aa r X i v : . [ phy s i c s . g e n - ph ] J un NONLINEAR CONFORMALELECTROMAGNETISM AND GRAVITATION
J.-F. PommaretCERMICS, Ecole des Ponts ParisTech, [email protected](http://cermics.enpc.fr/ ∼ pommaret/home.html) ABSTRACT
In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approachto elasticity (EL), with the only experimental need to measure the EL constants. In a modernlanguage, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janetsequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl,our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoiddefining the conformal group of space-time in order to provide the physical foundations of bothelectromagnetism (EM) and gravitation, with the only experimental need to measure the EM con-stant in vacuum and the gravitational constant. With a manifold of dimension n , the difficultyis to deal with the n nonlinear transformations that have been called ”elations” by E. Cartan in1922. Using the fact that dimension n = 4 has very specific properties for the computation of theSpencer cohomology, we prove that there is no conceptual difference between the Cosserat EL fieldor induction equations and the Maxwell EM field or induction equations. As a byproduct, the wellknown field/matter couplings (piezzoelectricity, photoelasticity, ...) can be described abstractly,with the only experimental need to measure the corresponding coupling constants. In the sudy ofgravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but atthe central attractive mass and the inversion law of the subgroupoid made by strict second orderjets transforms attraction into repulsion. KEY WORDS
Nonlinear differential sequences; Linear differential sequences; Lie groupoids; Lie algebroids;Conformal geometry; Spencer cohomology; Maxwell equations; Cosserat equations.1 ) INTRODUCTION
Let us start this paper with a personal but meaningful story that has oriented my researchduring the last fourty years or so, since the french ”
Grandes Ecoles ” created their own researchlaboratories. Being a fresh permanent researcher of Ecole Nationale des Ponts et Chauss´ees inParis, the author of this paper has been asked to become the scientific adviser of a young studentin order to introduce him to research. As General Relativity was far too much difficult for some-body without any specific mathematical knowledge while remembering his own experience at thesame age, he asked the student to collect about 50 books of Special Relativity and classify themalong the way each writer was avoiding the use of the conformal group of space-time implied bythe Michelson and Morley experiment, only caring about the Poincar´e or Lorentz subgroups. Aftersix months, the student (like any reader) arrived at the fact that most books were almost copyingeach other and could be nevertheless classified into three categories: •
30 books, including the original 1905 paper ([9],[23]) by Einstein, were at once, as a workingassumption, deciding to restrict their study to a linear group reducing to the Galil´ee group whenthe speed of light was going to infinity. It must be noticed that people did believe that Einsteinhad not been influenced in 1905 by the Michelson and Morley experiment of 1887 till the discoveryof hand written notes taken during lectures given by Einstein in Chicago (1921) and Kyoto (1922). •
15 books were trying to ” prove ” that the conformal factor was indeed reduced to a constantequal to 1 when space-time was supposed to be homogeneous and isotropic. • only were claiming that the conformal factor could eventually depend on the propertyof space-time, adding however that, if there was no surrounding electromagnetism or gravitation,the situation should be reduced to the preceding one but nothing was said otherwise.The sudent was so disgusted by such a state of affair that he decided to give up on research andto become a normal civil engineer. As a byproduct, if group theory must be used, the underlyinggroup of transformations of space-time must be related to the propagation of light by itself ratherthan by considering tricky signals between observers, thus must have to do with the biggest groupof invariance of Maxwell equations ([22],[54]). However, at the time we got the solution of thisproblem with the publication of ([27) in 1988 (See [46] for recent results), a deep confusion wasgoing on which is still not acknowledged though it can be explained in a few lines ([14]). Usingstandard notations of differential geometry, if the 2-form F ∈ ∧ T ∗ describing the EM field issatifying the first set of Maxwell equations, it amounts to say that it is closed, that is killed bythe exterior derivative d : ∧ T ∗ → ∧ T ∗ . The EM field can be thus (locally) parametrized by theEM potential 1-form A ∈ T ∗ with dA = F where d : T ∗ → ∧ T ∗ is again the exterior derivative,because d = d ◦ d = 0. Now, if E is a vector bundle over a manifold X of dimension n , thenwe may define its adjoint vector bundle ad ( E ) = ∧ n T ∗ ⊗ E ∗ where E ∗ is obtained from E byinverting the transition rules, like T ∗ is obtained from T = T ( X ) and such a construction canbe extended to linear partial differential operators between (sections of) vector bundles. When n = 4, it follows that the second set of Maxwell equations for the EM induction is just describedby ad ( d ) : ∧ T ∗ ⊗ ∧ T → ∧ T ∗ ⊗ T , independently of any Minkowski constitutive relation betweenfield and induction . Using Hodge duality with respect to the volume form dx = dx ∧ ... ∧ dx ,this operator is isomorphic to d : ∧ T ∗ → ∧ T ∗ . It follows that both the first set and second set ofMaxwell equations are invariant by any diffeomorphism and that the conformal group of space-timeis the biggest group of transformations preserving the Minkowski constitutive relations in vacuum where the speed of light is trully c as a universal constant. It was thus natural to believe thatthe mathematical structure of electromagnetism and gravitation had only to do with such a grouphaving: 4 translations + 6 rotations + 1 dilatation + 4 elations = 15 parameters the main difficulty being to deal with these later non-linear tranformations. Of course, such a chal-lenge could not be solved without the help of the non-linear theory of partial differential equationsand Lie pseudogroups combined with homological algebra, that is before 1995 at least ([28]).2 rom a purely physical point of view , these new nonlinear methods have been introduced for thefirst time in 1909 by the brothers E. and F. Cosserat for studying the mathematical foundations ofEL ([1],[7],[8],[18],[29-30],[50]). We have presented their link with the nonlinear Spencer differentialsequences existing in the formal theory of Lie pseudogroups at the end of our book ” DifferentialGalois Theory ” published in 1983 ([26]). Similarly, the conformal methods have been introducedby H. Weyl in 1918 for revisiting the mathematical foundations of EM ([54]). We have presentedtheir link with the above approach through a unique differential sequence only depending on thestructure of the conformal group in our book ” Lie Pseudogroups and Mechanics ” published in1988 ([27]). However, the Cosserat brothers were only dealing with translations and rotations whileWeyl was only dealing with dilatation and elations . Also, as an additional condition not fulfilledby the classical Einstein-Fokker-Nordstr¨om theory ([10]), if the conformal factor has to do withgravitation, it must be defined everywhere but at the central attractive mass as we already said. From a purely mathematical point of view , the concept of a finite length differential sequence ,now called
Janet sequence , has been first described as a footnote by M. Janet in 1920 ([16]). Then,the work of D. C. Spencer in 1970 has been the first attempt to use the formal theory of systems ofpartial differential equations that he developped himself in order to study the formal theory of Liepseudogroups ([12-13],[19],[49]). However, the nonlinear Spencer sequences for Lie pseudogroups ,though never used in physics, largely supersede the ”
Cartan structure equations ” introduced byE.Cartan in 1905 ([4-6],[17]) and are different from the ”
Vessiot structure equations ” introducedby E. Vessiot in 1903 ([51-52]) for the same purpose but still not known today after more than acentury because they have never been acknowledged by Cartan himself or even by his successors.The purpose of the present difficult paper is to apply these new methods for studying thecommon nonlinear conformal origin of gravitation and electromagnetism, in a purely mathematicalway , by constructing explicitly the corresponding nonlinear Spencer sequence. All the physicalconsequences will be presented in another paper.
2) GROUPOIDS AND ALGEBROIDS
Let us now turn to the clever way proposed by Vessiot in 1903 ([51]) and 1904 ([52]). Ourpurpose is only to sketch the main results that we have obtained in many books ([25-28], we do notknow other references) and to illustrate them by a series of specific examples, asking the reader toimagine any link with what has been said. We break the study into 9 successive steps.1) If E = X × X , we shall denote by Π q = Π q ( X, X ) the open subfibered manifold of J q ( X × X )defined independently of the coordinate system by det ( y ki ) = 0 with source projection α q : Π q → X : ( x, y q ) → ( x ) and target projection β q : Π q → X : ( x, y q ) → ( y ). We shall sometimes in-troduce a copy Y of X with local coordinates ( y ) in order to avoid any confusion between thesource and the target manifolds. In order to construct another nonlinear sequence, we need a fewbasic definitions on Lie groupoids and
Lie algebroids that will become substitutes for Lie groupsand Lie algebras. The first idea is to use the chain rule for derivatives j q ( g ◦ f ) = j q ( g ) ◦ j q ( f )whenever f, g ∈ aut ( X ) can be composed and to replace both j q ( f ) and j q ( g ) respectively by f q and g q in order to obtain the new section g q ◦ f q . This kind of ”composition” law can be written ina pointwise symbolic way by introducing another copy Z of X with local coordinates ( z ) as follows: γ q : Π q ( Y, Z ) × Y Π q ( X, Y ) → Π q ( X, Z ) : (( y, z, ∂z∂y , ... ) , ( x, y, ∂y∂x , ... ) → ( x, z, ∂z∂y ∂y∂x , ... )We may also define j q ( f ) − = j q ( f − ) and obtain similarly an ”inversion” law. DEFINITION 2.1 : A fibered submanifold R q ⊂ Π q is called a system of finite Lie equations or a Lie groupoid of order q if we have an induced source projection α q : R q → X , target pro-jection β q : R q → X , composition γ q : R q × X R q → R q , inversion ι q : R q → R q and identity j q ( id ) = id q : X → R q . In the sequel we shall only consider transitive Lie groupoids such that themap ( α q , β q ) : R q → X × X is an epimorphism and we shall denote by R q = id − ( R q ) the isotropyLie group bundle of R q . Also, one can prove that the new system ρ r ( R q ) = R q + r obtained bydifferentiating r times all the defining equations of R q is a Lie groupoid of order q + r .3et us start with a Lie pseudogroup Γ ⊂ aut ( X ) defined by a system R q ⊂ Π q of order q . Roughlyspeaking, if f, g ∈ Γ ⇒ g ◦ f, f − ∈ Γ) but such a definition is totally meaningless in actual practiceas it cannot be checked most of the time. In all the sequel we shall suppose that the system isinvolutive ([25-28],[30]) and that Γ is transitive that is ∀ x, y ∈ X, ∃ f ∈ Γ , y = f ( x ) or, equivalently,the map ( α q , β q ) : R q → X × X : ( x, y q ) → ( x, y ) is surjective.2) The Lie algebra Θ ⊂ T of infinitesimal transformations is then obtained by linearization, setting y = x + tξ ( x ) + ... and passing to the limit t → R q = id − q ( V ( R q )) ⊂ J q ( T ) by reciprocal image with Θ = { ξ ∈ T | j q ( ξ ) ∈ R q } . We define the isotropy Lie algebra bundle R q ⊂ J q ( T ) by the short exact sequence 0 → R q → R q π q −→ T → prolong the vertical infinitesimal transformations η = η k ( y ) ∂∂y k to the jet coordinates up to order q in order to obtain: η k ( y ) ∂∂y k + ( ∂η k ∂y r y ri ) ∂∂y ki + ( ∂ η k ∂y r ∂y s y ri y sj + ∂η k ∂y r y rij ) ∂∂y kij + ... where we have replaced j q ( f )( x ) by y q , each component beeing the ”formal” derivative of theprevious one .4) As [Θ , Θ] ⊂ Θ, we may use the Frobenius theorem in order to find a generating fundamental setof differential invariants { Φ τ ( y q ) } up to order q which are such that Φ τ (¯ y q ) = Φ τ ( y q ) by using thechain rule for derivatives whenever ¯ y = g ( y ) ∈ Γ acting now on Y . Specializing the Φ τ at id q ( x )we obtain the Lie form Φ τ ( y q ) = ω τ ( x ) of R q .Of course, in actual practice one must use sections of R q instead of solutions and we now prove whythe use of the Spencer operator becomes crucial for such a purpose. Indeed, using the algebraicbracket { j q +1 ( ξ ) , j q +1 ( η ) } = j q ([ ξ, η ]) , ∀ ξ, η ∈ T , we may obtain by bilinearity a differential bracket on J q ( T ) extending the bracket on T :[ ξ q , η q ] = { ξ q +1 , η q +1 } + i ( ξ ) Dη q +1 − i ( η ) Dξ q +1 , ∀ ξ q , η q ∈ J q ( T )which does not depend on the respective lifts ξ q +1 and η q +1 of ξ q and η q in J q +1 ( T ). This bracketon sections satisfies the Jacobi identity and we set ([25-28]): DEFINITION 2.2 : We say that a vector subbundle R q ⊂ J q ( T ) is a system of infinitesimal Lieequations or a Lie algebroid if [ R q , R q ] ⊂ R q , that is to say ξ q , η q ∈ R q ⇒ [ ξ q , η q ] ∈ R q . Such adefinition can be tested by means of computer algebra. We shall also say that R q is transitive if wehave the short exact sequence 0 → R q → R q π q → T →
0. In that case, a splitting of this sequence,namely a map χ q : T → R q such that π q ◦ χ q = id T or equivalently a section χ q ∈ T ∗ ⊗ R q over id T ∈ T ∗ ⊗ T , is called a R q - connection and its curvature κ q ∈ ∧ T ∗ ⊗ R q is defined by κ q ( ξ, η ) = [ χ q ( ξ ) , χ q ( η )] − χ q ([ ξ, η ]) , ∀ ξ, η ∈ T . PROPOSITION 2.3 : If [ R q , R q ] ⊂ R q , then [ R q + r , R q + r ] ⊂ R q + r , ∀ r ≥ Proof : When r = 1, we have ρ ( R q ) = R q +1 = { ξ q +1 ∈ J q +1 ( T ) | ξ q ∈ R q , Dξ q +1 ∈ T ∗ ⊗ R q } and we just need to use the following formulas showing how D acts on the various brackets (See[25],[28] and [40] for more details): i ( ζ ) D { ξ q +1 , η q +1 } = { i ( ζ ) Dξ q +1 , η q } + { ξ q , i ( ζ ) Dη q +1 } , ∀ ζ ∈ Ti ( ζ ) D [ ξ q +1 , η q +1 ] = [ i ( ζ ) Dξ q +1 , η q ] + [ ξ q , i ( ζ ) Dη q +1 ]+ i ( L ( η ) ζ ) Dξ q +1 − i ( L ( ξ ) ζ ) Dη q +1 because the right member of the second formula is a section of R q whenever ξ q +1 , η q +1 ∈ R q +1 .The first formula may be used when R q is formally integrable. Q.E.D.4 XAMPLE 2.4 : With n = 1 , q = 3 , X = R and evident notations, the components of [ ξ , η ] atorder zero, one, two and three are defined by the totally unusual successive formulas:[ ξ, η ] = ξ∂ x η − η∂ x ξ ([ ξ , η ]) x = ξ∂ x η x − η∂ x ξ x ([ ξ , η ]) xx = ξ x η xx − η x ξ xx + ξ∂ x η xx − η∂ x ξ xx ([ ξ , η ]) xxx = 2 ξ x η xxx − η x ξ xxx + ξ∂ x η xxx − η∂ x ξ xxx For affine transformations, ξ xx = 0 , η xx = 0 ⇒ ([ ξ , η ]) xx = 0 and thus [ R , R ] ⊂ R .For projective transformations, ξ xxx = 0 , η xxx = 0 ⇒ ([ ξ , η ]) xxx = 0 and thus [ R , R ] ⊂ R . THEOREM 2.5 : ( prolongation/projection (PP) procedure ) If an arbitrary system R q ⊆ J q ( E ) isgiven, one can effectively find two integers r, s ≥ R ( s ) q + r is formally integrableor even involutive. COROLLARY 2.6 : The bracket is compatible with the PP procedure:[ R q , R q ] ⊂ R q ⇒ [ R ( s ) q + r , R ( s ) q + r ] ⊂ R ( s ) q + r , ∀ r, s ≥ EXAMPLE 2.7 : With n = m = 2 and q = 1, let us consider the Lie pseudodogroup Γ ⊂ aut ( X )of finite transformations y = f ( x ) such that y dy = x dx = ω = ( x , ∈ T ∗ . Setting y = x + tξ ( x ) + ... and linearizing, we get the Lie operator D ξ = L ( ξ ) ω where L is the Liederivative because it is well known that [ L ( ξ ) , L ( η )] = L ( ξ ) ◦ L ( η ) − L ( η ) ◦ L ( ξ ) = L ([ ξ, η ]) in theoperator sense. The system R ⊂ J ( T ) of linear infinitesimal Lie equations is: x ∂ ξ + ξ = 0 , ∂ ξ = 0Replacing j ( ξ ) by a section ξ ∈ J ( T ), we have: ξ = − x ξ , ξ = 0Let us choose the two sections: ξ = { ξ = 0 , ξ = − x , ξ = 1 , ξ = 0 , ξ = 0 , ξ = 0 } ∈ R η = { η = x , η = 0 , η = 0 , η = − x , η = 0 , η = 1 } ∈ R We let the reader check that [ ξ , η ] ∈ R . However, we have the strict inclusion R (1)1 ⊂ R defined by the additional equation ξ + ξ = 0 and thus ξ , η / ∈ R (1)1 though we have indeed[ R (1)1 , R (1)1 ] ⊂ R (1)1 , a result not evident at all because the sections ξ and η have nothing to do with solutions. The reader may proceed in the same way with x dx − x dx and compare.5) The main discovery of Vessiot, as early as in 1903 and thus fifty years in advance, has beento notice that the prolongation at order q of any horizontal vector field ξ = ξ i ( x ) ∂∂x i commuteswith the prolongation at order q of any vertical vector field η = η k ( y ) ∂∂y k , exchanging there-fore the differential invariants. Keeping in mind the well known property of the Jacobian de-terminant while passing to the finite point of view, any (local) transformation y = f ( x ) can belifted to a (local) transformation of the differential invariants between themselves of the form u → λ ( u, j q ( f )( x )) allowing to introduce a natural bundle F over X by patching changes of coor-dinates ¯ x = ϕ ( x ) , ¯ u = λ ( u, j q ( ϕ )( x )). A section ω of F is called a geometric object or structure on X and transforms like ¯ ω ( f ( x )) = λ ( ω ( x ) , j q ( f )( x )) or simply ¯ ω = j q ( f )( ω ). This is a wayto generalize vectors and tensors ( q = 1) or even connections ( q = 2). As a byproduct we haveΓ = { f ∈ aut ( X ) | Φ ω ( j q ( f )) = j q ( f ) − ( ω ) = ω } as a new way to write out the Lie form and wemay say that Γ preserves ω . We also obtain R q = { f q ∈ Π q | f − q ( ω ) = ω } . Coming back to theinfinitesimal point of view and setting f t = exp ( tξ ) ∈ aut ( X ) , ∀ ξ ∈ T , we may define the ordinaryLie derivative with value in F = ω − ( V ( F )) by introducing the vertical bundle of F as a vector5undle over F and the formula : D ξ = L ( ξ ) ω = ddt j q ( f t ) − ( ω ) | t =0 ⇒ Θ = { ξ ∈ T |L ( ξ ) ω = 0 } while we have x → x + tξ ( x ) + ... ⇒ u τ → u τ + t∂ µ ξ k L τµk ( u ) + ... where µ = ( µ , ..., µ n ) is amulti-index as a way to write down the system R q ⊂ J q ( T ) of infinitesimal Lie equations in the Medolaghi form : Ω τ ≡ ( L ( ξ ) ω ) τ ≡ − L τµk ( ω ( x )) ∂ µ ξ k + ξ r ∂ r ω τ ( x ) = 0 EXAMPLE 2.8 : With n = 1, let us consider the Lie group of projective transformations y = ( ax + b ) / ( cx + d ) as a lie pseudogroup. Differentiating three times in order to eliminatethe parameters, we obtain the third order Schwarzian
OD equation and its linearization over y = x : R ⊂ Π Ψ ≡ y xxx y x −
32 ( y xx y x ) = 0 R ⊂ J ( T ) ξ xxx = 0Accordingly, the prolongation ♯ ( η )of any η ∈ J ( T ( Y )) over Y such that η yyy = 0 becomes: η ( y ) ∂∂y + η y ( y )( y x ∂∂y x + y xx ∂∂y xx + y xxx ∂∂y xxx ) + η yy ( y )(( y x ) ∂∂y xx + 3 y x y xx ∂∂y xxx )It follows that Ψ is a generating third order differential invariant and R is in Lie form.Now, we have:¯ x = ϕ ( x ) ⇒ y x = y ¯ x ∂ x ϕ, y xx = y ¯ x ¯ x ( ∂ x ϕ ) + y ¯ x ∂ xx ϕ, y xxx = y ¯ x ¯ x ¯ x ( ∂ x ϕ ) + 3 y ¯ x ¯ x ∂ x ϕ∂ xx ϕ + y ¯ x ∂ xxx ϕ and the natural bundle F is thus defined by the transition rules:¯ x = ϕ ( x ) , u = ¯ u ( ∂ x ϕ ) + ( ∂ xxx ϕ∂ x ϕ −
32 ( ∂ xx ϕ∂ x ϕ ) )The general Lie form of R is: y xxx y x −
32 ( y xx y x ) + ω ( y )( y x ) = ω ( x )We obtain R ⊂ J ( T ) in Medolaghi form as follows:Ω ≡ L ( ξ ) ω ≡ ∂ xxx ξ + 2 ω ( x ) ∂ x ξ + ξ∂ x ω ( x ) = 0Using a section ξ ∈ J ( T ), we finally get the formal Lie derivative:Ω ≡ L ( ξ ) ω ≡ ξ xxx + 2 ω ( x ) ξ x + ξ∂ x ω ( x ) = 0and let the reader ckeck directly that [ L ( ξ ) , L ( η )] = L ([ ξ , η ]) , ∀ ξ , η ∈ J ( T ), a result abso-lutely not evident at first sight ([47]).7) By analogy with ”special” and ”general” relativity, we shall call the given section special andany other arbitrary section general . The problem is now to study the formal properties of the linearsystem just obtained with coefficients only depending on j ( ω ), exactly like L.P. Eisenhart did for F = S T ∗ when finding the constant Riemann curvature condition for a metric ω with det ( ω ) = 0([11],[28], Example 10, p 246 to 256). Indeed, if any expression involving ω and its derivatives is ascalar object, it must reduce to a constant because Γ is assumed to be transitive and thus cannotbe defined by any zero order equation. Now one can prove that the CC for ¯ ω , thus for ω too,only depend on the Φ and take the quasi-linear symbolic form v ≡ I ( u ) ≡ A ( u ) u x + B ( u ) = 0,allowing to define an affine subfibered manifold B ⊂ J ( F ) over F . Now, if one has two sections ω and ¯ ω of F , the equivalence problem is to look for f ∈ aut ( X ) such that j q ( f ) − ( ω ) = ¯ ω . Whenthe two sections satisfy the same CC, the problem is sometimes locally possible (Lie groups oftransformations, Darboux problem in analytical mechanics,...) but sometimes not ([23], p 333).6) Instead of the CC for the equivalence problem, let us look for the integrability conditions (IC) forthe system of infinitesimal Lie equations and suppose that, for the given section, all the equationsof order q + r are obtained by differentiating r times only the equations of order q , then it wasclaimed by Vessiot ([50] with no proof, see [28], p 207-211) that such a property is held if and onlyif there is an equivariant section c : F → F : ( x, u ) → ( x, u, v = c ( u )) where F = J ( F ) / B isa natural vector bundle over F with local coordinates ( x, u, v ). Moreover, any such equivariantsection depends on a finite number of constants c called structure constants and the IC for the Vessiot structure equations I ( u ) = c ( u ) are of a polynomial form J ( c ) = 0. EXAMPLE 2.9 : Comig back to Example 2 . ω = ( α, β ) ∈ T ∗ ⊗ X ∧ T ∗ must satisfy the Vessiot structure equation dα = c β with a single Vessiot structure constant c = − α = x dx and β = dx ∧ dx (See ([40]) for other examples and applications). As a byproduct, there is no concep-tual difference between such a constant and the constant appearing in the constant Riemanniancurvature condition of Eisenhart ([11]).9) Finally, when Y is no longer a copy of X , a system A q ⊂ J q ( X × Y ) is said to be an automorphicsystem for a Lie pseudogroup Γ ⊂ aut ( Y ) if, whenever y = f ( x ) and ¯ y = ¯ f ( x ) are two solutions,then there exists one and only one transformation ¯ y = g ( y ) ∈ Γ such that ¯ f = g ◦ f . Explicit testsfor checking such a property formally have been given in ([26],[42]) and can be implemented oncomputer in the differential algebraic framework.
3) NONLINEAR SEQUENCES
Contrary to what happens in the study of Lie pseudogroups and in particular in the study ofthe algebraic ones that can be found in mathematical physics, nonlinear operators do not in gen-eral admit
CC, unless they are defined by differential polynomials, as can be seen by consideringthe two following examples with m = 1 , n = 2 , q = 2. With standard notations from differentialalgebra, if we are dealing with a ground differential field K , like Q in the next examples, we denoteby K { y } the ring (which is even an integral domain) of differential polynomials in y with coeffi-cients in K and by K < y > = Q ( K { y } ) the corresponding quotient field of differential rationalfunctions in y . Then, if u, v ∈ K < y > , we have the two towers K ⊂ K < u > ⊂ K < y > and K ⊂ K < v > ⊂ K < y > of extensions, thus the tower K ⊂ K < u, v > ⊂ K < y > . Accordingly,the differential extension
K < u, v > /K is a finitely generated differential extension. If we consider u and v as new indeterminates, then K < u > and
K < v > are both differential transcendentalextensions of K and the kernel of the canonical differential morphism K { u }⊗ K K { v } → K < y > is a prime differential ideal in the differential integral domain
K < u > ⊗ K K < v > , a way todescribe by residue the smallest differential field containing
K < u > and
K < v > in K < y > .Of course, the true difficulty is to find out such a prime differential ideal.
EXAMPLE 3.1 : First of all, let us consider the following nonlinear system in y with secondmember ( u, v ): P ≡ y −
13 ( y ) = u, Q ≡ y −
12 ( y ) = v ⇒ y = u − v v The differential ideal a generated by P and Q in Q { y } is prime because d Q + d P − y d Q = 0and thus Q { y } / { P, Q } ≃ Q [ y, y , y , y , y , ... ] is an integral domain.We may consider the following nonlinear involutive system with two equations: (cid:26) y − ( y ) = 0 y − ( y ) = 0 1 21 • We have also the linear inhomogeneous finite type second order system with three equations: y = u + ( u − v v ) y = v + ( u − v v ) y = u − v v • • a priori two CC, we let the reader prove, as a delicate exercise, that there is onlythe single nonlinear second order CC obtained from the bottom dot.: d ( u − v v ) − d ( v + ( u − v v ) ) = 0 EXAMPLE 3.2 : On the contrary, if we consider the following new nonlinear system: P ≡ y −
12 ( y ) = u, Q ≡ y − y = v ⇒ ( y − y = v + v − u = w we obtain successively: d Q + d Q − d P ≡ ( y − y y ( d Q + d Q − d P ) − y ( d Q + d Q − d P ) = ( y ) The symbol at order 3 is thus not a vector bundle and no direct study as above can be used becausethe differential ideal generated by (
P, Q ) is not perfect as it contains ( y ) without containing y (See [26] and [41] for more details). The following nonlinear system is not involutive: (cid:26) y − ( y ) = 0 y − y = 0 1 21 • We have the following four generic nonlinear additional finite type third order equations: y − y ( v + wy − ) = u y − y wy − = u y − wy − = v y − wy − = 0 1 21 • • • Though we have now a priori three CC and thus three additional equations because the systemis not involutive, setting y − z ⇒ y = z , y = z , there is only the single additionalnonlinear second order equation: v z + ( w − w ) z + v w = 0Differentiating once and using the relation zz = w , we get: v z + ( w − w ) z + ( v w + 3 v w ) z + ( w − w ) w = 0a result leading to a tricky resultant providing a third order differential polynomial in ( u, v ).However, the kernel of a linear operator D : E → F is always taken with respet to the zerosection of F , while it must be taken with respect to a prescribed section by a double arrow fora nonlinear operator. Keeping in mind the linear Janet sequence and the examples of Vessiotstructure equations already presented, one obtains: THEOREM 3.3 : There exists a nonlinear Janet sequence associated with the Lie form of aninvolutive system of finite Lie equations: Φ ◦ j q I ◦ j → Γ → aut ( X ) ⇒ F ⇒ F ω ◦ α f → Φ ◦ j q ( f ) = Φ( j q ( f )) = j q ( f ) − ( ω ) is taken with respectto the section ω of F while the kernel of the second operator ω → I ( j ( ω )) ≡ A ( ω ) ∂ x ω + B ( ω ) istaken with respect to the zero section of the vector bundle F over F . COROLLARY 3.4 : By linearization at the identity, one obtains the involutive
Lie operator D : T → F : ξ → L ( ξ ) ω with kernel Θ = { ξ ∈ T |L ( ξ ) ω = 0 } ⊂ T satisfying [Θ , Θ] ⊂ Θ and the8orresponding linear Janet sequence :0 → Θ → T D −→ F D −→ F where F = F = ω − ( V ( F )) and F = ω − ( F ).Now we notice that T is a natural vector bundle of order 1 and J q ( T ) is thus a natural vectorbundle of order q + 1. Looking at the way a vector field and its derivatives are transformed underany f ∈ aut ( X ) while replacing j q ( f ) by f q , we obtain: η k ( f ( x )) = f kr ( x ) ξ r ( x ) ⇒ η ku ( f ( x )) f ui ( x ) = f kr ( x ) ξ ri ( x ) + f kri ( x ) ξ r ( x )and so on, a result leading to: LEMMA 3.5 : J q ( T ) is associated with Π q +1 = Π q +1 ( X, X ) that is we can obtain a new section η q = f q +1 ( ξ q ) from any section ξ q ∈ J q ( T ) and any section f q +1 ∈ Π q +1 by the formula: d µ η k ≡ η kr f rµ + ... = f kr ξ rµ + ... + f kµ +1 r ξ r , ∀ ≤ | µ | ≤ q where the left member belongs to V (Π q ). Similarly R q ⊂ J q ( T ) is associated with R q +1 ⊂ Π q +1 .More generally, looking now for transformations ”close” to the identity, that is setting y = x + tξ ( x ) + ... when t ≪ t →
0, wemay linearize any (nonlinear) system of finite Lie equations in order to obtain a (linear) systemof infinitesimal Lie equations R q ⊂ J q ( T ) for vector fields. Such a system has the property that,if ξ, η are two solutions, then [ ξ, η ] is also a solution. Accordingly, the set Θ ⊂ T of its solutionssatisfies [Θ , Θ] ⊂ Θ and can therefore be considered as the Lie algebra of Γ.More generally, the next definition will extend the classical Lie derivative : L ( ξ ) ω = ( i ( ξ ) d + di ( ξ )) ω = ddt j q ( exp tξ ) − ( ω ) | t =0 . DEFINITION 3.6 : We say that a vector bundle F is associated with R q if there exists a firstorder differential operator L ( ξ q ) : F → F called formal Lie derivative and such that:1) L ( ξ q + η q ) = L ( ξ q ) + L ( η q ) ∀ ξ q , η q ∈ R q .2) L ( f ξ q ) = f L ( ξ q ) ∀ ξ q ∈ R q , ∀ f ∈ C ∞ ( X ).3) [ L ( ξ q ) , L ( η q )] = L ( ξ q ) ◦ L ( η q ) − L ( η q ) ◦ L ( ξ q ) = L ([ ξ q , η q ]) ∀ ξ q , η q ∈ R q .4) L ( ξ q )( f η ) = f L ( ξ q ) η + ( ξ.f ) η ∀ ξ q ∈ R q , ∀ f ∈ C ∞ ( X ) , ∀ η ∈ F . LEMMA 3.7 : If E and F are associated with R q , we may set on E ⊗ F : L ( ξ q )( η ⊗ ζ ) = L ( ξ q ) η ⊗ ζ + η ⊗ L ( ξ q ) ζ ∀ ξ q ∈ R q , ∀ η ∈ E, ∀ ζ ∈ F If Θ ⊂ T denotes the solutions of R q , then we may set L ( ξ ) = L ( j q ( ξ )) , ∀ ξ ∈ Θ but no explicitcomputation can be done when Θ is infinite dimensional. However, we have:
PROPOSITION 3.8 : J q ( T ) is associated with J q +1 ( T ) if we define: L ( ξ q +1 ) η q = { ξ q +1 , η q +1 } + i ( ξ ) Dη q +1 = [ ξ q , η q ] + i ( η ) Dξ q +1 and thus R q is associated with R q +1 . Proof : It is easy to check the properties 1, 2, 4 and it only remains to prove property 3 as follows.9 L ( ξ q +1 ) , L ( η q +1 )] ζ q = L ( ξ q +1 )( { η q +1 , ζ q +1 } + i ( η ) Dζ q +1 ) − L ( η q +1 )( { ξ q +1 , ζ q +1 } + i ( ξ ) Dζ q +1 )= { ξ q +1 , { η q +2 , ζ q +2 }} − { η q +1 , { ξ q +2 , ζ q +2 }} + { ξ q +1 , i ( η ) Dζ q +2 } − { η q +1 , i ( ξ ) Dζ q +2 } + i ( ξ ) D { η q +2 , ζ q +2 } − i ( η ) D { ξ q +2 , ζ q +2 } + i ( ξ ) D ( i ( η ) Dζ q +2 ) − i ( η ) D ( i ( ξ ) Dζ q +2 )= {{ ξ q +2 , η q +2 } , ζ q +1 } + { i ( ξ ) Dη q +2 , ζ q +1 }−{ i ( η ) Dξ q +2 , ζ q +1 } + i ([ ξ, η ]) Dζ q +1 = { [ ξ q +1 , η q +1 ] , ζ q +1 } + i ([ ξ, η ]) Dζ q +1 by using successively the Jacobi identity for the algebraic bracket and the last proposition.Q.E.D. EXAMPLE 3.9 : T and T ∗ both with any tensor bundle are associated with J ( T ). For T wemay define L ( ξ ) η = [ ξ, η ] + i ( η ) Dξ = { ξ , j ( η ) } . We have ξ r ∂ r η k − η s ∂ s ξ k + η s ( ∂ s ξ k − ξ ks ) = − η s ξ ks + ξ r ∂ r η k and the four properties of the formal Lie derivative can be checked directly. Ofcourse, we find back L ( ξ ) η = [ ξ, η ] , ∀ ξ, η ∈ T . We let the reader treat similarly the case of T ∗ . PROPOSITION 3.10 : There is a first nonlinear Spencer sequence :0 −→ aut ( X ) j q +1 −→ Π q +1 ( X, X ) ¯ D −→ T ∗ ⊗ J q ( T ) ¯ D ′ −→ ∧ T ∗ ⊗ J q − ( T )with ¯ Df q +1 ≡ f − q +1 ◦ j ( f q ) − id q +1 = χ q ⇒ ¯ D ′ χ q ( ξ, η ) ≡ Dχ q ( ξ, η ) −{ χ q ( ξ ) , χ q ( η ) } = 0. Moreover,setting χ = A − id ∈ T ∗ ⊗ T , this sequence is locally exact if det ( A ) = 0. Proof : There is a canonical inclusion Π q +1 ⊂ J (Π q ) defined by y kµ,i = y kµ +1 i and the composition f − q +1 ◦ j ( f q ) is a well defined section of J (Π q ) over the section f − q ◦ f q = id q of Π q like id q +1 .The difference χ q = f − q +1 ◦ j ( f q ) − id q +1 is thus a section of T ∗ ⊗ V (Π q ) over id q and we havealready noticed that id − q ( V (Π q )) = J q ( T ). For q = 1 we get with g = f − : χ k,i = g kl ∂ i f l − δ ki = A ki − δ ki , χ kj,i = g kl ( ∂ i f lj − A ri f lrj )We also obtain from Lemma 3.5 the useful formula f kr χ rµ,i + ... + f kµ +1 r χ r,i = ∂ i f kµ − f kµ +1 i allowingto determine χ q inductively.We refer to ([28], p 215-216) for the inductive proof of the local exactness, providing the onlyformulas that will be used later on and can be checked directly by the reader: ∂ i χ k,j − ∂ j χ k,i − χ ki,j + χ kj,i − ( χ r,i χ kr,j − χ r,j χ kr,i ) = 0 (1) ∂ i χ kl,j − ∂ j χ kl,i − χ kli,j + χ klj,i − ( χ r,i χ klr,j + χ rl,i χ kr,j − χ rl,j χ kr,i − χ r,j χ klr,i ) = 0 (2) ∂ i χ klr,j − ∂ j χ klr,i − χ klri,j + χ klrj,i − ( χ s,i χ klrs,j + χ sr,i χ kls,j + χ sl,i χ krs,j + χ slr,i χ ks,j − χ s,j χ klrs,i − χ sr,j χ kls,i − χ sl,j χ krs,i − χ slr,j χ ks,i ) = 0 (3)There is no need for double-arrows in this framework as the kernels are taken with respect tothe zero section of the vector bundles involved. We finally notice that the main difference with thegauge sequence is that all the indices range from to n and that the condition det ( A ) = 0 amountsto ∆ = det ( ∂ i f k ) = 0 because det ( f ki ) = 0 by assumption. Q.E.D. COROLLARY 3.11 : There is a restricted first nonlinear Spencer sequence :0 −→ Γ j q +1 −→ R q +1 ¯ D −→ T ∗ ⊗ R q ¯ D ′ −→ ∧ T ∗ ⊗ J q − ( T ) DEFINITION 3.12 : A splitting of the short exact sequence 0 → R q → R q π q → T → χ ′ q : T → R q such that π q ◦ χ ′ q = id T or equivalently a section of T ∗ ⊗ R q over id T ∈ T ∗ ⊗ T and is called a R q - connection . Its curvature κ ′ q ∈ ∧ T ∗ ⊗ R q is defined by10 ′ q ( ξ, η ) = [ χ ′ q ( ξ ) , χ ′ q ( η )] − χ ′ q ([ ξ, η ]). We notice that χ ′ q = − χ q is a connection with ¯ D ′ χ ′ q = κ ′ q ifand only if A = 0. In particular ( δ ki , − γ kij ) is the only existing symmetric connection for the Killingsystem. REMARK 3.13 : Rewriting the previous local formulas with A instead of χ we get: ∂ i A kj − ∂ j A ki − A ri χ kr,j + A rj χ kr,i = 0 (1 ∗ ) ∂ i χ kl,j − ∂ j χ kl,i − χ rl,i χ kr,j + χ rl,j χ kr,i − A ri χ klr,j + A rj χ klr,i = 0 (2 ∗ ) ∂ i χ klr,j − ∂ j χ klr,i − A s jχ klri,j + A si χ klrj,i − ( χ sr,i χ kls,j + χ sl,i χ krs,j + χ slr,i χ ks,j − χ sr,j χ kls,i − χ sl,j χ krs,i − χ slr,j χ ks,i ) = 0 (3 ∗ )When q = 1 , g = 0 and though surprising it may look like, we find back exactly all the formulaspresented by E. and F. Cosserat in ([8], p 123 and [16]). Even more strikingly, in the case of aRiemann structure, the last two terms disappear but the quadratic terms are left while, in the caseof screw and complex structures, the quadratic terms disappear but the last two terms are left . Wefinally notice that χ ′ q = − χ q is a R q -connection if and only if A = 0, a result contradicting the useof connections in physics . However, when A = 0, we have χ ′ ( ξ ) = ξ and thus:¯ D ′ χ q +1 = ( Dχ q +1 )( ξ, η ) − ([ χ q ( ξ ) , χ q ( η )] + i ( ξ ) D ( χ q +1 ( η )) − i ( η ) D ( χ q +1 ( ξ )))= − [ χ q ( ξ ) , χ q ( η )] − χ q ([ ξ, η ])= − ([ χ ′ q ( ξ ) , χ ′ q ( η )] − χ ′ q ([ ξ, η ]))= − κ ′ q ( ξ, η )does not depend on the lift of χ q . COROLLARY 3.14 : When det ( A ) = 0 there is a second nonlinear Spencer sequence stabilizedat order q : 0 −→ aut ( X ) j q −→ Π q ¯ D −→ C ( T ) ¯ D −→ C ( T )where ¯ D and ¯ D are involutive and a restricted second nonlinear Spencer sequence :0 −→ Γ j q −→ R q ¯ D −→ C D −→ C such that ¯ D and ¯ D are involutive whenever R q is involutive. Proof : With | µ | = q we have χ kµ,i = − g kl A ri f lµ +1 r + terms ( order ≤ q ). Setting χ kµ,i = A ri τ kµ,r , we ob-tain τ kµ,r = − g kl f lµ +1 r + terms ( order ≤ q ) and ¯ D : Π q +1 → T ∗ ⊗ J q ( T ) restricts to ¯ D : Π q → C ( T ).Finally, setting A − = B = id − τ , we obtain successively: ∂ i χ kµ,j − ∂ j χ kµ,i + terms ( χ q ) − ( A ri χ kµ +1 r ,j − A rj χ kµ +1 r ,i ) = 0 B ir B js ( ∂ i χ kµ,j − ∂ j χ kµ,i ) + terms ( χ q ) − ( τ kµ +1 r ,s − τ kµ +1 s ,r ) = 0We obtain therefore Dτ q +1 + terms ( τ q ) = 0 and ¯ D ′ : T ∗ ⊗ J q ( T ) → ∧ T ∗ ⊗ J q − ( T ) restricts to¯ D : C ( T ) → C ( T ).In the case of Lie groups of transformations, the symbol of the involutive system R q must be g q = 0providing an isomorphism R q +1 ≃ R q ⇒ R q +1 ≃ R q and we have therefore C r = ∧ r T ∗ ⊗ R q for r = 1 , ..., n like in the linear Spencer sequence. Q.E.D. REMARK 3.15 : In the case of the (local) action of a Lie group G on X , we may consider thegraph of this action, that is the morphism X × G → X × X : ( x, a ) → ( x, y = f ( x, a )). If q islarge enough, then there is an isomorphism X × G → R q ⊂ Π q : ( x, a ) → j q ( f )( x, a ) obtainedby eliminating the parameters and C r = ∧ r T ∗ ⊗ R q . If { θ τ } with 1 ≤ τ ≤ dim ( G ) is a basisof infinitesimal generators of this action, there is a morphism of Lie algebroids over X , namely11 × G → R q : λ τ ( x ) → λ τ ( x ) j q ( θ τ ) when q is large enough and the linear Spencer sequence R q D −→ T ∗ ⊗ R q D −→ ∧ T ∗ ⊗ R q D −→ ... is locally exact because it is locally isomoprphic to thetensor product by G of the Poincar´e sequence ∧ T ∗ d −→ ∧ T ∗ d −→ ∧ T ∗ d −→ ... where d is theexterior derivative ([28]).We may also consider similarly dy = dax = daa − y and dx = dbb − dx = − a − dax , depending onthe choice of the independent variable among the source x or the target y .Surprisingly, in the case of Lie pseudogroups or Lie groupoids, the situation is quite different.We recall the way to introduce a groupoid structure on Π q, ⊂ J (Π q ) from the groupoid structureon Π q when ∆ = det ( ∂ i f k ( x ) = 0, that is how to define j ( h q ) = j ( g q ◦ f q ) = j ( g q ) ◦ j ( f q ). Weget successively with y = f ( x ): h ( x ) = ( g ◦ f )( x ) = g ( f ( x )) ⇒ ∂h r ∂x i = ∂g r ∂y k ∂f k ∂x i ⇒ h ri ( x ) = g rk ( f ( x ) f ki ( x ) ∂h ri ∂x j = ∂g rk ∂y l f ki ∂f l ∂x j + g rk ∂f ki ∂x j ⇒ h rij ( x ) = g rkl ( f ( x )) f ki ( x ) f lj ( x ) + g rk ( f ( x )) f kij ( x ) ∂h rij ∂x s = ∂g rki ∂y u f ki f lj ∂f u ∂x s + g rkl ( ∂f ki ∂x s f lj + f ki ∂f lj ∂x s ) + ∂g rk ∂y u ∂f u ∂x s f kij + g rk ∂f kij ∂x s ⇒ h rijs = g rklu f ki f lj f us + g rkl ( f kis f lj + f ki f ljs ) + g rku f us f kij + g rk f kijs and so on with more and more involved formulas.Now, if we want to obtain objects over the source x according to the non-linear Spencer sequence,we have only two possibilities in actual practice, namely: χ q = f − q +1 ◦ j ( f q ) − id q +1 ∈ T ∗ ⊗ J q ( T ) ↔ ¯ χ q = j ( f q ) − ◦ f q +1 − id q +1 ∈ T ∗ ⊗ J q ( T )As we have already considered the first, we have now only to study the second. In J (Π q ), we have: χ q + id q +1 = ( A kr , χ ki,r , χ kij,r , ... ) and ¯ χ q + id q +1 = ( ¯ A kr , ¯ χ ki,r , ¯ χ kij,r , ... ) over ( x, x, δ, , ... ) LEMMA 3.16 : ¯ χ q is a quasi-linear rational function of χ q , ∀ q ≥
0. With more details, when q = 0, we have ¯ χ = ¯ A − id and χ = A − id with ¯ A = A − = B and when q ≥
1, we have¯ χ q ◦ A = − χ q , that is to say ¯ χ q = − τ q . Proof : In the groupoid framework, we have:( ¯ χ q + id q +1 ) ◦ ( χ q + id q +1 ) = id q +1 ∈ J (Π q )Doing the substitutions: ∂g r ∂y k → ¯ A rk , ∂g rk ∂y l → ¯ χ rk,l , ∂g rkl ∂y u → ¯ χ rkl,u ∂f k ∂x i → A ki , ∂f ki ∂x j → χ ki,j , ∂f kij ∂x s → χ kij,s while using the fact that f ki = δ ki , f kij = 0 , ... and g rk = δ rk , g rkl = 0 , .. , we obtain at once:¯ A rk A ki = δ ri , ¯ χ rk,l A lj + χ ki,j = 0 , ¯ χ rij,u A us + χ rij,s = 0 , ... Proceeding by induction, we finally obtain:¯ χ kµ,r A rs + χ kµ,i = 0that is to say ¯ χ kµ,i + τ kµ,i = 0 because ∆ = 0 ⇒ det ( A ) = 0, thus ¯ χ q ◦ A = − χ q or, equivalently,¯ χ q = − τ q . Q.E.D.12 EMARK 3.17 : The passage from χ q to τ q is exactly the one done by E. and F. Cosserat in ([8],p 190), even though it is based on a subtle misunderstanding that we shall correct later on. REMARK 3.18 : According to the previous results, the ” field ” must be a section of the naturalbundle F of geometric objects if we use the nonlinear Janet sequence or a section of the firstSpencer bundle C if we use the nonlinear Spencer sequence. The aim of this paper is to provethat the second choice is by far more convenient for mathematical physics.
4) VARIATIONAL CALCULUS
It remains to graft a variational procedure adapted to the previous results. Contrary to whathappens in analytical mechanics or elasticity for example, the main idea is to vary sections but notpoints . Hence, we may introduce the variation δf k ( x ) = η k ( f ( x )) and set η k ( f ( x )) = ξ i ∂ i f k ( x )( x )along the ” vertical machinery ” but notations like δx i = ξ i or δy k = η k have no meaning at all .As a major result first discovered in specific cases by the brothers Cosserat in 1909 and by Weylin 1916, we shall prove and apply the following key result:THE PROCEDURE ONLY DEPENDS ON THE LINEAR SPENCER OPERATOR AND ITSFORMAL ADJOINT.In order to prove this result, if f q +1 , g q +1 , h q +1 ∈ Π q +1 can be composed in such a way that g ′ q +1 = g q +1 ◦ f q +1 = f q +1 ◦ h q +1 , we get:¯ Dg ′ q +1 = f − q +1 ◦ g − q +1 ◦ j ( g q ) ◦ j ( f q ) − id q +1 = f − q +1 ◦ ¯ Dg q +1 ◦ j ( f q ) + ¯ Df q +1 = h − q +1 ◦ f − q +1 ◦ j ( f q ) ◦ j ( h q ) − id q +1 = h − q +1 ◦ ¯ Df q +1 ◦ j ( h q ) + ¯ Dh q +1 Using the local exactness of the first nonlinear Spencer sequence or ([23], p 219), we may state:
LEMMA 4.1 : For any section f q +1 ∈ R q +1 , the finite gauge transformation : χ q ∈ T ∗ ⊗ R q −→ χ ′ q = f − q +1 ◦ χ q ◦ j ( f q ) + ¯ Df q +1 ∈ T ∗ ⊗ R q exchanges the solutions of the field equations ¯ D ′ χ q = 0.Introducing the formal Lie derivative on J q ( T ) by the formulas: L ( ξ q +1 ) η q = { ξ q +1 , η q +1 } + i ( ξ ) Dη q +1 = [ ξ q , η q ] + i ( η ) Dξ q +1 ( L ( j ( ξ q +1 )) χ q )( ζ ) = L ( ξ q +1 )( χ q ( ζ )) − χ q ([ ξ, ζ ]) LEMMA 4.2 : Passing to the limit over the source with h q +1 = id q +1 + tξ q +1 + ... for t →
0, weget an infinitesimal gauge transformation leading to the infinitesimal variation : δχ q = Dξ q +1 + L ( j ( ξ q +1 )) χ q (3)which does not depend on the parametrization of χ q . Setting ¯ ξ q +1 = ξ q +1 + χ q +1 ( ξ ), we get: δχ q = D ¯ ξ q +1 − { χ q +1 , ¯ ξ q +1 } (3 ∗ ) LEMMA 4.3 : Passing to the limit over the target with χ q = ¯ Df q +1 and g q +1 = id q +1 + tη q +1 + ... ,we get the other infinitesimal variation where Dη q +1 is over the target : δχ q = f − q +1 ◦ Dη q +1 ◦ j ( f q ) (4)which depends on the parametrization of χ q . 13 XAMPLE 4.4 : We obtain for q = 1: δχ k,i = ( ∂ i ξ k − ξ ki ) + ( ξ r ∂ r χ k,i + χ k,r ∂ i ξ r − χ r,i ξ kr )= ( ∂ i ¯ ξ k − ¯ ξ ki ) + ( χ kr,i ¯ ξ r − χ r,i ¯ ξ kr ) δχ kj,i = ( ∂ i ξ kj − ξ kij ) + ( ξ r ∂ r χ kj,i + χ kj,r ∂ i ξ r + χ kr,i ξ rj − χ rj,i ξ kr − χ r,i ξ kjr )= ( ∂ i ¯ ξ kj − ¯ ξ kij ) + ( χ krj,i ¯ ξ r + χ kr,i ¯ ξ rj − χ rj,i ¯ ξ kr − χ r,i ¯ ξ kjr )Introducing the inverse matrix B = A − , we obtain therefore equivalently: δA ki = ξ r ∂ r A ki + A kr ∂ i ξ r − A ri ξ kr ⇔ δB ik = ξ r ∂ r B ik − B rk ∂ r ξ i + B ir ξ rk both with: δχ kj,i = ( ∂ i ξ kj − A ri ξ kjr ) + ( ξ r ∂ r χ kj,i + χ kj,r ∂ i ξ r + χ kr,i ξ rj − χ rj,i ξ kr )For the Killing system R ⊂ J ( T ) with g = 0, these variations are exactly the ones that canbe found in ([8], (50)+(49), p 124 with a printing mistake corrected on p 128) when replacing a3 × The last unavoidable Proposition is thusessential in order to bring back the nonlinear framework of finite elasticity to the linear framewokof infinitesimal elasticity that only depends on the linear Spencer operator .For the conformal Killing system ˆ R ⊂ J ( T ) (see next section) we obtain: δχ rr,i = ( ∂ i ξ rr − ξ rri ) + ( ξ r ∂ r χ ss,i + χ ss,r ∂ i ξ r − χ s,i ξ rrs )but χ rr,i ( x ) dx i is far from being a 1-form. However, ( χ kj,i + γ kjs χ s,i ) ∈ T ∗ ⊗ T ∗ ⊗ T and thus( α i = χ rr,i + γ rrs χ s,i ) ∈ T ∗ is a pure 1-form if we replace ( χ rr,i , χ r,i ) by ( α i , α ( ζ ) is a scalarfor any ζ ∈ T and we have L ( ξ )( α ( ζ )) − α ([ ξ, ζ ]) = ( α r ∂ i ξ r + ξ r ∂ r α i ) ζ i . As we shall see in section V.A , we have ( L ( ξ ) γ ) kij = ξ kij for any section ξ ∈ J ( T ) and we obtain therefore successively: δα i = ( ∂ i ξ rr − ξ rri ) + ( α r ∂ i ξ r + ξ r ∂ r α i ) ϕ ij = ∂ i α j − ∂ j α i ⇒ δϕ ij = ( ∂ j ξ rri − ∂ i ξ rrj ) + ( ϕ rj ∂ i ξ r + ϕ ir ∂ j ξ r + ξ r ∂ r ϕ ij )These are exactly the variations obtained by Weyl ([54], (76), p 289) who was assuming im-plicitly A = 0 when setting ¯ ξ rr = 0 ⇔ ξ rr = − α i ξ i by introducing a connection. Accordingly, ξ rri is the variation of the EM potential itself, that is the δA i of engineers used in order to exhibitthe Maxwell equations from a variational principle ([54], §
26) but the introduction of the Spenceroperator is new in this framework.The explicit general formulas of the two lemma cannot be found somewhere else (The readermay compare them to the ones obtained in [19] by means of the so-called ” diagonal ” method thatcannot be applied to the study of explicit examples). The following unusual difficult propositiongeneralizes well known variational techniques used in continuum mechanics and will be cruciallyused for applications:
PROPOSITION 4.5 : The same variation is obtained whenever η q = f q +1 ( ξ q + χ q ( ξ )) with χ q = ¯ Df q +1 , a transformation only depending on j ( f q ) and invertible if and only if det ( A ) = 0. Proof : First of all, setting ¯ ξ q = ξ q + χ q ( ξ ), we get ¯ ξ = A ( ξ ) for q = 0, a transformation which isinvertible if and only if det ( A ) = 0. In the nonlinear framework, we have to keep in mind thatthere is no need to vary the object ω which is given but only the need to vary the section f q +1 aswe already saw, using η q ∈ R q ( Y ) over the target or ξ q ∈ R q over the source . With η q = f q +1 ( ξ q ),we obtain for example: δf k = η k = f kr ξ r δf ki = η ku f ui = f kr ξ ri + f kri ξ r δf kij = η kuv f ui f vj + η ku f uij = f kr ξ rij + f kri ξ rj + f krj ξ ri + f krij ξ r and so on. Introducing the formal derivatives d i for i = 1 , ..., n , we have: δf kµ = ζ kµ ( f q , η q ) = d µ η k = η ku f uµ + ... = f kr ξ rµ + ... + f kµ +1 r ξ r
14e shall denote by ♯ ( η q ) = ζ kµ ( y q , η q ) ∂∂y kµ ∈ V ( R q ) with ζ k = η k the corresponding vertical vectorfield, namely: ♯ ( η q ) = 0 ∂∂x i + η k ( y ) ∂∂y k + ( η ku ( y ) y ui ) ∂∂y ki + ( η kuv ( y ) y ui y vj + η ku ( y ) y uij ) ∂∂y kij + ... However, the standard prolongation of an infinitesimal change of source coordinates described bythe horizontal vector field ξ , obtained by replacing all the derivatives of ξ by a section ξ q ∈ R q over ξ ∈ T , is the vector field: ♭ ( ξ q ) = ξ i ( x ) ∂∂x i + 0 ∂∂y k − ( y kr ξ ri ( x )) ∂∂y ki − ( y kr ξ rij ( x ) + y krj ξ ri ( x ) + y kri ξ rj ( x )) ∂∂y kij + ... It can be proved that [ ♭ ( ξ ) q , ♭ ( ξ ′ q ] = ♭ ([ ξ q , ξ ′ q ]) , ∀ ξ q , ξ ′ q ∈ R q over the source , with a similar propertyfor ♯ ( . ) over the target ([25]). However, ♭ ( ξ q ) is not a vertical vector field and cannot therefore becompared to ♯ ( η q ).The solution of this problem explains a strange comment made by Weyl in ([53],p 289 + (78), p 290) and which became a founding stone of classical gauge theory. Indeed, ξ rr is not a scalar because ξ ki is not a 2-tensor. However, when A = 0, then − χ q is a R q -connection and¯ ξ rr = ξ rr + χ rr,i ξ i is a true scalar that may be set equal to zero in order to obtain ξ rr = − χ rr,i ξ i , afact explaining why the EM-potential is considered as a connection in quantum mechanics insteadof using the second order jets ξ rri of the conformal system, with a shift by one step in the physicalinterpretation of the Spencer sequence (See [27] for more historical details).The main idea is to consider the vertical vector field T ( f q )( ξ ) − ♭ ( ξ q ) ∈ V ( R q ) whenever y q = f q ( x ).Passing to the limit t → g q ◦ f q = f q ◦ h q , we first get g ◦ f = f ◦ h ⇒ f ( x ) + tη ( f ( x )) + ... = f ( x + tξ ( x ) + ... ). Using the chain rule for derivatives and substituting jets,we get successively: δf k ( x ) = ξ r ∂ r f k , δf ki = ξ r ∂ r f ki + f kr ξ ri , δf kij = ξ r ∂ r f kij + f krj ξ ri + f kri ξ rj + f kr ξ rij and so on, replacing ξ r f kµ +1 r by ξ r ∂ r f kµ in η q = f q +1 ( ξ q ) in order to obtain: δf kµ = η kr f rµ + ... = ξ i ( ∂ i f kµ − f kµ +1 i ) + f kµ +1 r ξ r + ... + f kr ξ rµ where the right member only depends on j ( f q ) when | µ | = q .Finally, we may write the symbolic formula f q +1 ( χ q ) = j ( f q ) − f q +1 = Df q +1 ∈ T ∗ ⊗ V ( R q ) inthe explicit form: f kr χ rµ,i + ... + f kµ +1 r χ r,i = ∂ i f kµ − f kµ +1 i Substituting in the previous formula provides η q = f q +1 ( ξ q + χ q ( ξ )) and we just need to replace q by q + 1 in order to achieve the proof.Checking directly the proposition is not evident even when q = 0 as we have:( ∂η k ∂y u − η ku ) ∂ i f u = f kr [( ∂ i ¯ ξ r − ¯ ξ ri ) − ( χ s,i ¯ ξ rs − χ rs,i ¯ ξ s )]but cannot be done by hand when q ≥
1. Q.E.D.For an arbitrary vector bundle E and involutive system R q ⊆ J q ( E ), we may define the r - prolongations ρ r ( R q ) = R q + r = J r ( R q ) ∩ J q + r ( E ) ⊂ J r ( J q ( E )) and their respective symbols g q + r = ρ r ( g q ) defined from g q ⊆ S q T ∗ ⊗ E where S q T ∗ is the vector bundle of q -symmetriccovariant tensors. Using the Spencer δ -map, we now recall the definition of the Spencer bundles : C r = ∧ r T ∗ ⊗ R q /δ ( ∧ r − T ∗ ⊗ g q +1 ) ⊆ ∧ r T ∗ ⊗ J q ( E ) /δ ( ∧ r − T ∗ ⊗ S q +1 ) T ∗ ⊗ E ) = C r ( E )and of the Janet bundles : F r = ∧ r T ∗ ⊗ J q ( E ) / ( ∧ r T ∗ ⊗ R q + δ ∧ r − T ∗ ⊗ S q +1 T ∗ ⊗ E )15hen D = Φ ◦ j q , we may obtain by induction on r the following fundamental diagram I relatingthe second linear Spencer sequence to the linear Janet sequence with epimorphisms Φ = Φ , ..., Φ n :0 0 0 0 ↓ ↓ ↓ ↓ → Θ j q −→ C D −→ C D −→ C D −→ ... D n → C n → ↓ ↓ ↓ ↓ → E j q −→ C ( E ) D −→ C ( E ) D −→ C ( E ) D −→ ... D n −→ C n ( E ) → k ↓ Φ ↓ Φ ↓ Φ ↓ Φ n → Θ → E D −→ F D −→ F D −→ F D −→ ... D n −→ F n → ↓ ↓ ↓ ↓ C if and only if theJanet sequence is locally exact at F because the central sequence is locally exact (See [25],[28],[30]for more details). In the present situation, we shall always have E = T . The situation is muchmore complicate in the nonlinear framewok and we provide details for a later use .Let ¯ ω be a section of F satisfying the same CC as ω , namely I ( j ( ω )) = 0. As F is a quotientof Π q , we may find a section f q ∈ Π q such that:Φ ◦ f q ≡ f − q ( ω ) = ¯ ω ⇒ ρ (Φ) ◦ j ( f q ) ≡ j ( f − q )( j ( ω )) = j ( f − q ( ω )) = j (¯ ω )Similarly, as F is a natural bundle of order q , then J ( F ) is a natural bundle of order q + 1 andwe can find a section f q +1 ∈ Π q +1 such that: ρ (Φ) ◦ f q +1 ≡ f − q +1 ( j ( ω )) = j (¯ ω )and we are facing two possible but quite different situations: • Eliminating ¯ ω , we obtain: j ( f − q )( j ( ω )) = f − q +1 ( j ( ω )) ⇒ ( f q +1 ◦ j ( f − q )) − ( j ( ω )) − j ( ω ) = L ( σ q ) ω = 0and thus σ q = ¯ Df − q +1 ∈ T ∗ ⊗ R q = − f q +1 ◦ χ q ◦ j ( f ) − over the target if we set χ q = ¯ Df q +1 = f − q +1 ◦ j ( f q ) − id q +1 over the source , even if f q +1 may not be a section of R q +1 . As σ q is killed by¯ D ′ , we have related cocycles at F in the Janet sequence over the source with cocycles at T ∗ ⊗ R q or C over the target . • Eliminating ω , we obtain successively:( f − q +1 ◦ j ( f q ))( j (¯ ω )) − j (¯ ω ) = − ( f − q +1 ◦ j ( f q ))[ f − q +1 ◦ j ( f q )) − ( j (¯ ω ) − j (¯ ω )]= − ( f − q +1 ◦ j ( f q )) L ( χ q )¯ ω where we have over the source : L ( χ q )¯ ω = ( ¯Ω τi ≡ − L τµk (¯ ω ( x )) χ kµ,i + χ r,i ∂ r ¯ ω τ ( x )) ∈ T ∗ ⊗ F However, we know that F is associated with R q and is thus not affected by f − q +1 ◦ j ( f q ) whichprojects onto f − q ◦ f q = id q . Hence, only T ∗ is affected by f − ◦ j ( f ) = A in a covariant way andwe obtain therefore over the source :( f − q +1 ◦ j ( f q ))( j (¯ ω )) − j (¯ ω ) = − BL ( χ q )¯ ω = − L ( τ q )¯ ω = 0where B = A − . It follows that χ q ∈ T ∗ ⊗ R q (¯ ω ) with ¯ D ′ χ q = 0 in the first non-linear Spencersequence for R q (¯ ω ) ⊂ J q ( T ).We invite the reader to follow all the formulas involved in these technical results on the nextexamples. Of course, whenever R q is formally integrable and f q +1 ∈ R q +1 is a lift of f q ∈ R q ,16hen we have ¯ ω = ω and ξ q ∈ T ∗ ⊗ R q because R q ( ω ) = R q . EXAMPLE 4.6 : In the case of Riemannian structures, we have
F ∈ S T ∗ because we deal witha non-degenerate metric ω = ( ω ij ) ∈ S T ∗ with det ( ω ) = 0 and may introduce ω − = ( ω ij ) ∈ S T .We have by definition ω kl ( f ( x )) f ki ( x ) f lj ( x ) = ¯ ω ij ( x ) that we shall simply write ω kl ( f ) f ki f lj = ¯ ω ij ( x )and obtain therefore: ω kl ( f ) f lj ∂ r f ki + ω kl ( f ) f ki ∂ r f lj + ∂ω kl ∂y u ( f ) f ki f lj ∂ r f u − ∂ r ¯ ω ij ( x ) = 0Our purpose is now to compute the expression: ω kl ( f ) f lj f kir + ω kl ( f ) f ki f ljr + ∂ω kl ∂y u ( f ) f ki f lj f ur − ∂ r ¯ ω ij ( x ) = 0In order to eliminate the derivatives of ω over te target we may multiply the first equation by B and substract from the second while using the fact that ω kl ( f ) = ¯ ω ij ( x ) g ik g jl with χ = A − id T ⇒ τ = Bχ = id T − B in order to get: − (¯ ω sj τ si,r + ¯ ω is τ sj,r + τ s,r ∂ s ¯ ω ij ) = − ( L ( τ )¯ ω ) ij,r These results can be extended at once to any tensorial geometric object but the conformal caseneeds more work and we let the reader treat it as an exercise. He will discover that the standardelimination of a conformal factor is not the best way to use in order to understand the conformalstructure which has to do with a tensor density and no longer with a tensor.
REMARK 4.7 : In the non-linear case, the non-linear CC of the system R q defined by Φ( y q ) =¯ ω ( x ) only depend on the differential invariants and are exactly the ones satisfied by ω in the sensethat they have the same Vessiot structure constants whenever R q is formally integrable, in par-ticular involutive as shown in Example 2 .
7. Accordingly, we can always find f q +1 over f q . In thelinear case, the procedure is similar but slightly simpler. Indeed, if D : T → F is an involutive Lieoperator, we may consider only the initial part of the fundamental diagram I : SP EN CER ↓ ↓ ↓ → Θ j q → C D → C D → C ↓ ↓ ↓ → T j q → C ( T ) D → C ( T ) D → C ( T ) k ↓ Φ ↓ Φ → Θ → T D → F D → F ↓ ↓ JAN ET ↓ ↓ → g q +1 − δ → δ ( g q +1 ) → ↓ ↓ → Θ j q +1 → R q +1 D → T ∗ ⊗ R q k ↓ ↓ → Θ j q → R q D → C ↓ ↓ D ξ = Ω with Ω ∈ F and D Ω = 0. If wepick up any lift ξ q ∈ C ( T ) = J q ( T ) of Ω and chase, we notice that X = D ξ q ∈ C ⊂ C ( T ) issuch that D X = 0. In the Example 2 .
7, using the involutive system R ′ = R (1)1 ⊂ R ⊂ J ( T ),17e have m = n = 2 , q = 1 and the fiber dimensions:0 0 0 ↓ ↓ ↓ → Θ j → D → D → → ↓ ↓ k → j → D → D → → k ↓ Φ ↓ Φ ↓ → Θ → D → D → → ↓ ↓ ↓ ↓ → − δ → → ↓ ↓ → Θ j → D → k ↓ ↓ → Θ j → D → ↓ ↓ α = ( α , α ), denote its variation by A = ( A , A ) andconsider only the linear inhomogeneous system D ξ = L ( ξ ) α = A with no CC for A . If theground differential field is K = Q ( x , x ) with commuting derivations ( d , d ), let us choose α = x dx = ( x , , A = ( x , x ). As a lift ξ ∈ J ( T ) of A , we let the reader check that wemay choose in K : ξ = 0 , ξ = 0 , ξ = 1 , ξ = x x , ξ = 0 , ξ = 0Using one prolongation, we have: d A ≡ x ξ + ξ = 0 , d A ≡ x ξ + ξ + ξ = 1 , d A ≡ x ξ = 1 , d A ≡ x ξ + ξ = 0If β = − dα = dx ∧ dx , we may denote its variation by B and get at once B = d A − d A ≡ ξ + ξ = 0. Such a result is contradicting our inital choice 1 + 0 = 1 and we cannot therefore find alift ξ of j ( A ). Hence, we have to introduce the new geometric object ω = ( α, β ) with Ω = ( A, B )and CC dα + β = 0 leading to d A − d A + B = 0 while using the previous diagrams. We cantherefore lift Ω = ( A, B ) to ξ ∈ J ( T ) by choosing in K : ξ = 0 , ξ = 0 , ξ = 1 , ξ = x x , ξ = 0 , ξ = − d B ≡ ξ + ξ = 0 , d B ≡ ξ + ξ = 0and lift j (Ω) to ξ ∈ J ( T ) over ξ ∈ J ( T ) by choosing in K : ξ = 0 , ξ = 1 x , ξ = − x ( x ) , ξ = 0 , ξ = 0 , ξ = − x The image of the Spencer operator is X = Dξ = j ( ξ ) − ξ that is to say: X , = − , X , = − x x , X , = 0 , X , = 1 ,X , = 0 , X , = 0 , X , = − x , X , = 0 , X , = 0 , X , = 0 , X , = 0 , X , = 1 x X ∈ T ∗ ⊗ R , namely: x X ,i + X ,i = 0 , X ,i = 0 , X ,i + X ,i = 0 , ∀ i = 1 ,
2a result which is not evident at first sight and has no meaning in any classical approach becausewe use sections and not solutions .Now, if f q +1 , f ′ q +1 ∈ Π q +1 are such that f − q +1 ( j ( ω )) = f ′− q +1 ( j ( ω )) = j (¯ ω ), it follows that( f ′ q +1 ◦ f − q +1 )( j ( ω )) = j ( ω ) ⇒ ∃ g q +1 ∈ R q +1 such that f ′ q +1 = g q +1 ◦ f q +1 and the new σ ′ q = ¯ Df ′− q +1 differs from the initial σ q = ¯ Df − q +1 by a gauge transformation.Conversely, let f q +1 , f ′ q +1 ∈ Π q +1 be such that σ q = ¯ Df − q +1 = ¯ Df ′− q +1 = σ ′ q . It follows that¯ D ( f − q +1 ◦ f ′ q +1 ) = 0 and one can find g ∈ aut ( X ) such that f ′ q +1 = f q +1 ◦ j q +1 ( g ) providing¯ ω ′ = f ′− q ( ω ) = ( f q ◦ j q ( g )) − ( ω ) = j q ( g ) − ( f − q ( ω )) = j q ( g ) − (¯ ω ). PROPOSITION 4.8 : Natural transformations of F over the source in the nonlinear Janet se-quence correspond to gauge transformations of T ∗ ⊗ R q or C over the target in the nonlinearSpencer sequence. Similarly, the Lie derivative D ξ = L ( ξ ) ω ∈ F in the linear Janet sequencecorresponds to the Spencer operator Dξ q +1 ∈ T ∗ ⊗ R q or D ξ q ∈ C in the linear Spencer sequence.With a slight abuse of language δf = η ◦ f ⇔ δf ◦ f − = η ⇔ f − ◦ δf = ξ when η = T ( f )( ξ ) andwe get j q ( f ) − ( ω ) = ¯ ω ⇒ j q ( f + δf ) − ( ω ) = ¯ ω + δ ¯ ω that is j q ( f − ◦ ( f + δf )) − (¯ ω ) = ¯ ω + δ ¯ ω ⇒ δ ¯ ω = L ( ξ )¯ ω and j q (( f + δf ) ◦ f − ◦ f ) − ( ω ) = j q ( f ) − ( j q (( f + δf ) ◦ f − ) − ( ω )) ⇒ δ ¯ ω = j q ( f ) − ( L ( η ) ω ).Passing to the infinitesimal point of view, we obtain the following generalization of Remark 3.12which is important for applications (See [2] for details). COROLLARY 4.9 : ¯Ω = δ ¯ ω = L ( ξ q )¯ ω = f − q ( L ( η q ) ω ) ⇒ δ ¯ ω = L ( ξ )¯ ω = j q ( f ) − ( L ( η ) ω ).Recapitulating the results so far obtained concerning the links existing between the source andthe target points of view, we may set in a symbolic way: δf q ( f q ) ←→ η q ( f q +1 ) ←→ ¯ ξ q ( χ q ) ←→ ξ q In order to help the reader maturing the corresponding nontrivial formulas, we compute explicitlythe case n = 1 , q = 1 , n arbitrary left to the reader as a difficult exercise thatcannot be achieved by hand when q ≥ EXAMPLE 4.10 : Using the previous formulas, we have δf ( x ) = η ( f ( x )) , δf x ( x ) = η y ( f ( x )) f x ( x )and : η = f ( ¯ ξ ) ⇒ ( η ( f ( x )) = f x ( x ) ¯ ξ ( x ) , η y ( f ( x ) f x ( x ) = f x ( x ) ¯ ξ x ( x ) + f xx ( x ) ¯ ξ ( x ))The delicate point is that we have successively: χ ,x = ∂ x ff x − A − , χ x,x = 1 f x ( ∂ x f x − ∂ x ff x f xx )¯ ξ = ξ + χ ,x ( ξ ) = ∂ x ff x ξ = Aξ, ¯ ξ x = ξ x + χ x,x ξ ⇒ η = ∂ x f ξ, η y = ξ x + ∂ x f x f x ξf x η yy = ξ xx + f xx f x ξ x + ( ∂ x f xx f x − f xx ( f x ) ∂ x f x ) ξ When z = g ( y ) , y = f ( x ) ⇒ z = ( g ◦ f )( x ) = h ( x ), we obtain therefore the simple groupoidcomposition formulas h x ( x ) = g y ( f ( x )) f x ( x ) and thus: ζ = ∂ x hξ = ∂ y gη = ∂ y g∂ x f ξ, ζ z = η y + ∂ y g y g y η = ξ x + ( ∂ y g y g y ∂ x f + ∂ x f x f x ) ξ = ξ x + ∂ x h x h x ξ η k = f kr ¯ ξ r , η ku f ui = f kr ¯ ξ ri + f kri ¯ ξ r η k ⇒ η k = ξ r ∂ r f k , η ku f ui = f ks ( ξ si + g su ( ∂ r f ui − A tr f uti ) ξ r ) + f kti A tr ξ r η ku = g iu f ks ξ si + ξ r g iu ∂ r f ki ⇒ η kk = ξ rr + ξ r g iu ∂ r f ui As a very useful application, we obtain successively:∆( x ) = det ( ∂ i f k ( x )) ⇒ δ ∆ = ∆ ∂η k ∂y k = ∆ ∂ r ξ r + ξ r ∂ r ∆ = ∂ r ( ξ r ∆) δdet ( A ) = det ( A )( ∂η k ∂y k − η kk ) = det ( A )( ∂ r ξ r − ξ rr ) + ξ r ∂ r det ( A )where sections of jet bundles are used in an essential way, and the important lemma: LEMMA 4.11 : When the transformation y = f ( x ) is invertible with inverse x = g ( y ), we havethe fundamental identity over the source or over the target: ∂∂x i (∆( x ) ∂g i ∂y k ( f ( x ))) ≡ , ∀ x ∈ X ⇔ ∂∂y k ( 1∆( g ( y )) ∂f k ∂x i ( g ( y ))) ≡ , ∀ y ∈ Y EXAMPLE 4.12 : We proceed the same way for studying the links existing between χ q = ¯ Df q +1 over the source, χ q − = σ q = ¯ Df − q +1 over the target and the nonlinear Spencer operator. First ofall, we notice that: σ q = f q +1 ◦ j ( f q − ) − id q +1 = f q +1 ◦ ( id q +1 − f q +1 − j ( f q )) ◦ j ( f q ) − = − f q +1 ◦ χ q ◦ j ( f q ) − and the components of σ q thus factor through linear combinations of the components of χ q . Aftertedious computations, we get successively when m = n = 1: χ ,x = ∂ x ff x − A − f x ( ∂ x f − f x ) χ x,x = 1 f x ( ∂ x f x − ∂ x ff x f xx ) = 1 f x ( ∂ x f x − f xx ) − f xx ( f x ) ( ∂ x f − f x ) χ xx,x = f x ( ∂ x f xx − ∂ x ff x f xxx ) − f xx ( f x ) ( ∂ x f x − ∂ x ff x f xx )= f x ( ∂ x f xx − f xxx ) − f xx ( f x ) ( ∂ x f x − f xx ) + (2 ( f xx ) ( f x ) − f xxx ( f x ) )( ∂ x f − f x )These formulas agree with the successive constructive/inductive identities: χ ,x f x = ∂ x f − f x χ x,x f x + χ ,x f xx = ∂ x f x − f xx χ xx,x f x + 2 χ x,x f xx + χ ,x f xxx = ∂ x f xx − f xxx showing that χ q is linearly depending on Df q +1 and we finally get: σ ,y = − ( ∂ x f − f x ) ∂ x f = f x ∂ x f − − f x χ ,x ∂ x f σ y,y = − f x ( ∂ x f x − f xx ) ∂ x f = − ( χ x,x + f xx f x χ ,x ) ∂ x f σ yy,y = − (( f x ) ( ∂ x f xx − f xxx ) − f xx ( f x ) ( ∂ x f x − f xx )) ∂ x f = − ( f x χ xx,x + f xx ( f x ) χ x,x + ( f xxx ( f x ) − ( f xx ) ( f x ) ) χ ,x ) ∂ x f while using successively the relations g y f x = 1 , ∂ y g∂ x f = 1 , g yy ( f x ) + g y f xx = 0 and so on when x = g ( y ) is the inverse of y = f ( x ), in a coherent way with the action of f on J ( T ) which isdescribed as follows: η = f x ξη y = ξ x + f xx f x ξη yy = f x ξ xx + f xx ( f x ) ξ x + ( f xxx ( f x ) − ( f xx ) ( f x ) ) ξ y xx = 0 ⇒ ξ xx = 0, we have thus y xx = 0 , y xxx = 0 ⇒ f xx = 0 , f xxx = 0. It follows that η = f x ξ, η y = ξ x , η yy = f x ξ xx = 0on one side and χ xx,x = 0 ⇔ σ yy,y = 0 in a coherent way. It is finally important to notice thatthese results are not evident, even when m = n = 1, as soon as second order jets are involved.We shall use all the preceding formulas in the next example showing that, contrary to whathappens in elasticity theory where the source is usually identified with the Lagrange variables, inboth the Vessiot/Janet and the Cartan/Spencer approaches, the source must be identified with theEuler variables without any possible doubt . EXAMPLE 4.13 : With n = 1 , q = 1 , F = T ∗ and the finite OD Lie equation ω ( y ) y x = ω ( x ) with ω ∈ T ∗ and corresponding Lie operator D ξ ≡ L ( ξ ) ω = ω ( x ) ∂ x ξ + ξ∂ x ω ( x ) over the source , we have: ω ( f ( x )) f x ( x ) = ¯ ω ( x ) , ω ( f ( x )) f xx ( x ) + ∂ y ω ( f ( x )) f x ( x ) = ∂ x ¯ ω ( x )Differentiating once the first equation and substracting the second, we obtain therefore: ωσ y,y + σ ,y ∂ y ω ≡ − ω (1 /f x )( ∂ x f x − f xx )(1 /∂ x f ) + (( f x /∂ x f ) − ∂ y ω = 0whenever y = f ( x ). Finally, setting ω ( f ( x )) ∂ x f ( x ) = ¯ ω ( x ), we get over the target : δ ¯ ω = ω ( f ( x )) ∂η∂y ∂ x f ( x ) + ∂ x f ( x ) ∂ω∂y ( f ( x )) η = ∂ x f L ( η ) ω Differentiating η = ξ∂ x f in order to obtain ∂η∂y = ∂ x ξ + ξ ( ∂ xx f /∂ x f ), we get over the source : δ ¯ ω = ¯ ω∂ x ξ + ξ∂ x ¯ ω = L ( ξ )¯ ω We may summarize these results as follows: δ ¯ ω = L ( ξ )¯ ω ( j ( f )) −→ δ ¯ ω = ∂ x f L ( η ) ω We invite the reader to extend this result to an arbitrary dimension n ≥ EXAMPLE 4.14 : The case of an affine stucture needs more work with n = m = 1 , q = 2. Indeed,let us consider the action of the affine Lie group of transformations ¯ y = ay + b with a, b = cst acting on the target y ∈ Y considered as a copy of the real line X . We obtain the prolongationsup to order 2 of the 2 infinitesimal generators of the action: a → y ∂∂y + y x ∂∂y x + y xx ∂∂y xx , b → ∂∂y + 0 ∂∂y x + 0 ∂∂y xx There cannot be any differential invariant of order 1 and the only generating one of order 2 canbe Φ ≡ y xx /y x . When ¯ x = ϕ ( x ) we get successively y x = y ¯ x ∂ x ϕ, y xx = y ¯ x ¯ x ( ∂ x ϕ ) + y ¯ x ∂ xx ϕ and Φtransforms like u = ∂ x ϕ ¯ u + ∂ xx ϕ∂ x ϕ a result providing the bundle of geometric objects F with localcoordinates ( x, u ) and corresponding transition rules. For any section γ , we get the Vessiot generalsystem R ⊂ Π of second order finite Lie equations y xx y x + γ ( y ) y x = γ ( x ) which is already in Lieform and relates the jet coordinates ( x, y, y x , y xx ) of order 2. The special section is γ = 0 and wemay consider the automorphic system Φ ≡ y xx y x = ¯ γ ( x ) obtained by introducing any second ordersection f ( x ) = ( f ( x ) , f x ( x ) , f xx ( x )), for example f = j ( f ) providing ( f ( x ) , ∂ x f ( x ) , ∂ xx f ( x )). Itis not at all evident, even on such an elementary example , to compute the variation ¯Γ = δ ¯ γ inducedby the previous formulas and to prove that, like any field quantity, it only depends on ¯ γ on thecondition to use only source quantities. For this, setting f xx ( x ) f x ( x ) = ¯ γ ( x ), varying and substituting,we obtain: ¯Γ = δ ¯ γ = δf xx f x − f xx ( f x ) δf x = f x η yy = ξ xx + ¯ γξ x + ξ∂ x ¯ γ Now, linearizing the preceding Lie equation over the identity transformation y = x , we get theMedolaghi equation: L ( ξ ) γ ≡ ξ xx + γ ( x ) ξ x + ξ∂ x γ ( x ) = 0 , ∀ ξ ∈ R ⊂ J ( T )21nd the striking formula ¯Γ = δ ¯ γ = L ( ξ )¯ γ over the source for an arbitrary ξ ∈ J ( T ). We finallypoint out the fact that, as we have just shown above and contrary to what the brothers Cosserat hadin mind, the first order operators involved in the nonlinear Spencer sequence have strictly nothingto do with the operators involved in the nonlinear Janet sequence whenever q ≥
2. For example, inthe present situation, χ ,x = ∂ x ff x − ≡ f xx f x . Similarly, using Remark 4 . D : ( ξ, ξ x ) → ( ∂ x ξ − ξ x , ∂ x ξ x )on one side and the second order Lie operator D : ξ → ∂ xx ξ on the other side.The next delicate example proves nevertheless that target quantities may also be used. EXAMPLE 4.15 : In the last example, depending on the way we use ¯ γ ( x ) on the source or γ ( y )on the target, we may consider the two (very different) Medolaghi equations: ξ xx + ¯ γ ( x ) ξ x + ξ∂ x ¯ γ ( x ) = 0 ↔ η yy + γ ( y ) η y + η∂ y γ ( y ) = 0Now, starting from the single OD equation f xx f x = ¯ γ ( x ) in sectional notations, we may successively differentiate and prolongate once in order to get: ∂ x f xx f x − f xx ( f x ) ∂ x f x = ∂ x ¯ γ ( x ) ↔ f xxx f x − ( f xx f x ) = ∂ x ¯ γ ( x )Substracting the second from the first as a way to eliminate ¯ γ , we obtain a linear relation involv-ing only the components of the nonlinear Spencer operator in a coherent way with the theory ofnonlinear systems ([30],[41]), namely:1 f x ( ∂ x f xx − f xxx ) − f xx ( f x ) ( ∂ x f x − f xx ) = 0At first sight it does not seem possible to know whether we have a linear combination of thecomponents of χ or of the components of σ . However, if we come back to the original situation f − q ( ω ) = ¯ ω , we have eliminated j (¯ γ ) over the source and we are thus only left with j ( γ ) overthe target. Hence it can only depend on σ and we find indeed the striking relation: − f x [ 1 f x ( ∂ x f xx − f xxx ) − f xx ( f x ) ( ∂ x f x − f xx )] 1 ∂ x f = σ yy,y = 0provided by the simple second order Medolaghi equation γ = 0 ⇒ η yy = 0 over the target. Itis essential to notice that no classical technique can provide these results which are essentiallydepending on the nonlinear Spencer operator and are thus not known today. EXAMPLE 4.16 : The above methods can be applied to any explicit example. The reader maytreat as an exercise the case of the pseudogroup of isometries of a non degenerate metric which canbe found in any textbook of continuum mechanics or elasticity theory, though in a very differentframework with methods only valid for tensors. With the previous notations, let ω ∈ S T ∗ with det ( ω ) = 0 and consider the following nonlinear system ω kl ( f ( x )) ∂ i f k ( x ) ∂ j f l ( x ) = ¯ ω ij ( x ) with1 ≤ i, j, k, l ≤ n . One obtains therefore: δ ¯ ω ij = ¯ ω rj ∂ i ξ r + ¯ ω ir ∂ i ξ u + ξ r ∂ r ¯ ω ij = ∂ i f k ∂ j f l ( ω ul ∂η u ∂y k + ω ku ∂η u ∂y l + η u ∂ω kl ∂y u )and thus the same recapitulating formulas linking the source and target variations: δ ¯ ω = L ( ξ )¯ ω ( j ( f )) −→ δ ¯ ω = ( ∂ i f k ∂ j f l ( L ( η ) ω ) kl )It is also difficult to compute or compare the variational formulas over the source and target inthe nonlinear Spencer sequence, even when m = n = 1 and q = 0 , EXAMPLE 4.17 : Let us prove that the explicit computation of the gauge transformation is atthe limit of what can be done with the hand, even when m = n = 1 , q = 1. We have successively: χ ,x = ∂ x ff x − , χ x,x = 1 f x ( ∂ x f x − ∂ x ff x f xx )22 ′ ( x ) = g ( f ( x )) ⇒ f ′ x = g y f x , f ′ xx = g yy ( f x ) + g y f xx and thus: χ ′ ,x = ∂ x f ′ f ′ x − ∂ y g∂ x fg y f x − χ ,y + 1) ∂ x ff x − ∂ x ff x χ ,y + ( ∂ x ff x − χ ′ x,x = f ′ x ( ∂ x f ′ x − ∂ x f ′ f ′ x f ′ xx )= g y f x ( ∂ y g y ( ∂ x f ) f x + g y ∂ x f x − ∂ y g∂ x fg y f x ( g yy ( f x ) + g y f xx ))= g y ∂ y g y ∂ x f + ∂ x f x f x − ∂ y g∂ x fg yy ( g y ) − ∂ y g∂ x ff xx g y ( f x ) = ( ∂ x f χ y,y − ∂ x ff xx ( f x ) χ ,y ) + f x ( ∂ x f x − ∂ x ff x f xx )Setting f = id + tξ + ... and passing to the limit when t →
0, we finally obtain: δχ ,x = ( ∂ x ξ − ξ x ) + ( ξ∂ x χ ,x + χ ,x ∂ x ξ − χ ,x ξ x ) δχ x,x = ( ∂ x ξ x − ξ xx ) + ( ξ∂ x χ x,x + χ x,x ∂ x ξ − χ ,x ξ xx )If we use the standard euclidean metric ω = 1 ⇒ γ = 0, we may thus introduce the pure 1-form α = χ x,x + γχ ,x . We should consider the defining formula χ ′ = f − ◦ χ ◦ j ( f ) + ¯ Df and haveto introduce the addidtional term f − ( γ ) χ ,x which is only leading to the additional infinitesimalterm ( L ( ξ ) γ ) χ ,x = ξ xx χ ,x because γ = 0. We finally obtain: δα = δχ x,x + ξ xx χ ,x + γδχ ,x = ( ∂ x ξ x − ξ xx ) + ( ξ∂ x α + α∂ x ξ )and this result can be easily extended to an arbitrary dimension with the formula: α i = χ rr,i + γ ssr χ r,i ⇒ ( δα ) i = ( ∂ i ξ rr − ξ rri ) + ( ξ r ∂ r α i + α r ∂ i ξ r )Comparing this procedure with the one we have adopted in the previous exampes, we have: χ ,x = ∂ x ff x − A − ⇒ δχ ,x = ∂ x δff x − ∂ x f ( f x ) δf x = 1 f x ( ∂η∂y − η y ) ∂ x f However, taking into account the formulas η = ξ∂ x f and η y = ξ x + ∂ x f x f x ξ , we also get: δχ ,x = f x ( ∂ x ξ∂ x f + ξ∂ xx f ) − ∂ x f ( f x ) ( ξ x f x + ξ∂ x f x )= A ( ∂ x ξ − ξ x ) + ξ∂ x χ ,x = ( ∂ x ξ − ξ x ) + ( ξ∂ x χ ,x + χ ,x ∂ x ξ − χ ,x ξ x )Working over the target is more difficult. Indeed, we have successively ( care to the first step ): σ ,y = f x ∂ x f − ⇒ δσ ,y + η ∂σ ,y ∂y = δf x ∂ x f − f x ( ∂ x f ) ∂ x δf = − f x ( ∂ x f ) ( ∂η∂y − η y ) δσ ,y = − [ f x ∂ x f ( ∂η∂y − η y ) + η ∂σ ,y ∂y ]= − [( ∂η∂y − η y ) + ( η ∂σ ,y ∂y + σ ,y ∂η∂y − σ ,y η y )]More generaly, we let the reader prove that the variation of σ q over the target (respectively thesource) is described by ” minus ” the same formula as the variation of χ q over the source (respec-tively the target). In any case, the reader must not forget that the word ”variation” just meansthat the section f q +1 is changed, not that the source is moved. Accordingly, getting in mind thisexample and for simplicity, we shall always prefer to work with vertical bundles over the source,closely following the purely mathematical definitions, contrary to Weyl ([54], §
28, formulas (17) to(27), p 233-236). The reader must be now ready for comparing the variations of χ x,x and σ y,y .In order to conclude this section, we provide without any proof two results and refer the readerto ([28]) for details. PROPOSITION 4.18 : Changing slightly the notation while setting σ q − = ¯ D ′ χ q , we have: χ ′ q = f − q +1 ◦ χ q ◦ j ( f ) + ¯ Df q +1 ⇒ σ ′ q − = f − q ◦ σ q − ◦ j ( f )23here f − q acts on J q − ( T ) and j ( f ) acts on ∧ T ∗ . It follows that gauge transformations exchangethe solutions of ¯ D ′ among themselves. COROLLARY 4.19 : Denoting by C () the cyclic sum, we have the so-called Bianchi identity : Dσ q − ( ξ, η, ζ ) + C ( ξ, η, ζ ) { σ q − ( ξ, η ) , χ q − ( ζ ) = 0
5) APPLICATIONS
Before studying in a specific way electromagnetism and gravitation, we shall come back toExample 4 .
10 and provide a technical result which, though looking like evident at first sight, is atthe origin of a deep misunderstanding done by the brothers Cosserat and Weyl on the variationalprocedure used in the study of physical problems (Compare to [20] and [50]).Setting dx = dx ∧ ... ∧ dx n for simplicity while using Lemma 4 .
11 and the fact that the standardLie derivative is commuting with any diffeomorphism, we obtain at once: y = f ( x ) ⇒ dy = det ( ∂ i f k ( x )) dx = ∆( x ) dxη = T ( f ) ξ ⇒ L ( η ) dy = L ( ξ )(∆( x ) dx ) ⇒ δ ∆ = ∆ div y ( η ) = ∆ div x ( ξ ) + ξ r ∂ r ∆The interest of such a presentation is to provide the right correspondence between the source/targetand the Euler/Lagrange choices. Indeed, if we use the way followed by most authors up to nowin continuum mechanics, we should have source=Lagrange, target=Euler, a result leading to theconservation of mass dm = ρdy = ρ dx = dx when ρ is the original initial mass per unit volume.We may set ρ = 1 and obtain therefore ρ ( f ( x )) = 1 / ∆( x ), a choice leading to: δρ + η k ∂ρ∂y k = − δ ∆ ⇒ δρ = − ρ ∂η k ∂y k − η k ∂ρ∂y k = − ρ ∂ξ r ∂x r ⇒ δρ = − ∂ ( ρη k ) ∂y k but the concept of ” variation ” is not mathematically well defined, though this result is coherentwith the classical formulas that can be found for example in ([7],[27]) or ([53], (17) and (18) p 233,(20) to (21) p 234, (76) p 289, (78) p 290) where ” points are moved ”.On the contrary, if we adopt the unusual choice source=Euler, target=Lagrange, we should get ρ ( x ) = ∆( x ), a choice leading to δρ = δ ∆ and thus: δρ = ρ ∂η k ∂y k = ρ ∂ξ r ∂x r + ξ r ∂ r ρ = ∂ r ( ρξ r )which is the right choice agreeing, up to the sign, with classical formulas but with the importantimprovement that this result becomes a purely mathematical one, obtained from a well definedvariational procedure involving only the so-called ” vertical ” machinery. This result fully explainswhy we had doubts about the sign involved in the variational formulas of ([27], p 383) but withoutbeing able to correct them at that time. We may finally revisit Lemma 4 .
11 in order to obtain the fundamental identity over the source : ∂∂x i (∆( x ) ∂g i ∂y k ( f ( x ))) ≡ , ∀ x ∈ X which becomes the conservation of mass when n = 4 and k = 4.In addition, as many chases will be used through many diagrams in the sequel, we invite thereader not familiar with these technical tools to consult the books ([3],[21],[48]) that we consideras the best references for learning about homological algebra. A more elementary approach can befound in ([32]) that has been used during many intensive courses on the applications of homologicalalgebra to control theory. As for differential homological algebra , one of the most difficult toolsexisting in mathematics today, and its link with applications, we refer the reader to the various24eferences provided in ([37],[45]).Finally, for the reader interested by a survey on more explicit applications, we particularlyrefer to ([2],[24],[34],[42]) for analytical mechanics and hydrodynamics, ([31],[33],[46]) for couplingphenomenas, ([36],[38],[42],[55-56]) for the foundations of Gauge Theory, ([35],[39],[41],[42-43]) forthe foundations of General Relativity, ([40],[44]) for unusual explicit computations of compatibilityconditions (CC) for linear differential operators. A) POINCARE, WEYL AND CONFORMAL GROUPS
When constructing inductively the Janet and Spencer sequences for an involutive system R q ⊂ J q ( E ), we have to use the following commutative and exact diagrams where we have set F = J q ( E ) /R q and used a diagonal chase :0 ↓ ↓ ↓ ↓ → δ ( ∧ r − T ∗ ⊗ g q +1 ) → ∧ r T ∗ ⊗ R q → C r → ↓ ↓ ↓ → δ ( ∧ r − T ∗ ⊗ S q +1 T ∗ ⊗ T ) → ∧ r T ∗ ⊗ J q ( E ) → C r ( E ) → ↓ ↓ ց ↓ → ∧ r T ∗ ⊗ R q + δ ( ∧ r − T ∗ ⊗ S q +1 T ∗ ⊗ E ) → ∧ r T ∗ ⊗ F → F r → ↓ ↓ ↓ → C r → C r ( E ) Φ r −→ F r → two involutive systems 0 ⊂ R q ⊂ ˆ R q ⊂ J q ( E ),it follows that the kernels of the induced canonical epimorphisms F r → ˆ F r → cokernels of the canonical monomorphisms 0 → C r → ˆ C r ⊂ C r ( E ) and we may say that Janet and Spencer play at see-saw because we have the formula dim ( C r ) + dim ( F r ) = dim ( C r ( E )).When dealing with applications, we have set E = T and considered systems of finite typeLie equations determined by Lie groups of transformations. Accordingly, we have obtained inparticular C r = ∧ r T ∗ ⊗ R ⊂ ∧ r T ∗ ⊗ ˆ R = ˆ C r ⊂ C r ( T ) when comparing the classical and con-formal Killing systems, but these bundles have never been used in physics . However, instead ofthe classical Killing system R ⊂ J ( T ) defined by the infinitesimal first order PD Lie equationsΩ ≡ L ( ξ ) ω = 0 and its first prolongations R ⊂ J ( T ) defined by the infinitesimal additionalsecond order PD Lie equations Γ ≡ L ( ξ ) γ = 0 or the conformal Killing system ˆ R ⊂ J ( T ) definedby Ω ≡ L ( ξ ) ω = 2 A ( x ) ω and Γ ≡ L ( ξ ) γ = ( δ ki A j ( x ) + δ kj A i ( x ) − ω ij ω ks A s ( x )) ∈ S T ∗ ⊗ T but wemay also consider the formal Lie derivatives for geometric objects:Ω ij ≡ ( L ( ξ ) ω ) ij ≡ ω rj ξ ri + ω ir ξ rj + ξ r ∂ r ω ij = 0Γ kij ≡ ( L ( ξ ) γ ) kij ≡ ξ kij + γ krj ξ rj + γ kir ξ rj − γ rij ξ rk + ξ r ∂ r γ kij = 0We may now introduce the intermediate differential system ˜ R ⊂ J ( T ) defined by L ( ξ ) ω =2 A ( x ) ω and Γ ≡ L ( ξ ) γ = 0, for the Weyl group obtained by adding the only dilatation withinfinitesimal generator x i ∂ i to the Poincar´e group. We have the relations R ⊂ ˜ R = ˆ R and thestrict inclusions R ⊂ ˜ R ⊂ ˆ R when R = ρ ( R ) , ˜ R = ρ ( ˜ R ) , ˆ R = ρ ( ˆ R ) but we have tonotice that we must have ∂ i A − A i = 0 for the conformal system and thus A i = 0 ⇒ A = cst if we do want to deal again with an involutive second order system ˜ R ⊂ J ( T ). However, wemust not forget that the comparison between the Spencer and the Janet sequences can only bedone for involutive operators, that is we can indeed use the involutive systems R ⊂ ˜ R but wehave to use ˆ R even if it is isomorphic to ˆ R . Finally, as ˆ g ≃ T ∗ and ˆ g = 0 , ∀ n ≥
3, the firstSpencer operator ˆ R D −→ T ∗ ⊗ ˆ R is induced by the usual Spencer operator ˆ R D −→ T ∗ ⊗ ˆ R :250 , , ξ rrj , ξ rrij = 0) → (0 , ∂ i − ξ rri , ∂ i ξ rrj −
0) and thus projects by cokernel onto the induced opera-tor T ∗ → T ∗ ⊗ T ∗ . Composing with δ , it projects therefore onto T ∗ d → ∧ T ∗ : A → dA = F as inEM and so on by using he fact that D and d are both involutive or the composite epimorphismsˆ C r → ˆ C r / ˜ C r ≃ ∧ r T ∗ ⊗ ( ˆ R / ˜ R ) ≃ ∧ r T ∗ ⊗ ˆ g ≃ ∧ r T ∗ ⊗ T ∗ δ −→ ∧ r +1 T ∗ . The main result we haveobtained is thus to be able to increase the order and dimension of the underlying jet bundles andgroups as we have ([47],[53-55]): P OIN CARE GROU P ⊂ W EY L GROU P ⊂ CON F ORM AL GROU P that is 10 < <
15 when n = 4 and our aim is now to prove that the mathematical structures ofelectromagnetism and gravitation only depend on the second order jets .With more details, the Killing system R ⊂ J ( T ) is defined by the infinitesimal Lie equationsin Medolaghi form with the well known
Levi-Civita isomorphism ( ω, γ ) ≃ j ( ω ) for geometric ob-jects: (cid:26) Ω ij ≡ ω rj ξ ri + ω ir ξ rj + ξ r ∂ r ω ij = 0Γ kij ≡ γ krj ξ ri + γ kir ξ rj − γ rij ξ kr + ξ r ∂ r γ kij = 0We notice that R (¯ ω ) = R ( ω ) ⇔ ¯ ω = a ω, a = cst, ¯ γ = γ and refer the reader to ([LAP]) formore details about the link between this result and the deformation theory of algebraic structures.We also notice that R is formally integrable and thus R is involutive if and only if ω has constantRiemannian curvature along the results of L. P. Eisenhart ([11]). The only structure constant c appearing in the corresponding Vessiot structure equations is such that ¯ c = c/a and the normalizerof R is R if and only if c = 0. Otherwise R is of codimension 1 in its normalizer ˜ R as we shallsee below by adding the only dilatation. In all what follows, ω is assumed to be flat with c = 0and vanishing Weyl tensor.The Weyl system ˜ R ⊂ J ( T ) is defined by the infinitesimal Lie equations: (cid:26) ω rj ξ ri + ω ir ξ rj + ξ r ∂ r ω ij = 2 A ( x ) ω ij ξ kij + γ krj ξ ri + γ kri ξ rj − γ rij ξ kr + ξ r ∂ r γ kij = 0and is involutive if and only if ∂ i A = 0 ⇒ A = cst . Introducing for convenience the metric density ˆ ω ij = ω ij / ( | det ( ω ) | ) n , we obtain the Medolaghi form for (ˆ ω, γ ) with | det (ˆ ω ) | = 1 : (cid:26) ˆΩ ij ≡ ˆ ω rj ξ ri + ˆ ω ir ξ rj − n ˆ ω ij ξ rr + ξ r ∂ r ˆ ω ij = 0Γ kij ≡ ξ kij + γ krj ξ ri + γ kri ξ rj − γ rij ξ kr + ξ r ∂ r γ kij = 0Finally, the conformal system ˆ R ⊂ J ( T ) is defined by the following infinitesimal Lie equations: (cid:26) ω rj ξ ri + ω ir ξ rj + ξ r ∂ r ω ij = 2 A ( x ) ω ij ξ kij + γ krj ξ ri + γ kri ξ rj − γ rij ξ kr + ξ r ∂ r γ kij = δ ki A j ( x ) + δ kj A i ( x ) − ω ij ω kr A r ( x )and is involutive if and only if ∂ i A − A i = 0 or, equivalently, if ω has vanishing Weyl tensor.However, introducing again the metric density ˆ ω while substituting, we obtain after prolonga-tion and division by ( | det ( ω | ) n the second order system ˆ R ⊂ J ( T ) in Medolaghi form ad theLevi-Civita isomorphim ( ω, γ ≃ j ( ω ) restricts to an isomorphism (ˆ ω, ˆ γ ) ≃ j (ˆ ω ) if we set:ˆ γ kij = γ kij − n ( δ ki γ rrj + δ kj γ rri − ω ij ω ks γ rrs ) ⇒ ˆ γ rri = 0 ( tr (ˆ γ ) = 0) ( ˆΩ ij ≡ ˆ ω rj ξ ri + ˆ ω ir ξ rj − n ˆ ω ij ξ rr + ξ r ∂ r ˆ ω ij = 0 ⇒ ω ij ¯Ω ij = 0ˆΓ kij ≡ ξ kij − n ( δ ki ξ rrj + δ kj ξ rri − ˆ ω ij ˆ ω kr ξ srs ) + ˆ γ krj ξ ri + ˆ γ kri ξ rj − ˆ γ rij ξ kr + ξ r ∂ r ˆ γ kij = 0 ⇒ ˆΓ rri = 0Contracting the first equations by ˆ ω ij we notice that ξ rr is no longer vanishing while, contractig in k and j the second equations, we now notice that ξ rri is no longer vanishing . It is also essential tonotice that the symbols ˆ g and ˆ g only depend on ω and not on any conformal factor.26he following Proposition does not seem to be known: PROPOSITION 5.A.1 : ( id, − ˆ γ ) is the only symmetric ˆ R -connection wih vanishing trace. Proof : Using a direct substitution, we have to study: − ˆ ω ir ˆ γ rjt − ˆ ω rj ˆ γ rit + 2 n ˆ ω ij ˆ γ rrt + ∂ t ˆ ω ij Multiplying by ( | det ( ω ) | ) n , we have to study: − ω ir ˆ γ rjt − ω rj ˆ γ rit + 2 n ω ij ˆ γ rrt + ( | det ( ω ) | ) n ∂ t ˆ ω ij or equivalently: − ω ir ˆ γ rjt − ω rj ˆ γ rit + 2 n ω ij ˆ γ rrt + ∂ t ω ij − n ω ij ( | det ( ω ) | ) − ∂ t ( | det ( ω ) | )that is to say: − ω ir ˆ γ rjt − ω rj ˆ γ rit + ∂ t ω ij − n ω ij γ sst Now, we have: − ω ir ( γ rjt − n ( δ rj γ sst + δ rt γ ssj − ω jt ω ru γ ssu )) = − ω ir γ rjt + 1 n ω ij γ sst + 1 n ω it γ ssj − n ω jt γ ssi Finally, taking into account that ( id, − γ ) is a R -connection, we have: − ω ir γ rjt − ω rj γ rit + ∂ t ω ij = 0Hence, collecting all the remaining terms, we are left with n ω ij γ sst − n ω ij γ sst = 0.As for the unicity, it comes from a chase in the commutative and exact diagram:0 0 0 ↓ ↓ ↓ −→ ˆ g δ −→ T ∗ ⊗ ˆ g δ −→ ∧ T ∗ ⊗ T −→ ↓ ↓ k −→ S T ∗ ⊗ T δ −→ T ∗ ⊗ T ∗ ⊗ T δ −→ ∧ T ∗ ⊗ T −→ ↓ dim (ˆ g ) = ( n ( n − /
2) + 1 = ( n − n + 2) / dim (ˆ g ) = n while checking that − n + n ( n − n +2) − n ( n − / δ -sequence 0 → S T ∗ δ −→ T ∗ ⊗ T ∗ δ −→ ∧ T ∗ → ∧ T ∗ ⊗ T ≃ T ∗ ⊗ g → T ∗ ⊗ ˆ g .Q.E.D. COROLLARY 5.A.2 : The R -connection ( id, − γ ) is alo a ˆ R -connection. Proof : This result first follows from the fact that ( id, − γ ) ∈ T ∗ ⊗ R is over id ∈ T ∗ ⊗ T and R ⊂ ˆ R . However, we may also check such a property directly. Indeed, mutiplying − ˆ ω rj γ rit − ˆ ω ir γ rrt + n ˆ ω ij γ rrt + ∂ t ˆ ω ij by ( | det ( ω ) | ) n as in the last Proposition, we obtain: − ω rj γ rit − ω ir γ rjt + 2 n ω ij γ rrt + ∂ t ˆ ω ij = − ω rj γ rit − ω rj γ rjt + ∂ t ω ij = 0because ( id, − γ ) is a R -connection. Q.E.D.27 EMARK 5.A.3 : If one is using ( id, − γ ), then ( L ( ξ ) γ ) kij ξ kij when γ = 0 locally and wehave ( δα ) i = ( ∂ i ξ rr − ξ rri ) + ( α r ∂ i ξ r + ξ r ∂ r α − i ) as the simplest variation . However, we have f − ( γ ) = ¯ γ = γ and we cannot thus split the Spencer operator over the target by means of apull-back. Nevertheless, if one is using ( id, − ˆ γ ), then L ( ξ )ˆ γ = 0 when ξ ∈ ˆ R and the variation( δα ) i contains an additional term ξ ssr χ r,i but f − (ˆ γ ) = ˆ γ and one can split the Spencer operatorover the source and over the target with the help of ˆ γ but we have to point out that γ = 0 ⇒ ˆ γ = 0 locally . Q.E.D.We let the reader exhibit similarly the finite Lie forms of the previous equations that will bepresented when needed. We have to distinguish the strict inclusions Γ ⊂ ˜Γ ⊂ ˆΓ ⊂ aut ( X ) with: • The Lie pseudogroup Γ ⊂ aut ( X ) of isometries which is preserving the metric ω ∈ S T ∗ with det ( ω ) = 0 and thus also γ . • The Lie pseudogroup ˜Γ which is preserving ˆ ω and γ . • The Lie pseudogroup ˆΓ of conformal isometries which is preserving ˆ ω and thus also ˆ γ with: g kl ( x )( f lij ( x ) + γ lrs ( f ()) f ri ( x ) f sj ( x )) = ¯ γ kij ( x ) = γ kij ( x ) + δ ki a j ( x ) + δ ki a j ( x ) − ω ij ( x ) ω kr ( x ) a r ( x )where a i ( x ) dx i ∈ T ∗ and thus ¯ γ − γ ∈ ˆ g ⊂ S T ∗ ⊗ T ∗ ⊗ T . B) ELECTROMAGNETISM
The key idea, still never ackowledged, is that, even if we shall prove that electromagnetism onlydepends on the elations of the conformal group which are clearly non-linear transformations, weshall see that electromagnetism has ” by chance ” a purely linear behaviour .Indeed, setting as we already did χ = A − id and defining χ klr,j = A sj τ klr,s , we may rewrite thedefining equation of the second non-linear Spencer operator ¯ D ′ in the form: ∂ i A kj − ∂ j A ki = A ri χ kr,j − A rj χ kr,i = A ri A sj ( τ kr,s − τ ks,r ) ∂ i χ kl,j − ∂ j χ kl,i − χ rl,i χ kr,j + χ rl,j χ kr,i = A ri χ klr,j − A rj χ klr,i = A ri A sj ( τ klr,s − τ kls,r )Hence, contracting in k and l , the quadratic terms in χ disappear and we get: ∂ i χ rr,j − ∂ j χ rr,i = A ri A sj ( τ kkr,s − τ kks,r )By analogy with EM it should be tempting to introduce α i = χ rr,i and denote by ϕ ij the rightmember of the last formula but the relation ∂ i α j − ∂ j α i = ϕ ij thus obtained has no intrinsicmeaning because α is far from being a 1-form while ϕ is far from being a 2-form. REMARK 5.B.1 : The target ” y ” could be called ” hidden variable ” as it is just used in orderto construct objects over the source ” x ”. As a byproduct, the changes of local coordinates are ofthe form ¯ x = ϕ ( x ) , ¯ y = ψ ( y ) but the second one does not appear through the implicit summationsover the target because the first order transition rules are:¯ y lj ∂ϕ j ∂x i ( x ) = ∂ψ l ∂y k ( y ) y ki ⇒ ¯ f lj ( ϕ ( x )) ∂ϕ j ∂x i ( x ) = ∂ψ l ∂y k ( f ( x )) f ki ( x )It follows therefore that A ∈ T ∗ ⊗ T indeed and is thus a well defined object over the source. LEMMA 5.B.2 : The short exact δ -sequence 0 −→ S T ∗ δ −→ T ∗ ⊗ T ∗ δ −→ ∧ T ∗ −→ Proof : The splitting of the above sequence is obtained by setting ( τ i,j ) ∈ T ∗ ⊗ T ∗ → ( ( τ i,j + τ j,i )) ∈ S T ∗ in such a way that ( τ i,j = τ j,i = τ ij ) ∈ S T ∗ ⇒ ( τ ij + τ ji ) = τ ij .Similarly, ( ϕ ij = − ϕ ji ) ∈ ∧ T ∗ → ( ϕ ij ) ∈ T ∗ ⊗ T ∗ and ( ϕ ij − ϕ ji ) = ( ϕ ij ) ∈ ∧ T ∗ .28.E.D.We shall revisit the previous results by showing that, in fact , all the maps and splittings exist-ing for the Killing operator are coming from maps and splittings existing for the conformal Killingoperator, though surprising it may look like . As these results are based on a systematic use of theSpencer δ -map, they are neither known nor acknowledged.We now recall the commutative diagrams allowing to define the (analogue) of the first Janetbundle and their dimensions when n = 4: ↓ ↓ ↓ → g → S T ∗ ⊗ T → S T ∗ ⊗ F → F → ↓ ↓ ↓ → T ∗ ⊗ g → T ∗ ⊗ S T ∗ ⊗ T → T ∗ ⊗ T ∗ ⊗ F → ↓ ↓ ↓ → ∧ T ∗ ⊗ g → ∧ T ∗ ⊗ T ∗ ⊗ T → ∧ T ∗ ⊗ F → ↓ ↓ ↓ → ∧ T ∗ ⊗ T = ∧ T ∗ ⊗ T → ↓ ↓ ↓ ↓ → → → → ↓ ↓ → → → ↓ ↓ → → → → ↓ ↓ ↓ →
16 = 16 → ↓ ↓ PROPOSITION 5.B.3 : Recalling that we have F = H ( g ) = Z ( g ) in the Killing case andˆ F = H (ˆ g ) = Z (ˆ g ) in the conformal Killing case, we have the unusual commutative diagram:0 0 0 0 ↓ ↓ ↓ ↓ → Z ( g ) → Z (ˆ g ) ⊂ Z ( T ∗ ⊗ T ) → S T ∗ ↓ ↓ ↓ ↓ δ → ∧ T ∗ ⊗ g → ∧ T ∗ ⊗ ˆ g ⊂ ∧ T ∗ ⊗ T ∗ ⊗ T → T ∗ ⊗ T ∗ ↓ δ ↓ δ ↓ δ ↓ δ → ∧ T ∗ ⊗ T = ∧ T ∗ ⊗ T = ∧ T ∗ ⊗ T → ∧ T ∗ ↓ ↓ ↓ ↓ Proof : First of all, we must point out that the surjectivity of the bottom δ in the central columnis well known from the exactness of the δ -sequence for S T ∗ and thus also after tensoring by T .However, the surjectivity of the bottom δ in the left column is not evident at all as it comes froma delicate circular chase in the preceding diagram, using the fact that the Riemann and Weyloperators are second order operators. Then, setting ϕ ij = ρ rr,ij = − ϕ ji and ρ ij = ρ ri,rj = ρ ji , wemay define the right central horizontal map by ρ kl,ij → ρ ij − ϕ ij and the right bottom horizontalmap by ω ⊗ ξ → − i ( ξ ) ω by introducing the interior product i (). We obtain at once: − ( ρ rr,ij + ρ ri,jr + ρ rj,ri ) = − ϕ ij + ρ ij − ρ ji = ( ρ ij − ϕ ij ) − ( ρ ji − ϕ ji )29nd the right bottom diagram is commutative, clearly inducing the upper map. If we restrict tothe Killing symbol, then ϕ ij = 0 and we obtain ρ ij − ρ ji = 0 ⇒ ( ρ ij = ρ ji ) ∈ S T ∗ , that isthe classical contraction allowing to obtain the Ricci tensor from the Riemann tensor but there isno way to go backwards with a canonical lift . A similar comment may be done for the conformalKilling symbol and the coefficient. Q.E.D.Using the previous diagram allowing to define both F = H ( g ) = Z ( g )) if we use ω orˆ F = H (ˆ g ) = Z (ˆ g ) /δ ( T ∗ ⊗ ˆ g ) if we use ˆ ω while taking into account that dim (ˆ g /g ) = 1 andˆ g ≃ T ∗ , we obtain the crucial theorem which is in fact only depending on ω : THEOREM 5.B.4 : We have the commutative and exact ” fundamental diagram II ”:0 ↓ S T ∗ ↓ ↓ −→ Z ( g ) −→ H ( g ) −→ ↓ ↓ ↓ −→ T ∗ ⊗ ˆ g δ −→ Z (ˆ g ) −→ H (ˆ g ) −→ ↓ ↓ ↓ −→ S T ∗ δ −→ T ∗ ⊗ T ∗ δ −→ ∧ T ∗ −→ ↓ ↓ all the classical formulas of both Riemannian and conformalgeometry in a totally unusual framework not depending on any conformal factor : THEOREM 5.B.5 : All the short exact sequences of the preceding diagram split in a canonicalway, that is in a way compatible with the underlying tensorial properties of the vector bundlesinvolved. With more details: T ∗ ⊗ T ∗ ≃ S T ∗ ⊕ ∧ T ∗ ⇒ Z (ˆ g ) ≃ Z ( g ) + δ ( T ∗ ⊗ ˆ g ) ≃ Z ( g ) ⊕ ∧ T ∗ ⇒ H ( g ) ≃ H (ˆ g ) ⊕ S T ∗ ⇒ F ≃ ˆ F ⊕ S T ∗ Proof : First of all, we recall that: g = { ξ ki ∈ T ∗ ⊗ T | ω rj ξ ri + ω ir ξ rj = 0 } ⊂ ˆ g = { ξ ki ∈ T ∗ ⊗ T | ω rj ξ ri + ω ir ξ rj − n ω ij ξ rr = 0 }⇒ g ⊂ ˆ g = { ξ kij ∈ S T ∗ ⊗ T | nξ kij = δ ki ξ rrj + δ kj ξ rri − ω ij ω ks ξ rrs } Now, if ( τ kli,j ) ∈ T ∗ ⊗ ˆ g , then we have: nτ kli,j = δ kl τ rri,j + δ ki τ rrl,j − ω li ω ks τ rrs,j and we may set τ rri,j = τ i,j = τ j,i with ( τ i,j ) ∈ T ∗ ⊗ T and such a formula does not depend on anyconformal factor. Taking into account Proposition 4.B.5, we have: δ ( τ kli,j ) = ( τ kli,j − τ klj,i ) = ( ρ kl,ij ) ∈ B (ˆ g ) ⊂ Z (ˆ g )with: Z (ˆ g ) = { ( ρ kl,ij ) ∈ ∧ T ∗ ⊗ ˆ g ) | δ ( ρ kl,ij ) = 0 } ⇒ ϕ ij = ρ rr,ij = 0 δ ( ρ kl,ji ) = ( C ( l,i,j ) ρ kl,ij = ρ kl,ij + ρ ki,jl + ρ kj,li ) ∈ ∧ T ∗ ⊗ T The splitting of the central vertical column is obtained from a lift of the epimorphism Z (ˆ g ) →∧ T ∗ → ϕ ij ) ∈ ∧ T ∗ to ( ϕ ij ) ∈ T ∗ ⊗ T ∗ , setting τ rri,j = ϕ ij andapplying δ to obtain ( τ rri,j − τ rrj,i = ϕ ij − ϕ ji = ϕ ij ) ∈ B (ˆ g ) ⊂ Z (ˆ g ). • Now, let us define ( ρ i,j = ρ ri,rj = ρ j,i ) ∈ T ∗ ⊗ T ∗ . Hence, elements of Z ( g ) are such that: ϕ ij = ρ rr,ij = 0 , ϕ ij − ρ i,j + ρ j,i = 0 ⇒ ( ρ ij = ρ i,j = ρ j,i = ρ ji ) ∈ S T ∗ while elements of Z (ˆ g ) are such that:( ρ rr,ij = ϕ ij = ρ i,j − ρ j,i = τ i,j − τ j,i = 0) ∈ ∧ T ∗ Accordingly, ( ρ i,j − ϕ ij = ρ j,i − ϕ ji ) ∈ S T ∗ . More generally, we may consider ρ kl,ij − ( τ kli,j − τ klj,i )with τ rri,j = ϕ ij . Such an element is killed by δ and thus belongs to Z (ˆ g ) because each memberof the difference is killed by δ . However, we have ρ rr,ij − ( τ rri,j − τ rrj,i ) = ϕ ij − ϕ ij = 0 and theelement does belong indeed to Z ( g ), providing a lift Z (ˆ g ) → Z ( g ) → • Of course, the most important result is to split the right column . As this will be the hard step,we first need to describe the monomorphism 0 → S T ∗ → H ( g ) which is in fact produced by anorth-east diagonal snake type chase. Let us choose ( τ ij = τ i,j = τ j,i = τ ji ) ∈ S T ∗ ⊂ T ∗ ⊗ T ∗ .Then, we may find ( τ kli,j ) ∈ T ∗ ⊗ ˆ g by deciding that τ rri,j = τ i,j = τ j,i = τ rrj,i in Z (ˆ g ) and apply δ in order to get ρ kl,ij = τ kli,j − τ kk,lj,i such that ρ rr,ij = ϕ ij = 0 and thus ( ρ kl,ij ) ∈ Z ( g ) = H ( g ).We obtain: nρ kl,ij = δ kl τ rri,j − δ kl τ rrj,i + δ ki τ rrl,j − δ kj τ rrli − ω ks ( ω li τ rrs,j − ω lj τ rrs,i )= ( δ ki τ lj − δ kj τ li ) − ω ks ( ω li τ sj − ω lj τ si )Contracting in k and i while setting simply tr ( τ ) = ω ij τ ij , tr ( ρ ) = ω ij ρ ij , we get: nρ ij = nτ ij − τ ij − τ ij + ω ij tr ( τ ) = ( n − τ ij + ω ij tr ( τ ) = nρ ji ⇒ ntr ( ρ ) = 2( n − tr ( τ )Substituting, we finally obtain τ ij = n ( n − ρ ij − n n − n − ω ij tr ( ρ ) and thus the tricky formula: ρ kl,ij = 1( n −
2) (( δ ki ρ lj − δ kj ρ li ) − ω ks ( ω li ρ sj − ω lj ρ si )) − n − n −
2) ( δ ki ω lj − δ kj ω li ) tr ( ρ )Contracting in k and i , we check that ρ ij = ρ ij indeed, obtaining therefore the desired canonical lift H ( g ) → S T ∗ → ρ ki,lj → ρ ri,rj = ρ ij . Finally, using again Proposition 3.4, the epimorphism H ( g ) → H (ˆ g ) → σ kl,ij = ρ kl,ij − n − (( δ ki ρ lj − δ kj ρ li ) − ω ks ( ω li ρ sj − ω lj ρ si )) + n − n − ( δ ki ω lj − δ kj ω li ) tr ( ρ ) which is just the way to define the Weyl tensor. We notice that σ rr,ij = ρ rr,ij = 0 and σ ri,rj = 0 byusing indices or a circular chase showing that Z (ˆ g ) = Z ( g ) + δ ( T ∗ ⊗ ˆ g ). This purely algebraicresult only depends on the metric ω and does not depend on any conformal factor. In actualpractice, the lift H ( g ) → S T ∗ is described by ρ kl,ij → ρ ri,rj = ρ ij = ρ ji but it is not evident atall that the lift H (ˆ g ) → H ( g ) is described by the strict inclusion σ kl,ij → ρ kl,ij = σ kl,ij providinga short exact sequence as in Proposition 3 . ρ ij = ρ ri,rj = σ ri,rj = 0 by composition.Q.E.D. PROPOSITION 5.B.6 : We have the following comutative and exact diagram:0 0 ↓ ↓ −→ ˆ g −→ T ∗ −→ ↓ ↓ k −→ ˜ R −→ ˆ R −→ T ∗ −→ ↓ ↓ ↓ −→ ˜ R = ˆ R −→ ↓ ↓ −→ T ∗ ⊗ ˜ R −→ T ∗ ⊗ ˆ R −→ T ∗ ⊗ T ∗ −→ T ∗ ⊗ T ∗ ≃ S T ∗ ⊕ ∧ T ∗ . Proof : According to the definition of the Christoffel symbols γ for te metric ω , we have:2 ω rk γ kij = ∂ i ω rj + ∂ j ω ri − ∂ r ω ij ⇔ ω kj γ kir + ω ik γ kjr − ∂ r ω ij = 0It follows that − γ ( care ) is the unique symmetric R -connection, that is a map T → R consideredas an element of T ∗ ⊗ R projecting onto id T ∈ T ∗ ⊗ T . Accordingly, any χ ∈ T ∗ ⊗ J ( T ) provides( χ kj,i + γ kjr χ r,i ) ∈ T ∗ ⊗ T ∗ ⊗ T and thus a true 1-form ( α i = χ rr,i + γ rr,s χ s,i ) ∈ T ∗ . However, suchan approach cannot be extended to higher orders and we prefer to consider half of the morphismdefining the Killing operator, namely the morphism J ( T ) → S T ∗ : ξ → L ( ξ ) ω , tensor it by T ∗ and contract it by ω − in order to get:12 ω st ( ω rt χ rs,i + ω sr χ rt,i + χ r,i ∂ r ω st ) = χ rr,i + 12 χ r,i ω st ∂ r ω st = α i where we notice that:2 γ rri = ω st ∂ r ω st = (1 /det ( ω )) ∂ i det ( ω ) ⇒ ∂ i γ rrj − ∂ j γ rr,i = 0Similarly, there is a well defined map J ( T ) → S T ∗ ⊗ T : ξ → L ( ξ ) γ that can be tensored by T ∗ and restricted to T ∗ ⊗ ˆ R in order to obtain a well defined map T ∗ ⊗ ˆ R → T ∗ ⊗ S T ∗ ⊗ T thatcan be contracted to T ∗ ⊗ T ∗ according to the following local formulas: β klr,s = τ klr,s + γ kur τ ul,s + γ klu τ ur,s − γ ulr τ ku,s + τ u,s ∂ u γ klr β kkr,s = τ kkr,s + γ kku τ ur,s + τ u,s ∂ u γ kkr We can ” twist ” by A and apply δ : T ∗ ⊗ S T ∗ ⊗ T → ∧ T ∗ ⊗ T ∗ ⊗ T that can be contracted to ∧ T ∗ according to the following local formulas: ϕ kl,ij = A ri A sj ( β klr,s − β kls,r ) ⇒ ϕ ij = ϕ rr,ij = A ri A sj ( β kkr,s − β kks,r )Q.E.D.As ϕ ∈ ∧ T ∗ though it comes from the 1-form χ ∈ T ∗ ⊗ ˆ R , we obtain the following crucialtheorem ([27-28]): THEOREM 5.B.7 : The non-linear Spencer sequence for the conformal group of transformationsprojects onto a part of the Poincar´e sequence for the exterior derivative according to the followingcommutative and locally exact diagram:0 −→ ˆΓ j −→ ˆ R D −→ T ∗ ⊗ ˆ R D −→ ∧ T ∗ ⊗ ˆ R ↓ ւ ↓ ↓ T ∗ d −→ ∧ T ∗ d −→ ∧ T ∗ α dα = ϕ dϕ = 0Accordingly, this purely mathematical result contradicts classical gauge theory . Proof : Substituting the previous results in the last formula, we obtain successively: ϕ ij = A ri A sj ( τ kkr,s − τ kks,r ) + γ kku A ri A sj ( τ ur,s − τ us,r ) + A ri A sj ( τ u,s ∂ u γ kkr − τ u,r ∂ u γ kks )= ( ∂ i χ rr,j − ∂ j χ rr,i ) + γ rru ( ∂ i A uj − ∂ j A ui ) + ( δ ri + χ r,i ) χ s,j ∂ r γ kks − ( δ sj + χ s,j ) χ r,i ∂ s γ kkr = ( ∂ i χ rr,j − ∂ j χ rr,i ) + γ rrs ( ∂ i χ s,j − ∂ j χ s,i ) + ( χ s,j ∂ i γ rrs − χ s,i ∂ j γ rrs )= ∂ i α j − ∂ j α i because ∂ i γ rrj − ∂ j γ rri = 0. It follows that dα = ϕ ∈ ∧ T ∗ and thus dϕ = 0, that is ∂ i ϕ jk + ∂ j ϕ ki + ∂ k ϕ ij = 0, has an intrinsic meaning in ∧ T ∗ . It is important to notice that the corresponding32M Lagrangian is defined on sections of ˆ C killed by ¯ D but not on ˆ C , contrary to gauge theory .Finally, the south west arrow in the left square is the composition: f ∈ ˆ R D −→ χ ∈ T ∗ ⊗ ˆ R π −→ χ ∈ T ∗ ⊗ ˆ R γ ) −→ α ∈ T ∗ Accordingly, though α is a potential for ϕ , it can also be considered as a part of the field butthe important fact is that the first set of ( linear ) Maxwell equations dϕ = 0 is induced by the( nonlinear ) operator ¯ D . The linearized framework will explain this point.One of the most important but difficult result of this paper will be the following direct proof ofthe existence of the right square in the previous diagram.Supposing for simplicity that ω is a (locally) constant metric (in fact the Minkowski metric !)and thus γ = 0. When we are considering the conformal group of space-time, it first follows thatthe jets of order three vanish and the formula (3 ∗ ) can be now written: ∂ i χ klr,j − ∂ j χ klr,i − ( χ sr,i χ kls,j + χ sl,i χ krs,j + χ slr,i χ ks,j − χ sr,j χ kls,i − χ sl,j χ krs,i − χ slr,j χ ks,i ) = 0Contracting in k = l = u and replacing r by t , we obtain the simple formula: ∂ i χ uut,j − ∂ j χ uut,i − χ st,i χ uus,j + χ st,j χ uus,i = 0Multiplyig by A tk the two last terms and replacing χ by τ , we get for these terms only : A ri A sj A tk ( τ vt,s τ uuv,r − τ vt,r τ uuv,s )Now, denoting by C ( i, j, k ) the cyclic sum on the permutation ( i, j, k ) → ( j, k, i ) → ( k, i, j ) andproceeding in this way on the last formula, we obtain easily: C ( i, j, k ) A ri A sj A tk ( τ vt,s − τ vs,t ) τ uuv,r or, equivalently: A ri A sj A tk C ( r, s, t )( τ vt,s − τ vs,t ) τ uuv,r = A ri A sj A tk C ( r, s, t )( τ vs,r − τ vr,s ) τ uuv,t Let us now similarly consider only the two first terms . After multiplication by A tk and integra-tion by part, we get for the first: A tk ( ∂ i ( A sj τ ut,s ) = ∂ i ( A sj A tk τ uut,s ) − A sj τ uut,s ∂ i A tk Applying the same procedure to the second term and considering the sum C ( i, j, k ) while rearrang-ing the six terms of the summation two by two, we obtain: C ( i, j, k )( ∂ k ( A tj A si τ uut,s − A ti A rj τ uut,r ) + A tk τ uur,t ∂ j A ri − A tk τ uut,r ∂ i A rj )Exchanging the dumb indices between themselves, we finally obtain: C ( i, j, k )( ∂ k ( A ri A sj ( τ uus,r − τ uur,s ) + A tk τ uur,t ( ∂ i A rj − ∂ j A ri ))that is to say, taking into account the equations (1 ∗ ): −C ( i, j, k )( ∂ k ϕ ij ) + C ( i, j, k )( A ri A sj A tk ( τ vr,s − τ vs,r ) τ uuv,t )or, equivalently: −C ( i, j, k )( ∂ k ϕ ij ) + A ri A sj A tk C ( r, s, t )( τ vr,s − τ vs,r ) τ uuv,t )Collecting all the results, we are only left, up to sign, with C ( i, j, k )( ∂ k ϕ ij ) = 0 as we wished.Q.E.D.33 OROLLARY 5.B.8 : The linear Spencer sequence for the conformal group of transformationsprojects onto a part of the Poincar´e sequence for the exterior derivative according to the followingcommutative and locally exact diagram:0 −→ ˆΘ j −→ ˆ R D −→ T ∗ ⊗ ˆ R D −→ ∧ T ∗ ⊗ ˆ R ↓ ւ ↓ ↓ T ∗ d −→ ∧ T ∗ d −→ ∧ T ∗ A dA = F dF = 0Accordingly, this purely mathematical result also contradicts classical gauge theory because it provesthat EM only depends on the structure of the conformal group of space-time but not on U (1). Proof : Considering ω and γ as geometric objects, we obtain at once the formulas:¯ ω ij = e a ( x ) ω ij ⇒ ¯ γ rri = γ rri + ∂ i a Though looking like the key formula (69)in ([54], p 286), this transformation is quite differentbecause the sign is not coherent and the second object has nothing to do with a 1-form. More-over, if we use n = 2 and set L ( ξ ) ω = 2 Aω for the standard euclidean metric, we should have( ∂ + ∂ ) A = 0, contrary to the assumption that A is arbitrary which is only agreeing with thefollowing jet formulas improving the ones already provided in ([38],[42],[47]) in order to point outthe systematic use of the Spencer operator: L ( ξ ) ω = 2 Aω ⇒ ( ξ rr + γ rri ξ i ) = nA, ( L ( ξ ) γ ) rri = nA i , ∀ ξ ∈ ˆ R Now, if we make a change of coordinates ¯ x = ϕ ( x ) on a function a ∈ ∧ T ∗ , we get:¯ a ( ϕ ( x )) = a ( x ) ⇒ ∂ ¯ a∂ ¯ x j ∂ϕ j ∂x i = ∂a∂x i We obtain therefore an isomorphism J ( ∧ T ∗ ) ≃ ∧ T ∗ × X T ∗ , a result leading to the followingcommutative diagram:0 −→ R −→ ˆ R −→ J ( ∧ T ∗ ) −→ ↓ D ↓ D ↓ D −→ T ∗ ⊗ R −→ T ∗ ⊗ ˆ R −→ T ∗ −→ D : ( A, A i ) −→ ( ∂ i A − A i )on the right is induced by the central Spencer operator, a result that could not have been evenimagined by Weyl and followers. This result provides a good transition towards the conformalorigin of electromagnetism.As the nonlinear aspect has been already presented, we restrict our study to the linear framework.A first problem to solve is to construct vector bundles from the components of the image of D .Using the corresponding capital letter for denoting the linearization, let us introduce:( B kl,i = X kl,i + γ kls X s,i ) ∈ T ∗ ⊗ T ∗ ⊗ T ⇒ ( B rr,i = B i ) ∈ T ∗ ( B klj,i = X klj,i + γ ksj X sl,i + γ kls X sj,i − γ slj X ks,i + X r,i ∂ r γ klj ) ∈ T ∗ ⊗ S T ∗ ⊗ T ⇒ ( B rri,j − B rrj,i = F ij ) ∈ ∧ T ∗ We obtain from the relations ∂ i γ rrj = ∂ j γ rri and the previous results: F ij = B rri,j − B rrj,i = X rri,j − X rrj,i + γ rrs X si,j − γ rrs X sj,i + X r,j ∂ r γ ssi − X r,i ∂ r γ ssj = ∂ i X rr,j − ∂ j X rr,i + γ rrs ( X si,j − X sj,i ) + X r,j ∂ i γ ssr − X r,i ∂ j γ ssr = ∂ i ( X rr,j + γ rrs X s,j ) − ∂ j ( X rr,i + γ rrs X ss,i )= ∂ i B j − ∂ j B i Now, using the contracted formula ξ rri + γ rrs ξ si + ξ s ∂ s γ rri = nA i from section A , we obtain: B i = ( ∂ i ξ rr − ξ rri ) + γ rrs ( ∂ i ξ s − ξ si )= ∂ i ξ rr + γ rrs ∂ i ξ s + ξ s ∂ s γ rri − nA i = ∂ i ( ξ rr + γ rrs ξ s ) − nA i = n ( ∂ i A − A i )34nd we finally get F ij = n ( ∂ j A i − ∂ i A j ) which is no longer depending on A , a result fully solvingthe dream of Weyl. Of course, when n = 4 and ω is the Minkowski metric, then we have γ = 0 inactual practice and the previous formulas become particularly simple.It follows that dB = F ⇔ − ndA = F in ∧ T ∗ and thus dF = 0, that is ∂ i F jk + ∂ j F ki + ∂ k F ij = 0,has an intrinsic meaning in ∧ T ∗ . It is finally important to notice that the usual EM Lagrangianis defined on sections of ˆ C killed by D but not on ˆ C . Finally, the south west arrow in the leftsquare is the composition: ξ ∈ ˆ R D −→ X ∈ T ∗ ⊗ ˆ R π −→ X ∈ T ∗ ⊗ ˆ R γ ) −→ ( B i ) ∈ T ∗ ⇔ ξ ∈ ˆ R → ( nA i ) ∈ T ∗ Accordingly, though A and B are potentials for F , then B can also be considered as a part of the field but the important fact is that the first set of ( linear ) Maxwell equations dF = 0 is induced bythe ( linear ) operator D because we are only dealing with involutive and thus formally integrableoperators, a fact justifying the commutativity of the square on the left of the diagram.Q.E.D. REMARK 5.B.9 : Taking the determinant of each term of the non-linear second order PD equa-tions defining ˆΓ, we obtain successively: det ( ω )( det ( f ki ( x ))) = e na ( x ) det ( ω ) ⇒ det ( f ki ( x )) = e na ( x ) in such a way that we may define b ( f ( x )) = a ( x ) ⇔ b ( y ) = a ( g ( y )) and set Θ( y ) = e − b ( y ) > → → ∞ of the dilatationgroup. The problem is thus to change at the same time the numerical value of the section and /orthe nature of the geometric object cosifered, passing therefore from a (metric) tensor to a (metric)tensor density, exactly what also happens with the contact structure when it was necessary to passfrom a 1-form to a 1-form density ([27-28],[39],[45]). In a more specific way, the idea has been toconsider successively the two non-linear systems of finite defining Lie equations: ω kl ( y ) y ki y lj = ω ij ( x ) → ˆ ω kl y ki y lj ( det ( y ki )) − n = ˆ ω ij ( x )Now, with γ = 0 we have χ rr,i = g sk ( ∂ i f ks − A ri f krs ) and: g sk ∂ i f ks = (1 /det ( f ki )) ∂ i det ( f ki ) = n∂ i a, g sk f krs = na r ( x )Finally, we have the jet compositions and contractions: g rk f ki = δ ri ⇒ g rk f kij = − g rkl f ki f lj ⇒ n a i ( x ) = g sk f kis = − f ki f lr g rkl = − n f ki ( x ) b k ( f ( x ))It follows that α i = n ( ∂ i a ( x ) − A ri a r ( x )) but we may also set a i ( x ) = f ki ( x ) b k ( f ( x )) in order toobtain α i = n ( ∂b∂y k − b k ) ∂ i f k as a way to pass from source to target (Compare to [28]). We have: PROPOSITION 5.B.10 : EM does not depend on the choice between source and target.
Proof : Replacing the groupoid by its inverse in each formula, we may introduce: α = α i ( x ) dx i , α i = n ( ∂ i a − A ri a r ) ⇔ β = β k ( y ) dy k , β k = n ( ∂b∂y k − b k )and compare: x a −→ ( α, ϕ ) ⇔ y b −→ ( β, ψ )while setting ψ kl = ∂β l ∂y k − ∂β k ∂y l . We have successively: ϕ ij = ∂ i α j − ∂ j α i = − n ( ∂ i ( A sj a s ) − ∂ j ( A ri a r ))= − n ( ∂ i ( b l ∂ j f l ) − ∂ j ( b k ∂ i f k ))= − n ( ∂b l ∂y k − ∂b k ∂y l ) ∂ i f k ∂ j f l = ( ∂β l ∂y k − ∂β k ∂y l ) ∂ i f k ∂ j f l = ψ kl ∂ i f k ∂ j f l ϕ does not depend any longer on a while ψ does not depend any longer on b .Accordingly, we have the equivalences: N O EM ⇔ ϕ = 0 ⇔ ψ = 0 ⇔ ∂b l ∂y k − ∂b k ∂y l = 0 ⇔ ∂ i ( A rj a r ) − ∂ j ( A ri a r ) = 0Q.E.D. REMARK 5.B.11 : If we use only the conformal group, we must use the metric density ˆ ω insteadof the metric ω . However, if we can define ˆ ω from ω by setting ˆ ω ij = ω ij / ( | det ( ω ) | ) n , we cannotrecover ω from ˆ ω . The way to escape from such a situation is to notice that: ω → e a ( x ) ω ⇒ γ kij → γ kij + δ ki ∂ j a ( x ) + δ kj ∂ i a ( x ) − ω ij ω kr ∂ r a ( x ) ⇒ γ rri → γ rri + n∂ i a ( x )a result showing that the conformal symbols ˆ g and ˆ g do not depend on any conformal factor. REMARK 5.B.12 : In fact, our purpose is quite different now though it is also based on thecombined use of group theory and the Spencer operator. The idea is to notice that the brothersare always dealing with the same group of rigid motions because the lines, surfaces or media theyconsider are all supposed to be in the same 3-dimensional background/surrounding space whichis acted on by the group of rigid motions, namely a group with 6 parameters (3 translations +3 rotations ). In 1909 it should have been strictly impossible for the two brothers to extend theirapproach to bigger groups, in particular to include the only additional dilatation of the Weyl groupthat will provide the virial theorem and, a fortiori , the elations of the conformal group consideredlater on by H.Weyl ([47],[53]). In order to explain the reason for using Lie equations, we providethe explicit form of the n finite elations and their infinitesimal counterpart with 1 ≤ r, s, t ≤ n : y = x − x b − bx ) + b x ⇒ θ s = − x δ rs ∂ r + ω st x t x r ∂ r ⇒ ∂ r θ rs = nω st x t , [ θ s , θ t ] = 0where the underlying metric is used for the scalar products x , bx, b involved. It is easy to checkthat ξ ∈ S T ∗ ⊗ T defined by ξ kij ( x ) = λ s ( x ) ∂ ij θ ks ( x ) belongs to ˆ g with A i = ω si λ s . In view ofthese local formulas, we understand how important it is to use ” equations ” rather than ” solutions ”. REMARK 5.B.13 : Setting σ q − = ¯ D ′ χ q ∈ ∧ T ∗ ⊗ J q − ( T ), we let the reader prove, as anexercise, that the following so-called Bianchi identities hold ([28], p 221): Dσ q − ( ξ, η, ζ ) + C ( ξ, η, ζ ) { σ q − ( ξ, η ) , χ q − ( ζ ) } = 0 , ∀ ξ, η, ζ ∈ T In the nonlinear conformal framework, it follows that the first set of Maxwell equations has only todo with ¯ D ′ in the nonlinear Spencer sequence and thus nothing to do with the Bianchi identities,contrary to what happens with U (1) in classical gauge theory. Similarly, in the linear conformalframework, the first set of Maxwell equations has only to do with D and thus nothing to do with D in the linear Spencer sequence. Indeed, the EM potential A is a section of ˆ C while the EMfield F is a section of ˆ C killed by D . This ” shift by one step to the left ” is the most importantresult of this section and could not be even imagined with any other approach. C) GRAVITATION
In the subsection B, we proved that the EM field ∧ T ∗ could be described by n ( n − / T ∗ ⊗ ˆ g of 1-forms with value in the conformal symbol ˆ g which is a sub-bundleof the first Spencer bundle for the conformal group described by the bundle T ∗ ⊗ ˆ R of 1-forms withvalue in the Lie algebroid ˆ R , with no relation at all with the second Spencer bundle ∧ T ∗ ⊗ ˆ R that can be identified with the Cartan curvature. Similarly, in this subsection C, which is by farthe most difficult of the whole paper beause third order jets are involved, we shall prove that thesubstitute for the Riemann curvature is only described by n ( n + 1) / T ∗ ⊗ ˆ g ⊂ T ∗ ⊗ ˆ R in such a way that n ( n − / n ( n + 1) / n = dim ( T ∗ ⊗ ˆ g ).36et us start with a preliminary mathematical comment, independently of what has already beensaid in the previous subsection B, and explain the main differences existing between the initial partof the Janet sequence for a formally integrable system C = R q ⊂ J q ( E ) = C ( E ) with a 2-acyclicsymbol g q ⊂ S q T ∗ ⊗ E such that g q +1 = 0 and the initial part of the corresponding Spencersequence for the first order involutive system R q +1 ⊂ J ( R q ) (See [47] for examples). First of all,we recall the following commutative diagram with short exact vertical sequences, only dependingon the left lower commutative square: 0 0 0 ↓ ↓ ↓ −→ Θ j q −→ C D −→ C D −→ C ↓ ↓ ↓ −→ E j q −→ C ( E ) D −→ C ( E ) D −→ C ( E ) k ↓ Φ ↓ Φ −→ Θ −→ E D −→ q F D −→ F ↓ ↓ is defined by the canonical projection Φ : J q ( E ) → J q ( E ) /R q = F with kernel R q and F = T ∗ ⊗ J q ( E ) / ( T ∗ ⊗ R q + δ ( S q +1 T ∗ ⊗ E )) is induced by Φ after onlyone (care) prolongation. As D is a first order operator because it is induced by the Spenceroperator, it is essential to notice that such a result is coming from the fact that D is of or-der 1 because R q is formally integrable and g q is 2-acyclic (See [28], p 116,120, 165). This verydelicate result cannot be extended to the right with D : F → F unless g q is involutive, asituation fulfilled by j q which is an involutive injective operator. Also the first order operator D : R q → J ( R q ) /R q +1 = C ≃ T ∗ ⊗ R q /δ ( g q +1 ) = T ∗ ⊗ R q is trivially involutive because g q +1 = 0 and C ⊂ C ( E ) while C = ∧ T ∗ ⊗ R q ⊂ C ( E ). Hence, the upper sequence is formallyexact, a result that can be extended to the right side ( See [40] for a nice counterexample). Froma snake chase in this diagram, it follows that the (local) cohomology at C in the upper sequenceis the same as the (local) cohomology at F in the lower sequence though there is no link at allbetween C and F from a purely group theoretical point of view. In the present situation, wehave an isomorphism R q +1 ≃ R q and obtain therefore D ξ q = Dξ q +1 , ∀ ξ q ∈ R q .For helping the reader, we provide the two long exact sequences allowing to define C and C in the Spencer sequence while proving the formal exactness of the upper sequence on the jet levelif we set J r ( E ) = 0 , ∀ r < J ( E ) = E for any vector bundle E :0 0 0 ↓ ↓ ↓ → S r +1 T ∗ ⊗ C → S r T ∗ ⊗ C → S r − T ∗ ⊗ C ↓ ↓ ↓ ↓ → R q + r +1 → J r +1 ( C ) → J r ( C ) → J r − ( C ) ↓ ↓ ↓ ↓ → R q + r → J r ( C ) → J r − ( C ) → J r − ( C ) ↓ ↓ ↓ δ -operator to the various upper symbol sequencesobtained by successive prolongations, starting from the case r = 0 already considered.Similarly, if we define F in the Janet sequence by the following commutative and exact diagram:37 0 0 0 ↓ ↓ ↓ ↓ → S q +2 T ∗ ⊗ C → S T ∗ ⊗ F → T ∗ ⊗ F → F → ↓ ↓ ↓ ↓ k → R q +2 → J q +2 ( E ) → J ( F ) → J ( F ) → F → ↓ ↓ ↓ ↓ ↓ → R q +1 → J q +1 ( E ) → J ( F ) → F → ↓ ↓ ↓ ↓ F ≃ C ( E ) /C . If we apply the Spencer δ -operator to the long symbol sequence:0 → S q +3 T ∗ ⊗ E → S T ∗ ⊗ F → S T ∗ ⊗ F → T ∗ ⊗ F we discover, through a standard snake diagonal chase, that such a sequence may not be exact at S T ∗ ⊗ F with a cohomology equal to H ( g q ) that may not vanish .With n = 4 , q = 2 and the conformal system ˆ R ⊂ J ( T ), we provide below the fiber dimensions:0 0 0 ↓ ↓ ↓ D −→ D −→ ↓ ↓ ↓ −→ j −→ D −→ D −→ k ↓ Φ ↓ Φ D −→ D −→ ↓ ↓ H (ˆ g ) = 0 when n = 4 as ˆ g is 3-acyclic only when n ≥ n = 4 ([38], p 26-28).The large infinitesimal equivalence principle initiated by the Cosserat brothers becomes naturalin this framework, namely an observer cannot measure sections of R q but can only measure theirimages by D or, equivalently, can only measure sections of C killed by D . Accordingly, for afree falling particle in a constant gravitational field, we have successively: ∂ ξ k − ξ k = 0 , ∂ ξ k − ξ k = 0 , ∂ i ξ k − , ≤ i, k ≤ χ kl,i = g ku ( ∂ i f ul − A ri f url ) ⇒ τ ks,r = B ir χ ks,i = g ku ( B ir ∂ i f us − f urs ) = g ku B ir ∂ i f us − Γ krs A ri A sj ( τ kr,s − τ ks,r ) = ∂ i A kj − ∂ j A ki ⇒ τ kr,s − τ ks,r = B ir B js ( ∂ i A kj − ∂ j A ki )and let the reader ckeck these formulas directly as an exercise. LEMMA 5.C.1 : Summing on k and r when γ = 0, we get successively:( τ ri,r − τ rr,i ) det ( A ) = B ri B jk ( ∂ r A kj − ∂ j A kr ) det ( A )= B ri ( B jk ∂ r A kj + A jr ∂ s B sj ) det ( A )= B ri ( B jk ∂ r A kj ) det ( A ) + det ( A ) ∂ r B ri = B ri ∂ r det ( A ) + det ( A ) ∂ r B ri = ∂ r ( B ri det ( A ))38 ω ij ∂ r ( B ri det ( A )) ξ ssj − ω ij τ rj,i det ( A ) ξ ssr = [ ω ij ( τ rr,i − τ ri,r ) ξ ssj − ω ij τ rj,i ξ ssr ] det ( A )= [ ω ij τ rr,i − ( ω ij τ ri,r + ω ir τ jr,i )] det ( A ) ξ ssj = [ ω ij τ rr,i − ( ω ij τ ri,r + ω ir τ ji,r )] det ( A ) ξ ssj = [ ω ij τ rr,i − n ω jr τ tt,r ] det ( A ) ξ ssj = ( n − n ω ij τ rr,i det ( A ) ξ ssj Using the ” vertical machinery ”, namely the isomorphism V ( J q ( E )) ≃ J q ( V ( E )), like in thepreceding sections, we shall vary the sections δf q = ( δf kµ ( x )) while setting δ ( ∂ i f kµ ( x )) = ∂ i δ ( f kµ ( x ))as it is done in analytical mechanics with the notations δ ˙ q = ˙ δq when studying the variation of aLagrangian L ( t, q, ˙ q ) ([26],[27],[28],[30]). LEMMA 5.C.2 : Let us compute directly the variation of the 1-form α over the target and overthe source, recalling that α = α i dx i with χ rr,i = g rk ∂ i f kr − A ri g sk f krs = n ( ∂ i a − A ri a r ), na i = g rk f kri and α i = χ rr,i + γ rrs χ s,i , even if γ = 0 locally. We have successively, exchanging source with target: δf k = η k = ξ r ∂ r f k , δf ki = η ku f ui = ξ r ∂ r f ki + f kr ξ ri δf kij = η kuv f ui f vj + η ku f uij = ξ r ∂ r f kij + f krj ξ ri + f kri ξ rj + f kr ξ rij nδa i = g rk δf kri + f kri δg rk = g rk ( η kuv f ui f vr + η ku f uir ) − f uri g rk η ku = f ri η ssr nδa i = g sk ( ξ r ∂ r f kis + f krs ξ ri + f kri ξ rs + f kr ξ rsi ) − f usi g sk ( g tu ( ξ r ∂ r f kt + f kr ξ rt ))= n ( ξ r ∂ r a i + a r ξ ri ) + ξ rri a i = f ki b k ⇒ nδa i = n ( ξ r ∂ r a i + a r ξ ri ) + ξ rri = nf ki ( δb k + η l ∂b k /∂y l ) + nb k ( ξ r ∂ r f ki + f kr ξ ri ) ⇒ nξ r ∂ r a i + ξ rri = nf ki ( δb k + ξ r ∂b k /∂x r ) + nb k ξ r ∂ r f ki ⇒ nδb k = g ik ξ rri Then, using the definition of a , namely det ( f ki ( x )) = e na ( x ) , we have over the source : nδa = (1 /det ( f ki )) δdet ( f ki ) = g ik δf ki = η ss = g ik ( ξ r ∂ r f ki + f kr ξ ri ) = n ξ r ∂ r a + ξ rr Using the variation δA ki = ξ r ∂ r A ki + A kr ∂ i ξ r − A ri ξ kr , we finally get when γ = 0: δχ rr,i = n δ∂ i a − nA ri δa r − n a r δA ri = ( ∂ i ξ rr − ξ rri ) + ( ξ r ∂ r χ rr,i + χ ss,r ∂ i ξ r ) − χ s,i ξ rrs ⇒ δα i = δχ rr,i + ξ rrs χ s,i = ( ∂ i ξ rr − ξ rri ) + ( α r ∂ i ξ r + ξ r ∂ r α i )The terms ∂ i ξ rr + ( α r ∂ i ξ r + ξ r ∂ r α i ) of the variation, including the variation of α = α i dx i as a1-form, are exactly the ones introduced by Weyl in ([54] formula (76), p 289). We also recognizethe variation δA i of the 4-potential used by engineers now expressed by means of second order jetsbut the use of the Spencer operator sheds a new light on EM.Similarly,when γ = 0, we have over the target: f kr A ri = ∂ i f k ⇒ f kr δA ri + A ri η ku f ur = ∂η k ∂y u ∂ i f u ⇒ δA ri = g rl ( ∂η l ∂y k − η lk ) ∂ i f k δχ rr,i = [ ∂η ss ∂y k − ng rl ( ∂η l ∂y k − η lk ) a r ] ∂ i f k − A ri f kr η ssk = [( ∂η ss ∂y k − η ssk ) − n b l ( ∂η l ∂y k − η lk )] ∂ i f k a result only depending on the components of the Spencer operator, in a coherent way with thegeneral variational formulas that could have been used otherwise. We notice that these formulas,which have been obtained with difficulty for second order jets, could not even be obtained by handfor third order jets. They show the importance and usefulness of the general formulas providingthe Spencer non-linear operators for an arbitrary order, in particular for the study of the conformalgroup which is defined by second order lie equations with a 2-acyclic symbol. When γ = 0 locally,39t is also important to notice that: α i ( x ) = n ( ∂b∂y k − b k ) ∂ i f k = β k ( f ( x )) ∂ i f k ( x ) ⇒ δα i = ( δβ k + η l ∂β k ∂y l + β r ∂η r ∂y k ) ∂ i f k and thus δβ does not only depend linearly on the Spencer operator, contrary to δα . LEMMA 5.C.3 : We have over the source : δdet ( A ) = det ( A ) B ik δA ki = det ( A ) B ik ( ξ r ∂ r A ki + A kr ∂ i ξ r − A ri ξ kr )= ξ r ∂ r ( det ( A )) + det ( A )( ∂ r ξ r − ξ rr )Now, we recall the identities: ∂ i χ kl,j − ∂ j χ kl,i − χ rl,i χ kr,j + χ rl,j χ kr,i − A ri χ klr,j + A rj χ klr,i = 0that we may rewrite in the equivalent form: τ klr,s − τ kls,r = B ir B js ( ∂ i χ kl,j − ∂ j χ kl,i − χ rl,i χ kr,j + χ rl,j χ kr,i )= B ir B js ( ∂ i χ kl,j − ∂ j χ kl,i ) − ( τ tl,r τ kt,s − τ tl,s τ kt,r )Looking only at the terms not containing the jets of order 2 in the right member, we have separately : B ir B js (( ∂ i ( g ku ∂ j f ul ) − ∂ j ( g ku ∂ i f ul )) = B ir B js (( ∂ i g ku )( ∂ j f ul ) − ( ∂ j g ku )( ∂ i f ul ))( g tu B ir ∂ i f ul )( g kv B is ∂ i f vt ) − ( r ↔ s ) = B ir B js (( g tu ∂ i f ul )( g kv ∂ j f vt )) − ( r ↔ s )= − ( B ir B js ( g tu ∂ i f ul )( f vt ∂ j g kv ) − ( r ↔ s ))= − ( B ir B js ( ∂ j g ku )( ∂ i f ul ) − ( r ↔ s ))and the total sum vanishes.Looking at the terms linear in the second order jets g ku f uij , we have separately (care to the sign): B ir B js ( ∂ j A ti − ∂ i A tj ) g ku f utl = ( τ tr,s − τ ts,r ) g ku f utl = g tv ( B is ∂ i f vr − B ir ∂ i f vs ) g ku f utl ( g tu B ir ( ∂ i f ul ) g kv f vst + g kv B js ( ∂ j f vt ) g tu f url ) − ( r ↔ s )The simplest and final checking concerns the derivatives of the second order jets. We get: B ir B js ( ∂ i χ kl,j − ∂ j χ kl,i ) = B ir B js ( A ti ∂ j ( g ku f utl ) − A tj ∂ i ( g ku f utl )) + ... = B js ∂ j ( g ku f url ) − B ir ∂ i ( g ku f usl ) + ... With y = f ( x ) ↔ x = g ( y ), it remains to substitute the formulas B ir = f kr ∂g i /∂y k while takinginto account that we have Γ kij = g ku f uij = δ ki a j + δ kj a i − ω ij ω kr a r because γ = 0 in the conformalcase which only depends on the Minkowski metric ω and not on a conformal factor.The novelty and most tricky point is to notice that we have now only n components for( τ kli,j ) ∈ T ∗ ⊗ ˆ g and no longer the n ( n − /
12 components of the classical Riemannian curva-ture. As we have already used the n ( n − / ϕ ij = τ rri,j − τ rrj,i = − ϕ ji , we may choosethe n ( n + 1) / τ ij = ( τ rri,j + τ rrj,i ) = τ ji that should involve the third orderjets which are only vanishing in the linear case but do not vanish at all in the non-linear case. Toavoid such a situation, we shall use the following key proposition that must be compared to theprocedure used in classical GR: PROPOSITION 5.C.4 : Defining ρ kl,ij = τ kli,j − τ klj,i it is just sufficient to study ρ i,j = ρ ri,rj = ρ j,i and tr ( ρ ) = ω ij ρ i,j or τ i,j = τ rri,j and tr ( τ ) = ω ij τ i,j . Setting: ρ ij = ( n − n τ ij + 1 n ω ij tr ( τ )in a way not depending on any conformal factor, we have the equivalences: τ kli,j = 0 ⇔ ρ kl,ij ⇔ ϕ ij = 0 ⊕ τ ij = 0 ⇔ ϕ ij = 0 ⊕ ρ ij = 040 roof : As ˆ g ≃ T ∗ , we have successively (Compare to Proposition 4.B.3): nρ kl,ij = δ kl τ rri,j + δ ki τ rrl,j − ω li ω ks τ rrs,j − δ kl τ rrj,i − δ kj τ rrl,i + ω lj ω ks τ rrs,i ⇒ ρ rr,ij = τ i,j − τ j,i nρ i,j = ( n − τ rri,j − τ rrj,i + ω ij ω st τ rrs,t = ( n − τ i,j − τ j,i + ω ij tr ( τ ) ⇒ ρ i,j − ρ j,i = τ i,j − τ j,i ntr ( ρ ) = 2( n − ω ij τ rri,j = 2( n − tr ( τ )When we suppose that there is no EM, that is: ϕ ij = 0 ⇔ τ i,j = τ j,i = τ ij = τ ji ⇔ ρ i,j = ρ j,i = ρ ij = ρ ji the above formulas become simpler with: nρ ij = ( n − τ ij + ω ij tr ( τ ) ⇔ τ ij = nn − ρ ij − n n − n − ω ij tr ( ρ )Surprisingly, in the general situation, we have: nρ i,j = ( n − τ i,j − τ j,i + ω ij tr ( τ ) , nρ j,i = ( n − τ j,i − τ i,j + ω ij tr ( τ )Summing, we discover that the same formula is still valid.We may thus express ρ kl,ij by means of ρ i,j or by means of τ i,j while using the relations ϕ ij = ρ rr,ij = τ i,j − τ j,i = ρ i,j − ρ j,i . As ˆ g is 2-acyclic when n ≥ −→ ˆ g δ −→ T ∗ ⊗ ˆ g δ −→ δ ( T ∗ ⊗ ˆ g ) −→ g = 0 when n ≥
3, we have an isomorphism T ∗ ⊗ ˆ g ≃ δ ( T ∗ ⊗ ˆ g ), both vector bundleshaving the same fiber dimension n = n ( n − + n ( n +1)2 when n ≥ τ kli,j = 0 ⇔ ρ kl,ij = 0.When there is no EM, that is when ϕ = 0, then one can express ρ kl,ij by means of ρ ij = ρ i,j = ρ j,i = ρ ji but there is no longer the Levi-Civita isomorphism ( ω, γ ) ≃ j ( ω ) in the Spencer sequence andthe above proposition is quite different from the concept of curvature in GR as it just amounts tothe vanishing of the Weyl tensor according to Theorem 5.B.5. Q.E.D.We notice that no one of the preceding results could be obtained by classical methods becausethey crucially depend on the Spencer δ -cohomology. As a byproduct, the same formulas provide: COROLLARY 5.C.5 : The corresponding Weyl tensor vanishes.Supposing again that there is no EM and looking for the derivatives of the second order jets,contracting in k and r while replacing l by i and s by j , we get with a i = f ki b k : ρ ij = ρ ji = τ rri,j − τ rij,r = B tj ∂ t ( g ru f uri ) − B tr ∂ t ( g ru f uij ) + ... = nB tj ∂ t a i − B tr ∂ t ( δ ri a j + δ rj a i − ω ij ω rs a s ) + ... = nf lj ∂a i ∂y l − f ki ∂a j ∂y k − f lj ∂a i ∂y l + ω ij ω rs f kr ∂a s ∂y k + ... = f ki f lj [( n − ∂b k ∂y l + ω kl ( y ) ω rs ( y ) ∂b s ∂y r ] + ... with bracket symmetric under the exchange of k and l . We have to take into account the followingterms linear in the b k , left aside in the derivations:[( n − f lj ∂f ki ∂y l + f li ∂f kj ∂y l + ω ij ω rs f lr ∂f ks ∂y l ] b k = [( n − B tj ∂ t f ki + B ti ∂ t f kj + ω ij ω rs f lr ∂ t f ks ] b k Under the same assumption, let us work out the quadratic terms in b k as follows:( τ tl,r τ kt,s − τ tl,s τ kt,r ) = ( g tu f url )( g kv f vst ) − ( g tu f usl )( g kv f vrt )Contracting in k and r as above while replacing l by i and s by j , we get:( τ ti,r τ rt,j − τ ti,j τ rt,r ) = ( g tu f uri )( g rv f vjt ) − ( g tu f uij )( g rv f vrt )41hat is: ( δ tr a i + δ ti a r − ω ri ω st a s )( δ rj a t + δ rt a j − ω jt ω rs a s ) − n ( δ ti a j + δ tj a i − ω ij ω st a s ) a t Effecting all the contractions, we get:( na i a j ) + (2 a i a j − ω ij ω rs a r a s ) − ( ω ij ω rs a r a s ) − n (2 a i a j − ω ij ω rs a r a s )and obtain the unexpected very simple formula: na i a j + 2 a i a j − ω ij ω rs a r a s − na i a j + nω ij ω rs a r a s = (2 − n ) a i a j + ( n − ω ij ω rs a r a s or, equivalently f ki f lj [(2 − n ) b k b l + ( n − ω kl ( y ) ω rs b r b s ]. Collecting these results, we finally get: THEOREM 5.C.6 : When there is no EM, we have over the target the formulae: ρ ij = f ki f lj [( n − ∂b k ∂y l + ω kl ( y ) ω rs ( y ) ∂b s ∂y r + ( n − b k b l − ( n − ω kl ( y ) ω rs b r b s ] τ ij = nf ki f lj [ ∂b k ∂y l + b k b l − ω kl ( y ) ω rs ( y ) b r b s ]that do not depend on any conformal factor for ω and thus simply: τ = n Θ [ ω kl ( y ) ∂b k ∂y l − ( n − ω kl ( y ) b k b l ] = n [¯ ω kl ( y ) ∂b k ∂y l − ( n − ¯ ω kl ( y ) b k b l ]that only depends on the new metric ¯ ω = Θ ω defined over the target. Proof : We have to prove the following technical result which is indeed the hardest step, namelythat ρ ij does not contain terms linear in b k over the target . The main problem is that, if we haveany derivative of the second order jets over the source , like ∂ r a i , we obtain therefore a term like ∂ r ( f ki b k ) = f ki ∂ r b k + ( ∂ r f ki ) b k which is bringing a term linear in the b k and we have to prove that such terms may not exist if we work only over the target .For this, let us set over the source: τ ks,r = T ks,r − Γ krs , T ks,r = g ku B ir ∂ i f us = T kr,s , Γ krs = δ kr a s + δ ks a r − ω rs ω kt a t = Γ ksr Looking for the derivatives of the second order jets, we already saw that they can only appearthrough the terms: B is ∂ i Γ krl − B ir ∂ i Γ ksl = f vs ∂ Γ krl ∂y v − f ur ∂ Γ ksl ∂y u Contracting in k and r , we get when there is no EM: f vs ∂ Γ rrl ∂y v − f ur ∂ Γ rsl ∂y u = f vs ∂∂y v ( na l ) − f ur ∂∂y u ( δ rs a l + δ rl a s − ω sl ω rt a t )= ( n − f vs ∂ ( f ul b u ) ∂y v − f ul ∂ ( f vs b v ) ∂y u + ω sl ω rt f ur ∂ ( f vt b v ) ∂y u = ( n − f ul f vs ∂b v ∂y u + ω sl ω rt f ur f vt ∂b v ∂y u + ... but we have to take into account the linear terms produced by an integration by parts:( n − f vs ∂f ul ∂y v b u − f ul ∂f vs ∂y u b v + ω ls ω rt f ur ∂f vt ∂y u b v that is to say, we have to substract :( n − g tu B is ∂ i f ul a t − g tv B il ∂ i f vs a t + ω ls ω rt g uv B ir ∂ i f vt a u = ( n − T tl,s a t − T ts,l a t − ω ls ω rt T ut,r a u Meanwhile, as we already saw, we have to compute ( care to the signs involved ):( T tr,s − T ts,r )Γ klt − ( T tl,r Γ kst + T kt,s Γ tlr ) + ( T tl,s Γ krt + T kt,r Γ tls )42nd to contract in k and r in order to get:( T tr,s − T ts,r )Γ rlt − ( T tl,r Γ rst + T rt,s Γ tlr ) + ( T tl,s Γ rrt + T rt,r Γ tls )However, two terms are disappearing and we are left with: − T ts,r Γ rlt − T tl,r Γ rst + ( T tl,s Γ rrt + T rt,r Γ tls )that is to say: − T ts,r ( δ rl a t + δ rt a l − ω lt ω ru a u ) − T tl,r ( δ rs a t + δ rt a s − ω st ω ru a u ))+ nT tl,s a t + T rt,r ( δ tl a s + δ ts a l − ω ls ω tu a u )and thus: − T ts,l − T rs,r a l + ω lt ω ru T tr,s a u − T tl,s a t − T tl,t a s + ω st ω ru T tl,r a u + nT tl,s a t + T rl,r a s + T rs,r a l − ω ls ω tu T rt,r a u The four terms containing a l and a s are disappearing and we are left with:( n − T tl,s a t − T ts,l a t ω lt ω ru T tr,s a u + ω st ω ru T tl,r a u − ω ls ω tu T rt,r a u Taking into account twice successively the conformal Killing equations, we obtain:( n − T tl,s a t − T ts,l a t + n ω ls ω ru T tt,r a u − ω ls ω tu T rt,r a u = ( n − T tl,s a t − T ts,l a t − ω ls ω rt T ut,r a u that is exactly the terms we had to substract and there is thus no term linear in a i in the Riccitensor over the target, a quite difficult result indeed because no concept of classical Riemanniangeometry could be used .We finally obtain from the definition of Θ while taking inverse matrices:Θ ω kl ( y ) f ki f lj = ω ij ( x ) ⇒ Θ − ω kl ( y ) g ik g jl = ω ij ( x ) ⇒ Θ − ω kl ( y ) = ω ij ( x ) f ki f lj and just need to set τ = ω ij τ ij in order to get the last formula. Q.E.D. REMARK 5.C.7 : When A ri = δ ri , we get ρ kl,ij = ∂ i χ kl,j − ∂ j χ kl,i − χ rl,i χ kr,j + χ rl,j χ kr,i with χ kj,i = g ku ∂ i f uj − g ku f uij . However, in such a situation, we have: ω kl ( f ( x )) f ki f lj = e a ( x ) ω ij ( x ) ⇒ ω kl ( f ( x )) ∂ i f k ( x ) ∂ j f l ( x ) = e a ( x ) ω ij ( x ) = ¯ ω ij ( x )Using the Minkowski metric ω which is locally constant and thus flat, it follows from the Vessiotstructure equations that ¯ ω must also be flat but we may have f = j ( f ) even though f = j ( f ).As ¯ ω is conformally equivalent to ω , then both metric have vanishing Weyl tensor and the integra-bility condition for ¯ ω is thus to have a vanishing Ricci tensor, that is to say, prolonging once thesystem j ( f ) − ( ω ) = ¯ ω , we get j ( f ) − ( γ ) = ¯ γ and obtain: γ = 0 ⇒ ¯ γ kij = δ ki ∂ j a + δ kj ∂ i a − ω ij ω kr ∂ r a ( n − ∂ ij a + ω ij ω rs ∂ rs a + ( n − ∂ i a∂ j a − ( n − ω ij ω rs ∂ r a∂ s a = 0This is a very striking result showing out for the first time that there may be links between thenon-linear Spencer sequence and classical conformal geometry as the above result is just the vari-ation of the classical Ricci tensor under a conformal change of the metric and the reason for whichwe introduced exponentials for describing conformal factors.With φ = GMr and thus φc ≪
1, we have thus been able to replace 1 − φc by 1 + φc , suppressingtherefore the horizon r = GM/c when G is the gravitational constant and M the central attractivemass, along with the following scheme: source inversion ←→ target T T RACT ION inversion ←→ REP U LSION
As it is based on the inversion rule for the second order jets of the conformal Lie groupoid, we get: such a procedure could not be even imagined in any classical framework dealing with Lie groups oftransformations . THEOREM 5.C.8 : We have the variation over the source: δτ j,i = B ri ∂ r ξ ssj + ξ r ∂ r τ j,i + τ j,r ξ ri + τ r,i ξ rj − τ rj,i ξ ssr Proof : Using the general variational formulas one obtains: δχ klj,i = ( ∂ i ξ klj − ξ klij ) + ξ r ∂ r χ klj,i + χ klj,r ∂ i ξ r + χ klr,i ξ rj + ( χ krj,i ξ rl − χ rlj,i ξ kr )+ χ kr,i ξ rlj − χ rl,i ξ krj − χ rj.i ξ klr − χ r,i ξ klrj where one must take into account that the third order jets of conformal vector fields vanish, thatis to say ξ klrj = 0. Contracting in k and l , we get: δχ ssj,i = ∂ i ξ ssj + ξ r ∂ r χ ssj,i + χ ssj,r ∂ i ξ r + χ ssr,i ξ rj − ξ rj.i ξ ssr − χ r,i ξ ssrj χ klj,i = A ri τ klj,r ⇒ δχ klj,i = A ri δτ klj,r + τ klj,r δA ri A ri δτ ssj,r = δχ ssj,i − τ j,r δA ri A ri δτ j,r = ∂ i ξ ssj + ξ r ∂ r χ ssj,i + χ ssj,r ∂ i ξ r − χ rj.i ξ ssr − τ j,r ( ξ s ∂ s A ri + A ri ∂ i ξ s − A si ξ rs ) δτ j,i = B ri ∂ r ξ ssj + ξ r ∂ r τ j,i + ( τ j,r ξ ri + τ r,i ξ rj ) + τ rj,i ξ ssr Q.E.D.Using the fact that ω is locally constant and not varied ( care ), we have at once: δτ = ω ij ( B ri ∂ r ξ ssj ) + ξ r ∂ r τ + ω ij ( τ j,r ξ ri + τ r,i ξ rj ) − ω ij τ rj,i ξ ssr = ω ij ( B ri ∂ r ξ ssj ) + ξ r ∂ r τ + τ r,s ( ω is ξ ri + ω js ξ rj ) − ω ij τ rj,i ξ ssr = ω ij ( B ri ∂ r ξ ssj ) + ξ r ∂ r τ + n ω rs τ r,s ξ tt − ω ij τ rj,i ξ ssr and thus: COROLLARY 5.C.9 : δτ = ω ij B ri ∂ r ξ ssj + ξ r ∂ r τ + n τ ξ rr − ω ij τ rj,i ξ ssr Combining this result with the three preceding Lemmas, we finally obtain:
COROLLARY 5.C.10 : The action variation over the source is: δ ( τ det ( A )) = ∂ r ( ξ r τ det ( A ) + ω ij ( x ) B ri det ( A ) ξ ssj ) − ( n − n τ det ( A ) ξ rr + ( n − n ω ij ( x ) τ rr,i det ( A ) ξ ssj Proof : According to Lemma 5.C.3, we have: δ ( τ det ( A )) = ( δτ ) det ( A ) + τ δdet ( A )= ω ij B ri det ( A ) ∂ r ξ ssj + ∂ r ( ξ r τ det ( A )) − ( n − n τ det ( A ) ξ rr − ω ij τ rj,i det ( A ) ξ ssj = ∂ r ( ξ r τ det ( A ) + ω ij ( x ) B ri det ( A ) ξ ssj ) − ( n − n τ det ( A ) ξ rr − ω ij ( ∂ r ( B ri det ( A )) ξ ssj − ω ij τ rj,i det ( A ) ξ ssr and we just need to use Lemma 5.C.1. Q.E.D.44 HEOREM 5.C.11 : We have the following Euler-Lagrange equations when n = 4 only : ξ rri → ∃ gravitational potentialξ rr → ∃ P oisson equationξ r → ∃ N ewton law
In particular τ rr,i = 0 ⇔ χ rr,i = 0 ⇔ α i = 0 ⇔ b k = − ∂ Θ ∂y k . Proof : For n arbitrary, we have: τ det ( A ) = n Θ ( n − ∆( ω kl ∂b k ∂y l − ( n − ω kl b k b l )= − n Θ ( n − ∆(Θ − ω kl ∂ Θ ∂y k ∂y l + ( n − Θ − ω kl ∂ Θ ∂y k ∂ Θ ∂y l )Hence, for n = 4 only , we have τ det ( A ) = − ω kl ∂ Θ ∂y k ∂y l . In the static case the gravity vector must be in first approximation g k ≃ − γ k = ω ω kl b l = − b k < ⇔ b k > , ∀ k = 1 , , φ = GMr where r is the distance at the central attractive mass M and G is the gravi-tational constant, then we have φc ≪ ∂ Θ ∂y <
0. The only coherent possibility is to set Θ = 1 + φc in orderto correct the value Θ = 1 − φc we found in ([28], p 450) and we have already explained theconfusion we made on the physical meaning of source and target . Hence, gravity in vacuum onlydepends on the conformal isotropy groupoid through the conformal factor but this new approachis quite different from the ideas of G. Nordstr¨om ([15],[53]), H. Weyl ([54]) or even Einstein-Fokker([10],[23]). Indeed, it has only to do with the nonlinear Spencer sequence and not at all with thenonlinear Janet sequence, contrary to all these theories, as we just said, and the conformal factorΘ is now well defined everywhere apart from the origin of coordinates where is the central attrac-tive mass. We have thus no longer any need to introduce the so-called horizon r = GM/c andgravitation only depends on the structure of the conformal group theory like electromagnetism,with the only experimental need to fix the gravitational constant. Such a ” philosophy ” has beenfirst proposed by the Cosserat brothers in ([1],[8],[18],[27]) for elasticity with the only experimentalneed to measure the elastic constants and extended to electromagnetism in the last section withthe same comments (See [27],[30] and [46],[50] for details). An additional dynamical term must beadded for the Newton law but this rather physical question will be studied in another paper as wealready said in the Introduction. Q.E.D. REMARK 5.C.12 : We shall find back the same Euler-Lagrange variational equations by usingthe variation over the target. With dy = ∆ dx by definition, we have indeed for n arbitrary: Z τ det ( A ) dx = Z n Θ ( n − [ ω kl ( y ) ∂b k ∂y l − ( n − ω kl ( y ) b k b l ] dy If we are only interested by the variation of the second order jets, we may equivalently vary the b k alone and get after integration by parts: δb l −→ ( n − ( n − ω kl ∂ Θ ∂y k + ( n − ( n − ω kl b k = 0 ⇒ b k = − ∂ Θ ∂y k Now, with dx = dx ∧ ... ∧ dx n and dy = dy ∧ ... ∧ dy n , we have: R τ det ( A ) dx = − R [ n Θ ( n − ω kl ( y ) ∂ Θ ∂y k ∂y l + n ( n − Θ ( n − ω kl ( y ) ∂ Θ ∂y k ∂ Θ ∂y l ] dy = − R ∂∂y l ( n Θ ( n − ω kl ( y ) ∂ Θ ∂y k ) dy − R n ( n − Θ ( n − ω kl ( y ) ∂ Θ ∂y k ∂ Θ ∂y l dy If we only vary the section y = f ( x ) of X × Y over X , we have dy = ∆ dx, δ ∆ = ∆ ∂η u ∂y u and :Θ n det ( f ki ( x )) = 1 ⇒ δ ( ∂ i Θ) = δ ( ∂ Θ ∂y k ) ∂ i f k + ∂ Θ ∂y u ∂η u ∂y k ∂ i f k ⇒ δ ( ∂ Θ ∂y k ) = − ∂ Θ ∂y u ∂η u ∂y k
45t follows that the variation of the last integral is: − Z n ( n − ( n − ω kl ( y )( ∂ Θ ∂y l ∂ Θ ∂y u ∂η u ∂y k − ∂ Θ ∂y k ∂ Θ ∂y l ∂η u ∂y u ) dy After integration by parts, we get, up to a divergence: − n ( n − Z ∂∂y k [Θ ( n − ( ω rk ( y ) ∂ Θ ∂y r ∂ Θ ∂y u − δ ku ω rs ( y ) ∂ Θ ∂y r ∂ Θ ∂y s )] η u ) dy When n = 4, the direct computation becomes simpler because a part of the integral disappears.We are left with τ det ( A ) = − ✷ Θ and we recognize the well known
Abraham tensor in thebracket ([28]), without any other assumption. Accordingly, we may finally say as in the previoussection:
The whole gravitational scheme only depends on the structure of the conformal group . REMARK 5.C.13 : Proceeding as in GR, we may consider the variation: δ Z g kl ( y )[ ∂b l ∂y k + b k b l − ω kl ( y ) ω rs ( y ) b r b s ] dy = 0Varying only the second order jets b k , we get equivalently through an integration by parts:(2 g kl − ω kl ω rs g rs ) b l = ∂ g kl ∂y l If we set b ( f ( x )) = a ( x ) and Θ( y ) = e − b ( y ) , then Θ ( y ) = e − b ( y ) and we have successively: ω kl ( y ) f ki f lj = e a ( x ) ω ij ( x ) ⇔ e − b ( y ) ω kl ( y ) f ki f lj = ω ij ( x ) ⇔ Θ ( y ) ω kl ( y ) f ki f lj = ω ij ( x )Inverting the matrices, we obtain equivalently:Θ − ( y ) ω kl ( y ) g ik g jl = ω ij ( x ) ⇔ Θ − ( y ) ω kl ( y ) = ω ij ( x ) f ki f lj and thus: Θ n det ( f ki ) = 1 ⇒ Θ n det ( f ki ) = 1 ⇒ det ( A ) = det ( ∂ i f k ) /det ( f ki ) = Θ n ∆Hence, if we set g kl ( y ) = Θ ( n − ( y ) ω kl ( y ), we finally obtain:( n − ( n − b k = − ( n − ( n − ∂ Θ ∂y k ⇒ b k = − ∂ Θ ∂y k in a coherent way with the logarithmic derivatives: β k = 0 ⇔ ∂b∂y k = − ∂ Θ ∂y k = b k ⇔ ∂ i a = − ∂ Θ ∂y k ∂ i f k = b k ∂ i f k = a r g rk ∂ i f k = A ri a r ⇔ α i = 046 ) CONCLUSION This paper is the achievement of a lifetime research work on the common conformal origin ofelectromagnetism and gravitation. Roughly speaking, the Cosserat brothers have only been dealingwith the 3 translations and 3 rotations of the group of rigid motions of space with 6 parameterswhile Weyl has only been dealing with the dilatation and the 4 elations of the conformal groupof space-time with now 4 + 6 + 1 + 4 = 15 parameters ([47]). Among the most striking resultsobtained from this conformal extension, we successively notice: • The generating nonlinear first order (care) compatibility conditions (CC) for the Cosserat fieldsare exactly described by the first order nonlinear second Spencer operator ¯ D . Accordingly, thereis no conceptual difference between these nonlinear CC and the first set d : ∧ T ∗ → ∧ T ∗ ofMaxwell equations where d is the exterior derivative. However, the classical CC of elasticity aredescribed by the nonlinear second order (care) Riemann operator existing in the nonlinear Janetsequence but this different canonical nonlinear differential sequence could not explain the existenceof field-matter couplings like piezzoelectricity or photoelasticity ([31],[46]). On the contrary, in theconformal approach, it is essential to notice that the elastic and electromagnetic fields are bothspecific sections of ˆ C = T ∗ ⊗ ˆ R killed by ¯ D . They can thus be coupled in a natural way but cannot be associated to the concept of curvature described by ˆ C . This shift by one step to the left ,even in the nonlinear framework, can be considered as the main novelty of this paper. • The linear Cosserat equations are exactly described by the formal adjoint ad ( D ) of the linear first Spencer operator D : ˆ C → ˆ C which is a first order operator ([33]). Accordingly, thereis no conceptual difference between these equations and the second set ad ( d ) of Maxwell equa-tions where d : T ∗ → ∧ T ∗ . This result explains why the Cosserat equations are quite differentfrom the
Cauchy equations which are described by the formal adjoint of the
Killing operator inthe Janet sequence used in classical elasticity, that is