Gravitational Waves and Parametrizations of Linear Differential Operators

Abstract

When $\mathcal{D}:\xi \to \eta$ is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator ${\mathcal{D}}_1:\eta \to \zeta$ such that $\mathcal{D}\xi =\eta$ implies ${\mathcal{D}}_1\eta =0$. Similarly, ${\mathcal{D}}_1\eta =\zeta$ may imply ${\mathcal{D}}_2\zeta =0$ and so on. Conversely, when ${\mathcal{D}}_1$ is given, a much more difficult “inverse problem” is to look for an operator $\mathcal{D}:\xi \to \eta$ with generating CC ${\mathcal{D}}_1\eta =0$. If this is possible, one shall say that the operator ${\mathcal{D}}_1$ is parametrized by $\mathcal{D}$. The parametrization is “minimum” if the differential module defined by $\mathcal{D}$ does not contain any free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test. The parametrization of the Cauchy stress operator in arbitrary dimension $n$ has attracted many famous scientists (G.B. Airy in 1863 for $n=2$, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for $n=3$, A. Einstein in 1915 for $n=4$). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized, and are thus all based on a confusion between the Cauchy operator, (adjoint of the Killing operator $\mathcal{D}$), and the div operator induced from the Bianchi operator ${\mathcal{D}}_2$ CC of the Riemann operator ${\mathcal{D}}_1$ parametrized by $\mathcal{D}$. This purely mathematical result deeply questions the origin and existence of gravitational waves that are solutions of the adjoint of the Ricci operator. We do believe that Einstein was aware of these previous works as the comparison needs no comment. The same methods are also used in order to revisit the mathematical foundations of electromagnetism.

1. Introduction

The problem of parametrizing the Einstein operator or, equivalently and by analogy with Maxwell equations for electromagnetism (EM), to decide about the existence of a potential for Einstein equations in vacuum, has been proposed for the first time as a 1000 dollars challenge by J. Wheeler while the author of this paper was a visiting student of D. C. Spencer in 1970 at Princeton university. No progress at all has been done during the next 25 years, till the author gave a negative answer in 1995, contrary to what the general relativity (GR) community was believing [1, 2]. Indeed, after teaching elasticity for 25 years to high-level students in some of the best french civil engineering schools, the author of this paper proposed an exercise explaining why a dam made with concrete is always vertical on the water side with a slope of about 42 degrees on the other free side in order to obtain a minimum cost and the auto-stability under cracking of the surface underwater (See ([3], p. 108) and the introduction of [4] for more details). Surprisingly, the main tool involved is the approximate computation of the Airy function inside the dam in this two-dimensional elasticity problem. The author discovered at that time that no one of the other teachers did know that the Airy parametrization was nothing else than the adjoint of the linearized Riemann operator used as generating CC for the deformation tensor by any engineer. Being involved in GR with A. Lichnerowicz at that time, he got for the first time the idea of using the adjoint of an operator in a systematic way. Giving a seminar in Paris in order to present this result, somebody in the audience told him about a possible link with the recently published master thesis of the Japanese student M. Kashiwara [5]. It has been a shock to discover this mixing up of differential geometry [6, 7] and homological algebra [8, 9, 10], now called “Differential Homological Algebra”, in particular the introduction of the Differential Extension Modules (See [4, 5, 11, 12, 13] for extensive references) (See also Zbl 1079.93001 for comments). It is only recently that he discovered GR could be considered as a way to parametrize the Cauchy operator and to introduce gravitational waves (GW) [14, 15]. It follows that exactly the same confusion has been done by Maxwell, Morera, Beltrami and Einstein because, in all these cases, the operator considered is self-adjoint. However, most of the pure mathematicians involved were proud not to be interested by applications and in any case unable to do any computation. Accordingly and until now, the GR community has never wanted to take these new tools into account and Ref. [16] is providing a good example of such a poor situation. This is the reason for which we have not been able to provide any other reference and why all the results presented are new.

For example, the fact that the Cauchy operator is the adjoint of the Killing operator for the Euclidean metric is apparently in the chapter “variational calculus” of any textbook of continuum mechanics, and the parametrization problem has been quoted by many famous authors, as we said in the Abstract, but only from a computational point of view. The same comment can be done for the two sets of Maxwell equations in electromagnetism [4, 13]. However, it is still not known that the adjoint of the 20 components of the Bianchi operator has been introduced by C. Lanczos between 1939 and 1962 [17] as we explained with details in [18] by using Spencer cohomology. The main trouble is that these two problems have never been treated in an intrinsic way and, in particular, changes of coordinates have never been considered. The same situation can be met for the Vessiot structure equations but is out of the scope of this paper [12, 19].

Lemma 1.1. When $y^k=f^k\left(x\right)$ is invertible with $\mathrm{\Delta }\left(x\right)=\mathit{det}\left({\partial }_i{f}^k\left(x\right)\right)\ne 0$ and inverse $x=g\left(y\right)$, then we have $n$ identities $\frac{\partial }{\partial y^k}\left(\frac{1}{\mathrm{\Delta }}{\partial }_i{f}^k\left(g\left(y\right)\right)\right)=0$ (See [4] p 490 and [20] for other applications).

Proof: Using the chain rule for derivatives, we get ${\partial }_i\mathrm{\Delta }={\partial }_{\mathit{ij}}{f}^k$ cofactor $\left({\partial }_j{f}^k\right)={\partial }_{\mathit{ij}}{f}^k\mathrm{\Delta }\frac{\partial g^j}{\partial y^k}$ and thus:

$$\frac{1}{\mathrm{\Delta }}{\partial }_{\mathit{ij}}{f}^k\frac{\partial g^j}{\partial y^k}-\frac{1}{{\mathrm{\Delta }}^2}{\partial }_i{f}^k\left(\frac{\partial g^j}{\partial y^k}{\partial }_j\mathrm{\Delta }\right)=\frac{1}{\mathrm{\Delta }}{\partial }_{\mathit{ij}}{f}^k\frac{\partial g^j}{\partial y^k}-\frac{1}{{\mathrm{\Delta }}^2}{\partial }_i\mathrm{\Delta }=0$$

$\square$

Proposition 1.2. The Cauchy operator is the adjoint of the Killing operator in arbitrary dimension $n$, up to sign. Similarly, when $n=4$, the Maxwell operator ${\wedge }^4{T}^{\ast }\otimes {\wedge }^{2}T\stackrel{\mathit{ad}\left(d\right)}{\to }{\wedge }^4{T}^{\ast }\otimes T:\left({\mathcal{F}}^{\mathit{ij}}\right)\to \left({\partial }_i{\mathcal{F}}^{\mathit{ij}}={\mathcal{J}}^j\right)$ is the adjoint of the parametrizing operator $T^{\ast }\stackrel{d}{\to }{\wedge }^2{T}^{\ast }:A\to \mathit{dA}=F$ in electromagnetism (EM), independently of the Minkowski constitutive relations $F\to \mathcal{F}$.

Proof: Let $X$ be a manifold of dimension $n$ with local coordinates $\left(x^1,\dots ,x^n\right)$, tangent bundle $T$_, and cotangent bundle $T^{\ast }$. If $\omega \in S_2{T}^{\ast }$ is a metric with $\mathit{det}\left(\omega \right)\ne 0$, we may introduce the standard L, that is, derivative in order to define the first order Killing operator $\xi \to \mathcal{L}\left(\xi \right)\omega$, namely:

$$\mathcal{D}:\xi \in T\to \mathrm{\Omega }=\left({\mathrm{\Omega }}_{\mathit{ij}}={\omega }_{\mathit{rj}}\left(x\right){\partial }_i{\xi }^r+{\omega }_{\mathit{ir}}\left(x\right){\partial }_j{\xi }^r+{\xi }^r{\partial }_r{\omega }_{\mathit{ij}}\left(x\right)\right)\in S_2{T}^{\ast }$$E1

Here starts the problem because, in our opinion at least, a systematic use of the (formal) adjoint operator has never been done in mathematical physics (continuum mechanics, EM, …) apart from a variational procedure. As will be seen later on, the purely intrinsic definition of the adjoint can only be done in the theory of differential modules by means of the so-called side-changing functor. From a purely differential geometric point of view, the idea is to associate to any vector bundle $E$ over $X$_, a new vector bundle $\mathit{ad}\left(E\right)={\wedge }^n{T}^{\ast }\otimes E^{\ast }$_, where $E^{\ast }$ is obtained from $E$ by patching local coordinates while inverting the transition matrices, exactly like $T^{\ast }$ is obtained from $T$ in tensor calculus. It follows that the stress tensor $\sigma =\left({\sigma }^{\mathit{ij}}\right)\in \mathit{ad}\left(S_2{T}^{\ast }\right)={\wedge }^n{T}^{\ast }\otimes S_{2}T$ is not a tensor but a tensor density, that transforms like a tensor up to a certain power of the Jacobian matrix. When $n=4$, the fact that such an object is called stress-energy tensor does not change anything as it cannot be related to the Einstein tensor which is a true tensor indeed. Of course, it is always possible in GR to use ${\left(\mathit{det}\left(\omega \right)\right)}^{\frac{1}{2}}$ but, as we shall see, the study of contact structures must be done without any reference to a background metric. In any case, we may define as usual:

$$\mathit{ad}\left(\mathcal{D}\right):{\wedge }^n{T}^{\ast }\otimes S_{2}T\to {\wedge }^n{T}^{\ast }\otimes T:\sigma \to \phi$$E2

Multiplying ${\mathrm{\Omega }}_{\mathit{ij}}$ by ${\sigma }^{\mathit{ij}}$ and integrating by parts, the factor of $-2{\xi }^k$ is easily seen to be:

$${\nabla }_i{\sigma }^{\mathit{ik}}={\partial }_i{\sigma }^{\mathit{ik}}+{\gamma }_{\mathit{ij}}^k{\sigma }^{\mathit{ij}}={\phi }^k$$E3

with well known Christoffel symbols ${\gamma }_{\mathit{ij}}^k=\frac{1}{2}{\omega }^{\mathit{kr}}\left({\partial }_i{\omega }_{\mathit{rj}}+{\partial }_j{\omega }_{\mathit{ir}}-{\partial }_r{\omega }_{\mathit{ij}}\right)$.

However, if the stress should be a tensor, we should get for the covariant derivative:

$${\nabla }_r{\sigma }^{\mathit{ij}}={\partial }_r{\sigma }^{\mathit{ij}}+{\gamma }_{\mathit{rs}}^i{\sigma }^{\mathit{sj}}+{\gamma }_{\mathit{rs}}^j{\sigma }^{\mathit{\text{is}}}\Rightarrow {\nabla }_i{\sigma }^{\mathit{ik}}={\partial }_i{\sigma }^{\mathit{ik}}+{\gamma }_{\mathit{ri}}^r{\sigma }^{\mathit{ik}}+{\gamma }_{\mathit{ij}}^k{\sigma }^{\mathit{ij}}$$

The difficulty is to prove that we do not have a contradiction because $\sigma$ is a tensor density.

If we have an invertible transformation like in the lemma, we have successively by using it:

$$\begin{array}{c}{\tau }^{\mathit{kl}}\left(f\left(x\right)\right)=\frac{1}{\mathrm{\Delta }}{\partial }_i{f}^k\left(x\right){\partial }_j{f}^l\left(x\right){\sigma }^{\mathit{ij}}\left(x\right)\\ \frac{\partial {\tau }^{\mathit{kl}}}{\partial y^k}=\left(\frac{\partial }{\partial y^k}\left(\frac{1}{\mathrm{\Delta }}{\partial }_i{f}^k\right)\right){\partial }_j{f}^l{\sigma }^{\mathit{ij}}+\frac{1}{\mathrm{\Delta }}{\partial }_i{f}^k\frac{\partial }{\partial y^k}\left({\partial }_j{f}^l\right){\sigma }^{\mathit{ij}}+\frac{1}{\mathrm{\Delta }}{\partial }_i{f}^k{\partial }_j{f}^l\frac{\partial }{\partial y^k}{\sigma }^{\mathit{ij}}\\ \Rightarrow \frac{\partial {\tau }^{\mathit{ku}}}{\partial y^k}=\frac{1}{\mathrm{\Delta }}\left({\partial }_{\mathit{ij}}{f}^u\right){\sigma }^{\mathit{ij}}+\frac{1}{\mathrm{\Delta }}{\partial }_j{f}^u{\partial }_i{\sigma }^{\mathit{ij}}\end{array}$$

Now, we recall the transformation law of the Christoffel symbols, namely:

$$\begin{array}{l}{\partial }_r{f}^u\left(x\right){\gamma }_{\mathit{ij}}^r\left(x\right)={\partial }_{\mathit{ij}}{f}^u\left(x\right)+{\partial }_i{f}^k\left(x\right){\partial }_j{f}^l\left(x\right){\overline{\gamma }}_{\mathit{kl}}^u\left(f\left(x\right)\right)\\ \Rightarrow \frac{1}{\mathrm{\Delta }}{\partial }_r{f}^u{\gamma }_{\mathit{ij}}^r{\sigma }^{\mathit{ij}}=\frac{1}{\mathrm{\Delta }}{\partial }_{\mathit{ij}}{f}^u{\sigma }^{\mathit{ij}}+{\overline{\gamma }}_{\mathit{kl}}^u\left(y\right){\tau }^{\mathit{kl}}\end{array}$$

Eliminating the second derivatives of $f$, we finally get:

$${\psi }^u=\frac{\partial {\tau }^{\mathit{ku}}}{\partial y^k}+{\overline{\gamma }}_{\mathit{kl}}^u{\tau }^{\mathit{kl}}=\frac{1}{\mathrm{\Delta }}{\partial }_r{f}^u\left({\partial }_i{\sigma }^{\mathit{ir}}+{\gamma }_{\mathit{ij}}^r{\sigma }^{\mathit{ij}}\right)=\frac{1}{\mathrm{\Delta }}{\partial }_r{f}^u{\phi }^r$$

This tricky technical result, which is not evident, explains why the additional term we had is just disappearing, in fact, when $\sigma$ is a density.

The case of EM is even simpler because ${\partial }_{\mathit{ij}}{f}^u{\mathcal{F}}^{\mathit{ij}}={\partial }_{\mathit{ji}}{f}^u{\mathcal{F}}^{\mathit{ji}}=-{\partial }_{\mathit{ij}}{f}^u{\mathcal{F}}^{\mathit{ij}}=0$ and $\gamma$ is not needed. The two sets of Maxwell equations are thus separately invariant by any diffeomorphism. Though surprising it may look like, the conformal group of space–time is only the maximum group of invariance of the Minkowski constitutive law in vacuum. Indeed, this law is not at all $F_{\mathit{ij}}={\mu }_0{\omega }_{\mathit{ir}}{\omega }_{\mathit{js}}{\mathcal{F}}^{\mathit{rs}}$_, where ${\mu }_0$ is the magnetic constant because such a relation is not tensorial as $F$ is a $2$-form, that is, a $2$-covariant tensor, but $\mathcal{F}$ is a $2$-contravariant tensor density. Hence, introducing the metric density ${\widehat{\omega }}_{\mathit{ij}}={\left(|det\left(\omega \right)|\right)}^{-1/n}{\omega }_{\mathit{ij}}$, we must set $F_{\mathit{ij}}={\mu }_0{\widehat{\omega }}_{\mathit{ir}}{\widehat{\omega }}_{\mathit{js}}{\mathcal{F}}^{\mathit{rs}}$. Accordingly, this constitutive law is only invariant by diffeomorphisms preserving $\widehat{\omega }$_, and this is exactly the definition of the Lie pseudogroup of conformal transformations [13].

$\square$

By chance, the control community has been interested during a while by these new techniques for dealing with PD control theory but mostly restricting to operators with constant coefficients [4, 20, 21]. The following example, coming from partial differential (PD) control theory, will allow the reader to become familiar with these new tools and to understand why they are related to the mathematics of GW. Accordingly, the “relative” parametrization of the Cauchy stress operator has thus nothing to do with the mathematical background of elasticity theory. In particular, the way to simplify the GW equations by bringing them back to the d’Alembert operator while adding a few differential constraints (Compare [14, 15, 22, 23, 24]) through a reference to the so-called “gauge invariance” is rather a physical argument and not a mathematical one.

Example 1.3. Let us consider the first order operator with two independent variables $\left(x^1,x^2\right)$:

$${\mathcal{D}}_1:\left({\eta }^1,{\eta }^2\right)\to d_2{\eta }^1-d_1{\eta }^2+x^2{\eta }^2=\zeta$$E4

The non-commutative ring of differential operators involved in a formal study is $D=K\left[d_1,d_2\right]$ with $K=\mathbb{Q}\left(x^1,x^2\right)=\mathbb{Q}\left(x\right)$ and the characters of this involutive system are $\left({\alpha }_1^1=2,{\alpha }_1^2=1\right)$ with ${\beta }_1^2=2-1=1$ as there is only one equation. Multiplying on the left by a test function $\lambda$ and integrating by parts, the corresponding adjoint operator is described by:

$$\mathit{ad}\left({\mathcal{D}}_1\right):\lambda \to \left(-d_2\lambda ={\mu }^1,d_1\lambda +x^2\lambda ={\mu }^2\right)$$

Using crossed derivatives, this operator is injective because $\lambda =d_2{\mu }^2+d_1{\mu }^1+x^2{\mu }^1$ and we even obtain a lift $\mathit{id}:\lambda \to \mu \to \lambda$. Substituting, we get the second order involutive operator $\mathit{ad}\left(\mathcal{D}\right):\left({\mu }^1,{\mu }^2\right)\to \left({\nu }^1,{\nu }^2\right)$ with characters $\left({\alpha }_2^1=1,{\alpha }_2^2=1\right)$, namely:

$$\left\{\boxed{\frac{d_{22}{\mu }^2+d_{12}{\mu }^1}{d_{12}{\mu }^2+d_{11}{\mu }^1}}\right.\begin{array}{ll}+x^2{d}_2{\mu }^1+2{\mu }^1& ={\nu }^1\\ +2x^2{d}_1{\mu }^1+x^2{d}_2{\mu }^2+{\left(x^2\right)}^2{\mu }^1-{\mu }^2& ={\nu }^2\end{array}\boxed{\begin{array}{ll}1& 2\\ 1& •\end{array}}$$

allowing to define a second order operator $\mathcal{D}$ by using the fact that $\mathit{ad}\left(\mathit{ad}\left(\mathcal{D}\right)\right)=\mathcal{D}$. This operator is involutive and the only corresponding generating CC is $d_2{\nu }^2-d_1{\nu }^1-x^2{\nu }^1=0$. Therefore, ${\nu }^2$ is differentially dependent on ${\nu }^1$ but ${\nu }^1$ is also differentially dependent on ${\nu }^2$. Multiplying on the left by a test function $\theta$ and integrating by parts, the corresponding adjoint operator is:

$${\mathcal{D}}_{-1}:\theta \to \left(d_1\theta -x^2\theta ={\xi }^1,-d_2\theta ={\xi }^2\right)$$

Multiplying now the first equation of $\mathit{ad}\left(\mathcal{D}\right)$ by the test function ${\xi }^1$, the second equation by the test function ${\xi }^2$, adding and integrating by parts, we get the second order operator:

$$\mathcal{D}:\left({\xi }^1,{\xi }^2\right)\to \left\{\boxed{\frac{d_{22}{\xi }^1+d_{12}{\xi }^2}{d_{12}{\xi }^1+d_{11}{\xi }^2}}\right.\begin{array}{lll}-x^2{d}_2{\xi }^2-2{\xi }^2& =& {\eta }^2\\ -x^2{d}_2{\xi }^1-2x^2{d}_1{\xi }^2+{\xi }^1+{\left(x^2\right)}^2{\xi }^2& =& {\eta }^1\end{array}\boxed{\begin{array}{cc}1& 2\\ 1& •\end{array}}$$E5

which is easily seen to be a parametrization of ${\mathcal{D}}_1$. This operator is involutive and the kernel of this parametrization has differential rank equal to 1 because ${\xi }^1$ or ${\xi }^2$ can be given arbitrarily.

As we are using the Lagrange multiplier $\left({\xi }^1,{\xi }^2\right)$, we consider in fact the PD equation ${\xi }^1{\nu }^1+{\xi }^2{\nu }^2$. Hence, we could indeed consider each term separately, that is using independently each equation as we shall see later on, ${\nu }^1$ for a certain $\xi$ and ${\nu }^2$ for a certain ${\xi }^{\text{'}}$. Equivalently (exactly like Morera and Maxwell did as we shall see later on), keeping ${\xi }^1=\xi$ while setting ${\xi }^2=0$, we now obtain the first second order minimal parametrization

$$\xi \to \left(d_{22}\xi ={\eta }^2,d_{12}\xi -x^2{d}_2\xi +\xi ={\eta }^1\right)$$E6

This system is again involutive, and the parametrization is minimal because the kernel of this parametrization has differential rank equal to 0. With a similar comment, setting now ${\xi }^1=0$ while keeping ${\xi }^2={\xi }^{\prime }$, we get the second second order minimal parametrization:

$${\xi }^{\prime }\to \left(d_{12}{\xi }^{\prime }-x^2{d}_2{\xi }^{\prime }-2{\xi }^{\prime }={\eta }^2,d_{11}{\xi }^{\prime }-2x^2{d}_1{\xi }^{\prime }+{\left(x^2\right)}^2{\xi }^{\prime }={\eta }^1\right),$$E7

which is again easily seen to be involutive by exchanging $x^1$ with $x^2$.

With again a similar comment, setting now ${\xi }^1=d_1\varphi ,{\xi }_2=-d_2\varphi$ in the canonical parametrization, we obtain the third different second order minimal parametrization:

$$\varphi \to \left(x^2{d}_{22}\varphi +2d_2\varphi ={\eta }^2,x^2{d}_{12}\varphi -{\left(x^2\right)}^2{d}_2\varphi +d_1\varphi ={\eta }^1\right)$$E8

We are now ready for understanding the meaning and usefulness of what we have defined and called “relative parametrization” in [24] by imposing the differential constraint $d_2{\xi }^1+d_1{\xi }^2=0$. First of all, we have to prove that such a constraint is compatible. For this, taking into account the constraint, we have the following first order system defined over $K$:

$$\left\{\begin{array}{lll}{d}_2{\xi }^1+d_1{\xi }^2& =& 0\\ d_2{\xi }^2+\frac{2}{x^2}{\xi }^2& =& -\frac{1}{x^2}{\eta }^2\\ d_1{\xi }^2-x^2{\xi }^2-\frac{1}{x^2}{\xi }^1& =& -\frac{1}{x^2}{\eta }^1\end{array}\right.\boxed{\begin{array}{cc}1& 2\\ 1& 2\\ 1& •\end{array}}$$

We let the reader prove that this first order system is involutive with full class 2 and characters $\left({\alpha }_1^1=1,{\alpha }_1^2=0\right)$ and that the only CC involved is the initial system for $\eta$ (The reader will discover that this checking is quite harder that what one could believe on such an elementary example).

We obtain therefore the new first order relative parametrization:

$$\left({\xi }^1,{\xi }^2\right)\to \left(-x^2{d}_2{\xi }^2-2{\xi }^2={\eta }^2,x^2{d}_2{\xi }^1+{\left(x^2\right)}^2{\xi }^2+{\xi }^1={\eta }^1\right)\mathit{mod}\left(d_2{\xi }^1+d_1{\xi }^2=0\right)$$

In a different way, we may add the differential constraint $d_1{\xi }^1+d_2{\xi }^2=0$ but we have to check similarly that it is compatible with the previous parametrization. For this, we have to consider the following second order involutive system with five equations, which are easily seen to be involutive, obtained by adding the constraint and its two derivatives to the system like before:

$$\left\{\begin{array}{lll}{d}_{22}{\xi }^2+d_{12}{\xi }^1& =& 0\\ d_{22}{\xi }^1+d_{12}{\xi }^2-x^2{d}_2{\xi }^2-2{\xi }^2& =& {\eta }^2\\ d_{12}{\xi }^2+d_{11}{\xi }^1& =& 0\\ d_{12}{\xi }^1+d_{11}{\xi }^2-x^2{d}_2{\xi }^1-2x^2{d}_1{\xi }^2+{\xi }^1+{\left(x^2\right)}^2{\xi }^2& =& {\eta }^1\\ d_2{\xi }^2+d_1{\xi }^1& =& 0\end{array}\right.\boxed{\begin{array}{cc}1& 2\\ 1& 2\\ 1& •\\ 1& •\\ •& •\end{array}}$$

The four generating CC only produce the desired system for $\left({\eta }^1,{\eta }^2\right)$ as we wished.

We could not impose the condition ${\mathcal{D}}_{-1}\theta =\xi$ already found as it should give the identity $0=\eta$.

It is, however, also important to notice that the long strictly exact sequence [25]:

$$0\to \theta \stackrel{{\mathcal{D}}_{-1}}{\to }\xi \stackrel{\mathcal{D}}{\to }\eta \stackrel{{\mathcal{D}}_1}{\to }\zeta \to 0$$

splits because we have a lift $\mathit{id}:\zeta \to \left(-{\partial }_1\zeta +x^2\zeta ={\eta }^1,-{\partial }_2{\eta }^2={\eta }^2\right)\to {\partial }_2{\eta }^1-{\partial }_1{\eta }^2+x^2{\eta }^2=\zeta$.

All the differential modules defined from the operators involved are projective, thus torsion-free, and we notice that ${\mathcal{D}}_1$ is parametrized by $\mathcal{D}$ which is again parametrized by ${\mathcal{D}}_{-1}$, exactly like div is parametrized by curl which is again parametrized by grad in vector geometry. Needless to say that such an approach has nothing to do with Lorenz gauge invariance in electromagnetism (EM) and we shall arrive to the same conclusion for GW in GR.

Going on along the historical survey 50 years ago, while the author of this paper was working in GR under the leadership of Prof. A. Lichnerowicz, he became familiar with the Lanczos problems. Since that time, he had no wish at all to enter this kind of game as this domain became the private garden of a few persons, each one writing after another one alternatively, claiming to have the full solution. Also, the papers were covered with “computations” involving awful technical formulas, one paper using Gröbner bases, another computer algebra, another Cartan exterior calculus or Janet bases and so on during these 50 years.

Later on, in 2001 and for different reasons, namely control theory as we just explained, being more familiar with differential homological algebra and the so-called “parametrization problem,” the way towards the Lanczos problems became easier as follows [18]. In dimension four, the only considered by Lanczos, the Lanczos “potential” $L_{\mathit{ij},k}=-L_{\mathit{ji},k}$ has $6\times 4=24$ components. As they must be related by the four relations $L_{\mathit{ij},k}+L_{\mathit{jk},i}+L_{\mathit{ki},j}=0$, we get 20 independent components, namely the number of (second) Bianchi identities. However, who is speaking about “potential” means “parametrization”, … of what?. Here comes the confusion of Lanczos, too much familiar with electromagnetism (EM) while using mainly quadratic Lagrangians with the Riemann tensor in place of the EM field $F$ and the Bianchi identities as differential constraint in the corresponding variational calculus with constraint. The operator to be parametrized was thus the adjoint of the Riemann operator that we called Beltrami operator while the parametrizing operator, that we called Lanczos operator, was just the adjoint of the Bianchi operator, going now backwards, that is from right to left in the adjoint sequence of the Killing resolution presented in the abstract. As for the extension to the conformal framework, it is clear that Lanczos did not even know the Weyl tensor when he lectured in France in 1962, invited by Lichnerowicz. Moreover, it is only recently in 2016 that the author of this paper proved that the analogue of the Bianchi identities is made by an operator of order two when $n=4$, such a result being tested through computer algebra by his former PhD student A. Quadrat (See [12, 13] and arXiv:1603.05030 for more details). Such a construction, based on difficult results of homological algebra, has been missed by Lanczos and all followers as such tools were only available after 1995 through the works of pure mathematicians not interested by applications. Therefore, the main idea is to replace technical formulas by diagram chasing without any formula. As a byproduct, we shall understand, without any computation, the confusion done between the Cauchy operator, adjoint of the Killing operator, and the Bianchi operator as explained in the abstract. We shall explain its historical origin as the names of many celebrated scientists are involved in this confusion as we also said in the abstract.

The story ended with a strange letter sent by J. Wheeler back to the author with a one dollar bill attached, refusing to admit the negative answer to his parametrization challenge and claiming that future quantum GR should find a positive answer (!). As a byproduct, the impossibility to parametrize Einstein equations in vacuum can only be found in books of control theory [20, 26].

After this rather historical introduction, the content of the paper is clear:

The second section presents the mathematical tools that are absolutely needed while the third section is dealing with the solution of the parametrization problem. The applications to Einstein equations and the corresponding GW equations is finally presented in the fourth section before concluding. In any case, we want to point out that no one of these methods have ever been used in GR, in particular for the study of GW.

2. Mathematical tools

We start recalling the basic tools from the formal theory of systems of ordinary differential (OD) or partial differential (PD) equations and differential modules needed in order to understand and solve the parametrization problem presented in the abstract. Then we provide the example of the system of infinitesimal lie equations defining contact transformations and conclude the paper with the general parametrization problem existing in continuum mechanics and general relativity for an arbitrary dimension of the ground manifold. As these new tools are difficult and not so well known as we already said, we advise the interested reader to follow them step by step on the explicit motivating examples illustrating this paper, while referring to [4, 12, 20, 22, 25, 27, 28, 29, 30] for more details, even though the paper is rather self-contained and uses standard notations. Parts of the present paper have already been published independently with slight differences in [2, 12, 13, 18, 20, 21, 22, 23, 24, 31] but the present paper is mainly revisiting the mathematical foundations of GW in the framework of differential homological algebra.

2.1 System theory

If $X$ is a manifold of dimension $n$ with local coordinates $\left(x\right)=\left(x^1,\dots ,x^n\right)$, we denote as usual by $T=T\left(X\right)$ the tangent bundle of $X$, by $T^{\ast }=T^{\ast }\left(X\right)$ the cotangent bundle, by ${\wedge }^r{T}^{\ast }$ the bundle of r-forms and by $S_q{T}^{\ast }$ the bundle of q-symmetric tensors. More generally, let $E$ be a vector bundle over $X$ with local coordinates $\left(x^i,y^k\right)$ for $i=1,\dots ,n$ and $k=1,\dots ,m$ simply denoted by $\left(x,y\right)$, projection $\pi :E\to X:\left(x,y\right)\to \left(x\right)$ and changes of local coordinate $\overline{x}=\phi \left(x\right),\overline{y}=A\left(x\right)y$. We shall denote by $E^{\ast }$ the vector bundle obtained by inverting the matrix $A$ of the changes of coordinates, exactly like $T^{\ast }$ is obtained from $T$. We denote by $f:X\to E:\left(x\right)\to \left(x,y=f\left(x\right)\right)$ a global section of $E$, that is a map such that $\pi \circ f={\mathit{id}}_X$ but local sections over an open set $U\subset X$ may also be considered when needed. Under a change of coordinates, a section transforms like $\overline{f}\left(\phi \left(x\right)\right)=A\left(x\right)f\left(x\right)$ and the changes of the derivatives can also be obtained with more work. We shall denote by $J_q\left(E\right)$ the q-jet bundle of $E$ with local coordinates $\left(x^i,y^k,y_i^k,y_{\mathit{ij}}^k,\dots \right)=\left(x,y_q\right)$ called jet coordinates and sections $f_q:\left(x\right)\to \left(x,f^k\left(x\right),f_i^k\left(x\right),f_{\mathit{ij}}^k\left(x\right),\dots \right)=\left(x,f_q\left(x\right)\right)$ transforming like the sections $j_q\left(f\right):\left(x\right)\to \left(x,f^k\left(x\right),{\partial }_i{f}^k\left(x\right),{\partial }_{\mathit{ij}}{f}^k\left(x\right),\dots \right)=\left(x,j_q\left(f\right)\left(x\right)\right)$ where both $f_q$ and $j_q\left(f\right)$ are over the section $f$ of $E$. For any $q\ge 0$, $J_q\left(E\right)$ is a vector bundle over $X$ with projection ${\pi }_q$_, while $J_{q+r}\left(E\right)$ is a vector bundle over $J_q\left(E\right)$ with projection ${\pi }_q^{q+r},\forall r\ge 0$.

Definition 2.A.1: A linear system of order $q$ on $E$ is a vector sub-bundle $R_q\subset J_q\left(E\right)$ and a solution of $R_q$ is a section $f$ of $E$ such that $j_q\left(f\right)$ is a section of $R_q$. With a slight abuse of language, the set of local solutions will be denoted by $\mathrm{\Theta }\subset E$.

Let $\mu =\left({\mu }_1,\dots ,{\mu }_n\right)$ be a multi-index with length $\mid \mu \mid ={\mu }_1+\dots +{\mu }_n$, class $i$ if ${\mu }_1=\dots ={\mu }_{i-1}=0,{\mu }_i\ne 0$ and $\mu +1_i=\left({\mu }_1,\dots ,{\mu }_{i-1},{\mu }_i+1,{\mu }_{i+1},\dots ,{\mu }_n\right)$. We set $y_q=\left\{y_{\mu }^k|1\le k\le m,0\le |\mu |\le q\right\}$ with $y_{\mu }^k=y^k$ when $\mid \mu \mid =0$. If $E$ is a vector bundle over $X$ and $J_q\left(E\right)$ is the $q$-jet bundle of $E$, then both sections $f_q\in J_q\left(E\right)$ and $j_q\left(f\right)\in J_q\left(E\right)$ are over the section $f\in E$. There is a natural way to distinguish them by introducing the Spencer operator $d:J_{q+1}\left(E\right)\to T^{\ast }\otimes J_q\left(E\right)$ with components ${\left({\mathit{df}}_{q+1}\right)}_{\mu ,i}^k\left(x\right)={\partial }_i{f}_{\mu }^k\left(x\right)-f_{\mu +1_i}^k\left(x\right)$. The kernel of $d$ consists of sections such that $f_{q+1}=j_1\left(f_q\right)=j_2\left(f_{q-1}\right)=\dots =j_{q+1}\left(f\right)$. Finally, if $R_q\subset J_q\left(E\right)$ is a system of order $q$ on $E$ locally defined by linear equations ${\mathrm{\Phi }}^{\tau }\left(x,y_q\right)\equiv a_k^{\mathit{\tau \mu }}\left(x\right)y_{\mu }^k=0$ and local coordinates $\left(x,z\right)$ for the parametric jets up to order $q$, the $r$-prolongation $R_{q+r}={\rho }_r\left(R_q\right)=J_r\left(R_q\right)\cap J_{q+r}\left(E\right)\subset J_r\left(J_q\left(E\right)\right)$ is locally defined when $r=1$ by the linear equations ${\mathrm{\Phi }}^{\tau }\left(x,y_q\right)=0,d_i{\mathrm{\Phi }}^{\tau }\left(x,y_{q+1}\right)\equiv a_k^{\mathit{\tau \mu }}\left(x\right)y_{\mu +1_i}^k+{\partial }_i{a}_k^{\mathit{\tau \mu }}\left(x\right)y_{\mu }^k=0$ and has symbol $g_{q+r}=R_{q+r}\cap S_{q+r}{T}^{\ast }\otimes E\subset J_{q+r}\left(E\right)$ if one looks at the top order terms. If $f_{q+1}\in R_{q+1}$ is over $f_q\in R_q$, differentiating the identity $a_k^{\mathit{\tau \mu }}\left(x\right)f_{\mu }^k\left(x\right)\equiv 0$ with respect to $x^i$ and substracting the identity $a_k^{\mathit{\tau \mu }}\left(x\right)f_{\mu +1_i}^k\left(x\right)+{\partial }_i{a}_k^{\mathit{\tau \mu }}\left(x\right)f_{\mu }^k\left(x\right)\equiv 0$, we obtain the identity $a_k^{\mathit{\tau \mu }}\left(x\right)\left({\partial }_i{f}_{\mu }^k\left(x\right)-f_{\mu +1_i}^k\left(x\right)\right)\equiv 0$ and thus the restriction $d:R_{q+1}\to T^{\ast }\otimes R_q$. More generally, we have the restriction:

$$\boxed{d:{\wedge }^s{T}^{\ast }\otimes R_{q+1}\to {\wedge }^{s+1}{T}^{\ast }\otimes R_q:\left(f_{\mu ,I}^k\left(x\right){\mathit{dx}}^I\right)\to \left(\left({\partial }_i{f}_{\mu ,I}^k\left(x\right)-f_{\mu +1_i,I}^k\left(x\right)\right){\mathit{dx}}^i\wedge {\mathit{dx}}^I\right)}$$E9

with standard multi-index notation for exterior forms and one can easily check that $d\circ d=0$. The restriction of $-d$ to the symbol is called the Spencer map $\delta$ in the sequences:

$$\boxed{{\wedge }^{s-1}{T}^{\ast }\otimes g_{q+r+1}\stackrel{\delta }{\to }{\wedge }^s{T}^{\ast }\otimes g_{q+r}\stackrel{\delta }{\to }{\wedge }^{s+1}{T}^{\ast }\otimes g_{q+r-1}}$$E10

because $\delta \circ \delta =0$ similarly, leading to the purely algebraic $\delta$-cohomology $H_{q+r}^s\left(g_q\right)$ at ${\wedge }^s{T}^{\ast }\otimes g_{q+r}$ (See [4, 6, 7, 12, 13, 25, 27, 28, 29, 31] for details and examples).

Definition 2.A.2: A system $R_q$ is said to be formally integrable (FI) when all the equations of order $q+r$ are obtained by $r$ prolongations only, $\forall r\ge 0$ or, equivalently, when the projections ${\pi }_{q+r}^{q+r+s}:R_{q+r+s}\to R_{q+r}^{\left(s\right)}\subseteq R_{q+r}$ are such that $R_{q+r}^{\left(s\right)}=R_{q+r}$, $\forall r,s\ge 0$.

Finding an intrinsic test has been achieved by D.C. Spencer in 1965–1970 [6, 7] along coordinate dependent lines sketched by M. Janet in 1920 [20, 25, 29]. The next procedure providing a Pommaret basis and where one may have to change linearly the independent variables if necessary, is intrinsic though it must be checked in a particular coordinate system called $\delta$-regular [4, 25, 27, 28, 29].

• Equations of class $n$: Solve the maximum number ${\beta }_q^n$ of equations with respect to the jets of order $q$ and class $n$. Then, call $\left(x^1,\dots ,x^n\right)$ multiplicative variables.

• Equations of class $i\ge 1$: Solve the maximum number ${\beta }_q^i$ of remaining equations with respect to the jets of order $q$ and class $i$. Then, call $\left(x^1,\dots ,x^i\right)$ multiplicative variables and $\left(x^{i+1},\dots ,x^n\right)$ non-multiplicative variables.

• Remaining equations equations of order $\le q-1$: Call $\left(x^1,\dots ,x^n\right)$ non-multiplicative variables.

In actual practice, we shall use a Janet tabular where the multiplicative “variables” are in upper left position while the non-multiplicative variables are represented by dots in lower right position.

Definition 2.A.3: A system of PD equations is said to be involutive if its first prolongation can be obtained by prolonging its equations only with respect to the corresponding multiplicative variables. In that case, we may introduce the characters ${\alpha }_q^i=m\frac{\left(q+n-i-1\right)!}{\left(q-1\right)!(\left(n-i\right)!}-{\beta }_q^i$ for $i=1,\dots ,n$ with ${\alpha }_q^1\ge \dots \ge {\alpha }_q^n\ge 0$ and we have $\mathit{dim}\left(g_q\right)={\alpha }_q^1+\dots +{\alpha }_q^n$ while $\mathit{dim}\left(g_{q+1}\right)={\alpha }_q^1+\dots +n{\alpha }_q^n$.

Remark 2.A.4: As long as the prolongation/projection (PP) procedure allowing to find two integers $r,s\ge 0$ such that the system $R_{q+r}^{\left(s\right)}$ is involutive, has not been achieved, nothing can be said about the CC (Fine examples can be found in [12, 16]).

When $R_q$ is involutive, the operator $\mathcal{D}:E\stackrel{j_q}{\to }{J}_q\left(E\right)\stackrel{\mathrm{\Phi }}{\to }{J}_q\left(E\right)/R_q=F_0$ of order $q$ is said to be involutive. Introducing the Janet bundles $F_r={\wedge }^r{T}^{\ast }\otimes J_q\left(E\right)/\left({\wedge }^r{T}^{\ast }\otimes R_q+\delta \left(S_{q+1}{T}^{\ast }\otimes E\right)\right)$, we obtain the linear Janet sequence (Introduced in [25]):

$$\boxed{0\to \mathrm{\Theta }\to E\underset{q}{\stackrel{\mathcal{D}}{\to }}{F}_0\underset{1}{\stackrel{{\mathcal{D}}_1}{\to }}{F}_1\underset{1}{\stackrel{{\mathcal{D}}_2}{\to }}\dots \underset{1}{\stackrel{{\mathcal{D}}_n}{\to }}{F}_n\to 0}$$E11

where each other operator is first order involutive and generates the CC of the preceding one.

Similarly, introducing the Spencer bundles $C_r={\wedge }^r{T}^{\ast }\otimes R_q/\delta \left({\wedge }^{r-1}{T}^{\ast }\otimes g_{q+1}\right)$_, we obtain the linear Spencer sequence induced by the Spencer operator, that can be linked to the Janet sequence [25, 29]:

$$\boxed{0\to \mathrm{\Theta }\stackrel{j_q}{\to }{C}_0\underset{1}{\stackrel{D_1}{\to }}{C}_1\underset{1}{\stackrel{D_2}{\to }}\dots \underset{1}{\stackrel{D_n}{\to }}{C}_n\to 0}$$E12

It must be noticed, as we shall see in Section 4, that the Killing operator/system is FI if and only if the metric $\omega$ has constant Riemanniann curvature (for example the flat Minkowski metric) but is not involutive and the Janet sequence cannot be exhibited. As for the conformal Killing operator/system obtained by eliminating the arbitrary function $A\left(x\right)$ in the inhomogeneous system $\mathcal{L}\left(\xi \right)\omega =A\left(x\right)\omega$ or by considering simply the system $\mathcal{L}\left(\xi \right)\widehat{\omega }=0$, it is FI if and only if $\omega$ has vanishing Weyl tensor but is not involutive and the order of the successive CC operators may change with the ground dimension $n$, the worst situation being for $n=4$ as the analogue of the Bianchi operator is now an operator of order two indeed [12, 13]. As these results highly depend on the Spencer $\delta$-cohomology, it is clear that they are neither known nor acknowledged and it follows that the mathematical foundations of conformal geometry must be entirely revisited.

2.2 Module theory

Let $K$ be a differential field with $n$ commuting derivations $\left({\partial }_1,\dots ,{\partial }_n\right)$ and consider the ring $D=K\left[d_1,\dots ,d_n\right]=K\left[d\right]$ of differential operators with coefficients in $K$ with $n$ commuting formal derivatives satisfying $d_{i}a={\mathit{ad}}_i+{\partial }_{i}a$ in the operator sense. If $P=a^{\mu }{d}_{\mu }\in D=K\left[d\right]$, the highest value of $\mid \mu \mid$ with $a^{\mu }\ne 0$ is called the order of the operator $P$ and the ring $D$ with multiplication $\left(P,Q\right)\to P\circ Q=\mathit{PQ}$ is filtred by the order $q$ of the operators. We have the filtration $0\subset K=D_0\subset D_1\subset \dots \subset D_q\subset \dots \subset D_{\infty }=D$. As an algebra, $D$ is generated by $K=D_0$ and $T=D_1/D_0$ with $D_1=K\oplus T$ if we identify an element $\xi ={\xi }^i{d}_i\in T$ with the vector field $\xi ={\xi }^i\left(x\right){\partial }_i$ of differential geometry, but with ${\xi }^i\in K$ now. It follows that $D{=}_D{D}_D$ is a bimodule over itself, being at the same time a left $D$-module by the composition $P\to \mathit{QP}$ and a right $D$-module by the composition $P\to \mathit{PQ}$. We define the adjoint functor $\mathit{ad}:D\to D^{\mathit{op}}:P=a^{\mu }{d}_{\mu }\to \mathit{ad}\left(P\right)={\left(-1\right)}^{\mid \mu \mid }{d}_{\mu }{a}^{\mu }$ and we have $\mathit{ad}\left(\mathit{ad}\left(P\right)\right)=P$ both with $\mathit{ad}\left(\mathit{PQ}\right)=\mathit{ad}\left(Q\right)\mathit{ad}\left(P\right),\forall P,Q\in D$. It follows that any operator can be considered as the adjoint of its own adjoint. Such a definition can be extended by linearity to any matrix of operators by using the transposed matrix of adjoint operators (See [4, 5, 11, 12, 13, 18, 20, 21, 24] for more details and applications to control theory or mathematical physics).

Accordingly, if $y=\left(y^1,\dots ,y^m\right)$ are differential indeterminates, then $D$ acts on $y^k$ by setting $d_i{y}^k=y_i^k\to d_{\mu }{y}^k=y_{\mu }^k$ with $d_i{y}_{\mu }^k=y_{\mu +1_i}^k$ and $y_0^k=y^k$. We may therefore use the jet coordinates in a formal way as in the previous section. Therefore, if a system of OD/PD equations is written in the form ${\mathrm{\Phi }}^{\tau }\equiv a_k^{\mathit{\tau \mu }}{y}_{\mu }^k=0$ with coefficients $a\in K$, we may introduce the free differential module $\mathit{Dy}={\mathit{Dy}}^1+\dots +{\mathit{Dy}}^m\simeq D^m$ and consider the differential module of equations $I=D\mathrm{\Phi }\subset \mathit{Dy}$, both with the residual differential module $M=\mathit{Dy}/D\mathrm{\Phi }$ or $D$-module and we may set $M{=}_{D}M$ if we want to specify the ring of differential operators. We may introduce the formal prolongation with respect to $d_i$ by setting $d_i{\mathrm{\Phi }}^{\tau }\equiv a_k^{\mathit{\tau \mu }}{y}_{\mu +1_i}^k+\left({\partial }_i{a}_k^{\mathit{\tau \mu }}\right)y_{\mu }^k$ in order to induce maps $d_i:M\to M:{\overline{y}}_{\mu }^k\to {\overline{y}}_{\mu +1_i}^k$ by residue with respect to $I$ if we use to denote the residue $\mathit{Dy}\to M:y^k\to {\overline{y}}^k$ by a bar like in algebraic geometry. However, for simplicity, we shall not write down the bar when the background will indicate clearly if we are in $\mathit{Dy}$ or in $M$. As a byproduct, the differential modules we shall consider will always be finitely generated ($k=1,\dots ,m<\infty$) and finitely presented ($\tau =1,\dots ,p<\infty$). Equivalently, introducing the matrix of operators $\mathcal{D}=\left(a_k^{\mathit{\tau \mu }}{d}_{\mu }\right)$ with $m$ columns and $p$ rows, we may introduce the morphism $D^p\stackrel{\mathcal{D}}{\to }{D}^m:\left(P_{\tau }\right)\to \left(P_{\tau }{\mathrm{\Phi }}^{\tau }\right)$ over $D$ by acting with $D$ on the left of these row vectors while acting with $\mathcal{D}$ on the right of these row vectors by composition of operators with $\mathit{im}\left(\mathcal{D}\right)=I$. The presentation of $M$ is defined by the exact cokernel sequence $D^p\stackrel{\mathcal{D}}{\to }{D}^m\to M\to 0$. We notice that the presentation only depends on $K,D$ and $\mathrm{\Phi }$ or $\mathcal{D}$, that is to say never refers to the concept of (explicit local or formal) solutions. It follows from its definition that $M$ can be endowed with a quotient filtration obtained from that of $D^m$ which is defined by the order of the jet coordinates $y_q$ in $D_{q}y$. We have therefore the inductive limit $0\subseteq M_0\subseteq M_1\subseteq \dots \subseteq M_q\subseteq \dots \subseteq M_{\infty }=M$ with $d_i{M}_q\subseteq M_{q+1}$ and $M={\mathit{DM}}_q$ for $q\gg 0$ with prolongations $D_r{M}_q\subseteq M_{q+r},\forall q,r\ge 0$. It may be sometimes quite difficult to work out $I_q$ or $M_q$ from a given presentation which is not involutive [4].

Definition 2.B.1: An exact sequence of morphisms finishing at $M$ is said to be a resolution of $M$. If the differential modules involved apart from $M$ are free, that is isomorphic to a certain power of $D$, we shall say that we have a free resolution of $M$.

Having in mind that $K$ is a left $D$-module with the action $\left(D,K\right)\to K:\left(d_i,a\right)\to {\partial }_{i}a$ and that $D$ is a bimodule over itself with $\mathit{PQ}\ne \mathit{QP}$, we have only two possible constructions:

Definition 2.B.2: We may define the right (care) differential module ${\mathit{hom}}_D\left(M,D\right)$ with $\left(\mathit{fP}\right)\left(m\right)=\left(f\left(m\right)\right)P\Rightarrow \left(\mathit{fPQ}\right)\left(m\right)=\left(\left(\mathit{fP}\right)\left(m\right)\right)Q=\left(\left(f\left(m\right)\right)P\right)Q=\left(f\left(m\right)\right)\mathit{PQ}$.

Definition 2.B.3: We define the system $R={\mathit{hom}}_K\left(M,K\right)$ and set $R_q={\mathit{hom}}_K\left(M_q,K\right)$ as the system of order $q$. We have the projective limit $R=R_{\infty }\to \dots \to R_q\to \dots \to R_1\to R_0$. It follows that $f_q\in R_q:y_{\mu }^k\to f_{\mu }^k\in K$ with $a_k^{\mathit{\tau \mu }}{f}_{\mu }^k=0$ defines a section at order $q$_, and we may set $f_{\infty }=f\in R$ for a section of $R$. For an arbitrary differential field $K$, such a definition has nothing to do with the concept of a formal power series solution (care).

Proposition 2.B.4: When $M$ is a left $D$-module, then $R$ is also a left $D$-module.

Proof: As $D$ is generated by $K$ and $T$ as we already said, let us define:

$$\begin{array}{c}\left(\mathit{af}\right)\left(m\right)=\mathit{af}\left(m\right)=f\left(\mathit{am}\right),\forall a\in K,\forall m\in M\\ \left(\mathit{\xi f}\right)\left(m\right)=\mathit{\xi f}\left(m\right)-f\left(\mathit{\xi m}\right),\forall \xi =a^i{d}_i\in T,\forall m\in M\end{array}$$

In the operator sense, it is easy to check that $d_{i}a={\mathit{ad}}_i+{\partial }_{i}a$ and that $\mathit{\xi \eta }-\mathit{\eta \xi }=\left[\xi ,\eta \right]$ is the standard bracket of vector fields. We finally get ${\left(d_{i}f\right)}_{\mu }^k=\left(d_{i}f\right)\left(y_{\mu }^k\right)={\partial }_i{f}_{\mu }^k-f_{\mu +1_i}^k$ and thus recover exactly the Spencer operator of the previous section though this is not evident at all. We also get ${\left(d_i{d}_{j}f\right)}_{\mu }^k={\partial }_{\mathit{ij}}{f}_{\mu }^k-{\partial }_i{f}_{\mu +1_j}^k-{\partial }_j{f}_{\mu +1_i}^k+f_{\mu +1_i+1_j}^k\Rightarrow d_i{d}_j=d_j{d}_i,\forall i,j=1,\dots ,n$ and thus $d_i{R}_{q+1}\subseteq R_q\Rightarrow d_{i}R\subset R$ induces a well defined operator $R\to T^{\ast }\otimes R:f\to {\mathit{dx}}^i\otimes d_{i}f$. This operator has been first introduced, up to sign, by F.S. Macaulay as early as in 1916 but this is still not ackowledged. For more details on the Spencer operator and its applications, the reader may look at [12, 13, 25, 27, 28, 29, 30].

$\square$

Definition 2.B.5: With any differential module $M$_, we shall associate the graded module $G=\mathit{gr}\left(M\right)$ over the polynomial ring $\mathit{gr}\left(D\right)\simeq K\left[\chi \right]$ by setting $G={\oplus }_{q=0}^{\infty }{G}_q$ with $G_q=M_q/M_{q-1}$_, and we get $g_q=G_q^{\ast }$ where the symbol $g_q$ is defined by the short exact sequences:

$$0\to M_{q-1}\to M_q\to G_q\to 0\iff 0\to g_q\to R_q\to R_{q-1}\to 0$$

We have the short exact sequences $0\to D_{q-1}\to D_q\to S_{q}T\to 0$ leading to ${\mathit{gr}}_q\left(D\right)\simeq S_{q}T$ and we may set as usual $T^{\ast }={\mathit{hom}}_K\left(T,K\right)$ in a coherent way with differential geometry.

The two following definitions, which are well known in commutative algebra, are also valid (with more work) in the case of differential modules (See [4, 20] for more details or the references [4, 8, 9, 10, 12, 13, 29] for an introduction to homological algebra and diagram chasing).

Definition 2.B.6: The set of elements $t\left(M\right)=\left\{m\in M|\exists 0\ne P\in D,\mathit{Pm}=0\right\}\subseteq M$ is a differential module called the torsion submodule of $M$. More generally, a module $M$ is called a torsion module if $t\left(M\right)=M$ and a torsion-free module if $t\left(M\right)=0$. In the short exact sequence $0\to t\left(M\right)\to M\to M^{\prime }\to 0$, the module $M^{\prime }$ is torsion-free. Its defining module of equations $I^{\prime }$ is obtained by adding to $I$ a representative basis of $t\left(M\right)$ set up to zero and we have thus $I\subseteq I^{\prime }$.

Definition 2.B.7: A differential module $F$ is said to be free if $F\simeq D^r$ for some integer $r>0$ and we shall define ${\mathit{rk}}_D\left(F\right)=r$. If $F$ is the biggest free differential module contained in $M$, then $M/F$ is a torsion differential module and ${\mathit{hom}}_D\left(M/F,D\right)=0$. In that case, we shall define the differential rank of $M$ to be ${\mathit{rk}}_D\left(M\right)={\mathit{rk}}_D\left(F\right)=r$. Accordingly, if $M$ is defined by a linear involutive operator of order $q$, then ${\mathit{rk}}_D\left(M\right)={\alpha }_q^n$.

Proposition 2.B.8: If $0\to M^{\prime }\to M\to M^{″}\to 0$ is a short exact sequence of differential modules and maps or operators, we have ${\mathit{rk}}_D\left(M\right)={\mathit{rk}}_D\left(M^{\prime }\right)+{\mathit{rk}}_D\left(M^{″}\right)$.

In the general situation, let us consider the sequence $M^{\prime }\stackrel{f}{\to }M\stackrel{g}{\to }{M}^{”}$ of modules which may not be exact. Then, we may define the coboundary submodule $B=\mathit{im}\left(f\right)\subseteq M$, the cocycle submodule $Z=\mathit{ker}\left(g\right)\subseteq M$ and the cohomology module $H=Z/B$.

Using the last (delicate) proposition, we may provide the following definitions that will be in the heart of the parametrization problem, successively in the operator and module frameworks.

Definition 2.B.9: When $\mathcal{D}=\mathrm{\Phi }\circ j_q:E\to F$ is a linear differential operator of order $q$ with coefficients in a differential field $K$, between the sections of two vector bundles $E$ with $\mathit{dim}\left(E\right)=m$ and $F$ with $\mathit{dim}\left(F\right)=p$, we shall define in a formal way the differential rank ${\mathit{rk}}_D\left(\mathcal{D}\right)=m-{\alpha }_q^n={\beta }_q^n={\mathit{rk}}_D\left(\mathit{im}\left(\mathcal{D}\right)\right)\le p$ by introducing the characters of the corresponding linear system $R_q=\mathit{ker}\left(\mathrm{\Phi }\right)\subseteq J_q\left(E\right)$ of order $q$ over $E$. We have thus ${\mathit{rk}}_D\left(\mathcal{D}\right)\le \mathit{inf}\left(m,p\right)$ (See [4, 20] for details). The order of an operator will be indicated under it arrow.

Definition 2.B.10: When a differential module $M$ is defined by the presentation $D^p\underset{q}{\stackrel{\mathcal{D}}{\to }}{D}^m\to M\to 0$, we shall introduce the differential module $I=\mathit{im}\left(\mathcal{D}\right)\subseteq D^m$ and set ${\mathit{rk}}_D\left(\mathcal{D}\right)={\mathit{rk}}_D\left(I\right)=m-{\mathit{rk}}_D\left(M\right)$ in a coherent way with the last proposition and definition.

We obtain the important theorem which is generalizing to operators the rank property of a $m\times p$ matrix, even when $\mathcal{D}$ and $\mathit{ad}\left(\mathcal{D}\right)$ are neither FI nor involutive ([4, 20, 29], p. 340):

Theorem 2.B.11: One has ${\mathit{rk}}_D\left(\mathcal{D}\right)={\mathit{rk}}_D\left(\mathit{ad}\left(\mathcal{D}\right)\right)\le \mathit{inf}\left(m,p\right)$.

In order to conclude this section, we may say that the main difficulty met when passing from the differential framework to the algebraic framework is the “inversion” of arrows. Indeed, when an operator is injective, that is when we have the exact sequence $0\to E\stackrel{\mathcal{D}}{\to }F$ with $\mathit{dim}\left(E\right)=m$, $\mathit{dim}\left(F\right)=p$, like in the case of the operator $0\to E\stackrel{j_q}{\to }{J}_q\left(E\right)$, on the contrary, using differential modules, we have the epimorphism $D^p\stackrel{\mathcal{D}}{\to }{D}^m\to 0$. The case of a formally surjective operator, like the div operator, described by the exact sequence $E\stackrel{\mathcal{D}}{\to }F\to 0$ is now providing the exact sequence of differential modules $0\to D^p\stackrel{\mathcal{D}}{\to }{D}^m\to M\to 0$ because $\mathcal{D}$ has no CC.

3. Parametrization problem

In this section, we shall set up and solve the minimum parametrization problem by comparing the differential geometric approach and the differential algebraic approach. In fact, both sides are essential because certain concepts, like “torsion”, are simpler in the module approach while others, like “involution” are simpler in the operator approach. However, the reader must never forget that the “extension modules” or the “side changing functor” are pure product of differential homological algebra with no system counterpart. The link between “differential duality” and the “adjoint operator” may not be evident at all, even for people familiar with mathematical physics [2, 4, 12, 13, 18, 22, 29].

Let us start with a given linear differential operator $\eta \stackrel{{\mathcal{D}}_1}{\to }\zeta$ between the sections of two given vector bundles $F_0$ and $F_1$ of respective fiber dimension $m$ and $p$. Multiplying the equations ${\mathcal{D}}_1\eta =\zeta$ by $p$ test functions $\lambda$ considered as a section of the adjoint vector bundle $\mathit{ad}\left(F_1\right)={\wedge }^n{T}^{\ast }\otimes F_1^{\ast }$ and integrating by parts as we did in the introduction, we may introduce the adjoint vector bundle $\mathit{ad}\left(F_0\right)={\wedge }^n{T}^{\ast }\otimes F_0^{\ast }$ with sections $\mu$ in order to obtain the adjoint operator $\mu \stackrel{\mathit{ad}\left({\mathcal{D}}_1\right)}{\leftarrow }\lambda$, writing on purpose the arrow backwards. More generally, let us consider a differential sequence:

$$\xi \stackrel{\mathcal{D}}{\to }\eta \stackrel{{\mathcal{D}}_1}{\to }\zeta$$E13

such that ${\mathcal{D}}_1$ generates the CC of $\mathcal{D}$ or, equivalently, such that ${\mathcal{D}}_1$ is parametrized by $\mathcal{D}$.

We may introduce the adjoint differential sequence:

$$\nu \stackrel{\mathit{ad}\left(\mathcal{D}\right)}{\leftarrow }\mu \stackrel{\mathit{ad}\left({\mathcal{D}}_1\right)}{\leftarrow }\lambda$$E14

As we have ${\mathcal{D}}_1\circ \mathcal{D}=0$, we obtain $\mathit{ad}\left(\mathcal{D}\right)\circ \mathit{ad}\left({\mathcal{D}}_1\right)=0$. However, if ${\mathcal{D}}_1$ generates the CC of $\mathcal{D}$, then $\mathit{ad}\left(\mathcal{D}\right)$ may not generate the CC of $\mathit{ad}\left({\mathcal{D}}_1\right)$. Such a situation may not be satisfied as we saw and the so-called extension modules have been introduced in order to measure these “gaps.”

In order to pass to the differential module framework, let us introduce the free differential modules $\mathit{D\xi }\simeq D^l,\mathit{D\eta }\simeq D^m,\mathit{D\zeta }\simeq D^p$. We have similarly the adjoint free differential modules $\mathit{D\nu }\simeq D^l,\mathit{D\mu }\simeq D^m,\mathit{D\lambda }\simeq D^p$, because $\mathit{dim}\left(\mathit{ad}\left(E\right)\right)=\mathit{dim}\left(E\right)$ and ${\mathit{hom}}_D\left(D^m,D\right)\simeq D^m$. Of course, in actual practice, the geometric meaning is totally different because we have volume forms in the dual framework. We have thus obtained the formally exact sequence of differential modules:

$$\begin{array}{rrrrrrrr}{D}^p& \stackrel{{\mathcal{D}}_1}{\to }& D^m& & \stackrel{\mathcal{D}}{\to }& & D^l& \to M\to 0\\ & & & \searrow & & \nearrow & & \\ & & & & M_1& & & \\ & & & \nearrow & & \searrow & & \\ & & 0& & & & 0& \end{array}$$E15

with ${\mathit{rk}}_D\left(\mathcal{D}\right)+{\mathit{rk}}_D\left({\mathcal{D}}_1\right)=m\iff {\mathit{rk}}_D\left(M\right)+{\mathit{rk}}_D\left(M_1\right)=l$. We have the adjoint sequence:

$$D^p\stackrel{\mathit{ad}\left({\mathcal{D}}_1\right)}{\leftarrow }{D}^m\stackrel{\mathit{ad}\left(\mathcal{D}\right)}{\leftarrow }{D}^l$$

with ${\mathit{rk}}_D\left(\mathit{ad}\left(\mathcal{D}\right)\right)+{\mathit{rk}}_D\left(\mathit{ad}\left({\mathcal{D}}_1\right)\right)=m$ and we may thus state [4, 9, 10, 20]:

Definition 3.1: We may define the zero differential extension module ${\mathit{ext}}^0\left(M\right)=\mathit{ker}\left(\mathit{ad}\left(\mathcal{D}\right)\right)$ and the first differential extension module ${\mathit{ext}}^1\left(M\right)$ to be the cohomology of this sequence at $D^m$. The latter is a torsion module because it has vanishing differential rank. They only depend on $M$.

Theorem 3.2: There is a constructive test in order to find out whether a differential operator ${\mathcal{D}}_1$ can be parametrized or not (Example 1.3 or the div operator with $n=3$).

Proof: The test has five steps along with the following diagram in operator language:

$\boxed{1}\Rightarrow$ Start with ${\mathcal{D}}_1:\eta \to \zeta$,

$\boxed{2}\Rightarrow$ Construct $\mathit{ad}\left({\mathcal{D}}_1\right):\lambda \to \mu$,

$\boxed{3}\Rightarrow$ Construct its CC as an operator $\mathit{ad}\left(\mathcal{D}\right):\mu \to \nu$,

$\boxed{4}\Rightarrow$ Construct its adjoint $\mathcal{D}=\mathit{ad}\left(\mathit{ad}\left(\mathcal{D}\right)\right):\xi \to \eta$,

$\boxed{5}\Rightarrow$ Construct the CC ${\mathcal{D}}_1^{\prime }$ of $\mathcal{D}:\eta \to {\zeta }^{\prime }$.

$$\begin{array}{ccccccccc}& & & & & & & {\zeta }^{\prime }& \boxed{5}\\ & & & & & & \stackrel{{\mathcal{D}}_1^{\prime }}{\nearrow }& & \\ \boxed{4}& & & \xi & \stackrel{\mathcal{D}}{\to }& \eta & \stackrel{{\mathcal{D}}_1}{\to }& \zeta & \boxed{1}\\ & & & & & & & & \\ & & & & & & & & \\ \boxed{3}& & & \nu & \stackrel{\mathit{ad}\left(\mathcal{D}\right)}{\leftarrow }& \mu & \stackrel{\mathit{ad}\left({\mathcal{D}}_1\right)}{\leftarrow }& \lambda & \boxed{2}\end{array}$$

We have $\mathit{ad}\left(\mathcal{D}\right)\circ \mathit{ad}\left({\mathcal{D}}_1\right)=0\Rightarrow {\mathcal{D}}_1\circ \mathcal{D}=0$, that is ${\mathcal{D}}_1$ is surely among the CC of $\mathcal{D}$ but other CC may also exist. The parametrization is thus existing if and only if we may have ${\mathcal{D}}_1^{\prime }={\mathcal{D}}_1$.

$\square$

Corollary 3.3: Each new CC eventually brought by ${\mathcal{D}}^{\prime }$ which is not already a differential consequence of ${\mathcal{D}}_1$ is providing a torsion element of the differential module $M_1$ determined by ${\mathcal{D}}_1$.

Example 3.4: If $\mathcal{D}:\xi \to \left(d_{22}\xi ={\eta }^2,d_{12}\xi ={\eta }^1\right)$ we have ${\mathcal{D}}_1=\left({\eta }^1,{\eta }^2\right)\to d_1{\eta }^2-d_2{\eta }^1=\zeta$ and the only first order generating CC of $\mathit{ad}\left({\mathcal{D}}_1\right):\lambda \to \left(d_2\lambda ={\mu }^1,-d_1\lambda ={\mu }^2\right)$ is $d_1{\mu }^1+d_2{\mu }^2={\nu }^{\text{'}}$ while $\mathit{ad}\left(\mathcal{D}\right):\left({\mu }^1,{\mu }^2\right)\to d_{12}{\mu }^1+d_{22}{\mu }^2=\nu$ is a second order operator like $\mathcal{D}$. Hence, if we should like to parametrize $\mathit{ad}\left(\mathcal{D}\right)$, we should successively find $\mathcal{D}$, ${\mathcal{D}}_1$, $\mathit{ad}\left({\mathcal{D}}_1\right)$ and finally get the additional first order CC $d_1{\mu }^1+d_2{\mu }^2={\nu }^{\text{'}}$ which is such that $\nu =0\Rightarrow d_2{\nu }^{\text{'}}=0$, that is ${\nu }^{\text{'}}$ is a torsion element for the system $d_{12}{\mu }^1+d_{22}{\mu }^2=0$.

Example 3.5: Many other examples can be found in ordinary differential control theory because it is known that a linear control system is controllable if and only if it is parametrizable (See [4, 20, 29] for more details and examples). In our opinion, the simplest one is provided by the double pendulum in which a rigid bar is able to move horizontally with reference position $x$ and has two pendulums attached with respective length $l_1$ and $l_2$ making the (small) angles ${\theta }_1$ and ${\theta }_2$ with the vertical. The corresponding control system does not depend on the mass of each pendulum and is:

$$d^{2}x+l_1{d}^2{\theta }^1+g{\theta }^1=0,d^{2}x+l_2{d}^2{\theta }^2+g{\theta }^2=0$$

where $g$ is the gravity. The classical approach is to prove that this control system is controllable if and only if $l_1\ne l_2$ by using a tedious computation through the standard Kalman test [20]. However, equivalently, but this way is still not acknowledged by the control community, the idea is to prove that the corresponding second order operator $\mathit{ad}\left({\mathcal{D}}_1\right)$ is injective. We let the reader realize the experiment, prove this result as an exercise and apply the previous theorem in order to work out the parametrizing operator $\mathcal{D}$ of order 4, namely:

$$\left\{\begin{array}{rrr}-l_1{l}_2{d}^4\varphi -g\left(l_1+l_2\right)d^2\varphi -g^2\varphi & =& x\\ l_2{d}^4\varphi +{\mathit{gd}}^2\varphi & =& {\theta }_1\\ l_1{d}^4\varphi +{\mathit{gd}}^2\varphi & =& {\theta }_2\end{array}\right.$$

Example 3.6: As a less academic example, the following diagram is proving that Einstein equations cannot be parametrized [1, 2]:

$$\begin{array}{ccccccccc}& & & 10& \stackrel{\mathit{\text{Riemann}}}{\to }& 20& \stackrel{\mathit{\text{Bianchi}}}{\to }& 20& \\ & & & \parallel & \nearrow & \downarrow & & \downarrow & \\ & 4& \stackrel{\mathit{\text{Killing}}}{\to }& 10& \stackrel{\mathit{\text{Einstein}}}{\to }& 10& \stackrel{\mathit{div}}{\to }& 4& \to 0\\ & & & & & & & & \\ 0\to & 4& \stackrel{\mathit{\text{Cauchy}}}{\leftarrow }& 10& \stackrel{\mathit{\text{Einstein}}}{\leftarrow }& 10& & & \end{array}$$

and that the Cauchy and Killing operators (left side) have strictly nothing to do with the Bianchi and div operators (right side). According to the last corollary, the $20-10=10$ new CC are generating the torsion submodule of the differential module defined by the Einstein operator. In the last section we shall explain why such a basis of the torsion module is made by the 10 independent components of the Weyl tensor, a result which is not evident, leading to the so-called Lichnerowicz waves (in France) [12, 13, 22, 32, 33].

In continuum mechanics, the Cauchy stress tensor may not be symmetric in the so-called Cosserat media where the Cauchy stress equations are replaced by the Cosserat couple-stress equations which are nothing else than the adjoint of the first Spencer operator, totally different from the third [27, 28, 34, 35]. When $n=2$, we shall see that the single Airy function has strictly nothing to do with any perturbation of the metric having three components.

Example 3.7: A similar comment can be done for electromagnetism through the exterior derivative as the first set of Maxwell equations can be parametrized by the EM potential 1-form because $\mathit{dA}=F\Rightarrow \mathit{dF}=0$ while the second set of Maxwell equations (adjoint of this parametrization) can be parametrized by the EM pseudo-potential, a 3 - skew-symmetric contravariant tensor density in ${\wedge }^4{T}^{\ast }\otimes {\wedge }^{3}T$ through the adjoint of the exterior derivative ${\wedge }^2{T}^{\ast }\stackrel{d}{\to }{\wedge }^3{T}^{\ast }$ [12, 13, 27, 28]. These results, which are deeply supporting the conformal origin of EM [31], are also strengthening the comments we shall make in Section 4 on the origin and existence of gravitational waves [13, 22, 23].

As a summarizing comment, we discover that not only it is sometimes not possible to parametrize a linear differential operator but that, whenever it is possible, not only it is not easy to have an idea about the number of potential functions needed but even more difficult to have any idea about the order of the parametrizing operator that may be unexpectedly quite high indeed for the double pendulum. In addition, we have provided in [4], examples showing that the case of variable coefficients is even much more difficult than the case of constant coefficients. Moreover, the mathematical tools involved are sometimes not accessible to intuition like this theorem (See [4, 5, 20] for details):

Theorem 3.8: If $M$ is a differential module, we have the exact sequence of differential modules:

$$0\to t\left(M\right)\to M\stackrel{\epsilon }{\to }{\mathit{hom}}_D\left({\mathit{hom}}_D\left(M,D\right),D\right)$$E16

where the map $\epsilon$ is defined by $\epsilon \left(m\right)\left(f\right)=f\left(m\right),\forall m\in M,f\in {\mathit{hom}}_D\left(M,D\right)$.

Theorem 3.9: When ${\mathcal{D}}_1$ can be parametrized or, equivalently, when $M_1$ is torsion-free and can be thus embedded into a free module $D^l$, we have thus ${\mathit{rk}}_D\left(M_1\right)=l^{\prime }\le l$. There is a constructive procedure in order to embed $M_1$ into $D^{l^{\prime }}$, that is to obtain a minimum parametrization.

Proof: The procedure with $4$ steps is as follows in the operator language (Example 1.3):

$\boxed{1}\Rightarrow$ Start with the formally exact parametrizing sequence already constructed by differential biduality. We have thus $\mathit{im}\left(\mathcal{D}\right)=\mathit{ker}\left({\mathcal{D}}_1\right)$ and the corresponding differential module $M_1$ defined by ${\mathcal{D}}_1$ is torsion-free by assumption.

$\boxed{2}\Rightarrow$ Construct the adjoint sequence which is also formally exact by assumption.

$\boxed{3}\Rightarrow$ Find a maximum set of differentially independent CC $\mathit{ad}\left({\mathcal{D}}^{\text{'}}\right):\mu \to {\nu }^{\text{'}}$ among the generating CC $\mathit{ad}\left(\mathcal{D}\right):\mu \to \nu$ of $\mathit{ad}\left({\mathcal{D}}_1\right)$ in such a way that $\mathit{im}\left(\mathit{ad}\left({\mathcal{D}}^{\prime }\right)\right)$ is a maximum free differential submodule of $\mathit{im}\left(\mathit{ad}\left(\mathcal{D}\right)\right)$ that is any element in $\mathit{im}\left(\mathit{ad}\left(\mathcal{D}\right)\right)$ is differentially algebraic over $\mathit{im}\left(\mathit{ad}\left({\mathcal{D}}^{\prime }\right)\right)$.

$\boxed{4}\Rightarrow$ Using differential duality, construct ${\mathcal{D}}^{\prime }=\mathit{ad}\left(\mathit{ad}\left({\mathcal{D}}^{\prime }\right)\right)$.

Let us prove that ${\mathcal{D}}_1$ generates the CC of ${\mathcal{D}}^{\prime }$ in the following double diagram:

$$\begin{array}{rrrrrrrrr}\boxed{4}& & & {\xi }^{\text{'}}& & & & & \\ & & & \uparrow & \stackrel{\mathcal{D}\text{'}}{\searrow }& & & & \\ & & & \xi & \stackrel{\mathcal{D}}{\to }& \eta & \stackrel{{\mathcal{D}}_1}{\to }& \zeta & \boxed{1}\\ & & & & & & & & \\ & & & & & & & & \\ & & & \nu & \stackrel{\mathit{ad}\left(\mathcal{D}\right)}{\leftarrow }& \mu & \stackrel{\mathit{ad}\left({\mathcal{D}}_1\right)}{\leftarrow }& \lambda & \boxed{2}\\ & & & \uparrow & \stackrel{\mathit{ad}\left({\mathcal{D}}^{\prime }\right)}{\swarrow }& & & & \\ \boxed{3}& & & {\nu }^{\prime }& & & & & \\ & & \swarrow & \uparrow & & & & & \\ & 0& & 0& & & & & \end{array}$$

First of all, we have by construction $\mathit{im}\left(\mathit{ad}\left({\mathcal{D}}^{\prime }\right)\right)=\mathit{im}\left(\mathit{ad}\left(\mathcal{D}\right)\right)$ in the bottom diagram and thus:

$${\mathit{rk}}_D\left(\mathit{ad}\left({\mathcal{D}}^{\prime }\right)\right)+{\mathit{rk}}_D\left(\mathit{ad}\left({\mathcal{D}}_1\right)\right)={\mathit{rk}}_D\left(\mathit{ad}\left(\mathcal{D}\right)\right)+{\mathit{rk}}_D\left(\mathit{ad}\left({\mathcal{D}}_1\right)\right)=m$$

Passing to the upper diagram, we have, therefore:

$${\mathit{rk}}_D\left({\mathcal{D}}^{\text{'}}\right)+{\mathit{rk}}_D\left({\mathcal{D}}_1\right)={\mathit{rk}}_D\left(\mathcal{D}\right)+{\mathit{rk}}_D\left({\mathcal{D}}_1\right)=m$$

We have $\mathit{ad}\left({\mathcal{D}}^{\prime }\right)\circ \mathit{ad}\left({\mathcal{D}}_1\right)=\mathit{ad}\left({\mathcal{D}}_1\circ {\mathcal{D}}^{\prime }\right)=0\Rightarrow {\mathcal{D}}_1\circ {\mathcal{D}}^{\prime }=0$ and ${\mathcal{D}}_1$ is surely among the CC of ${\mathcal{D}}^{\prime }$. Therefore, the differential sequence ${\xi }^{\prime }\stackrel{{\mathcal{D}}^{\prime }}{\to }\eta \stackrel{{\mathcal{D}}_1}{\to }\zeta$ on the operator level or the sequence $D^p\stackrel{{\mathcal{D}}_1}{\to }{D}^m\stackrel{{\mathcal{D}}^{\prime }}{\to }{D}^{l^{\prime }}$ on the differential module level may not be exact. In the latter, we have now $B=\mathit{im}\left({\mathcal{D}}_1\right)=\mathit{ker}\left(\mathcal{D}\right)\subseteq \mathit{ker}\left({\mathcal{D}}^{\text{'}}\right)=Z\subseteq D^m$. But we have also ${\mathit{rk}}_D\left(B\right)=m-{\mathit{rk}}_D\left(\mathcal{D}\right),\mathit{rk}\left(Z\right)=m-{\mathit{rk}}_D\left({\mathcal{D}}^{\prime }\right)\Rightarrow {\mathit{rk}}_D\left(H\right)={\mathit{rk}}_D\left(\mathcal{D}\right)-{\mathit{rk}}_D\left({\mathcal{D}}^{\prime }\right)=0$. Using the fact that $M_1=\mathit{\text{coker}}\left({\mathcal{D}}_1\right)$ while setting $M_1^{\prime }=\mathit{im}\left({\mathcal{D}}^{\prime }\right)\subseteq D^{l^{\prime }}$, we get the commutative and exact diagram:

$$\begin{array}{rrrrrrrrr}& & 0& & 0& & & & \\ & & \downarrow & & \downarrow & & & & \\ 0& \to & B& \to & D^m& \to & M_1& \to & 0\\ & & \downarrow & & \parallel & & \downarrow & & \\ 0& \to & Z& \to & D^m& \to & M_1^{\prime }& \to & 0\\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & H& & 0& & 0& & \\ & & \downarrow & & & & & & \\ & & 0& & & & & & \end{array}$$

A snake chase shows that the kernel of the induced epimorphism $M_1\to M_1^{\prime }\to 0$ is isomorphic to $H$ and is thus a torsion module because we have just proved that ${\mathit{rk}}_D\left(H\right)=0$. However, we know that $M_1$ is a torsion-free module and we reach a contradiction unless $H=0\iff M_1\simeq M_1^{\prime }$.

In actual practice, as shown in the contact case below, things are not so simple and we shall use the following commutative and exact diagram of differential modules:

$$\begin{array}{rrrrrrr}0\to \mathit{ker}\left(\mathit{ad}\left(\mathcal{D}\right)\right)\to & D^l& & \stackrel{\mathit{ad}\left(\mathcal{D}\right)}{\to }& & D^m& \to \mathit{\text{coker}}\left(\mathit{ad}\left(\mathcal{D}\right)\right)\to 0\\ & & \searrow & & \nearrow & & \\ & & & L& & & \\ & & \nearrow & \uparrow & \searrow & & \\ & 0& & D^{l^{\prime }}& & 0& \\ & & & \uparrow & & & \\ & & & 0& & & \end{array}$$

Setting $L=D^l/\mathit{ker}\left(\mathit{ad}\left(\mathcal{D}\right)\right)$ and introducing the biggest free differential module $D^{l^{\prime }}\subseteq L$ we have $l^{\prime }={\mathit{rk}}_D\left(D^{l^{\prime }}\right)={\mathit{rk}}_D\left(L\right)\le {\mathit{rk}}_D\left(D^l\right)\Rightarrow l^{\prime }\le l$, we may define the injective (care) operator $\mathit{ad}\left({\mathcal{D}}^{\prime }\right)$ by the composition of monomorphisms $D^{l^{\prime }}\to L\to D^m$ where the second is obtained by picking a basis of $D^{l^{\prime }}$, lifting it to $D^l$ and pushing it to $D^m$ by applying $\mathit{ad}\left(\mathcal{D}\right)$. We notice that $L$ can be viewed as the differential module defined by the generating CC of $\mathit{ad}\left(\mathcal{D}\right)$.

□

Example 3.10: Contact transformations.

With $m=n=3,K=\mathbb{Q}\left(x^1,x^2,x^3\right)=\mathbb{Q}\left(x\right)$, we may introduce the so-called contact $1$-form $\alpha ={\mathit{dx}}^1-x^3{\mathit{dx}}^2$. The system of infinitesimal Lie equations defining the infinitesimal contact transformations is obtained by eliminating the factor $\rho \left(x\right)$ in the equations $\mathcal{L}\left(\xi \right)\alpha =\mathit{\rho \alpha }$ where $\mathcal{L}$ is the standard Lie derivative. This system is thus only generated by ${\eta }^1$ and ${\eta }^2$ below but is not involutive and one has to introduce ${\eta }^3$ defined by the first order CC operator:

$$\left({\eta }^1,{\eta }^2,{\eta }^3\right)\to \left(d_3{\eta }^1-d_2{\eta }^2-x^3{d}_1{\eta }^2+{\eta }^3=\zeta \right)$$

in order to obtain the following involutive system with two equations of class 3 and one equation of class 2, a result leading to ${\beta }_1^3=2,{\beta }_1^2=1,{\beta }_1^1=0$ and the characters ${\alpha }_1^3=3-2=1<{\alpha }_1^2=3-1=2,{\alpha }_1^1=3-0=3$ with sum equal to $1+2+3=6=\mathit{dim}\left(g_1\right)=3\times 3-3$.

$$\left\{\begin{array}{lll}{d}_3{\xi }^3+d_2{\xi }^2+2x^3{d}_1{\xi }^2-d_1{\xi }^1& =& {\eta }^3\\ d_3{\xi }^1-x^3{d}_3{\xi }^2& =& {\eta }^2\\ d_2{\xi }^1-x^3{d}_2{\xi }^2+x^3{d}_1{\xi }^1-{\left(x^3\right)}^2{d}_1{\xi }^2-{\xi }^3& =& {\eta }^1\end{array}\right.\boxed{\begin{array}{lll}1& 2& 3\\ 1& 2& 3\\ 1& 2& •\end{array}}$$

The characters are thus ${\alpha }_1^3=3-2=1<{\alpha }_1^2=3-1=2,{\alpha }_1^1=3-0=3$ with sum equal to $1+2+3=6=\mathit{dim}\left(g_1\right)=3\times 3-3=6$ and we get $\mathit{dim}\left(g_2\right)=3+\left(2\times 2\right)+\left(3\times 1\right)=10=\mathit{dim}\left(S_2{T}^{\ast }\otimes T\right)-8$ along the Janet tabular [20].

In this situation, if $M$ is the differential module defined by this system or the corresponding operator $\mathcal{D}$, we know that ${\mathit{rk}}_D\left(M\right)={\alpha }_1^3=1=3-2={\mathit{rk}}_D\left(\mathit{D\xi }\right)-{\mathit{rk}}_D\left(\mathcal{D}\right)$. Of course, a differential transcendence basis for $\mathcal{D}$ can be the operator ${\mathcal{D}}^{\text{'}}:\xi \to \left\{{\eta }^2,{\eta }^3\right\}$ but, in view of the CC, we may equally choose any couple among $\left\{{\eta }^1,{\eta }^2,{\eta }^3\right\}$ and we obtain ${\mathit{rk}}_D\left({\mathcal{D}}^{\text{'}}\right)={\mathit{rk}}_D\left(\mathcal{D}\right)=2$ in any case, but now ${\mathcal{D}}^{\text{'}}$ is formally surjective, contrary to $\mathcal{D}$. The same result can also be obtained directly from the unique CC or the corresponding operator ${\mathcal{D}}_1$ defining the differential module $M_1$. Finally, we have ${\mathit{rk}}_D\left(M_1\right)=3-1=2={\mathit{rk}}_D\left(\mathit{D\eta }\right)-{\mathit{rk}}_D\left({\mathcal{D}}_1\right)$ and we check that we have indeed ${\mathit{rk}}_D\left(M\right)+{\mathit{rk}}_D\left(M_1\right)=1+2=3={\mathit{rk}}_D\left(\mathit{D\xi }\right)$.

It is well known that such a system can be parametrized by the injective parametrization (See [19, 29] for more details and the study of the general dimension $n=2p+1$):

$$-x^3{d}_3\varphi +\varphi ={\xi }^1,-d_3\varphi ={\xi }^2,d_2\varphi +x^3{d}_1\varphi ={\xi }^3\Rightarrow {\xi }^1-x^3{\xi }^2=\varphi$$

Noticing that $\mathcal{D}$ is generated by ${\mathcal{D}}^{\prime }:\xi \to \left({\eta }^1,{\eta }^2\right)$, we obtain the operator $\mathit{ad}\left({\mathcal{D}}^{\text{'}}\right)$:

$$-d_3{\mu }^2-d_2{\mu }^1-x^3{d}_1{\mu }^1={\nu }^1,x^3{d}_3{\mu }^2+{\mu }^2+x^3{d}_2{\mu }^1+{\left(x^3\right)}^2{d}_1{\mu }^1={\nu }^2,-{\mu }^1={\nu }^3$$

It follows that ${\mu }^1=-{\nu }^3,{\mu }^2={\nu }^2+x^3{\nu }^1$ and, substituing, the only CC:

$$\left(x^3{d}_3{\nu }^1\right)+\left(d_3{\nu }^2\right)+\left(d_2{\nu }^3+x^3{d}_1{\nu }^3\right)=0$$

which is exactly $\mathit{ad}\left({\mathcal{D}}_{-1}\right)$ as can be seen by multiplying by a test function $\varphi$ and integrating by parts. No such computations can be found in the literature on contact structures.

The associated differential sequence is:

$$\boxed{\begin{array}{rrrrrrrrr}0\to & \varphi & \stackrel{{\mathcal{D}}_{-1}}{\to }& \xi & \stackrel{\mathcal{D}}{\to }& \eta & \stackrel{{\mathcal{D}}_1}{\to }& \zeta & \to 0\\ 0\to & 1& \to & 3& \to & 3& \to & 1& \to 0\end{array}}$$

with Euler-Poincaré characteristic $1-3+3-1=0$ but is not a Janet sequence because ${\mathcal{D}}_{-1}$ is not involutive, its completion to involution being the trivially involutive operator $j_1:\varphi \to j_1\left(\varphi \right)$.

Introducing the ring $D=K\left[d_1,d_2,d_3\right]=K\left[d\right]$ of linear differential operators with coefficients in the differential field $K$, the corresponding differential module $M\simeq D$ is projective and even free, thus torsion-free or 0-pure [24], being defined by the split exact sequence of free differential modules:

$$0\to D\stackrel{{\mathcal{D}}_1}{\to }{D}^3\stackrel{\mathcal{D}}{\to }{D}^3\stackrel{{\mathcal{D}}_{-1}}{\to }D\to 0$$

We let the reader prove as an exercise that the adjoint sequence:

$$\begin{array}{l}0\leftarrow \theta \stackrel{\mathit{ad}\left({\mathcal{D}}_{-1}\right)}{\leftarrow }\nu \stackrel{\mathit{ad}\left(\mathcal{D}\right)}{\leftarrow }\mu \stackrel{\mathit{ad}\left({\mathcal{D}}_1\right)}{\leftarrow }\lambda \leftarrow 0\\ 0\leftarrow 1\leftarrow 3\leftarrow 3\leftarrow 1\leftarrow 0\end{array}$$

starting from the Lagrange multiplier $\lambda$ is also a split exact sequence of free differential modules.

We invite the reader to study, as a delicate exercise, the system of infinitesimal Lie equations defining the infinitesimal unimodular contact transformations preserving the 1-form $\alpha ={\mathit{dx}}^1-x^3{\mathit{dx}}^2$, thus also both the 2-form $\beta =\mathit{d\alpha }={\mathit{dx}}^2\wedge {\mathit{dx}}^3$ and the volume 3-form $\alpha \wedge \beta ={\mathit{dx}}^1\wedge {\mathit{dx}}^2\wedge {\mathit{dx}}^3$. Surprisingly, the Lie operator $\mathcal{D}:T\to {\wedge }^1{T}^{\ast }{\times }_X{\wedge }^2{T}^{\ast }:\xi \to \left(\mathcal{L}\left(\xi \right)\alpha =A,\mathcal{L}\left(\xi \right)\beta =B\right)$ for the geometric object $\omega =\left(\alpha ,\beta \right)$ is involutive if and only if $\mathit{d\alpha }=c^{\prime }\beta ,\mathit{d\beta }={c^{\prime }}^{\prime }\alpha \wedge \beta$ with $c^{\prime }{c^{\prime }}^{\prime }=0$. It provides the differential Janet sequence $3\to 6\to 4\to 1\to 0$ with Euler-Poincaré characteristic ${\mathit{rk}}_D\left(M\right)=3-6+4-1=0$. It follows that $\mathcal{D}$ cannot be parametrized. We have ${\mathcal{D}}_1:\left(A,B\right)\to \left(\mathit{dA}-c^{\prime }B=U,\mathit{dB}-{c^{\prime }}^{\prime }\left(A\wedge \beta +\alpha \wedge B\right)=V\right)$ and ${\mathcal{D}}_2:\left(U,V\right)\to \mathit{dU}+c^{\prime }V=W$ because $c^{\prime }{c^{\prime }}^{\prime }=0$. (See [12] and the recent [19] for more details).

4. Einstein equations

If $g_1\subset T^{\ast }\otimes T$ with $\mathit{dim}\left(g_1\right)=n\left(n-1\right)/2$ is the symbol of the Killing system $R_1\subset J_1\left(T\right)$ with $\mathit{dim}\left(R_1\right)=n\left(n+1\right)/2$, its first prolongation is $g_2=0$. We may introduce the Riemann tensor $\rho =\left({\rho }_{l,\mathit{ij}}^k\right)\in F_1=H_1^2\left(g_1\right)$ with $n^2\left(n^2-1\right)/12$ components in the short exact sequence [13, 18, 25, 27, 28, 29]:

$$0\to F_1\to {\wedge }^2{T}^{\ast }\otimes g_1\stackrel{\delta }{\to }{\wedge }^3{T}^{\ast }\otimes T\to 0$$E17

Linearizing the Ricci tensor $\left({\rho }_{\mathit{ij}}={\rho }_{i,\mathit{rj}}^r={\rho }_{\mathit{ji}}\right)\in S_2{T}^{\ast }$ over the Minkowski metric $\omega$, we obtain the usual second order homogeneous Ricci operator $\mathrm{\Omega }\to R$ with $4$ terms (care):

$$\begin{array}{c}2R_{\mathit{ij}}={\omega }^{\mathit{rs}}\left(d_{\mathit{rs}}{\mathrm{\Omega }}_{\mathit{ij}}+d_{\mathit{ij}}{\mathrm{\Omega }}_{\mathit{rs}}-d_{\mathit{ri}}{\mathrm{\Omega }}_{\mathit{sj}}-d_{\mathit{sj}}{\mathrm{\Omega }}_{\mathit{ri}}\right)=2R_{\mathit{ji}}\\ \mathit{tr}\left(\mathrm{\Omega }\right)={\omega }^{\mathit{ij}}{\mathrm{\Omega }}_{\mathit{ij}}\Rightarrow \mathit{tr}\left(R\right)={\omega }^{\mathit{ij}}{R}_{\mathit{ij}}={\omega }^{\mathit{ij}}{d}_{\mathit{ij}}\mathit{tr}\left(\mathrm{\Omega }\right)-{\omega }^{\mathit{ru}}{\omega }^{\mathit{sv}}{d}_{\mathit{rs}}{\mathrm{\Omega }}_{\mathit{uv}}\end{array}$$

We may define the $\mathit{\text{Einstein}}$ operator by setting ${\epsilon }_{\mathit{ij}}={\rho }_{\mathit{ij}}-\frac{1}{2}{\omega }_{\mathit{ij}}{\omega }^{\mathit{rs}}{\rho }_{\mathit{rs}}\Rightarrow E_{\mathit{ij}}=R_{\mathit{ij}}-\frac{1}{2}{\omega }_{\mathit{ij}}\mathit{tr}\left(R\right)$ and obtain the $6$ terms (care) [9]:

$$2E_{\mathit{ij}}={\omega }^{\mathit{rs}}\left(d_{\mathit{rs}}{\mathrm{\Omega }}_{\mathit{ij}}+d_{\mathit{ij}}{\mathrm{\Omega }}_{\mathit{rs}}-d_{\mathit{ri}}{\mathrm{\Omega }}_{\mathit{sj}}-d_{\mathit{sj}}{\mathrm{\Omega }}_{\mathit{ri}}\right)-{\omega }_{\mathit{ij}}\left({\omega }^{\mathit{rs}}{\omega }^{\mathit{uv}}{d}_{\mathit{rs}}{\mathrm{\Omega }}_{\mathit{uv}}-{\omega }^{\mathit{ru}}{\omega }^{\mathit{sv}}{d}_{\mathit{rs}}{\mathrm{\Omega }}_{\mathit{uv}}\right)$$

We have the (locally exact) differential sequence of operators acting on sections of vector bundles:

$$\begin{array}{c}T\underset{1}{\stackrel{\mathit{\text{Killing}}}{\to }}{S}_2{T}^{\ast }\underset{2}{\stackrel{\mathit{\text{Riemann}}}{\to }}{F}_1\underset{1}{\stackrel{\mathit{\text{Bianchi}}}{\to }}{F}_2\\ n\stackrel{\mathcal{D}}{\to }n\left(n+1\right)/2\stackrel{{\mathcal{D}}_1}{\to }{n}^2\left(n^2-1\right)/12\stackrel{{\mathcal{D}}_2}{\to }{n}^2\left(n^2-1\right)\left(n-2\right)/24\end{array}$$E18

where $F_2=H_1^3\left(g_1\right)$ is similarly defined by the short exact sequence:

$$0\to F_2\to {\wedge }^3{T}^{\ast }\otimes g_1\stackrel{\delta }{\to }{\wedge }^4{T}^{\ast }\otimes T\to 0$$E19

Our purpose is now to study the differential sequence onto which its right part is projecting:

$$\begin{array}{c}{S}_2{T}^{\ast }\underset{2}{\stackrel{\mathit{\text{Einstein}}}{\to }}{S}_2{T}^{\ast }\underset{1}{\stackrel{\mathit{div}}{\to }}{T}^{\ast }\to 0\\ n\left(n+1\right)/2\to n\left(n+1\right)/2\to n\to 0\end{array}$$

and the following adjoint sequence where we have set [13, 18, 25, 27, 28, 29]:

$$\mathit{ad}\left(T\right)\stackrel{\mathit{\text{Cauchy}}}{\leftarrow }\mathit{ad}\left(S_2{T}^{\ast }\right)\stackrel{\mathit{\text{Beltrami}}}{\leftarrow }\mathit{ad}\left(F_1\right)\stackrel{\mathit{\text{Lanczos}}}{\leftarrow }\mathit{ad}\left(F_2\right)$$E20

In this sequence, if $E$ is a vector bundle over the ground manifold $X$ with dimension $n$, we may introduce the new vector bundle $\mathit{ad}\left(E\right)={\wedge }^n{T}^{\ast }\otimes E^{\ast }$_, where $E^{\ast }$ is obtained from $E$ by inverting the transition rules exactly like $T^{\ast }$ is obtained from $T$. We have for example $\mathit{ad}\left(T\right)={\wedge }^n{T}^{\ast }\otimes T^{\ast }\simeq {\wedge }^n{T}^{\ast }\otimes T\simeq {\wedge }^{n-1}{T}^{\ast }$ because $T^{\ast }$ is isomorphic to $T$ by using the metric $\omega$. The $10\times 10$ Einstein operator matrix is induced from the $10\times 20$ Riemann operator matrix and the $10\times 4$ div operator matrix is induced from the $20\times 20$ Bianchi operator matrix. We advise the reader not familiar with the formal theory of systems or operators to follow the computation in dimension $n=2$ with the $1\times 3$ $\mathit{\text{Airy}}$ operator matrix, which is the formal adjoint of the $3\times 1$ Riemann operator matrix, and $n=3$ with the $6\times 6$ Beltrami operator matrix which is the formal adjoint of the $6\times 6$ Riemann operator matrix which is easily seen to be self-adjoint up to a change of basis.

• $n=2$: The stress equations become $d_1{\sigma }^{11}+d_2{\sigma }^{12}=0,d_1{\sigma }^{21}+d_2{\sigma }^{22}=0$. Their second order parametrization ${\sigma }^{11}=d_{22}\varphi ,{\sigma }^{12}={\sigma }^{21}=-d_{12}\varphi ,{\sigma }^{22}=d_{11}\varphi$ has been provided by George Biddell Airy in 1863 [36] and is well known [4]. We get the second order system:

$$\left\{\begin{array}{cc}{\sigma }^{11}& \equiv d_{22}\varphi =0\\ -{\sigma }^{12}& \equiv d_{12}\varphi =0\\ {\sigma }^{22}& \equiv d_{11}\varphi =0\end{array}\right.\boxed{\begin{array}{cc}1& 2\\ 1& •\\ 1& •\end{array}}$$

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 are just the Cauchy equations. Of course, the $\mathit{\text{Airy}}$ function (1 term) has absolutely nothing to do with the perturbation of the metric (3 terms). With more details, when $\omega$ is the Euclidean metric, we may consider the only component:

$$\mathit{tr}\left(R\right)=\left(d_{11}+d_{22}\right)\left({\mathrm{\Omega }}_{11}+{\mathrm{\Omega }}_{22}\right)-\left(d_{11}{\mathrm{\Omega }}_{11}+2d_{12}{\mathrm{\Omega }}_{12}+d_{22}{\mathrm{\Omega }}_{22}\right)=d_{22}{\mathrm{\Omega }}_{11}+d_{11}{\mathrm{\Omega }}_{22}-2d_{12}{\mathrm{\Omega }}_{12}$$

Multiplying by the Airy function $\varphi$ and integrating by parts, we get $\mathit{\text{Airy}}=\mathit{ad}\left(\mathit{\text{Riemann}}\right)$ and $\mathit{\text{Cauchy}}=\mathit{ad}\left(\mathit{\text{Killing}}\right)$ in the following differential sequences:

$$\begin{array}{l}2\underset{1}{\stackrel{\mathit{\text{Killing}}}{\to }}3\underset{2}{\stackrel{\mathit{\text{Riemann}}}{\to }}1\to 0\\ 0\leftarrow 2\underset{1}{\stackrel{\mathit{\text{Cauchy}}}{\leftarrow }}3\underset{2}{\stackrel{\mathit{\text{Airy}}}{\leftarrow }}1\end{array}$$

$•n=3$: It is quite more delicate to parametrize the 3 PD equations:

$$d_1{\sigma }^{11}+d_2{\sigma }^{12}+d_3{\sigma }^{13}=0,d_1{\sigma }^{21}+d_2{\sigma }^{22}+d_3{\sigma }^{23}=0,d_1{\sigma }^{31}+d_2{\sigma }^{32}+d_3{\sigma }^{33}=0$$

A direct computational approach has been provided by Eugenio Beltrami in 1892 [37, 38], James Clerk Maxwell in 1870 [39] and Giacinto Morera in 1892 [38, 40] by introducing the 6 stress functions ${\varphi }_{\mathit{ij}}={\varphi }_{\mathit{ji}}$ in the Beltrami parametrization. The corresponding system:

$$\left\{\begin{array}{ll}{\sigma }^{11}\equiv & d_{33}{\varphi }_{22}+d_{22}{\varphi }_{33}-2d_{23}{\varphi }_{23}=0\\ -{\sigma }^{12}\equiv & d_{33}{\varphi }_{12}+d_{12}{\varphi }_{33}-d_{13}{\varphi }_{23}-d_{23}{\varphi }_{13}=0\\ {\sigma }^{22}\equiv & d_{33}{\varphi }_{11}+d_{11}{\varphi }_{33}-2d_{13}{\varphi }_{13}=0\\ {\sigma }^{13}\equiv & d_{23}{\varphi }_{12}+d_{12}{\varphi }_{23}-d_{22}{\varphi }_{13}-d_{13}{\varphi }_{22}=0\\ -{\sigma }^{23}\equiv & d_{23}{\varphi }_{11}+d_{11}{\varphi }_{23}-d_{12}{\varphi }_{13}-d_{13}{\varphi }_{12}=0\\ {\sigma }^{33}\equiv & d_{22}{\varphi }_{11}+d_{11}{\varphi }_{22}-2d_{12}{\varphi }_{12}=0\end{array}\right.\boxed{\begin{array}{lll}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\\ 1& 2& •\\ 1& 2& •\\ 1& 2& •\end{array}}$$

is involutive with 3 equations of class 3, 3 equations of class 2 and no equation of class 1. The three characters are thus ${\alpha }_2^3=1\times 6-3=3<{\alpha }_2^2=2\times 6-3=9<{\alpha }_2^1=3\times 6-0=18$ and we have $\mathit{dim}\left(g_2\right)={\alpha }_2^1+{\alpha }_2^2+{\alpha }_2^3=18+9+3=30=\mathit{dim}\left(S_2{T}^{\ast }\otimes S_2{T}^{\ast }\right)-\mathit{dim}\left(S_2{T}^{\ast }\right)=6\times 6-6$ [22]. The 3 CC are 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 $n^2\left(n^2-1\right)/2=6$ independent components when $n=3$ [12, 13].

Breaking the canonical form of the six equations which is associated with the Janet tabular, we may rewrite the Beltrami parametrization of the Cauchy stress equations as follows, after exchanging the third row with the fourth row, keeping the ordering $\left\{\left(11\right)<\left(12\right)<\left(13\right)<\left(22\right)<\left(23\right)<\left(33\right)\right\}$:

$$\left(\begin{array}{cccccc}{d}_1& d_2& d_3& 0& 0& 0\\ 0& d_1& 0& d_2& d_3& 0\\ 0& 0& d_1& 0& d_2& d_3\end{array}\right)\left(\begin{array}{cccccc}0& 0& 0& d_{33}& -2d_{23}& d_{22}\\ 0& -d_{33}& d_{23}& 0& d_{13}& -d_{12}\\ 0& d_{23}& -d_{22}& -d_{13}& d_{12}& 0\\ d_{33}& 0& -2d_{13}& 0& 0& d_{11}\\ -d_{23}& d_{13}& d_{12}& 0& -d_{11}& 0\\ d_{22}& -2d_{12}& 0& d_{11}& 0& 0\end{array}\right)\equiv 0$$

as an identity where 0 on the right denotes the zero operator. However, if $\mathrm{\Omega }$ is a perturbation of the metric $\omega$, the standard implicit summation used in continuum mechanics is, when $n=3$:

$$\begin{array}{l}{\sigma }^{\mathit{ij}}{\mathrm{\Omega }}_{\mathit{ij}}={\sigma }^{11}{\mathrm{\Omega }}_{11}+2{\sigma }^{12}{\mathrm{\Omega }}_{12}+2{\sigma }^{13}{\mathrm{\Omega }}_{13}+{\sigma }^{22}{\mathrm{\Omega }}_{22}+2{\sigma }^{23}{\mathrm{\Omega }}_{23}+{\sigma }^{33}{\mathrm{\Omega }}_{33}\\ ={\mathrm{\Omega }}_{22}{d}_{33}{\varphi }_{11}+{\mathrm{\Omega }}_{33}{d}_{22}{\varphi }_{11}-2{\mathrm{\Omega }}_{23}{d}_{23}{\varphi }_{11}+\dots \\ +{\mathrm{\Omega }}_{23}{d}_{13}{\varphi }_{12}+{\mathrm{\Omega }}_{13}{d}_{23}{\varphi }_{12}-{\mathrm{\Omega }}_{12}{d}_{33}{\varphi }_{12}-{\mathrm{\Omega }}_{33}{d}_{12}{\varphi }_{12}+\dots \end{array}$$

because the stress tensor density $\sigma$ is supposed to be symmetric. Integrating by parts in order to construct the adjoint operator, we get:

$$\begin{array}{lll}{\varphi }_{11}& \to & d_{33}{\mathrm{\Omega }}_{22}+d_{22}{\mathrm{\Omega }}_{33}-2d_{23}{\mathrm{\Omega }}_{23}\\ {\varphi }_{12}& \to & d_{13}{\mathrm{\Omega }}_{23}+d_{23}{\mathrm{\Omega }}_{13}-d_{33}{\mathrm{\Omega }}_{12}-d_{12}{\mathrm{\Omega }}_{33}\end{array}$$

and so on. The identifications $\mathit{\text{Beltrami}}=\mathit{ad}\left(\mathit{\text{Riemann}}\right),\mathit{\text{Lanczos}}=\mathit{ad}\left(\mathit{\text{Bianchi}}\right)$ in the diagram:

$$\boxed{\begin{array}{rrrrrrrrr}& 3& \underset{1}{\stackrel{\mathit{\text{Killing}}}{\to }}& 6& \underset{2}{\stackrel{\mathit{\text{Riemann}}}{\to }}& 6& \underset{1}{\stackrel{\mathit{\text{Bianchi}}}{\to }}& 3& \to 0\\ 0\leftarrow & 3& \underset{1}{\stackrel{\mathit{\text{Cauchy}}}{\leftarrow }}& 6& \underset{2}{\stackrel{\mathit{\text{Beltrami}}}{\leftarrow }}& 6& \underset{1}{\stackrel{\mathit{\text{Lanczos}}}{\leftarrow }}& 3& \end{array}.}$$E21

prove that the Cauchy operator has nothing to do with the Bianchi operator [12, 13].

When $\omega$ is the Euclidean metric, the link between the two sequences is established by means of the elastic constitutive relations $2{\sigma }_{\mathit{ij}}=\mathit{\lambda tr}\left(\mathrm{\Omega }\right){\omega }_{\mathit{ij}}+2\mu {\mathrm{\Omega }}_{\mathit{ij}}$ with the Lamé elastic constants $\left(\lambda ,\mu \right)$ but mechanicians are usually setting ${\mathrm{\Omega }}_{\mathit{ij}}=2{\epsilon }_{\mathit{ij}}$. Using the standard Helmholtz decomposition $\overrightarrow{\xi }=\overrightarrow{\nabla }\phi +\overrightarrow{\nabla }\wedge \overrightarrow{\psi }$ and substituting in the dynamical equation $d_i{\sigma }^{\mathit{ij}}=\rho d^2/{\mathit{dt}}^2{\xi }^j$ where $\rho$ is the mass per unit volume, we get the longitudinal and transverse wave equations, namely $\mathrm{\Delta }\phi -\frac{\rho }{\lambda +2\mu }\frac{d^2}{{\mathit{dt}}^2}\phi =0$ and $\mathrm{\Delta }\overrightarrow{\psi }-\frac{\rho }{\mu }\frac{d^2}{{\mathit{dt}}^2}\overrightarrow{\psi }=0$, responsible for earthquakes, with respective speeds $c_L^2=\left(\lambda +2\mu \right)/\rho$ and $c_T^2=\mu /\rho$.

Then, taking into account, the factor 2 involved by multiplying the second, third and fifth row by 2, we get the new $6\times 6$ operator matrix with rank 3 which is clearly self-adjoint:

$$\boxed{\left(\begin{array}{cccccc}0& 0& 0& d_{33}& -2d_{23}& d_{22}\\ 0& -2d_{33}& 2d_{23}& 0& 2d_{13}& -2d_{12}\\ 0& 2d_{23}& -2d_{22}& -2d_{13}& 2d_{12}& 0\\ d_{33}& 0& -2d_{13}& 0& 0& d_{11}\\ -2d_{23}& 2d_{13}& 2d_{12}& 0& -2d_{11}& 0\\ d_{22}& -2d_{12}& 0& d_{11}& 0& 0\end{array}\right)}$$

Following Maxwell, we keep ${\varphi }_{11}=A,{\varphi }_{22}=B,{\varphi }_{33}=C$, set ${\varphi }_{12}={\varphi }_{23}={\varphi }_{31}=0$ and get:

$$\left\{\begin{array}{ll}{\sigma }^{11}\equiv & d_{33}B+d_{22}C=0\\ {\sigma }^{22}\equiv & d_{33}A+d_{11}C=0\\ -{\sigma }^{23}\equiv & d_{23}A=0\\ {\sigma }^{33}\equiv & d_{22}A+d_{11}B=0\\ -{\sigma }^{13}\equiv & d_{13}B=0\\ -{\sigma }^{12}\equiv & d_{12}C=0\end{array}\right.\boxed{\begin{array}{lll}1& 2& 3\\ 1& 2& 3\\ 1& 2& •\\ 1& 2& •\\ 1& •& •\\ 1& •& •\end{array}}$$

This system may not be involutive and no CC can be found “a priori” because the coordinate system is surely not $\delta$-regular. Effecting the linear change of coordinates ${\overline{x}}^1=x^1,{\overline{x}}^2=x^2,{\overline{x}}^3=x^3+x^2+x^1$ and taking out the bar for simplicity, we obtain the involutive system:

$$\left\{\begin{array}{l}{d}_{33}C+d_{13}C+d_{23}C+d_{12}C=0\\ d_{33}B+d_{13}B=0\\ d_{33}A+d_{23}A=0\\ d_{23}C+d_{22}C-d_{13}C-d_{13}B-d_{12}C=0\\ d_{23}A-d_{22}C+d_{13}B+2d_{12}C-d_{11}C=0\\ d_{22}A+d_{22}C-2d_{12}C+d_{11}C+d_{11}B=0\end{array}\right.\boxed{\begin{array}{lll}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\\ 1& 2& •\\ 1& 2& •\\ 1& 2& •\end{array}}$$

and it is easy to check that the 3 CC obtained just amount to the desired 3 stress equations when coming back to the original system of coordinates. However, the three characters are different as we have now ${\alpha }_2^3=3-3=0<{\alpha }_2^2=2\times 3-3=3<{\alpha }_2^1=3\times 3-0=9$ with sum equal to $\mathit{dim}\left(g_2\right)=6\times 3-6=18-6=12$. We have thus a minimum parametrization. Of course, if there is a geometrical background, this change of local coordinates is hiding it totally. Moreover, we notice that the stress functions kept in the procedure are just the ones on which $d_{33}$ is acting. The reason for such an apparently technical choice is related to very general deep arguments in the theory of differential modules that will only be explained at the end of the paper.

Following Morera, we keep ${\varphi }_{23}=L,{\varphi }_{13}=M,{\varphi }_{12}=N$, set ${\varphi }_{11}={\varphi }_{22}={\varphi }_{33}=0$_, and get:

$$\left\{\begin{array}{l}{d}_{23}L=0\\ d_{33}N-d_{13}L-d_{23}M=0\\ d_{13}M=0\\ d_{22}M-d_{23}N-d_{12}L=0\\ d_{11}L-d_{12}M-d_{13}N=0\\ d_{12}N=0\end{array}\right.$$

Using the same change of coordinates, we obtain the following involutive system:

$$\left\{\begin{array}{l}{d}_{33}N+d_{23}N+d_{13}N+d_{12}N=0\\ d_{33}M+d_{13}M=0\\ d_{33}L+d_{23}L=0\\ \left(d_{23}N+d_{23}M-d_{23}L\right)+\left(d_{13}N-d_{13}M+d_{13}L\right)+d_{12}N=0\\ 2d_{23}M+\left(d_{13}N-d_{13}M-d_{13}L\right)+d_{12}M-d_{11}L=0\\ d_{22}M+\left(d_{12}N-d_{12}M-d_{12}L\right)+d_{11}L=0\end{array}\right.\boxed{\begin{array}{lll}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\\ 1& 2& •\\ 1& 2& •\\ 1& 2& •\end{array}}$$

After elementary but tedious computations (that could not be avoided!), one can prove that the 3 CC corresponding to the 3 dots are effectively satisfied and that they correspond to the 3 Cauchy stress equations which are therefore parametrized. The parametrization is thus provided by an involutive operator defining a torsion module because the character ${\alpha }_2^3$ is vanishing in $\delta$-regular coordinates, just like before for the Maxwell parametrization. We have thus another minimum parametrization. Of course, such a result could not have been understood by Beltrami in 1892 because the work of Cartan could not be adapted easily in the language of exterior forms and the work of Janet appeared only in 1920 with no explicit reference to involution because only Janet bases are used while the Pommaret bases have only been introduced in 1978 (See [25, 29] for historical facts).

We may finally keeep $\left\{{\varphi }_{11},{\varphi }_{12},{\varphi }_{22}\right\}$, set ${\varphi }_{13}={\varphi }_{23}={\varphi }_{33}=0$ and get the different involutive system with the same characters, in particular with ${\alpha }_2^3=0$:

$$\left\{\begin{array}{lll}{\sigma }^{11}& \equiv & {\partial }_{33}{\varphi }_{22}=0\\ -{\sigma }^{12}& \equiv & {\partial }_{33}{\varphi }_{12}=0\\ {\sigma }^{22}& \equiv & {\partial }_{33}{\varphi }_{11}=0\\ {\sigma }^{13}& \equiv & {\partial }_{23}{\varphi }_{12}-{\partial }_{13}{\varphi }_{22}=0\\ -{\sigma }^{23}& \equiv & {\partial }_{23}{\varphi }_{11}-{\partial }_{13}{\varphi }_{12}=0\\ {\sigma }^{33}& \equiv & {\partial }_{22}{\varphi }_{11}+{\partial }_{11}{\varphi }_{22}-2{\partial }_{12}{\varphi }_{12}=0\end{array}\right.\boxed{\begin{array}{lll}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\\ 1& 2& •\\ 1& 2& •\\ 1& 2& •\end{array}}$$E22

So far, we have thus obtained three explicit local minimum parametrizations of the Cauchy stress equations with $n\left(n-1\right)/2=3$ stress potentials but there may be others.

$•n=4$: It just remains to explain the relation of the previous results with Einstein equations. The first suprising link is provided by the following technical proposition:

Proposition 4.1: The Beltrami parametrization is just described by the Einstein operator when $n=3$. The same confusion existing between the Bianchi operator and the Cauchy operator has been made by both Einstein and Beltrami because the Einstein operator is self-adjoint in arbitrary dimension $n\ge 3$, contrary to the Ricci operator.

Proof: The number of components of the Riemann tensor is $\mathit{dim}\left(F_1\right)=n^2\left(n^2-1\right)/12$. We have the combinatorial formula $n^2\left(n^2-1\right)/12-n\left(n+1\right)/2=n\left(n+1\right)\left(n+2\right)\left(n-3\right)/12$ expressing that the number of components of the Riemann tensor is always greater or equal to the number of components of the Ricci tensor whenever $n>2$. Also, we have shown in many books [12, 13, 25, 27, 28, 29] or papers [16, 18] that the number of Bianchi identities is equal to $n^2\left(n^2-1\right)\left(n-2\right)/24$, that is 3 when $n=3$ and 20 when $n=4$. Of course, it is well known that the div operator, induced as CC of the Einstein operator, has $n$ components in arbitrary dimension $n\ge 3$.

Accordingly, when $n=3$ we have $n^2\left(n^2-1\right)/12=n\left(n+1\right)/2=6$ and it only remains to prove that the Einstein operator reduces to the Beltrami operator and not just to the Ricci operator.

The following formulas can be found in any textbook on general relativity. In particular, the difference existing between $R_{\mathit{ij}}$ (4 terms only) and $E_{\mathit{ij}}$ (6 terms) can only be seen when ${\omega }_{i\ne j}=0$. In our situation with $n=3$ and the Euclidean metric, we obtain successively:

$$\begin{array}{l}2R_{12}=2E_{12}=\left(d_{11}+d_{22}+d_{33}\right){\mathrm{\Omega }}_{12}+d_{12}\left({\mathrm{\Omega }}_{11}+{\mathrm{\Omega }}_{22}+{\mathrm{\Omega }}_{33}\right)\\ -\left(d_{11}{\mathrm{\Omega }}_{12}+d_{12}{\mathrm{\Omega }}_{22}+d_{13}{\mathrm{\Omega }}_{23}\right)-\left(d_{12}{\mathrm{\Omega }}_{11}+d_{22}{\mathrm{\Omega }}_{12}+d_{23}{\mathrm{\Omega }}_{13}\right)\\ =d_{33}{\mathrm{\Omega }}_{12}+d_{12}{\mathrm{\Omega }}_{33}-d_{13}{\mathrm{\Omega }}_{23}-d_{23}{\mathrm{\Omega }}_{13}\end{array}$$

$$\begin{array}{l}2R_{11}=\left(d_{11}+d_{22}+d_{33}\right){\mathrm{\Omega }}_{11}+d_{11}\left({\mathrm{\Omega }}_{11}+{\mathrm{\Omega }}_{22}+{\mathrm{\Omega }}_{33}\right)\\ -2\left(d_{11}{\mathrm{\Omega }}_{11}+d_{12}{\mathrm{\Omega }}_{12}+d_{13}{\mathrm{\Omega }}_{13}\right)\\ =\left(d_{22}+d_{33}\right){\mathrm{\Omega }}_{11}+d_{11}\left({\mathrm{\Omega }}_{22}+{\mathrm{\Omega }}_{33}\right)-2\left(d_{12}{\mathrm{\Omega }}_{12}+d_{13}{\mathrm{\Omega }}_{13}\right)\end{array}$$

$$\begin{array}{ll}\mathit{tr}\left(R\right)=& \left(d_{11}{\mathrm{\Omega }}_{22}+d_{11}{\mathrm{\Omega }}_{33}+d_{22}{\mathrm{\Omega }}_{11}+d_{22}{\mathrm{\Omega }}_{33}+d_{33}{\mathrm{\Omega }}_{11}+d_{33}{\mathrm{\Omega }}_{22}\right)\\ & -2\left(d_{12}{\mathrm{\Omega }}_{12}+d_{13}{\mathrm{\Omega }}_{13}+d_{23}{\mathrm{\Omega }}_{23}\right)\\ 2E_{11}=& d_{22}{\mathrm{\Omega }}_{33}+d_{33}{\mathrm{\Omega }}_{22}-2d_{23}{\mathrm{\Omega }}_{23}\end{array}$$

In the light of modern differential geometry, comparing these results with the works of Maxwell, Morera, Beltrami and Einstein, it becomes clear that they have been confusing the div operator induced from the Bianchi operator with the Cauchy operator. However, it is also clear that they both obtained a possibility to parametrize the Cauchy operator by means of 3 arbitrary potential like functions in the case of Maxwell and Morera, 6 in the case of Beltrami explaining the previous choices, and 10 in the case of Einstein. Of course, as they were ignoring that the Einstein operator was self-adjoint whenever $n\ge 3$, they did not notice that we have $\mathit{\text{Cauchy}}=\mathit{ad}\left(\mathit{\text{Killing}}\right)$ and they were unable to compare their results with the Airy operator found as early as in 1870 for the same mechanical purpose when $n=2$. To speak in a rough way, the situation is similar to what could happen in the study of contact structures if one should confuse ${\mathcal{D}}_{-1}$ with ${\mathcal{D}}_1$ or $\mathcal{D}$ with ${\mathcal{D}}_2$ in the Killing sequence [16]. Finally, using the double differential duality test, we can choose a differential transcendence basis with $n\left(n-1\right)/2$ potentials that can be indexed by ${\varphi }_{\mathit{ij}}={\varphi }_{\mathit{ji}}$ with $i<j$ or $1\le i,j\le n-1$ or even $2\le i,j\le n$ when the dimension $n\ge 2$ is arbitrary (See [24, 29] for more details on differential algebra).

$\square$

Remark 4.2: The author of this paper is not an historian of sciences but a specialist of mathematical physics interested by the analogy existing between electromagnetism (EM), elasticity (EL), and gravitation (GR) by using the conformal group of space–time along the idea of H. Weyl [41] (See [4, 13, 18, 19, 28, 29, 30, 31] for related works). It is thus difficult to imagine that Einstein could not have been aware of the works of Maxwell and Beltrami on the foundations of EL and tensor calculus as they were quite famous when he started his research on GR. We also notice that the Mach-Lippmann analogy [4, 13, 28, 29, 35] was introduced at the same time and that the phenomenological law of piezzo-electricity has been discovered by… Maxwell [19, 20]. We do believe that classical variational calculus must be considered as a kind of “duality theory” that should only depend on new purely mathematical tools, namely “group theory” on one side and “differential homological algebra” on the other side (See [12, 13, 28] for the theory and [4] for the applications).

The two following crucial results, still neither known nor acknowledged today, are provided by the next proposition and corresponding corollary:

Proposition 4.3: The cauchy operator can be parametrized by the adjoint of the Ricci Operator

Proof: The Einstein operator $\mathrm{\Omega }\to E$ is defined by setting $E_{\mathit{ij}}=R_{\mathit{ij}}-\frac{1}{2}{\omega }_{\mathit{ij}}\mathit{tr}\left(R\right)$ that we shall write $\mathit{\text{Einstein}}=C\circ \mathit{\text{Ricci}}$ where $C:S_2{T}^{\ast }\to S_2{T}^{\ast }$ is a symmetric matrix only depending on $\omega$, which is invertible whenever $n\ge 3$. Surprisingly, we may also introduce the same linear transformation $C:\mathrm{\Omega }\to \overline{\mathrm{\Omega }}=\mathrm{\Omega }-\frac{1}{2}\omega \mathit{tr}\left(\mathrm{\Omega }\right)$ and the unknown composite operator $\mathcal{X}:\overline{\mathrm{\Omega }}\to \mathrm{\Omega }\to E$ in such a way that $\mathit{\text{Einstein}}=\mathcal{X}\circ C$ where $\mathcal{X}$ is defined by (See [15], 5.1.5, p. 134):

$$2E_{\mathit{ij}}={\omega }^{\mathit{rs}}{d}_{\mathit{rs}}{\overline{\mathrm{\Omega }}}_{\mathit{ij}}-{\omega }^{\mathit{rs}}{d}_{\mathit{ri}}{\overline{\mathrm{\Omega }}}_{\mathit{sj}}-{\omega }^{\mathit{rs}}{d}_{\mathit{sj}}{\overline{\mathrm{\Omega }}}_{\mathit{ri}}+{\omega }_{\mathit{ij}}{\omega }^{\mathit{ru}}{\omega }^{\mathit{sv}}{d}_{\mathit{rs}}{\overline{\mathrm{\Omega }}}_{\mathit{uv}}$$

Now, introducing the test functions ${\lambda }^{\mathit{ij}}$, we get:

$${\lambda }^{\mathit{ij}}{E}_{\mathit{ij}}={\lambda }^{\mathit{ij}}\left(R_{\mathit{ij}}-\frac{1}{2}{\omega }_{\mathit{ij}}\mathit{tr}\left(R\right)\right)=\left({\lambda }^{\mathit{ij}}-\frac{1}{2}{\lambda }^{\mathit{rs}}{\omega }_{\mathit{rs}}{\omega }^{\mathit{ij}}\right)R_{\mathit{ij}}={\overline{\lambda }}^{\mathit{ij}}{R}_{\mathit{ij}}$$

Integrating by parts while setting as usual $\square ={\omega }^{\mathit{rs}}{d}_{\mathit{rs}}$, we obtain:

$$\left(\square {\overline{\lambda }}^{\mathit{rs}}+{\omega }^{\mathit{rs}}{d}_{\mathit{ij}}{\overline{\lambda }}^{\mathit{ij}}-{\omega }^{\mathit{sj}}{d}_{\mathit{ij}}{\overline{\lambda }}^{\mathit{ri}}-{\omega }^{\mathit{ri}}{d}_{\mathit{ij}}{\overline{\lambda }}^{\mathit{sj}}\right){\mathrm{\Omega }}_{\mathit{rs}}={\sigma }^{\mathit{rs}}{\mathrm{\Omega }}_{\mathit{rs}}$$E23

Moreover, suppressing the “bar” for simplicity, we have:

$$d_r{\sigma }^{\mathit{rs}}={\omega }^{\mathit{ij}}{d}_{\mathit{rij}}{\lambda }^{\mathit{rs}}+{\omega }^{\mathit{rs}}{d}_{\mathit{rij}}{\lambda }^{\mathit{ij}}-{\omega }^{\mathit{sj}}{d}_{\mathit{rij}}{\lambda }^{\mathit{ri}}-{\omega }^{\mathit{ri}}{d}_{\mathit{rij}}{\lambda }^{\mathit{sj}}=0$$E24

As Einstein is a self-adjoint operator (contrary to the Ricci operator), we have the identities:

$$\mathit{ad}\left(\mathit{\text{Einstein}}\right)=\mathit{ad}\left(C\right)\circ \mathit{ad}\left(\mathcal{X}\right)\Rightarrow \mathit{\text{Einstein}}=C\circ \mathit{ad}\left(\mathcal{X}\right)\Rightarrow \mathit{ad}\left(\mathcal{X}\right)=\mathit{\text{Ricci}}\Rightarrow \mathcal{X}=\mathit{ad}\left(\mathit{\text{Ricci}}\right)$$

Indeed, $\mathit{ad}\left(C\right)=C$ because $C$ is a symmetric matrix and we know that $\mathit{ad}\left(\mathit{\text{Einstein}}\right)=\mathit{\text{Einstein}}$. Accordingly, the operator ad(Ricci) parametrizes the Cauchy equations, without any reference to the Einstein operator, which has no mathematical origin, in the sense that it cannot be obtained by any diagram chasing. The three terms after the Dalembert operator factorize through the divergence operator $d_i{\lambda }^{\mathit{ri}}$. We may thus add the differential constraints $d_i{\lambda }^{\mathit{ri}}=0$ without any reference to a gauge transformation in order to obtain a (minimum) relative parametrization (see [24] for details and explicit examples). When $n=4$ we finally obtain the adjoint sequences:

$$\begin{array}{rrrrrr}& 4& \stackrel{\mathit{\text{Killing}}}{\to }& 10& \stackrel{\mathit{\text{Ricci}}}{\to }& 10\\ & & & & & \\ 0\leftarrow & 4& \stackrel{\mathit{\text{Cauchy}}}{\leftarrow }& 10& \stackrel{\mathit{ad}\left(\mathit{\text{Ricci}}\right)}{\leftarrow }& 10\end{array}$$E25

without any reference to the Bianchi operator and the induced $\mathit{div}$ operator.

Finally, using Theorem 2.1 or Proposition 2.2, we may choose a differential transcendence basis made by $\left\{{\lambda }^{\mathit{ij}}|i<j\right\}$ or $\left\{{\lambda }^{\mathit{ij}}|1<i,j<n-1\right\}$ or even $\left\{{\lambda }^{\mathit{ij}}|2<i,j<n\right\}$ when $n\ge 2$.

$\square$

According to Theorem 3.2 and Example 3.6, the Einstein and thus also the Ricci operators cannot be parametrized. Now, according to Corollary 3.3, the differential module $N$ that they define is not torsion-free.

Corollary 4.4: $t\left(N\right)$ is generated by the 10 components of the Weyl tensor and each component is killed by the Dalembert operator.

Proof: We first recall the 5 steps of the double differential duality test in this framework:

$\boxed{1}\Rightarrow$ Start with the Einstein operator ${\mathcal{D}}_1:10\stackrel{\mathit{\text{Einstein}}}{\to }10$.

$\boxed{2}\Rightarrow$ Consider its formal adjoint: $\mathit{ad}\left({\mathcal{D}}_1\right):10\stackrel{\mathit{\text{Einstein}}}{\leftarrow }10$.

$\boxed{3}\Rightarrow$ Compute the generating CC, namely the Cauchy operator: $\mathit{ad}\left(\mathcal{D}\right):4\stackrel{\mathit{\text{Cauchy}}}{\leftarrow }10$.

$\boxed{4}\Rightarrow$ Consider its formal adjoint: $\mathcal{D}=\mathit{ad}\left(\mathit{ad}\left(\mathcal{D}\right)\right):4\stackrel{\mathit{\text{Killing}}}{\to }10$.

$\boxed{5}\Rightarrow$Compute the generating CC, namely the Riemann operator: ${\mathcal{D}}_1^{\prime }:10\stackrel{\mathit{\text{Riemann}}}{\to }20$.

With a slight abuse of language, we have the direct sum $\mathit{\text{Riemann}}=\mathit{\text{Ricci}}\oplus \mathit{\text{Weyl}}$ with $20=10+10$. It follows from differential homological algebra that the 10 additional CC in $\mathcal{D}{\text{'}}_1$ that are not in ${\mathcal{D}}_1$, are generating the torsion submodule $t\left(N\right)$ of the differential module $N$ defined by the Einstein or Ricci operator. If $K$ is a differential field and we consider the ring $D=K\left[d_1,\dots ,d_n\right]=K\left[d\right]$ of differential operators with coefficients in $K$, we know that ${\mathit{rk}}_D\left(\mathcal{D}\right)={\mathit{rk}}_D\left(\mathit{ad}\left(\mathcal{D}\right)\right)$ for any operator matrix $\mathcal{D}$ with coefficients in $K$. In the present situation, as the Minkowski metric has coefficients equal to $0,1,-1$, we may choose $K=\mathbb{Q}$. Hence, there must exist operators $\mathcal{P}$ and $\mathcal{Q}$ with ${\mathit{rk}}_D\left(\mathcal{P}\right)=10$ and:

$$\boxed{\mathcal{P}\circ \mathit{\text{Weyl}}=\mathcal{Q}\circ \mathit{\text{Ricci}}}$$E26

One may also notice that ${\mathit{rk}}_D\left(\mathit{\text{Einstein}}\right)={\mathit{rk}}_D\left(\mathit{\text{Ricci}}\right)$ with:

$${\mathit{rk}}_D\left(\mathit{\text{Einstein}}\right)=\frac{n\left(n+1\right)}{2}-n=\frac{n\left(n-1\right)}{2},{\mathit{rk}}_D\left(\mathit{\text{Riemann}}\right)=\frac{n\left(n+1\right)}{2}-n=\frac{n\left(n-1\right)}{2}$$

It is a pure chance that the differential ranks of the Einstein and Riemann operators are equal. Indeed, ${\mathit{rk}}_D\left(\mathit{\text{Einstein}}\right)$ has only to do with the div operator induced by contracting the Bianchi operator, while ${\mathit{rk}}_D\left(\mathit{\text{Riemann}}\right)$ has only to do with the classical Killing operator and the fact that the corresponding differential Killing module is a torsion module because we have a Lie group of transformations having $n+\frac{n\left(n-1\right)}{2}=\frac{n\left(n+1\right)}{2}$ parameters (translations + rotations). Hence, as the Riemann operator is a direct sum of the Weyl operator and the Einstein or Ricci operator according to the previous theorem, each component of the Weyl operator must be killed by a certain operator whenever the Einstein or Ricci equations in vacuum are satisfied. Equivalently, we have to prove that we obtain a torsion differential module if we set the 10 constraints $R_{\mathit{ij}}=0$ in the 20 equations $R_{\mathit{kl},\mathit{ij}}=0$. With more details, we may start from the long exact sequence:

$$\boxed{0\to \mathrm{\Theta }\to 4\underset{1}{\stackrel{\mathit{\text{Killing}}}{\to }}10\underset{2}{\stackrel{\mathit{\text{Riemann}}}{\to }}20\underset{1}{\stackrel{\mathit{\text{Bianchi}}}{\to }}20\underset{1}{\to }6\to 0}$$E27

This resolution of the set of Killing vector fields is not at all a Janet sequence because the Killing operator is not involutive as it is an operator of finite type with symbol of dimension $n\left(n-1\right)/2=6$ and one should need one prolongation for getting an involutive operator with vanishing symbol at order two. Splitting the Riemann operator, we get the commutative and exact diagram:

$$\begin{array}{cccccccccc}& & & & 0& & 0& & 0& \\ & & & & \downarrow & & \downarrow & & \downarrow & \\ & & 0& & 10& \to & 16& \to & 6& \to 0\\ & & \downarrow & & \downarrow \uparrow & & \downarrow & & \parallel & \\ 4& \stackrel{\mathit{\text{Killing}}}{\to }& 10& \stackrel{\mathit{\text{Riemann}}}{\to }& 20& \stackrel{\mathit{\text{Bianchi}}}{\to }& 20& \to & 6& \to 0\\ & & \parallel & & \downarrow \uparrow & & \downarrow & & \downarrow & \\ & & 10& \stackrel{\mathit{\text{Einstein}}}{\to }& 10& \stackrel{\mathit{div}}{\to }& 4& \to & 0& \\ & & \downarrow & & \downarrow & & \downarrow & & & \\ & & 0& & 0& & 0& & & \end{array}$$E28

Passing to the differential module point of view, we have the long exact sequence:

$$\boxed{0\to D^6\to D^{20}\stackrel{\mathit{\text{Bianchi}}}{\to }{D}^{20}\stackrel{\mathit{\text{Riemann}}}{\to }{D}^{10}\stackrel{\mathit{\text{Killing}}}{\to }{D}^4\to M\to 0}$$E29

Which is a resolution of the Killing differential module $M=\mathit{\text{coker}}\left(\mathit{\text{Killing}}\right)$_, and we have indeed the vanishing of the Euler-Poincaré characteristic with ${\mathit{rk}}_D\left(M\right)=4-10+20-20+6=0$.

Accordingly, we have $N^{\prime }=\mathit{\text{coker}}\left(\mathit{\text{Riemann}}\right)\simeq \mathit{im}\left(\mathit{\text{Killing}}\right)\subset D^4$ and thus $N^{\prime }$ is torsion-free with ${\mathit{rk}}_D\left(N^{\prime }\right)=4-0=4=n$ because ${\mathit{rk}}_D\left(M\right)=0$. The kernel $L$ of the epimorphism $N\to N^{\prime }$ is a torsion module because ${\mathit{rk}}_D\left(L\right)={\mathit{rk}}_D\left(N\right)-{\mathit{rk}}_D\left(N^{\prime }\right)=4-4=0$. As $D$ is an integral domain and $N^{\prime }\subset D^4$ is torsion-free, we have thus $L=t\left(N\right)$ in the following commutative and exact diagram where $N=\mathit{\text{coker}}\left(\mathit{\text{Einstein}}\right)$ is the differential module defined by Einstein equations in vacuum:

$$\begin{array}{rrrrrrrrrrr}& & & & & & & & & 0& \\ & & & & & & & & & \downarrow & \\ & & & 0& & 0& & 0& & t\left(N\right)& \\ & & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & D^4& \stackrel{\mathit{div}}{\to }& D^{10}& \stackrel{\mathit{\text{Einstein}}}{\to }& D^{10}& \to & N& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \parallel & & \downarrow & \\ 0\to & D^6& \to & D^{20}& \stackrel{\mathit{\text{Bianchi}}}{\to }& D^{20}& \stackrel{\mathit{\text{Riemann}}}{\to }& D^{10}& \to & N^{\prime }& \to 0\\ & \parallel & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & D^6& \to & D^{16}& \to & D^{10}& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & & & \\ & 0& & 0& & 0& & & & & \end{array}$$

A snake chase in the previous diagram allows to exhibit the long exact connecting sequence:

$$\boxed{0\to D^6\to D^{16}\to D^{10}\to t\left(N\right)\to 0}$$

We point out that, for $n=4$ (only), the CC of the Weyl operator are of order 2 and not 1 like the Bianchi CC for the Riemann operator (See [12, 18], Appendix 2 for a computer algebra checking by A. Quadrat). Accordingly, we have the conformal sequence in which $\widehat{\mathcal{D}}$ is the conformal Killing operator when $n=4$, with ${\mathit{rk}}_D\left(\widehat{M}\right)=4-9+10-9+4=0$:

$$\boxed{0\to \widehat{\mathrm{\Theta }}\to 4\underset{1}{\stackrel{\widehat{\mathcal{D}}}{\to }}9\underset{2}{\stackrel{{\widehat{\mathcal{D}}}_1}{\to }}10\underset{2}{\stackrel{{\widehat{\mathcal{D}}}_2}{\to }}9\underset{1}{\stackrel{{\widehat{\mathcal{D}}}_3}{\to }}4\to 0}$$E30

and one cannot find canonical morphisms between the classical and the conformal resolutions (!).

It follows from the last theorem that the short exact sequence $0\to D^{10}\to D^{20}\to D^{10}\to 0$ splits with $D^{20}\simeq D^{10}\oplus D^{10}$ but the existence of a canonical lift $D^{20}\to D^{10}\to 0$ in the above diagram does not allow to split the right column and thus $N\ne N^{\prime }\oplus t\left(N\right)$ as $N^{\prime }$ is not even free. As for the torsion elements, we have $t\left(N\right)=\mathit{\text{coker}}\left(D^{16}\to D^{10}\right)$ and we may thus represent them by the 10 components of the Weyl tensor. It is not at all evident that $\square$ is killing each component of the $\mathit{\text{Weyl}}$ tensor whenever $R_{\mathit{ij}}=0$. A tricky technical computation can be found in ([32], p. 206), ([33], Exercise 7.7) and ([13], p. 95).

Indeed, according to the double differential duality test, each additional CC in $\mathcal{D}{\text{'}}_1$ which is not already in ${\mathcal{D}}_1$ is a torsion element as it belongs to this module. Now, as ${\mathit{rk}}_D\left(\mathcal{D}\right)={\mathit{rk}}_D(\mathit{im}\left(\mathcal{D}\right)$ the differential ranks of the Einstein and Riemann operators are thus equal to $n\left(n-1\right)/2$, but this is a pure coincidence because ${\mathit{rk}}_D\left(\mathit{\text{Einstein}}\right)$ has only to do with the $\mathit{div}$ operator induced by contracting the Bianchi identities, while ${\mathit{rk}}_D\left(\mathit{\text{Riemann}}\right)$ has only to do with the classical Killing operator and the fact that the corresponding differential module is a torsion module because we have a Lie group of transformations having $n+\frac{n\left(n-1\right)}{2}=\frac{n(n+1}{2}$ parameters (translations + rotations). Hence, as the Riemann operator is a direct sum of the Weyl operator and the Einstein or Ricci operator according to the previous theorem, each component of the Weyl operator must be killed by a certain operator whenever the Einstein or Ricci equations in vacuum are satisfied in arbitrary dimension $n\ge 4$.

We prove directly that each component of the Weyl tensor is killed by the Dalembertian.

With Christoffel symbols $\gamma$ and the corresponding covariant derivative $\nabla$ we know that $\nabla \omega =0$ and we may thus move up and down the indices as needed. We provide this tricky computation using essentially the (second) Bianchi identities. We have successively:

$$\begin{array}{l}{\nabla }_r{\rho }_{l,\mathit{ij}}^k+{\nabla }_i{\rho }_{l,\mathit{jr}}^k+{\nabla }_j{\rho }_{l,\mathit{ri}}^k=0\Rightarrow {\nabla }^r{\rho }_{\mathit{rl},\mathit{ij}}-{\nabla }_i{\rho }_{\mathit{lj}}+{\nabla }_j{\rho }_{\mathit{li}}=0\\ \Rightarrow {\nabla }^r{\nabla }_r{\rho }_{\mathit{kl},\mathit{ij}}+{\nabla }^r{\nabla }_i{\rho }_{\mathit{kl},\mathit{jr}}+{\nabla }^r{\nabla }_j{\rho }_{\mathit{kl},\mathit{ri}}=0\\ \Rightarrow {\nabla }^r{\nabla }_r{\rho }_{\mathit{kl},\mathit{ij}}+{\nabla }_i{\nabla }^r{\rho }_{\mathit{kl},\mathit{jr}}+{\nabla }_j{\nabla }^r{\rho }_{\mathit{kl},\mathit{ri}}+\left[{\nabla }^r,{\nabla }_i\right]{\rho }_{\mathit{kl},\mathit{jr}}+\left[{\nabla }^r,{\nabla }_j\right]{\rho }_{\mathit{kl},\mathit{ri}}=0\\ \Rightarrow {\nabla }^r{\nabla }_r{\rho }_{\mathit{kl},\mathit{ij}}+{\nabla }_i{\nabla }^r{\rho }_{\mathit{kl},\mathit{jr}}+{\nabla }_j{\nabla }^r{\rho }_{\mathit{kl},\mathit{ri}}+\left(\sum \mathit{\text{quadratic}}\right)=0\end{array}$$

But we have also ${\rho }_{\mathit{kl},\mathit{ij}}={\rho }_{\mathit{ij},\mathit{kl}}$ and thus ${\nabla }^r{\rho }_{\mathit{ij},\mathit{rl}}={\nabla }_i{\rho }_{\mathit{lj}}-{\nabla }_j{\rho }_{\mathit{li}}\Rightarrow {\nabla }^r{\rho }_{\mathit{jr}}-\frac{1}{2}{\nabla }_j\rho =0$.

$$\square {\rho }_{\mathit{kl},\mathit{ij}}=\left({\nabla }_i\left({\nabla }_k{\rho }_{\mathit{lj}}-{\nabla }_l{\rho }_{\mathit{kj}}\right)\right)-\left(i\leftrightarrow j\right)+\left(\sum \mathit{\text{quadratic}}\right)$$

Linearizing at the Euclidean metric for $n=2,3$ or at the Minkowski metric for $n=4$, we get:

$$\square R_{\mathit{kl},\mathit{ij}}=d_i\left(d_k{R}_{\mathit{lj}}-d_l{R}_{\mathit{kj}}\right)-d_j\left(d_k{R}_{\mathit{li}}-d_l{R}_{\mathit{ki}}\right)$$E31

We may finally use the splitting formula for defining the Weyl tensor ${\sigma }_{l,\mathit{ij}}^k$ with ${\sigma }_{l,\mathit{rj}}^r=0$, namely:

$$\begin{array}{c}{\sigma }_{l,\mathit{ij}}^k={\rho }_{l,\mathit{ij}}^k-\frac{1}{\left(n-2\right)}\left({\delta }_i^k{\rho }_{\mathit{lj}}-{\delta }_j^k{\rho }_{\mathit{li}}+{\omega }^{\mathit{ks}}\left({\omega }_{\mathit{lj}}{\rho }_{\mathit{si}}-{\omega }_{\mathit{li}}{\rho }_{\mathit{sj}}\right)\right)+\frac{1}{\left(n-1\right)\left(n-2\right)}\left({\delta }_i^k{\omega }_{\mathit{lj}}-{\delta }_j^k{\omega }_{\mathit{li}}\right)\rho \\ {\sigma }_{l,\mathit{ij}}^k={\rho }_{l,\mathit{ij}}^k-\left(\sum {\rho }_{\mathit{rs}}\right)\Rightarrow {\mathrm{\Sigma }}_{l,\mathit{ij}}^k=R_{l,\mathit{ij}}^k-\left(\sum R_{\mathit{rs}}\right)\end{array}$$

At any moment, we could have used the Ricci operator in place of the Einstein operator.

$\square$

Finally, for the sake of completeness, we compute directly the characters of the Einstein system.

Using a direct checking with the ordering $11<12<13<14<22<\dots <34<44$, we obtain:

$$E_{33}={\omega }^{44}{d}_{44}{\mathrm{\Omega }}_{33}+\mathit{\text{lower\hspace{0.17em}\hspace{0.17em}terms}},E_{23}={\omega }^{44}{d}_{44}{\mathrm{\Omega }}_{23}+\dots$$

We are in the position to compute the characters of the Einstein operator but a similar procedure could be followed with the Ricci operator. We obtain at once:

$${\beta }_2^4=6,{\beta }_2^3=4,{\beta }_2^2=0,{\beta }_2^1=0\iff {\alpha }_2^4=4,{\alpha }_2^3=16,{\alpha }_2^3=30,{\alpha }_2^1=40$$

a result leading to $\mathit{dim}\left(g_2\right)={\alpha }_2^1+{\alpha }_2^2+{\alpha }_2^3+{\alpha }_2^4=90$ and $\mathit{dim}\left(g_3\right)={\alpha }_2^1+2{\alpha }_2^2+3{\alpha }_2^3+4{\alpha }_2^4=164$ in a coherent way with the long exact sequences:

$$0\to g_2\to S_2{T}^{\ast }\otimes S_2{T}^{\ast }\to S_2{T}^{\ast }\to 0,0\to g_3\to S_3{T}^{\ast }\otimes S_2{T}^{\ast }\to T^{\ast }\otimes S_2{T}^{\ast }\to T^{\ast }\to 0$$

Now, we have by definition $\mathit{div}=\left(d_1,d_2,d_3,d_4\right)$ and $\mathit{div}\circ \mathit{\text{Einstein}}=\left(\mathrm{0,0,0,0}\right)$. However, the $\mathit{\text{Einstein}}$ operator is a $10\times 10$ operator matrix which is self-adjoint up to a change of basis [2] because it is made with homogeneous second order terms. It is thus of rank 6 and we obtain therefore, with a slight abuse of language, $\mathit{det}\left(\mathit{\text{Einstein}}\right)=0$. This result which is not evident at first sight must be compared with the Poincaré situation when $n=3$:

$$\left(d_1{d}_2{d}_3\right)\left(\begin{array}{ccc}0& -d_3& d_2\\ d_3& 0& -d_1\\ -d_2& d_1& 0\end{array}\right)=\left(000\right)$$

This module interpretation may thus question the proper origin and existence of gravitational waves because the div operator on the upper left part of the diagram has strictly nothing to do with the $\mathit{\text{Cauchy}}=\mathit{ad}\left(\mathit{\text{Killing}}\right)$ operator which cannot even appear anywhere in this diagram. Also, looking back to Example 3.3, if we use ad(Einstein) or ad(Ricci) in order to parametrize the Cauchy operator, the 10 potentials involved are similar to the Airy or Maxwell/Morera potentials and have thus strictly nothing to do with a perturbation of the metric. Such a result is explaining the conceptual confusion announced in the abstract.

Corollary 4.5: When $\mathcal{D}$ is a Lie operator of finite type, that is when $\left[\mathrm{\Theta },\mathrm{\Theta }\right]\subset \mathrm{\Theta }$ under the ordinary bracket of vector fields and $g_{q+r}=0$ for $r$ large enough, then the Spencer sequence is locally isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by a finite dimensional Lie algebra $\mathcal{G}$, namely [29]:

$$0\to \mathrm{\Theta }\to {\wedge }^0{T}^{\ast }\otimes \mathcal{G}\stackrel{d}{\to }{\wedge }^1{T}^{\ast }\otimes \widehat{A}\mathcal{G}\stackrel{d}{\to }{\wedge }^2{T}^{\ast }\otimes \mathcal{G}\stackrel{d}{\to }\dots \stackrel{d}{\to }{\wedge }^n{T}^{\ast }\otimes \mathcal{G}\to 0.$$E32

It is thus formally exact both with its adjoint sequence. As it is known that the extension modules do not depend on the resolution used, this is the reason for which not only the Cauchy stress operator can be parametrized but also the Cosserat couple-stress operator $\mathit{ad}\left(D_1\right)$ can be parametrized by $\mathit{ad}\left(D_2\right)$, a result not evident at all (see [34, 35] for explicit computations). Similarly, in the case of the conformal Killing system ${\widehat{R}}_1\subset J_1\left(T\right)$ for $n=4$, the symbols do not depend on any conformal factor because ${\widehat{g}}_1$ is defined by ${\omega }_{\mathit{rj}}{\xi }_i^r+{\omega }_{\mathit{ir}}{\xi }_j^r-\frac{2}{n}{\omega }_{\mathit{ij}}{\xi }_r^r=0$, ${\widehat{g}}_2\simeq T^{\ast }$ (the so-called elations of E. Cartan) is defined by ${\xi }_{\mathit{ij}}^k-\frac{1}{n}({\delta }_i^k{\xi }_{\mathit{rj}}^r+{\delta }_j^k{\xi }_{\mathit{ri}}^r-{\omega }_{\mathit{ij}}{\omega }^{\mathit{ks}}{\xi }_{\mathit{rs}}^r)=0$ (with the Kronecker notation) and finally ${\widehat{g}}_3=0,\forall n\ge 3$. The EM field is thus a section $F\in \delta \left(T^{\ast }\otimes {\widehat{g}}_2\right)=\delta \left(T^{\ast }\otimes T^{\ast }\right)={\wedge }^2{T}^{\ast }$ killed by the exterior derivative $d$. It follows that EM only depends on the conformal group and not on $U\left(1\right)$ [30, 31] in a coherent way with the dream of H. Weyl because $T^{\ast }\otimes T^{\ast }\simeq S_2{T}^{\ast }\oplus {\wedge }^2{T}^{\ast }=\left(R_{\mathit{ij}}\right)\oplus \left(F_{\mathit{ij}}\right)$ [29, 41].

Remark 4.6: A similar situation is well known for the Cauchy- Riemann equations when $n=2$. Indeed, any infinitesimal complex transformation $\xi$ must be solution of the linear first order homogeneous system ${\xi }_2^2-{\xi }_1^1=0,{\xi }_2^1+{\xi }_1^2=0$ of infinitesimal Lie equations which is defining a torsion differential module because we have ${\xi }_{11}^1+{\xi }_{22}^1=0,{\xi }_{11}^2+{\xi }_{22}^2=0$, that is ${\xi }^1$ and ${\xi }^2$ are separately killed by the Laplace operator $\mathrm{\Delta }=d_{11}+d_{22}$.

Remark 4.7: A similar situation is also well known for the EM field $F$ in electromagnetism. Indeed, starting with the first set of Maxwell equations $\mathit{dF}=0$ ($M_1$) and using the Minkowski constitutive law in vacuum with electric constant ${\epsilon }_0$ and magnetic constant ${\mu }_0$_, such that ${\epsilon }_0{\mu }_0{c}^2=1$ for the second set of Maxwell equations ($M_2$) for the induction $\mathcal{F}$, a standard tricky differential elimination allows to avoid the Lorenz (no “t”) gauge condition for the EM potential. Indeed, from $M_1$ we get formally $d_r{F}_{\mathit{ij}}+d_i{F}_{\mathit{jr}}+d_j{F}_{\mathit{ri}}=0$ and from $M_2$ we get $d_r{\mathcal{F}}^{\mathit{ri}}=0\Rightarrow d^r{F}_{\mathit{ri}}=0$. We finally obtain directly $\square F=d^r{d}_r{F}_{\mathit{ij}}=d^r\left(d_i{F}_{\mathit{rj}}-d_j{F}_{\mathit{ri}}\right)=0$ [4].

Example 4.8: (Lorenz condition for EM)

We prove that the Lorenz gauge condition for EM is just amounting to a relative minimum parametrization. Using an Euclidean metric instead of the Minkowski metric in order to have the Minkowski constitutive relation $\mathcal{F}=F$ in vacuum between the EM field and the EM induction, we have the parametrization $d_i{A}_j-d_j{A}_i=F_{\mathit{ij}}$ and obtain the conservation of current through the composition:

$$d_i\left(d^i{A}^j-d^j{A}^i\right)=d_i{d}^i{A}^j-d^j\left(d_i{A}^i\right)={\mathcal{J}}^j\Rightarrow d_j\left(d_i{d}^i{A}^j-d^j{d}_i{A}^i\right)=d_j{\mathcal{J}}^j=0$$

with implicit summation on $i$ and $j$. The differential module defined by the involutive system $d_{\mathit{ii}}{A}_j-d_{\mathit{ij}}{A}_i=0$ has a differential rank equal to 1 as there is only one CC. Adding the Lorenz condition $d_i{A}^i=0$ is bringing the rank to zero. The idea is just to prove that the inhomogeneous system $d_i{d}^i{A}^j={\mathcal{J}}^j,d_i{A}^i=0$ has again the only CC $d_j{\mathcal{J}}^j=0$ like in Example 1.3.

5. Conclusion

After teaching elasticity theory during 25 years to students in some of the best french civil engineering schools, the author of this paper still keeps in mind one of the most fascinating but most difficult exercises that he has set up. The purpose was to explain why the dome of the cathedral St Paul in London is called “whispering cupola”, that is why if you go up to the gallery at one point, you can listen to a friend whispering 30 meters away on the opposite side. This striking phenomena has been first studied by Lord Rayleigh in 1878 and he introduced in 1910 the “surface elastic waves” now called Rayleigh waves (See [3] p. 199). In fact the molecules close to the free surface are moving along small ellipses as a combination of standard longitudinal (L) and transverse (T) waves while propagating at a slightly smaller speed. Such waves cannot propagate in liquid but may travel all around the earth surface many times after earthquakes. If $\lambda$ and $\mu$ are the elastic Lamé constants of the material with mass $\rho$ per unit volume, the respective speeds are such that $c_L^2=\left(\lambda +2\mu \right)/\rho ,c_T^2=\mu /\rho$ and the speed of the Rayleigh wave is $c_R^2=\chi \left(1-2\nu \right)/\left(2\left(1-\nu \right)\right)$ where $\nu =\lambda /\left(2\left(\lambda +\mu \right)\right)$ is the Poisson coefficient. After (very) tedious computations one may find that $\chi$ must be the only real root of the cubic equation ${\chi }^3-8{\chi }^2+8\frac{2-\nu }{1-\nu }\chi -\frac{8}{1-\nu }=0$ existing in the interval [0, 1] because $0<\nu <0,5$.

Accordingly, elasticity can be considered as a way to parametrize the Cauchy operator when $n=3$ while GR can be considered as a way to parametrize the Cauchy operator when $n=4$. Hence, the situations existing with the Cauchy stress equations, with the Cosserat couple-stress equations and with the Maxwell equations are similar, only the constitutive laws are different. Meanwhile, we have shown why the mathematical foundations of conformal geometry must be revisited in this new framework which is valid in arbitrary dimension and could provide an intrinsic way to unify EM and GR along the dream of Weyl [12, 31].

The situation of the gravitational waves equations seems quite different but the least that can be said is that it is not coherent with differential double duality. However, it follows that exactly the same confusion has been done by Maxwell, Morera, Beltrami, and Einstein because, in all these cases, the operator considered is self-adjoint. Like the Michelson and Morley experiment, we do believe that Einstein already knew the previous works of all these researchers who were quite famous at the time he was active. In any case, the comparison of the various parametrizations described in this paper needs no comment.