The Mathematical Foundations of Elasticity and Electromagnetism Revisited

The first purpose of this striking but difficult paper is to revisit the mathematical foundations of Elasticity (EL) and Electromagnetism (EM) by comparing the structure of these two theories and examining with details their known couplings, in particular piezoelectricity and photoelasticity. Despite the strange Helmholtz and Mach-Lippmann analogies existing between them, no classical technique may provide a common setting. However, unexpected arguments discovered independently by the brothers E. and F. Cosserat in 1909 for EL and by H. Weyl in 1918 for EM are leading to construct a new differential sequence called Spencer sequence in the framework of the formal theory of Lie pseudo groups and to introduce it for the conformal group of space-time with 15 parameters. Then, all the previous explicit couplings can be deduced abstractly and one must just go to a laboratory in order to know about the coupling constants on which they are depending, like in the Hooke or Minkowski constitutive relations existing respectively and separately in EL or EM. We finally provide a new combined experimental and theoretical proof of the fact that any 1-form with value in the second order jets (elations) of the conformal group of space-time can be uniquely decomposed into the direct sum of the Ricci tensor and the electromagnetic field. This result ques-tions the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT). In particular, the Einstein operator (6 terms) must be thus replaced by the adjoint of the Ricci operator (4 terms only) in the study of gravitational waves.

knew each other, were both looking for the possibility to interpret thermostatic and electric phenomena by exhibiting a common macroscopic mechanical origin through a kind of variational calculus similar to the one used in analytical mechanics for getting Euler-Lagrange equations. As a byproduct, it is not possible to separate the Mach-Lippmann analogy from the Helmholtz analogy that we now recall.
In analytical mechanics, if ( ) , , L t q q  is the Lagrangian of a mechanical system, one easily gets the Hamiltonian L H q L q where t is time, q represents a certain number of dependent variables or generalized position, allowing to define the position of the various rigid bodies constituting the system (coordinates of center of gravity, relative angles, ...) and q  is the derivative with respect to time or generalized speed. There are two ideas behind such a construction. The first is to introduce the energy as in the movement of a point of mass m with Cartesian coordinates ( , , x y z vertical) or ( 1 2 3 , , x x x vertical) in the gravitational field g where ( ) . The second is to take into account the well known implied by the variational condition ( ) , , d 0 L t q q t δ = ∫  and to obtain therefore: that is the conservation of energy along the trajectories whenever L does not contain t explicitly.
Similarly, in thermostatics, if F is the free energy of a system at absolute temperature T, we may obtain, in general, the internal energy U by the formula We explain the underlying difficulty in the case of a perfect gas with pressure P, volume V and entropy S for one mole. The first principle of thermostatics says that the sum of the exchange of work To avoid such a situation, Helmholtz postulated the possibility for any system to choose "normal" state variables such that dT should not appear in W δ . Therefore, if one could introduce V and T on an equal geometric footing, then should already contain, in a built-in manner, not only the first and second principle but also the well defined possibility to recover U from F as before. In the case of continuum mechanics that we shall study later on, V must be replaced by the deformation tensor, as we shall see later on, which is a function of the first order derivatives of the actual (Euler) position x at time t with respect to the initial (Lagrange) position 0 x at time 0 t . Accordingly, the idea of Helmholtz has been to compare the relations L H → and F U → and to notice that they should become indeed similar if one could set L F = − and q T =  for a certain q. However, despite many attempts [1], nobody knows any variable q such that its derivative with respect to time should be the absolute temperature T of the system considered.
We now present the work done by Lippmann in a modern setting. The basic idea is to compare two kinds of conceptual experiments, namely a Carnot cycle for a steam engine working between the absolute temperatures 1 T and 2 T with 2 1 T T > on one side, and a cycle of charge and discharge of a spherical condenser (say a soap buble) of radius r, moving in between two plates at constant electric potentials 1 V and 2 V with 2 1 V V > on the other side ([2] [3] [4] [5]). In the first case, let the system receive the heat 2 0 Q > from the hot source and the heat 1 0 Q < from the cold source through corresponding isothermal evolutions, while receiving the work 0 W < from the surroundings in a cycle completed by two adiabatic evolutions.
The vanishing of the cycle integral: coming from the first principle of thermostatics leads to the relation Then, the vanishing of the cycle integral coming from the second principle of thermostatics: leads to the Clausius formula and the computation of the efficiency ν : Now, in the second case, things are quite more subtle. Recalling the formula q CV = relating the charge q to the potential V of a condenser with for a sphere of radius r, the electric energy should be: Whenever C remains constant, the exchange of work done by the sources should be d W V q δ ′ = because, by definition, sources are at constant potential, and we have d However, the situation is completely different whenever C depends on r and we do not believe that Lippmann was very conscious about this fact. Let us suppose that the bubble receives the work 2 0 W ′ > from the source at potential 2 V for having its charge changing at constant potential 2 V and similarly the work 1 0 W < from the source at constant potential 1 V for having its charge changing at constant potential 1 V , while receiving the (mechanical) work 0 W < from the surroundings for changing C in a cycle where the geometry of the system may vary (change of radius or distance). The problem is now to construct the cycle in order to be able to copy the procedure used for thermostatics. In the evolution at constant potential we have d W V q δ ′ = , as already said, and therefore, comparing with d Q T S δ = , the remaining evolution must be at constant charge, a situation happily realized in the experiment proposed by Lippmann, during the transport of the bubble from one plate to the other. Now, taking into account the expression d W V q δ ′ = already introduced and allowing C to vary (through r in our case), we have the formula: if we express E as a function of C and q. In our case Copying the use of the first principle of thermostatics, the vanishing of the cycle integral provides: Lippmann then notices that the conservation of entropy now becomes the conservation of charge and the vanishing of the cycle integral provides: analogous to the Clausius formula with similar efficiency ν , a result he called "Principe de conservation de l'électricité " or "Second principe de la théorie des phénomènes électriques".
One must notice the formula: if we express E as a function of C and V. Also the analogue of the free energy should be E qV E − = − expressed as a function of C and V. Hence it is not evident, at first sight, to know whether the more "geometric" quantity is q or V.
Finally, the analogy between T and V in the corresponding "second principles" is clear and constitutes the Mach-Lippmann analogy. However, the reader may find strange that T, which is just defined up to a change of scale because of the existence of a reference absolute zero, should be put in correspondence with V which is defined up to an additive constant. In fact, the formula for the spherical condenser (Gauss theorem) is only true if the potential at infinity is chosen to be zero, as a zero charge on the sphere is perfectly detectable by counting the number of electrons on the surface. Accordingly, the two previous dimensionless ratios are perfectly well defined, independently of any unit chosen for T or V.
However, such an analogy is perfectly coherent with the existence of thermocouples where the gradient of T is proportional to the gradient of V, that is we have for the electric field ( ) and the latter difficulty entirely disappears.
We recall that the thermoelectric effect, that is the existence of an electric current circulating in two different metal threads A and B with soldered ends at different temperatures 1 T and 2 1 T T > , has been discovered in 1821 by the physicist Seebeck from the Netherlands. Also cutting one of the threads to set a condenser and integrating along the circuit, the difference of potential becomes: T T ≠ and tables of coefficients can be found in the literature. It is the French physicist Becquerel who got the idea in 1830 to use such a property for measuring temperature and Le Chatelier in 1905 who set up the platine thermocouple still used today. Meanwhile, J. Peltier proved that, when an electric current is passing in a thermocouple circuit with soldered joints at the same temperature, then one of the joints absorbs heat while the other produces heat. Also W. Thomson proved that an electric current passing in a piece of homogeneous conductor in thermal equilibrium gives a difference of potential at the ends whenever they are not at the same temperature.
We end this presentation of the Mach-Lippmann analogy with the main problem that it raises. From the special relativity of A. Einstein in 1905 [6] it is known that space cannot be separated from time and that one of the best examples is given by the relativistic formulation of EM. Indeed, instead of writing down separately the first set of Maxwell equations for the electric field E and the magnetic field B under their classical form, ne may introduce local coordinates ( ) where c is the speed of light and consider the 2-form with standard notations: in order to obtain: is the exterior derivative. Similarly, introducing the electromagnetic potential A and the electric potential V in the 1-form is the time component, we obtain: though, surprisingly, V has been introduced in thermostatics. Hence, even if we may accept and understand an analogy between T and V, we cannot separate V from A in the 4-potential A and a good conceptual analogy should be between T and ( ) The surprising fact is that almost nobody knows about the Mach-Lippmann analogy today but many persons are using it through finite element computations and thus any engineer working with finite elements knows that elasticity, heat and electromagnetism, though being quite different theories at first sight, are organized along the same scheme and cannot be separated because of the existence of the following couplings that we shall study with more details in the next Section.
 THERMOELASTICITY (Elasticity/Heat): When a bar of metal is heated, its length is increasing and, conversely, its length is decreasing when it is cooled down. It is a perfectly reversible phenomenon.  PIEZOELECTRICITY, PHOTOELASTICITY (Elasticity/Electromagnetism): When a crystal is pinched between the two plates of a condenser, it produces a difference of potential between the plates and conversely, in a purely reversible way. Piezoelectric lighters are of common use in industry. Similarly, when a transparent homogeneous isotropic dielectric is deformed, piezoelectricity cannot appear but the index of refraction becomes different along the three orthogonal proper directions common to both the strain and stress tensors. Here we recall that a material is called "homogeneous" if a property does not depend on the point in the material and it is called "isotropic" if a property does not depend on the direction in the material. Accordingly, a light ray propagating along one of these directions may have its electric field decomposed along the two others and the two components propagate with different speeds. Hence, after crossing the material, they recompose with production of an interference pattern, a fact leading to optical birefringence. Such a property has been used in order to get information on the stress inside the material, say a bridge or a building, by using reduced transparent plastic models. This phenomenon was discovered by Brewster in 1815 but the phenomenological law that we shall prove in the next section, We have already spoken about this coupling which, nevertheless, can only be understood today within the framework of the phenomenological Onsager relations for irreversible phenomena. If we want to make the Fourier law T χ = q ∇ between heat flux and gradient of temperature more precise, we may suppose that the heat conductivity χ also depends on the magnetic field and we may obtain "a priori" the additional term when ω is the space Euclidean metric and we have ( ) We have thus been able to recover the Righi-Leduc effect in a purely macroscopic way.
Hence, as a very restrictive conclusion, we discover that the Mach-Lippmann analogy must be at least set up in a clear picture of the analogy existing between elasticity, heat and electromagnetism that must also be coherent with the above couplings.

Elasticity versus Electromagnetism
The rough idea is to make the constitutive law of an homogeneous isotropic di- where D is the electric induction and σ σ λ − = that we shall demonstrate and apply to the study of a specific beam. In this formula 1 2 , σ σ are the two eigenvalues of the symmetric stress tensor along directions orthogonal to the ray, k is a relative integer fixing the lines of interference, λ is the wave length, e is the thickness of the transparent beam and C is the photoelastic constant of the material.
With more details, the infinitesimal deformation tensor of elasticity theory is equal to half of the Lie derivative ξ ω Ω = Ω = Ω =  of the euclidean metric ω with respect to the displacement vector ξ . Hence, a general quadratic lagrangian may contain, apart from its standard purely elastic or electrical parts well known by engineers in finite element computations, a coupling part and is therefore modified by an electric polarization k ijk ij P c = Ω , brought by the deformation of the medium. In all these formulas and in the forthcoming ones the indices are raised or lowered by means of the euclidean metric. If this medium is homogeneous, the components of the 3-tensor c are constants and the corresponding coupling, called piezoelectricity, is only existing if the medium is non-isoptropic (like a crystal), because an isotropic 3-tensor vanishes identically.
In the case of an homogeneous isotropic medium (like a transparent plastic), one must push the coupling part to become cubic by adding from Curie's law. The corresponding coupling, called photoelasticity, has been discovered by T. J. Seebeck in 1813 and D.
Brewster in 1815. With δ β γ = + , the new electric induction is: As Ω is a symmetric tensor, we may choose an orthogonal frame at each point of the medium in such a way that the deformation tensor becomes diagonal with ( ) where the third direction is orthogonal to the elastic plate. We get: 1 Ω  , we obtain in first approximation: µ is the magnetic constant of vacuum, c is the speed of light in vacuum and n is the refraction index. The speed of light in the medium becomes i c n and therefore depends on the polarization of the beam. As the light is crossing the plate of thickness e put between two polarized filters at right angle, the entering monochromatic beam of light may be decomposed along the two proper directions into two separate beams recovering together after crossing with a time delay equal to: providing interferences and we find back the Maxwell phenomenological law of 1850: where σ is the stress tensor, k is an integer, λ is the wave length of the light used and C is the photoelastic constant of the medium ivolved in the experience.
Looking at the picture, let F be the vertical downwards force acting on the upper left side of the beam like on the picture, at a distance D from the center of the vertical beam on the right. We may consider this vertical beam as a dense sheaf of juxtaposed thin beams with young modulus E. Choosing orthogonal axes (Oxyz) such that Ox is horizontal towards the right with origin O in the geometric center of the vertical beam on the right which has a thickness in such a way that we have the equilibrium equation for couples: Finally, the distance d between two lines is such that k is modified by 1, that is Therefore the line " 0 k = " cannot exist. As for the lines " , a property that can be checked on the picture but cannot be even imagined.
We have thus explained, in a perfectly coherent way with the picture, why the interference lines are parallel and equidistant from each other in the right vertical part of the beam, on both sides of an (almost) central line which, surprisingly, stops at the upper and lower corner, even though, by continuity, we could imagine that it could be followed in the upper and lower horizontal parts of the beam. Also, we understand now the reason for which the lines in these parts of the beam look like symmetric hyperbolas. cannot appear at this level as we shall see, the main purpose of this paper is to prove that another differential sequence must be used, namely the Spencer sequence. The idea has been found totally independently, by the brothers E. and F. Cosserat in 1909 ( [12]) for revisting elasticity theory and by H. Weyl in 1916 ([13]) for revisiting electromagnetism by using the conformal group of space time, but the first ones were only dealing with the translations and rotations while the second was only dealing with the dilatation and the non-linear elations of this group, with no real progress during the last hundred years.
Extending the space ( ) , , x x x or ( ) , , x y z to space-time as before, the speed is now extended from ( ) of space at the same "time". Accordingly, the deformation tensor  , which is dimensionless, is extended by ( ) (Euler theorem) and is extended by setting where ρ is the mass per unit volume. Dealing with the rest-frame and using the (small) dilatation relation ρ is the value of ρ in the initial position where the body is supposed to be homogeneous, isotropic and unstressed, that is, 0 ρ is supposed to be a constant. The Hooke law is now extended by setting: in a way compatible with the conservation of mass and we suddenly discover that there is no conceptual difference between the Lamé constants ( ) We now understand that couplings are in fact more general constitutive laws taking into account the tensorial nature of the various terms involved through the Curie principle.

General Relativity versus Gauge Theory
Let A be a unitary ring, that is 1, , ,   These results have been used in control theory and it is now known that a control system is controllable if and only if it is parametrizable (See [18] [19] [30] [31] for more details). As a byproduct, and though it is still not acknowledged by engineers, controllability is a "built in" property that does not depend at all on the choice of the inputs and outputs among the control variables.
Keeping the same "operational" notations for simplicity, we may state ( [18] Accordingly, the corresponding differential sequence, which is formally exact by definition, is also locally exact. is not injective. There is therefore no lift and thus no splitting.
Multilying by a test function φ and integrating by parts, we obtain the parametrization :φ ξ →  in the form and "a" minimum involutive parametrization (but there can be others!): We get the long formally and locally exact differential sequence  which is involutive with one equation of class 2, 2 equations of class 1 and it is easy to check that the 2 corresponding first order CC is just the stress equations. Now, multiplying the Cauchy stress equations respectively by test functions 1 ξ and 2 ξ , then integrating by parts, we discover that (up to sign and a factor 2) the Cauchy operator is the formal adjoint of the Killing operator defined by ( ) , introducing the standard Lie derivative of the (non-degenerate) euclidean metric ω with respect to ξ and using the fact that we have 11   and ω is the euclidean metric, we get a single component that can be chosen to be the scalar curvature 11  There is no relation at all between the Airy stress function φ and the deformation Ω of the metric ω .  11  11  33  23  22  12  12  33  23  13  12  13  13  23  22  13  12  22  22  33  13  11  23  23  23  13  12  11  33  33  22 12 11 It is involutive with 3 equations of class 3, 3 equations of class 2 and no equation of class 1. The 3 CC is describing the stress equations which admit therefore a parametrization, but without any geometric framework, in particular without any possibility to imagine that the above second order operator is nothing else but the formal adjoint of the Riemann operator, namely the (linearized) Riemann tensor with ( ) J.-F. Pommaret [9] [10] [11]. We may rewrite the Beltrami parametrization of the Cauchy stress equations as follows, after exchanging the third row with the fourth row and using formal notations: 33 23 22   33  23  13  12  1  2  3  23  22  13  12  1  2  3  33  13  11  1  2  3  23  13  12  11   22 12 11 as an identity where 0 on the right denotes the zero operator. However, the standard implicit summation used in continuum mechanics (See [36] for more details) is, when 3 n = : between the (linearized ) Riemann tensor and the Beltrami parametrization.
As we already said, the brothers E. and F. Cosserat proved in 1909 that the assumption ij ji σ σ = may be too strong because it only takes into account density of forces and ignores density of couples, and the Cauchy stress equations must be replaced by the so-called Cosserat couple-stress equations ( [9] [10] [12] [37] [38]). In any case, taking into account the factor 2 involved by multiplying the second, third and fifth row by 2, we get the new 6 6 × matrix with rank 3: This is a symmetric matrix and the corresponding second order operator with constant coefficients is thus self-adjoint.
Surprisingly, the Maxwell parametrization is obtained by keeping only . Surprisingly, the Einstein operator is self adjoint while the Ricci operator is not and "Einstein equations are just a way to parametrize the Cauchy stress equations" because of the well known contraction of the  [44]). We now prove that only the use of differential homological algebra, a mixture of differential geometry (differential sequences, formal adjoint) and homological algebra (module theory, double duality, extension modules) totally unknown by physicists, is able to explain why the Einstein operator (with 6 terms) defined above is useless as it can be replaced by the Ricci operator (with 4 terms) in the search for gravitational waves equations. Indeed, denoting by The basic idea used in GR has been to simplify these equations by adding the differential constraints We have the following fiber dimensions for the classical Killing case and arbitrary dimension n: with fiber dimensions when Using the previous diagram, we obtain the isomorphisms ξ ξ → , in order to describe the cokernel of the left vertical monomorphism, we obtain the following commutative and exact diagram which is only depending on the first order jets of T: Prolonging twice to the jets of order 3 of T, we obtain the commutative and exact diagram: providing the same short exact sequence as in the Theorem but without any possibility to establish a link between Passing now to electromagnetism and the original Gauge Theory (GT) which is still, up to now, the only known way to establish a link between EM and group theory, the first idea is to introduce the nonlinear gauge sequence: where X is a manifold, G is a Lie group with identity e not acting on X, was the only possibility existing before 1970 to get a pure 1-form A (EM potential) and a pure 2-form F (EM field) when G is abelian. However, this result is not coherent at all with elasticity theory as we saw and, a fortiori, with the analytical mechanics of rigid bodies where the Lagrangian is a quadratic expression of such 1-forms when  [45] and [46]  Looking for the first order generating compatibility conditions (CC) 1  of the corresponding second order operator  just described, we may then look for the generating CC 2  of 1  and so on, exactly like in the differential sequence made successively by the Killing, Riemann, Bianchi, ... operators. We may proceed similarly for the injective operator It follows that "Spencer and Janet play at see-saw", the dimension of each Janet bundle being decreased by the same amount as the dimension of the corresponding Spencer bundle is increased, this number being the number of additional parameters multiplied by The linear Spencer Sequence is locally isomorphic to the linear gauge sequence for Lie groups, with the main difference that the group is now acting on the manifold, contrary to the previous situation.
We may pick up a section of r F , lift it up to a section of ( )   and we discover exactly the group scheme used through this paper, both with the need to shift by one step to the left the physical interpretation of the various differential sequences used. Indeed, as  "Beyond the mirror" of the classical approach to apparently well known and established theories, there is a totally new interpretation of these theories and the corresponding field/matter couplings by means of the Spencer sequence for the conformal Killing operator.  The purely mathematical results of Section 3 perfectly agree with the origin and existence of elastic and electromagnetic waves but question the origin and existence of gravitational waves because the parametrization of the Cauchy operator can be simply done by the adjoint of the Ricci operator without any reference to the Einstein or even Bianchi operators. We believe that such a confusion mainly came from the fact that it had never been noticed that the Einstein operator was self-adjoint.
 They prove that the concept of "field" in a physical theory must not be related with the concept of "curvature" because it is a 1-form with value in a Lie algebroid (first Spencer bundle) and not a 2 form with value in a Lie algebra (second Spencer bundle). The "shift by one step" in the physical interpretation of a differential sequence is thus the main feature of this new mathematical framework.  They also prove that gravitation and electromagnetism have a common conformal origin. In particular, electromagnetism has only to do with the conformal group of space-time and not with ( ) 1 U as it is still believed today in Gauge Theory.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.