle E with sections ξ and its adjoint ad ( E ) = ∧ n T ∗ ⊗ E ∗ with sections ν . We have thus obtained theformally exact differential sequence: ν ad ( D ) ←− µ ad ( D ) ←− λ and its formaly exac adjoint: ξ D −→ η D −→ ζ providing a parametrization if and only if D generates the CC of D , a situation that may notbe satisfied but that we shall assume from now on because otherwise D cannot be parametrizedaccording to the double differential duality test, for example in the case of the Einstein equations([32],[50]) or the extension to the conformal group and other Lie groups of transformations ([33],[37-39],[42-44]). Nevertheless, for the interested reader only, we provide the following key result onwhich this procedure is based (See [12],[28],[29] and [34] for more details): THEOREM 2.1 : If M is a differential module, we have the exact sequence of differential modules:0 → t ( M ) → M ǫ −→ hom D ( hom D ( M, D ) , D )where the map ǫ is defined by ǫ ( m )( f ) = f ( m ) , ∀ m ∈ M, f ∈ hom D ( M, D ). Moreover, if N is thedifferential module defined by ad ( D ), then t ( M ) = ext D ( N, D ).In order to pass to the differential module framework, let us introduce the free differentialmodules Dξ ≃ D l , Dη ≃ D m , Dζ ≃ D p . We have similarly the adjoint free differential modules Dν ≃ D l , Dµ ≃ D m , Dλ ≃ D p , because dim ( ad ( E )) = dim ( E ) and hom D ( D m , D ) ≃ D m . Ofcourse, in actual practice, the geometric meaning is totally different because we have volume formsin the dual framework. We have thus obtained the formally exact sequence of differential modules: D p D −→ D m D −→ D l and the formally exact adjoint sequence: D p ad ( D ) ←− D m ad ( D ) ←− D l The procedure with 4 steps is as follows in the operator language: • STEP 1: Start with the formally exact parametrizing sequence already constructed by differentialbiduality. We have thus im ( D ) = ker ( D ) and the corresponding differential module M defined6y D is torsion-free by assumption. • STEP 2: Construct the adjoint sequence which is also formally exact by assumption. • STEP 3: Find a maximum set of differentially independent CC ad ( D ′ ) : µ → ν ′ among thegenerating CC ad ( D ) : µ → ν of ad ( D ) in such a way that im ( ad ( D ′ )) is a maximum free dif-ferential submodule of im ( ad ( D )) that is any element in im ( ad ( D )) is differentially algebraic over im ( ad ( D ′ )). • STEP 4: Using differential duality, construct D ′ = ad ( ad ( D ′ )).It remains to prove that D generates the CC of D ′ in the following diagram:4 ξ ′ ↑ D ′ ց ξ D −→ η D −→ ζ ν ad ( D ) ←− µ ad ( D ) ←− λ ↑ ad ( D ′ ) ւ ν ′ ւ ↑ PROPOSITION 2.2 : D ′ is a minimum parametrization of D . Proof : Let us denote the number of potentials ξ by l (respectively ξ ′ by l ′ ), the number of unknowns η by m and the number of given equations ζ by p . As ad ( D ′ ) has no CC by construction, then ad ( D ′ ) : µ → ν ′ is a formally surjective operator. On the differential module level, we have theinjective operator ad ( D ′ ) : D l ′ → D m because there are no CC. Applying hom D ( • , D ) or duality,we get an operator D m → D l ′ with a cokernel which is a torsion module because it has rank l ′ − rk D ( D ′ ) = l ′ − rk D ( ad ( D ′ )) = l ′ − l ′ = 0.However, in actual practice as will be seen in the contact case, things are not so simple andwe shall use the following commutative and exact diagram of differential modules based on a long ker/coker long exact sequence (Compare to [35], and [46]):0 → ker ( ad ( D )) → D l ad ( D ) −→ D m → coker ( ad ( D )) → ց ր L ր ↑ ց D l ′ ↑ L = D l /ker ( ad ( D )) and introducing the biggest free differential module D l ′ ⊆ L we have rk D ( D l ′ ) = rk D ( L ) ≤ rk D ( D l ) ⇒ l ′ ≤ l , we may define the injective ( care ) operator ad ( D ′ ) by thecomposition of monomorphisms D l ′ → L → D m where the second is obtained by picking a basisof D l ′ , lifting it to D l and pushing it to D m by applying ad ( D ). We notice that L can be viewedas the differential module defined by the generating CC of ad ( D ) that could also be used as in ([35]).Then we have ad ( D ′ ) ◦ ad ( D ) = ad ( D ◦ D ′ ) = 0 ⇒ D ◦ D ′ = 0 and thus D is surely among the CC of D ′ . Therefore, the differential sequence ξ ′ D ′ −→ η D −→ ζ on the operatorlevel or the sequence D p D −→ D m D ′ −→ D l ′ on the differential module level may not be exactand we can thus apply the previous Lemma. Changing slightly the notations, we have now7 = im ( D ) = ker ( D ) ⊆ ker ( D ′ ) = Z . But we have also rk D ( B ) = m − rk D ( D ) , rk ( Z ) = m − rk D ( D ′ ) ⇒ rk D ( H ) = rk D ( D ) − rk D ( D ′ ) = 0 by construction.Taking into account the previous Lemma, we may set coim ( D ) = M ⊆ D l by assumption andconsider im ( D ′ ) = M ′ ⊆ D l ′ in order to obtain the short exact sequence of differential modules0 → H → M → M ′ →
0. As H is a torsion module and the differential module M defined by D is torsion-free by assumption, the only possibility is that H = 0 and thus im ( D ) = ker ( D ′ ),that is D ′ is a minimum parametrization of D with l ′ ≤ l potentials. Q.E.D. EXAMPLE 2.3 : Contact transformations
With m = n = 3 , K = Q ( x , x , x ) = Q ( x ), we may introduce the so-called contact α = dx − x dx . The system of infinitesimal Lie equations defining the infinitesimal contacttransformations is obtained by eliminating the factor ρ ( x ) in the equations L ( ξ ) α = ρα where L is the standard Lie derivative. This system is thus only generated by η and η below but is notinvolutive and one has to introduce η defined by the first order CC: ζ ≡ ∂ η − ∂ η − x ∂ η + η = 0in order to obtain the following involutive system with two equations of class 3 and one equationof class 2, a result leading to β = 2 , β = 1 , β = 0: η ≡ ∂ ξ + ∂ ξ + 2 x ∂ ξ − ∂ ξ = 0 η ≡ ∂ ξ − x ∂ ξ = 0 η ≡ ∂ ξ − x ∂ ξ + x ∂ ξ − ( x ) ∂ ξ − ξ = 0 1 2 31 2 31 2 • The characters are thus α = 3 − < α = 3 − , α = 3 − dim ( g ) = 3 × −
3. In this situation, if M is the differential moduledefined by this system or the corresponding operator D , we know that rk D ( M ) = α = 1 =3 − rk D ( Dξ ) − rk D ( D ). Of course, a differential trancendence basis for D can be the operator D ′ : ξ → { η , η } but, in view of the CC, we may equally choose any couple among { η , η , η } andwe obtain rk D ( D ′ ) = rk D ( D ) = 2 in any case, but now D ′ is formally surjective, contrary to D .The same result can also be obtained directly from the unique CC or the corresponding operator D defining the differential module M . Finally, we have rk D ( M ) = 3 − rk D ( Dη ) − rk D ( D )and we check that we have indeed rk D ( M ) + rk D ( M ) = 1 + 2 = 3 = rk D ( Dξ ).It is well known that such a system can be parametrized by the injective parametrization (See[23] and [24] for more details and the study of the general dimension n = 2 p + 1): − x ∂ φ + φ = ξ , − ∂ φ = ξ , ∂ φ + x ∂ φ = ξ ⇒ ξ − x ξ = φ It is however not so well known and quite striking that such a parametrization can be recoveredidependently by using the parametrization of the differential module defined by η = 0 with po-tentials ξ and ξ while setting:( ξ , ξ ) −→ ξ = ∂ ξ − x ∂ ξ + x ∂ ξ − ( x ) ∂ ξ Taking into account the differential constraint η ≡ ∂ ξ − x ∂ ξ = 0, that is ξ = − ∂ ( ξ − x ξ )and substituting in η = 0, we get no additional constraint. We finally only need to modify thepotentials while ” defining ” now φ = ξ − x ξ = ¯ ξ as before.The associated differential sequence is:0 → φ D − −→ ξ D −→ η D −→ ζ → → −→ −→ −→ → − − not a Janet sequence because D − isnot involutive, its completion to involution being the trivially involutive operator j : φ → j ( φ ).Introducing the ring D = K [ d , d , d ] = K [ d ] of linear differential operators with coefficients in8he differential field K , the corresponding differential module M ≃ D is projective and even free,thus torsion-free or 0-pure, being defined by the split exact sequence of free differential modules:0 → D D −→ D D −→ D D − −→ D → ← θ ad ( D − ) ←− ν ad ( D ) ←− µ ad ( D ) ←− λ ← ← ←− ←− ←− ← λ is also a split exact sequence of free differential modules.We finaly prove that the situation met for the contact structure is exactly the same as the onethat we shall meet in the metric structure, namely that one can identify D − not with D of coursebut with ad ( D ). For this, let us modify the ”basis” linearly by setting ( ¯ ξ = ξ − x ξ , ¯ ξ = ξ , ¯ ξ = ξ ) and suppressing the bar for simplicity, we obtain the new injective parametrization: φ = ξ , − ∂ φ = ξ , ∂ φ + x ∂ φ = ξ and may eliminate φ in order to consider the new involutive system, renumbering the equationsthrough a cyclic permutation of (1 , , η ≡ ∂ ξ + ∂ ξ + x ∂ ξ − ∂ ξ = 0 η ≡ ∂ ξ + ξ = 0 η ≡ ∂ ξ + x ∂ ξ − ξ = 0 1 2 31 2 31 2 • with the unique first order CC defining D : ζ ≡ ∂ η − ∂ η − x ∂ η + η = 0Multiplying by λ and integrating by parts, we obtain for ad ( D ): η → λ = µ , η → − ∂ λ = µ , η → ∂ λ + x ∂ λ = µ obtaining therefore D − = ad ( D ) ⇔ D = ad ( D − ) exactly .As for D ξ = η , we obtain the formal operator matrix: − d d + x d d d + x d − d Similarly, for ad ( D ) we obtain the formal operator matrix: d − ( d + x d ) − d − ( d + x d ) 0 1 − d − and finally discover that ad ( D ) = −D , a striking result showing that both operators have the sameCC and parametrization even though D is not self-adjoint.
3) EINSTEIN EQUATIONS
Linearizing the
Ricci tensor ρ ij over the Minkowski metric ω , we obtain the usual second orderhomogeneous Ricci operator Ω → R with 4 terms:2 R ij = ω rs ( d rs Ω ij + d ij Ω rs − d ri Ω sj − d sj Ω ri ) = 2 R ji tr ( R ) = ω ij R ij = ω ij d ij tr (Ω) − ω ru ω sv d rs Ω uv
9e may define the
Einstein operator by setting E ij = R ij − ω ij tr ( R ) and obtain the 6 terms ([7]):2 E ij = ω rs ( d rs Ω ij + d ij Ω rs − d ri Ω sj − d sj Ω ri ) − ω ij ( ω rs ω uv d rs Ω uv − ω ru ω sv d rs Ω uv )We have the (locally exact) differential sequence of operators acting on sections of vector bundleswhere the order of an operator is written under its arrow.: T Killing −→ S T ∗ Riemann −→ F Bianchi −→ F n D −→ n ( n + 1) / D −→ n ( n − / D −→ n ( n − n − / S T ∗ Einstein −→ S T ∗ div −→ T ∗ → n ( n + 1) / −→ n ( n + 1) / −→ n → Cauchy = ad ( Killing ) , Beltrami = ad ( Riemann ) , Lanczos = ad ( Bianchi ) ad ( T ) Cauchy ←− ad ( S T ∗ ) Beltrami ←− ad ( F ) Lanczos ←− ad ( F )In this sequence, if E is a vector bundle over the ground manifold X with dimension n , we mayintroduce the new vector bundle ad ( E ) = ∧ n T ∗ ⊗ E ∗ where E ∗ is obtained from E by invertingthe transition rules exactly like T ∗ is obtained from T . We have for eample ad ( T ) = ∧ n T ∗ ⊗ T ∗ ≃∧ n T ∗ ⊗ T ≃ ∧ n − T ∗ because T ∗ is isomorphic to T by using the metric ω . The 10 × Einstein operator matrix is induced from the 10 × Riemann operator matrix and the 10 × div operatormatrix is induced from the 20 × Bianchi operator matrix. We advise the reader not familar withthe formal theory of systems or operators to follow the computation in dimension n = 2 with the1 × Airy operator matrix, which is the formal adjoint of the 3 × Riemann operator matrix, and n = 3 with the 6 × Beltrami operator matrix which is the formal adjoint of the 6 × Riemann operator matrix which is easily seen to be self-adjoint up to a change of basis.With more details, we have: • n = 2: The stress equations become d σ + d σ = 0 , d σ + d σ = 0. Their second orderparametrization σ = d φ, σ = σ = − d φ, σ = d φ has been provided by George BiddellAiry in 1863 ([2]) and is well known ([28]). We get the second order system: σ ≡ d φ = 0 − σ ≡ d φ = 0 σ ≡ d φ = 0 1 21 • • which is involutive with one equation of class 2, 2 equations of class 1 and it is easy to check thatthe 2 corresponding first order CC are just the Cauchy equations. Of course, the
Airy function(1 term) has absolutely nothing to do with the perturbation of the metric (3 terms). With moredetails, when ω is the Euclidean metric, we may consider the only component: tr ( R ) = ( d + d )(Ω + Ω ) − ( d Ω + 2 d Ω + d Ω )= d Ω + d Ω − d Ω Multiplying by the Airy function φ and integrating by parts, we discover that: Airy = ad ( Riemann ) ⇔ Riemann = ad ( Airy )in the following differential sequences:2
Killing −→ Riemann −→ −→ ←− Cauchy ←− Airy ←− • n = 3: It is more delicate to parametrize the 3 PD equations: d σ + d σ + d σ = 0 , d σ + d σ + d σ = 0 , d σ + d σ + d σ = 0A direct computational approach has been provided by Eugenio Beltrami in 1892 ([3],[14]), JamesClerk Maxwell in 1870 ([19]) and Giacinto Morera in 1892 ([14],[20]) by introducing the 6 stressfunctions φ ij = φ ji in the Beltrami parametrization . The corresponding system: σ ≡ d φ + d φ − d φ = 0 − σ ≡ d φ + d φ − d φ − d φ = 0 σ ≡ d φ + d φ − d φ = 0 σ ≡ d φ + d φ − d φ − d φ = 0 − σ ≡ d φ + d φ − d φ − d φ = 0 σ ≡ d φ + d φ − d φ = 0 1 2 31 2 31 2 31 2 • • • is involutive with 3 equations of class 3, 3 equations of class 2 and no equation of class 1. Thethree characters are thus α = 1 × − < α = 2 × − < α = 3 × − dim ( g ) = α + α + α = 18 + 9 + 3 = 30 = dim ( S T ∗ ⊗ S T ∗ ) − dim ( S T ∗ ) == 6 × − nothing else but the formal adjoint of the Riemann operator , namelythe (linearized) Riemann tensor with n ( n − / n = 3 ([35]).Breaking the canonical form of the six equations which is associated with the Janet tabular, we mayrewrite the Beltrami parametrization of the Cauchy stress equations as follows, after exchangingthe third row with the fourth row, keeping the ordering { (11) < (12) < (13) < (22) < (23) < (33) } : d d d d d d
00 0 d d d d − d d − d d d − d d − d − d d d − d d − d d d − d d − d d ≡ ω , the standard implicit summation used in continuum mechanics is, when n = 3: σ ij Ω ij = σ Ω + 2 σ Ω + 2 σ Ω + σ Ω + 2 σ Ω + σ Ω = Ω d φ + Ω d φ − d φ + ... +Ω d φ + Ω d φ − Ω d φ − Ω d φ + ... because the stress tensor density σ is supposed to be symmetric . Integrating by parts in order toconstruct the adjoint operator, we get: φ −→ d Ω + d Ω − d Ω φ −→ d Ω + d Ω − d Ω − d Ω and so on, obtaining therefore the striking identification: Riemann = ad ( Beltrami ) ⇐⇒ Beltrami = ad ( Riemann )between the (linearized ) Riemann tensor and the Beltrami parametrization.Taking into account the factor 2 involved by multiplying the second, third and fifth row by 2, weget the new 6 × d − d d − d d d − d d − d − d d d − d d − d d d − d d − d d Surprisingly , the Maxwell parametrization is obtained by keeping φ = A, φ = B, φ = C while setting φ = φ = φ = 0 in order to obtain the system: σ ≡ d B + d C = 0 σ ≡ d A + d C = 0 − σ ≡ d A = 0 σ ≡ d A + d B = 0 − σ ≡ d B = 0 − σ ≡ d C = 0 1 2 31 2 31 2 • • • • • • However, this system may not be involutive and no CC can be found ” a priori ” because thecoordinate system is surely not δ -regular. Indeed, effecting the linear change of coordinates¯ x = x , ¯ x = x , ¯ x = x + x + x and taking out the bar for simplicity, we obtain the newinvolutive system: d C + d C + d C + d C = 0 d B + d B = 0 d A + d A = 0 d C + d C − d C − d B − d C = 0 d A − d C + d B + 2 d C − d C = 0 d A + d C − d C + d C + d B = 0 1 2 31 2 31 2 31 2 • • • and it is easy to check that the 3 CC obtained just amount to the desired 3 stress equations whencoming back to the original system of coordinates. However, the three characters are different aswe have now α = 3 − < α = 2 × − < α = 3 × − dim ( g ) = 6 × − − We have thus a minimum parametrization .Again, if there is a geometrical background, this change of local coordinates is hidding it totally .Moreover, we notice that the stress functions kept in the procedure are just the ones on which ∂ is acting. The reason for such an apparently technical choice is related to very general deeparguments in the theory of differential modules that will only be explained at the end of the paper.The Morera parametrization is obtained similarly by keeping now φ = L, φ = M, φ = N while setting φ = φ = φ = 0, namely: d L = 0 d N − d L − d M = 0 d M = 0 d M − d N − d L = 0 d L − d M − d N = 0 d N = 0Using now the same change of coordinates as the one already done for the Maxwell parametriza-tion, we obtain the following system with 3 equations of (full) class 3 and 3 equations of class 2 inthe Pommaret basis corresponding to the Janet tabular: d N + d N + d N + d N = 0 d M + d M = 0 d L + d L = 0( d N + d M − d L ) + ( d N − d M + d L ) + d N = 02 d M + ( d N − d M − d L ) + d M − d L = 0 d M + ( d N − d M − d L ) + d L = 0 1 2 31 2 31 2 31 2 • • • After elementary but tedious computations ( that could not be avoided !), one can prove that the 3CC corresponding to the 3 dots are effectively satisfied and that they correspond to the 3 Cauchystress equations which are therefore parametrized. The parametrization is thus provided by aninvolutive operator defining a torsion module because the character α is vanishing in δ -regular12oordinates, just like before for the Maxwell parametrization. We have thus another minimumparametrization . Of course, such a result could not have been understood by Beltrami in 1892because the work of Cartan could not be adapted easily in the language of exterior forms and thework of Janet appeared only in 1920 with no explicit reference to involution because only Janetbases are used ([11]) while the Pommaret bases have only been introduced in 1978 ([22]).On a purely computational level, we may also keeep only { φ , φ , φ } and obtain the differentinvolutive system with the same characters and, in particular , α = 0: σ ≡ ∂ φ = 0 − σ ≡ ∂ φ = 0 σ ≡ ∂ φ = 0 σ ≡ ∂ φ − ∂ φ = 0 − σ ≡ ∂ φ − ∂ φ = 0 σ ≡ ∂ φ + ∂ φ − ∂ φ = 0 1 2 31 2 31 2 31 2 • • • So far, we have thus obtained three explicit local minimumm parametrizations of the Cauchystress equations with n ( n − / • n = 4: It just remains to explain the relation of the previous results with Einstein equations.The first suprising link is provided by the following technical proposition: PROPOSITION 3.1 : The Beltrami parametrization is just described by the
Einstein operatorwhen n = 3. The same confusion existing between the Bianchi operator and the
Cauchy operatorhas been made by both Einstein and Beltrami because the
Einstein operator and the
Beltrami operator are self-adjoint in arbitrary dimension n ≥
3, contrary to the
Ricci operator.
Proof : The number of components of the Riemann tensor is dim ( F ) = n ( n − /
12. We havethe combinatorial formula n ( n − / − n ( n + 1) / n ( n + 1)( n + 2)( n − /
12 expressing thatthe number of components of the Riemann tensor is always greater or equal to the number of com-ponents of the Ricci tensor whenever n >
2. Also, we have shown in many books ([22-25],[37],[38])or papers ([42-45]) that the number of Bianchi identities is equal to n ( n − n − /
24, that is3 when n = 3 and 20 when n = 4. Of course, it is well known that the div operator, induced asCC of the Einstein operator, has n components in arbitrary dimension n ≥ n = 3 we have n ( n − /
12 = n ( n + 1) / Einstein operator reduces to the
Beltrami operator and not just to the Ricci operator .The following formulas can be found in any textbook on general relativity:Hence the difference can only be seen when ω i = j = 0. In our situation with n = 3 and theEuclidean metric, we have:2 R = 2 E = ( d + d + d )Ω + d (Ω + Ω + Ω ) − ( d Ω + d Ω + d Ω ) − ( d Ω + d Ω + d Ω )= d Ω + d Ω − d Ω − d Ω R = ( d + d + d )Ω + d (Ω + Ω + Ω ) − d Ω + d Ω + d Ω = ( d + d )Ω + d (Ω + Ω ) − d Ω + d Ω ) tr ( R ) = ( d Ω + d Ω + d Ω + d Ω + d Ω + d Ω ) − d Ω + d Ω + d Ω )2 E = d Ω + d Ω − d Ω In the light of modern differential geometry, comparing these results with the works of bothMaxwell, Morera, Beltrami and Einstein, it becomes clear that they have been confusing the div operator induced from the
Bianchi operator with the
Cauchy operator. However, it is also clearthat they both obtained a possibility to parametrize the
Cauchy operator by means of 3 arbi-trary potential like functions in the case of Maxwell and Morera, 6 in the case of Beltrami whoexplains the previous choices, and 10 in the case of Einstein. Of course, as they were ignoring13hat the
Einstein operator was self-adjoint whenever n ≥
3, they did not notice that we have
Cauchy = ad ( Killing ) and they were unable to compare their reslts with the
Airy operator foundas early as in 1870 for the same mechanical purpose when n = 2. To speak in a rough way, thesituation is similar to what could happen in the study of contact structures if one should confuse D − with D ([43]). Finally, using Theorem 2.1 or Proposition 2.2, we can choose a differentialtranscendence basis with n ( n − / φ ij = φ ji with i < j or1 ≤ i, j ≤ n − ≤ i, j ≤ n when the dimension n ≥ REMARK 3.2 : In the opinion of the author of this paper who is not an historian of sciencesbut a specialist of mathematical physics interested by the analogy existing between electromag-netism (EM), elasticity (EL) and gravitation (GR) by using the conformal group of space-time(See [6],[24],[27],[30],[33],[40],[43-45] for related works), it is ifficult to imagine that Einstein couldnot have been aware of the works of Maxwell and Beltrami on the foundations of EL and tensorcalculus. Indeed, not only they were quite famous when he started his research work but it mustalso be noticed that the Mach-Lippmann analogy ([1],[15],[16],[18]) was introduced at the sametime (See [24] and [40] for more details on the field-matter couplings and the phenomenologicallaw discovered by ... Maxwell too). The main idea is that classical variational calculus using aLagrangian formalism must be considered as the basic scheme of a more general and powerful” duality theory ” that only depends on new purely mathematical tools, namely ” group theory ”and ” differential homological algebra ” (See [25] or [38] for the theory and [42] for the applications).The two following crucial results, still neither known nor acknowledged today, are provided bythe next proposition and corresponding corollary ([36]):
PROPOSITION 3.3 : The
Cauchy operator can be parametrized by the formal adjoint of the
Ricci operator (4 terms) and the
Einstein operator (6 terms) is thus useless. The so-called gravita-tional waves equations are thus nothing else than the formal adjoint of the linearized
Ricci operator.
Proof : The
Einstein operator Ω → E is defined by setting E ij = R ij − ω ij tr ( R ) that weshall write Einstein = C ◦ Ricci where C : S T ∗ → S T ∗ is a symmetric matrix only dependingon ω , which is invertible whenever n ≥ Surprisingly , we may also introduce the same lineartransformation C : Ω → ¯Ω = Ω − ω tr (Ω) and the unknown composite operator X : ¯Ω → Ω → E in such a way that Einstein = X ◦ C where X is defined by (See [GR], 5.1.5 p 134):2 E ij = ω rs d rs ¯Ω ij − ω rs d ri ¯Ω sj − ω rs d sj ¯Ω ri + ω ij ω ru ω sv d rs ¯Ω uv Now, introducing the test functions λ ij , we get: λ ij E ij = λ ij ( R ij − ω ij r ( tr ( R ) = ( λ ij − λ rs ω rs ω ij ) R ij = ¯ λ ij R ij Integrating by parts while setting as usual ✷ = ω rs d rs , we obtain:( ✷ ¯ λ rs + ω rs d ij ¯ λ ij − ω sj d ij ¯ λ ri − ω ri d ij ¯ λ sj )Ω rs = σ rs Ω rs Moreover, suppressing the ”bar ” for simplicity, we have: d r σ rs = ω ij d rij λ rs + ω rs d rij λ ij − ω sj d rij λ ri − ω ri d rij λ sj = 0As Einstein is a self-adjoint operator (contrary to the Ricci operator), we have the identities: ad ( Einstein ) = ad ( C ) ◦ ad ( X ) ⇒ Einstein = C ◦ ad ( X ) ⇒ ad ( X ) = Ricci ⇒ X = ad ( Ricci )Indeed, ad ( C ) = C because C is a symmetric matrix and we know that ad ( Einstein ) =
Einstein .Accordingly, the operator ad ( Ricci ) parametrizes the
Cauchy equations, without any reference to the
Einstein operator which has no mathematical origin, in the sense that it cannot be ob-tained by any diagram chasing. The three terms after the
Dalembert operator factorize throughthe divergence operator d i λ ri . We may thus add the differential constraints d i λ ri = 0 without any eference to a gauge transformation in order to obtain a (minimum) relative parametrization (see[31] and [34] for details and explicit examples). When n = 4 we finally obtain the adjoint sequences:4 Killing −→ Ricci −→ ← Cauchy ←− ad ( Ricci ) ←− without any reference to the Bianchi operator and the induced div operator.Finally, using Theorem 2.1 or Proposition 2.2, we may choose a differential transcendence basismade by { λ ij | i < j } or { λ ij | < i, j < n − } or even { λ ij | < i, j < n } when the dimension n ≥ COROLLARY 3.4 : The differential module N defined by the Ricci or the
Einstein operator isnot torsion-free and cannot therefore be parametrized. Its torsion submodule is generated by the10 components of the linearized
W eyl tensor that are killed by the
Dalembert operator.
Proof : In order to avoid using extension modules, we present the 5 steps of the double differentialduality test in this framework:Step 1: Start with the
Einstein operator D : 10 Einstein −→ ad ( D ) : 10 Einstein ←− Cauchy operator: ad ( D ) : 4 Cauchy ←− D = ad ( ad ( D )) : 4 Killing −→ Riemann operator: D ′ : 10 Riemann −→ Riemann = Ricci ⊕ W eyl with 20 = 10+10.It follows from differential homological algebra that the 10 additional CC in D ′ that are not in D ,are generating the torsion submodule t ( N ) of the differentil module N defined by the Einstein or Ricci operator. In general, if K is a differential field with commuting derivations ∂ , ..., ∂ n , we wayconsider the ring D = K [ d , ..., d n ] = K [ d ] of differential operators with coefficients in K and it isknow that rk D ( D ) = rk D ( ad ( D )) for any operator matrix D with coefficients in K . In the presentsituation, as the M inkowski metric has coefficients equal to 0 , , −
1, we may choose the grounddifferential field to be K = Q . Hence, there exists operators P and Q such that we have an identity: P ◦
W eyl = Q ◦
Ricci
One may also notice that rk D ( Einstein ) = rk D ( Ricci ) with: rk D ( Einstein ) = n ( n + 1)2 − n = n ( n − , rk D ( Riemann ) = n ( n + 1)2 − n = n ( n − this is a purecoincidence because rk D ( Einstein ) has only to do with the div operator induced by contractingthe
Bianchi operator, while rk D ( Riemann ) has only to do with the classical
Killing operatorand the fact that the corresponding differential module is a torsion module because we have aLie group of transformations having n + n ( n − = n ( n +1)2 parameters (translations + rotations).Hence, as the Riemann operator is a direct sum of the
W eyl operator and the
Einstein or Ricci operator according to the previous theorem, each component of the
W eyl operator must be killedby a certain operator whenever the
Einstein or Ricci equations in vacuum are satisfied.
It is notat all evident that we have P = ✷ acting on each component of the W eyl operator. A direct trickycomputation can be found in ([5], p 206]), ([10], exercise 7.7]) and ([37], p 95). With more details,we may start from the long exact sequence:0 → Θ → Killing −→ Riemann −→ Bianchi −→ → → Killing vector fields is not a Janet sequence because the
Killing op-erator is not involutive as it is an operator of finite type with symbol of dimension n ( n − / Riemann operator we get the commutative and exact diagram:0 0 0 ↓ ↓ ↓ −→ → → ↓ ↓↑ ↓ k Killing −→ Riemann −→ Bianchi −→ → → k ↓↑ ↓ ↓ Einstein −→ div −→ → ↓ ↓ ↓ → D −→ D Bianchi −→ D Riemann −→ D Killing −→ D → M → Killing differential module M = coker ( Killing ) and we check thatwe have indeed the vanishing of the
Euler-Poincar´e characteristic −
20 + 20 −
10 + 4 = 0. Ac-cordingly, we have N ′ = coker ( Riemann ) ≃ im ( Killing ) ⊂ D and thus N ′ is torsion-free with rk D ( N ′ ) = 4 − n because rk D ( M ) = 0.We have the following commutative and exact diagram where N = coker ( Einstein ):0 ↓ t ( N ) ↓ ↓ ↓ ↓ −→ D div −→ D Einstein −→ D −→ N → ↓ ↓ ↓ k ↓ → D −→ D Bianchi −→ D Riemann −→ D −→ N ′ → k ↓ ↓ ↓ ↓ → D −→ D −→ D ↓ ↓ ↓ L is the kernel of the epimorphism N → N ′ , it is a torsion module because rk D ( L ) = rk D ( N ) − rk D ( N ′ ) = 4 − L ⊆ t ( N ) in the following commutative andexact diagram: 0 0 ↓ ↓ → L −→ t ( N ) ↓ ↓ → N = N → ↓ ↓ N ′ −→ N/t ( N ) → ↓ ↓ N/t ( N ) is a torsion-free module by definition. A snake chase allows to prove that the cok-ernel of the monomorphism L → t ( N ) is isomorphic to the kernel of the induced epimorphism N ′ → N/t ( N ) and must be therefore, at the same time, a torsion module because rk D ( L ) = rk D ( t ( N )) = 0 and a torsion-free module because N ′ ⊂ D , a result leading to a contradictionunless it is zero and thus L = t ( N ). A snake chase in the previous diagram allows to exhibit thelong exact connecting sequence:0 → D −→ D −→ D −→ t ( N ) → n = 4 ( only ), the CC of the Weyl16perator are of order 2 and not → D −→ D −→ D → D ≃ D ⊕ D but the existence of a canon-ical lift D → D → N = N ′ ⊕ t ( N ) as N ′ is not even free. Hence, one can only say that the space of solutions of Einsteinequations in vacuum contains the generic solutions of the Riemann operator which are parametrizedby arbitrary vector fields. As for the torsion elements, we have t ( N ) = coker ( D → D ) and wemay thus represent them by the components of the Weyl tensor, killed by the Dalembertian. Thismodule interpretation may thus question the proper origin and existence of gravitational wavesbecause the div operator on the upper left part of the diagram has strictly nothing to do with the Cauchy = ad ( Killing ) operator which cannot appear anywhere in this diagram.Q.E.D.
COROLLARY 3.5 : More generally, when D is a Lie operator of finite type , that is when[Θ , Θ] ⊂ Θ under the ordinary bracket of vector fields and g q + r = 0 for r large enough, thenthe Spencer sequence is locally isomorphic to the tensor product of the Poincar´e sequence for theexterior derivative by a finite dimensional Lie algebra. It is thus formally exact both with itsadjoint sequence. As it is known that the extension modules do not depend on the resolutionused, this is the reason for which not only the Cauchy operator can be parametrized but also the Cosserat couple-stress equations ad ( D ) can be parametrized by ad ( D ), a result not evident at all(see [6] and [30] for explicit computations). REMARK 3.6 : A similar situation is well known for the
Cauchy - Riemann equations when n = 2. Indeed, any infinitesimal complex transformation ξ must be solution of the linear firstorder homogeneous system ξ − ξ = 0 , ξ + ξ = 0 of infinitesimal Lie equations though we obtain ξ + ξ = 0 , ξ + ξ = 0, that is ξ and ξ are separately killed by the second order Laplace operator ∆ = d + d . REMARK 3.7 : A similar situation is also well known for the wave equations for the EM field F inelectromagnetism. Indeed, starting with the first set of M axwell equations dF = 0 and using the M inkowski constitutive law in vacuum with electric constant ǫ and magnetic constant µ suchthat ǫ µ c = 1 for the seconf set of M axwell equations, a standard tricky differential eliminationallows to avoid the
Lorenz ( no ”t”) gauge condition for the EM potential and to obtain directly ✷ F = 0 (See [36] or [38] for the details).Using computer algebra or a direct checking with the ordering 11 < < < < < E = ω d Ω + lower termsE = ω d Ω ........... We have therefore the following Janet tabular:1 2 3 41 2 3 41 2 3 41 2 3 41 2 3 41 2 3 41 2 3 • • • • we are in the position to compute the characters of the Einstein operator but a similar procedurecould be followed with the
Ricci operator. We obtain at once:17 = 6 ⇒ α = (10 × − β = 4 ⇒ α = (10 × − β = 0 ⇒ α = (10 × − β = 0 ⇒ α = (10 × − dim ( g ) = α + α + α + α = 90 and dim ( g ) = α + 2 α + 3 α + 4 α = 164along with the long exact sequences:0 → g → S T ∗ ⊗ S T ∗ → S T ∗ → → g → S T ∗ ⊗ S T ∗ → T ∗ ⊗ S T ∗ → T ∗ → σ χ ( div ) = (cid:0) χ , χ , χ , χ (cid:1) and: σ χ (cid:0) div (cid:1) σ χ (cid:0) Einstein (cid:1) = (0 , , , Einstein operator is self-adjoint 10 ×
10 operator matrix up to a change of basis ([32]),we obain therefore det ( σ χ ( Einstein )) = 0 a result not evident at first sight that we shall now refine. χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ E E E E E E E E E E that must be compared with the Poincar´e situation when n = 3, namely:( χ χ , χ ) − χ χ χ − χ − χ χ = (0 , ,
4) CONCLUSION
After teaching elasticity during 25 years to high level students in some of the best french civilengineering schools, the author of this paper still keeps in mind one of the most fascinating exercisesthat he has set up. The purpose was to explain why a dam made with concrete is always vertical onthe water-side with a slope of about 42 degrees on the other free side in order to obtain a minimumcost and the auto-stability under cracking of the surface under water (See the introduction of [K2]for more details). Surprisingly, the main tool involved is the approximate computation of the Airyfunction inside the dam. The author discovered at that time that no one of the other teachersdid know that the Airy parametrization is nothing else than the adjoint of the linearized Riemannoperator used as generating CC for the deformation tensor by any engineer. Being involved inGeneral Relativity (GR) at that time, it took him 25 years (1970-1995) to prove that the Einsteinequations could not be parametrized ([26],[50]). However, nobody is a prophet in his own countryand it is only now that he discovered that GR could be considered as a way to parametrize theCauchy operator. It follows that exactly the same confusion has been done by Maxwell, Morera,Beltrami and Einstein because, in all these cases, the operator considered is self-adjoint. As abyproduct, the variational formalism cannot allow to discover it as no engineer could have hadin mind to confuse the deformation tensor with its CC in the Lagrangian used for finite elementscomputations. It is thus an open historical problem to know whether Einstein knew any one ofthe previous works done as all these researchers were quite famous at the time he was active. Inour opinion at least, the comparison of the various parametrizations described in this paper needsno comment as we have only presented facts , just facts .18 ) BIBLIOGRAPHY) BIBLIOGRAPHY