1. Introduction
The purpose of this paper is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of ordinary differential (OD) or partial differential (PD) equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity (GR). More elementary engineering examples (elasticity theory, electromagnetism (EM)) will also be considered in order to illustrate the quoted three fundamental results that we shall provide. The paper, based on the material of two lectures given at the department of mathematics of the university of Montpellier 2, France, in may 2013 and Firenze, Italy, in june 2013, is divided into three parts corresponding to the different formal methods used.
PART 1: In 1880 S. Lie (1842-1899) studied the groups of transformations depending on a finite number of parameters and now called Lie groups of transformations. Ten years later he discovered that these groups are only examples of groups of transformations solutions of linear or nonlinear systems of ordinary differential (OD) or partial differential (PD) equations which may even be of high order and are now called Lie pseudogroups of transformations. During the next fifty years the latter groups have only been studied by two frenchmen, namely Elie Cartan (1869-1951) who is quite famous today, and Ernest Vessiot (1865-1952) who is almost ignored today. We have proved in many books and papers that the Cartan structure equations have nothing to do with the Vessiot structure equations still not known today. Accordingly, the quadratic terms appearing in the Riemann tensor must not be identified with the quadratic terms appearing in the well known Maurer-Cartan equations for Lie groups. In particular, curvature + torsion (Cartan) must not be considered as a generalization of curvature alone (Vessiot).
PART 2: Though we consider that the first formal work on systems of PD equations is dating back to Maurice Janet (1888-1983) who introduced as early as in 1920 a differential sequence now called Janet sequence, it is only around 1970 that Donald Spencer (1912-2001) developped, in a quite independent way, the formal theory of systems of PD equations in order to study Lie pseudogroups, exactly like E. Cartan did with exterior systems. However, all the physicists who tried to understand the only book “Lie equations” that he published in 1972 with A. Kumpera, have been stopped by the fact that the examples of the Introduction (Janet sequence) have nothing to do with the core of the book (Spencer sequence). We may say that the work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is not known today. Accordingly, the Spencer sequence for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but general relativity (GR) is not coherent with this result because we shall prove that the Ricci tensor only depends on the nonlinear transformations (called elations by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters).
PART 3: At the same time, mixing differential geometry and homological algebra but always supposing that the reader knows a lot about the work of Spencer, V.P. Palamodov (constant coefficients) and M. Kashiwara (variable coefficients) developped “algebraic analysis” in order to study the formal properties of finitely generated differential modules that do not depend on their presentation or even on a corresponding differential resolution, namely the algebraic analogue of a differential sequence. Using double duality theory, we prove that, contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be “parametrized”, that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970.
The new methods involve tools from differential geometry (jet theory, Spencer operator,
-cohomology) and homological algebra (diagram chasing, snake theorem, extension modules, double duality). The reader may just have a look to the book [1] (review in Zbl 1079. 93001) in order to understand the amount of mathematics needed from many domains.
The following diagram summarizes at the same time the historical background and the difficulties presented in the introduction:
![](https://www.scirp.org/html/22-7501431\06adbac0-0c79-4926-a395-8a3a698f9893.jpg)
Roughly, Cartan and followers have not been able to “quotient down to the base manifold” [2,3], a result only obtained by Spencer in 1970 through the nonlinear Spencer sequence [4-7] but in a way quite different from the one followed by Vessiot in 1903 for the same purpose [8,9]. Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework, though striking it may look like for certain apparently well established theories such as EM (J. C. Maxwell, 1864) and GR (A. Einstein, 1915).
2. First Part: From Lie Groups to Lie Pseudogroups
If
is a manifold with local coordinates
for
, let
be a fibered manifold over
, that is a manifold with local coordinates
for
and
simply denoted by
, projection
![](https://www.scirp.org/html/22-7501431\fe705f0a-3fb6-4c8d-a755-c9696fbaac7d.jpg)
and changes of local coordinates
.
If
and
are two fibered manifolds over
with respective local coordinates
and
, we denote by
the fibered product of
and
over
as the new fibered manifold over
with local coordinates
. We denote by
![](https://www.scirp.org/html/22-7501431\70ab0bdc-42aa-48f0-80f8-0d6f38302d4c.jpg)
a global section of
, that is a map such that
but local sections over an open set
may also be considered when needed. We shall use for simplicity the same notation for a fibered manifold and its set of sections while setting
. Under a change of coordinates, a section transforms like
![](https://www.scirp.org/html/22-7501431\52c09bab-1660-4e33-bb0a-54aaf07e2835.jpg)
and the derivatives transform like:
.
We may introduce new coordinates
transforming like:
.
We shall denote by
the q-jet bundle of ![](https://www.scirp.org/html/22-7501431\321f2c02-a8c7-4fd1-b01a-7f2cb617a5cf.jpg)
with local coordinates
![](https://www.scirp.org/html/22-7501431\21d16b1e-aff2-4a82-9cee-33077e60312c.jpg)
called jet coordinates and sections
![](https://www.scirp.org/html/22-7501431\013ccb8e-005d-4a6b-a82c-cd3eec56be44.jpg)
transforming like the sections
![](https://www.scirp.org/html/22-7501431\41b00823-679d-4884-aa33-d7b02d8f69c2.jpg)
where both
and
are over the section
of
. Of course
is a fibered manifold over ![](https://www.scirp.org/html/22-7501431\a4fdeb16-57c2-45b1-b354-3a90576edf1c.jpg)
with projection
while
is a fibered manifold over
with projection
.
DEFINITION 1.1: A (nonlinear) system of order
on
is a fibered submanifold
and a solution of
is a section
of
such that
is a section of
.
DEFINITION 1.2: When the changes of coordinates have the linear form
, we say that
is a vector bundle over
. Vector bundles will be denoted by capital letters
and will have sections denoted by
. In particular, we shall denote as usual by
the tangent bundle of
, by
the cotangent bundle, by
the bundle of r-forms and by
the bundle of qsymmetric covariant tensors. When the changes of coordinates have the form
![](https://www.scirp.org/html/22-7501431\fe161ed9-e409-4c30-8d1c-6605b2c49a13.jpg)
we say that
is an affine bundle over
and we define the associated vector bundle
over
by the local coordinates
changing like ![](https://www.scirp.org/html/22-7501431\49ee84f6-5947-485a-b70e-1432639dcff2.jpg)
. Finally, If
, we shall denote by
the open subfibered manifold of
defined independently of the coordinate system by
with source projection
![](https://www.scirp.org/html/22-7501431\adf5e3f2-397c-4776-ac06-bad703651b0c.jpg)
and target projection
.
DEFINITION 1.3: If the tangent bundle
has local coordinates
changing like
we may introduce the vertical bundle
as a vector bundle over
with local coordinates
obtained by setting
and changes
.
Of course, when
is an affine bundle over
with associated vector bundle
over
, we have
.
For a later use, if
is a fibered manifold over
and
is a section of
, we denote by
the reciprocal image of
by
as the vector bundle over
obtained when replacing
by
in each chart. A similar construction may also be done for any affine bundle over
.
We now recall a few basic geometric concepts that will be constantly used through this paper. First of all, if
, we define their bracket
by the local formula
![](https://www.scirp.org/html/22-7501431\9ae112e6-0591-4898-bde8-cbb511107261.jpg)
leading to the Jacobi identity
![](https://www.scirp.org/html/22-7501431\9acf16ef-9c72-4ec2-9702-6d0fd4685992.jpg)
allowing to define a Lie algebra and to the useful formula
![](https://www.scirp.org/html/22-7501431\9ab0f937-6a87-4dc5-a279-556287fee3ef.jpg)
where
is the tangent mapping of a map
.
When
is a multi-index, we may set
![](https://www.scirp.org/html/22-7501431\0928bdd0-b4d5-4ac4-80d9-947fb83b5e96.jpg)
for describing
by means of a basis and introduce the exterior derivative
![](https://www.scirp.org/html/22-7501431\bc721b17-4beb-44d3-a735-0475e4156221.jpg)
with
in the Poincaré sequence:
.
The Lie derivative of an
-form with respect to a vector field
is the linear first order operator
linearly depending on
and uniquely defined by the following three properties:
1)
.
2)
.
3) ![](https://www.scirp.org/html/22-7501431\f314159b-4859-4698-9de8-57efa758ccb4.jpg)
.
It can be proved that
![](https://www.scirp.org/html/22-7501431\ab4a0238-507d-42b7-8c19-f3db04d8f95f.jpg)
where
is the interior multiplication
![](https://www.scirp.org/html/22-7501431\be97cc24-c04b-478e-93c1-ceca1c044afa.jpg)
and that
![](https://www.scirp.org/html/22-7501431\c470ce37-7bef-45eb-b50c-6f9b2239cdae.jpg)
We now turn to group theory and start with two basic definitions:
Let
be a Lie group, that is a manifold with local coordinates
for
called parameters, a composition
an inverse
and an identity
satisfying:
![](https://www.scirp.org/html/22-7501431\2f74bd4e-dd42-447e-866f-e4c66019ac0c.jpg)
DEFINITION 1.4:
is said to act on
if there is a map
such that
and we shall say that we have a Lie group of transformations of
. In order to simplify the notations, we shall use global notations even if only local actions are existing. It is well known that the action of
onto itself allows to introduce a purely algebraic bracket on its Lie algebra
.
DEFINITION 1.5: A Lie pseudogroup of transformations
is a group of transformations solutions of a system of OD or PD equations such that, if
and
are two solutions, called finite transformations, that can be composed, then
![](https://www.scirp.org/html/22-7501431\49750b3d-2d62-4efe-a7a5-14b9961a5ca1.jpg)
and
![](https://www.scirp.org/html/22-7501431\ad02fc54-6b93-444d-b115-08f844101c62.jpg)
are also solutions while
is the identity solution denoted by
and we shall set
. In all the sequel we shall suppose that
is transitive that is
.
It becomes clear that Lie groups of transformations are particular cases of Lie pseudogroups of transformations as the system defining the finite transformations can be obtained by eliminating the parameters among the equations
![](https://www.scirp.org/html/22-7501431\7173f383-13c9-41a9-b43f-4f2190ca9271.jpg)
when
is large enough. The underlying system may be nonlinear and of high order. Looking for transformations “close” to the identity, that is setting
when
is a small constant parameter and passing to the limit
, we may linearize the above nonlinear system of finite Lie equations in order to obtain a linear system of infinitesimal Lie equations of the same order for vector fields. Such a system has the property that, if
are two solutions, then
is also a solution. Accordingly, the set
of solutions of this new system satisfies
and can therefore be considered as the Lie algebra of
.
EXAMPLE 1.6: While the affine transformations
are solutions of the second order linear system
, the projective transformations
![](https://www.scirp.org/html/22-7501431\9f76756b-3c51-49fb-b679-8a67b7cbf034.jpg)
are solutions of the third order nonlinear system
.
The sections of the corresponding linearized systems are respectively satisfying
and
. The generating differential invariant
of the affine case transforms like
![](https://www.scirp.org/html/22-7501431\db8730a6-ad79-42e2-ae51-deb6a3279ccd.jpg)
when
while
transforms like
.
We now sketch the discovery of Vessiot [8,9] still not known today after more than a century for reasons which are not scientific at all. Roughly, a Lie pseudogroup
is made by finite transformations
solutions of a (possibly nonlinear) system
while the infinitesimal transformations
are solutions of the linearized system
![](https://www.scirp.org/html/22-7501431\3e41580a-51e3-4def-aa22-3f70be1d9402.jpg)
as we have
.
When
is transitive, there is a canonical epimorphism
. Also, as changes of source
commute with changes of target
, they exchange between themselves any generating set of differential invariants
as in the previous example.Then one can introduce a natural bundle
over
, also called bundle of geomeric objects, by patching changes of coordinates of the form
![](https://www.scirp.org/html/22-7501431\85f91194-63be-4fa3-9971-164dd3eace22.jpg)
thus obtained. A section
of
is called a geometric object or structure on
and transforms like
![](https://www.scirp.org/html/22-7501431\7f7f9ff4-8bd7-42c2-b125-9d319d0dfc6f.jpg)
or simply
. This is a way to generalize vectors and tensors
or even connections
. As a byproduct we have
![](https://www.scirp.org/html/22-7501431\c789e758-5fdf-49ce-b571-f6366cbb95f1.jpg)
and we may say that
preserves
. Replacing
by
, we also obtain
.
Coming back to the infinitesimal point of view and setting
we may define the ordinary Lie derivative with value in the vector bundle
by the formula:
![](https://www.scirp.org/html/22-7501431\cd5980e0-0762-4559-a5af-31f73adf53ce.jpg)
and we say that
is a Lie operator because
as we already saw.
Differentiating
times the equations of
that only depend on
, we may obtain the
- prolongation
.
The problem is then to know under what conditions on
all the equations of order
are obtained by
prolongations only,
or, equivalently,
is formally integrable (FI). The solution, found by Vessiot, has been to exhibit another natural vector bundle
with local coordinates
over
with local coordinates
and to prove that an equivariant section
only depends on a finite number of constants called structure constants. The integrability conditions (IC) of
, called Vessiot structure equations, are of the form
and are invariant under any change of source.
We provide in a self-contained way and parallel manners the following five striking examples which are among the best nontrivial ones we know and invite the reader to imagine at this stage any possible link that could exist between them (A few specific definitions will be given later on).
EXAMPLE 1.7: Coming back to the last example, we show that Vessiot structure equations may even exist when
. For this, if
is the geometric object of the affine group
and
is a
-form, we consider the object
and get at once the two second order Medolaghi equations:
![](https://www.scirp.org/html/22-7501431\4e28a83a-6d4c-4f8e-83e5-0b0e2be15121.jpg)
Differentiating the first equation and substituting the second, we get the zero order equation:
![](https://www.scirp.org/html/22-7501431\deae9aa0-ca98-447d-9b80-310bd8e93253.jpg)
and the Vessiot structure equation
. Alternatively, setting
we get
.
With
![](https://www.scirp.org/html/22-7501431\0ee2954a-b9e3-43a0-8d97-80e3a0f461bc.jpg)
we get the translation subgroup
while, with
![](https://www.scirp.org/html/22-7501431\8dfef4fb-d630-43dc-94e1-0208c1180a52.jpg)
we get the dilatation subgroup
. Similarly, if
is the geometric object of the projective group and we consider the new geometric object
, we get the only Vessiot structure equation
![](https://www.scirp.org/html/22-7501431\93c9a192-b8a1-4a5a-a935-8e942f5f3f82.jpg)
without any structure constant.
EXAMPLE 1.8: (Principal homogeneous structure) When
is the Lie group of transformations made by the constant translations
for
of a manifold
with
, the characteristic object invariant by
is a family
![](https://www.scirp.org/html/22-7501431\dc67d97a-932d-45b7-b271-8de17f2100cf.jpg)
of
-forms with
in such a way that
![](https://www.scirp.org/html/22-7501431\85cd8f8d-f8c1-4620-996d-aa7980b4fe7a.jpg)
where
denotes the pseudogroup of local diffeomorphisms of
,
denotes the derivatives of
up to order
and
acts in the usual way on covariant tensors. For any vector field
the tangent bundle to
, introducing the standard Lie derivative
of forms with respect to
, we may consider the
first order Medolaghi equations:
.
The particular situation is found with the special choice
that leads to the involutive system
. Introducing the inverse matrix
, the above equations amount to the bracket relations
and, using crossed derivatives on the solved form
we obtain the
zero order equations:
.
The integrability conditions (IC), that is the conditions under which these equations do not bring new equations, are thus the
Vessiot structure equations:
![](https://www.scirp.org/html/22-7501431\27fb0d9e-d9db-49f0-9118-0ba0fa508173.jpg)
with
structure constants
.
When
, these equations can be identified with the Maurer-Cartan equations (MC) existing in the theory of Lie groups, on the condition to change the sign of the structure constants involved because we have
.
Writing these equations in the form of the exterior system
and closing this system by applying once more the exterior derivative
, we obtain the quadratic IC:
![](https://www.scirp.org/html/22-7501431\eebdacd8-6a41-45fb-ac9e-09690bc521fc.jpg)
also called Jacobi relations
.
EXAMPLE 1.9: (Riemann structure) If
![](https://www.scirp.org/html/22-7501431\8ce7d457-6632-4759-9cd4-db4acd509183.jpg)
is a metric on a manifold
with
such that
, the Lie pseudogroup of transformations preserving
is
![](https://www.scirp.org/html/22-7501431\56c1ac51-1542-48da-9a0d-ebab7168b7e3.jpg)
and is a Lie group with a maximum number of
parameters. A special metric could be the Euclidean metric when
as in elasticity theory or the Minkowski metric when
as in special relativity [10]. The first order Medolaghi equations:
![](https://www.scirp.org/html/22-7501431\d5813bf4-a064-4a40-90e3-6919f27e4a46.jpg)
are also called classical Killing equations for historical reasons. The main problem is that this system is not involutive unless we prolong it to order two by differentiating once the equations. For such a purpose, introducing
as usual, we may define the Christoffel symbols:
![](https://www.scirp.org/html/22-7501431\ea8528ca-8b12-4493-95fb-7ce251e1c911.jpg)
This is a new geometric object of order 2 providing the Levi-Civita isomorphism
of affine bundles and allowing to obtain the second order Medolaghi equations:
![](https://www.scirp.org/html/22-7501431\3debb572-2b34-4ce5-8aad-404d3aec4999.jpg)
Surprisingly, the following expression called Riemann tensor:
![](https://www.scirp.org/html/22-7501431\36993772-9b54-42bb-9299-81d4656ab4e3.jpg)
is still a first order geometric object and even a
-tensor with
independent components satisfying the purely algebraic relations:
.
Accordingly, the IC must express that the new first order equations
are only linear combinations of the previous ones and we get the Vessiot structure equations:
![](https://www.scirp.org/html/22-7501431\039a868f-afaf-463e-b5fa-a334232fb61c.jpg)
with the only structure constant
describing the constant Riemannian curvature condition of Eisenhart [11,12]. One can proceed similarly for the conformal Killing system
and obtain that the Weyl tensor must vanish, without any structure constant [12].
EXAMPLE 1.10: (Contact structure) We only treat the case
as the case
needs much more work [6]. Let us consider the so-called contact
-form
and consider the Lie pseudogroup
of (local) transformations preserving
up to a function factor, that is
![](https://www.scirp.org/html/22-7501431\bf779334-1c94-45dc-97d7-c9bdb251b7d7.jpg)
where again
is a symbolic way for writing out the derivatives of
up to order
and
transforms like a
-covariant tensor. It may be tempting to look for a kind of “object” the invariance of which should characterize
. Introducing the exterior derivative
as a
-form, we obtain the volume
-form
. As it is well known that the exterior derivative commutes with any diffeomorphism, we obtain sucessively:
![](https://www.scirp.org/html/22-7501431\1e674c5f-eead-4913-9c8a-694763f94f61.jpg)
As the volume
-form
transforms through a division by the Jacobian determinant
![](https://www.scirp.org/html/22-7501431\d49369eb-73a8-48cd-9034-285466dd55d9.jpg)
of the transformation
with inverse
the desired object is thus no longer a 1-form but a 1- form density
transforming like a 1- form but up to a division by the square root of the Jacobian determinant. It follows that the infinitesimal contact transformations are vector fields
the tangent bundle of
, satisfying the 3 so-called first order Medolaghi equations:
.
When
, we obtain the special involutive system:
![](https://www.scirp.org/html/22-7501431\92db01b6-b092-4c5f-8087-0fce7130e276.jpg)
with 2 equations of class 3 and 1 equation of class 2 (see later on for a precise definition) and thus only 1 compatibility conditions (CC) for the second members.
For an arbitrary
, we may ask about the differential conditions on
such that all the equations of order
are only obtained by differentiating
times the first order equations, exactly like in the special situation just considered where the system is involutive. We notice that, in a symbolic way,
is now a scalar
providing the zero order equation
and the condition is
. The integrability condition (IC) is the Vessiot structure equation:
![](https://www.scirp.org/html/22-7501431\567a0818-90f4-4c48-84c5-6c85aa107d65.jpg)
involving the only structure constant
.
For
, we get
. If we choose
leading to
, we may define
![](https://www.scirp.org/html/22-7501431\64a70e96-6ce1-4024-9e79-769d137d68fb.jpg)
with infinitesimal transformations satisfying the involutive system:
![](https://www.scirp.org/html/22-7501431\63647062-baff-4e24-ad71-f49503b5baa3.jpg)
with again 2 equations of class 3 and 1 equation of class 2.
EXAMPLE 1.11: (Unimodular contact structure) With similar notations, let us again set
![](https://www.scirp.org/html/22-7501431\6235701e-33c7-468f-ac8d-9e3bbbcfa1c5.jpg)
but let us now consider the new Lie pseudogroup of transformations preserving
and thus
too, that is preserving the mixed object
![](https://www.scirp.org/html/22-7501431\682f8b2d-9d65-40e7-b816-e0423777658f.jpg)
made up by a
-form
and a
-form
with
and
.
Then
is a Lie subpseudogroup of the one just considered in the previous example and the corresponding infinitesimal transformations now satisfy the involutive system:
![](https://www.scirp.org/html/22-7501431\76f18f0e-4dbd-481e-8633-bec4d9c6b8b1.jpg)
with 3 equations of class 3, 2 equations of class 2 and
equation of class 1 if we exchange
with
, a result leading now to 4 CC.
More generally, when
where
is a 1- form and
is a
-form satifying
, we may study the same problem as before for the general system
.
We let the reader provide the details of the tedious computation involved as it is at this point that computer algebra may be used [13]. The result, not evident at first sight, is that the 2-form
must be proportional to the 2-form
, that is
and thus
.
As
, we must have
and thus
. Similarly, we get
and obtain finally the
Vessiot structure equations
![](https://www.scirp.org/html/22-7501431\a21e7671-befe-46fb-b8f2-96d0ad01768b.jpg)
involving 2 structure constants
. Contrary to the previous situation (but like in the Riemann case!) we notice that we have now 2 structure equations not containing any constant (called first kind by Vessiot) and 2 structure equations with the same number of different constants (called second kind by Vessiot), namely
.
Finally, closing this system by taking once more the exterior derivative, we get
![](https://www.scirp.org/html/22-7501431\7e7dd6ab-6179-4565-b33e-9fa4b09b32e9.jpg)
and thus the unexpected purely algebraic Jacobi condition
. For the special choice
![](https://www.scirp.org/html/22-7501431\88e6ee59-3d8b-4953-b2db-74598d0e37b9.jpg)
we get
, for the second special choice
![](https://www.scirp.org/html/22-7501431\b74b91df-bd63-4700-9664-8deeb6de30d0.jpg)
we get
and for the third special choice
![](https://www.scirp.org/html/22-7501431\daa156d4-48b8-4338-b0ff-163e5a41ec0f.jpg)
we get
.
FIRST FUNDAMENTAL RESULT: Comparing the various Vessiot structure equations containing structure constants that we have just presented and that we recall below in a symbolic way, we notice that these structure constants are absolutely on equal footing though they have in general nothing to do with any Lie algebra.
![](https://www.scirp.org/html/22-7501431\2a5e3495-5e20-4b21-bc0f-e2e17c7940aa.jpg)
![](https://www.scirp.org/html/22-7501431\91606c79-4107-4a7a-8eba-6b315e24e5ef.jpg)
.
Accordingly, the fact that the ones appearing in the MC equations are related to a Lie algebra is a coincidence and the Cartan structure equations have nothing to do with the Vessiot structure equations. Also, as their factors are either constant, linear or quadratic, any identification of the quadratic terms appearing in the Riemann tensor with the quadratic terms appearing in the MC equations is definitively not correct [7]. We also understand why the torsion is automatically combined with curvature in the Cartan structure equations but totally absent from the Vessiot structure equations, even though the underlying group (translations + rotations) is the same.
HISTORICAL REMARK 1.12: Despite the prophetic comments of the italian mathematician Ugo Amaldi in 1909 [12], it has been a pity that Cartan deliberately ignored the work of Vessiot at the beginning of the last century and that the things did not improve afterwards in the eighties with Spencer and coworkers (Compare MR 720863 (85 m: 12004) and MR 954613 (90e: 58166)).
3. Second Part: The Janet and Spencer Sequences
Let
be a multi-index with length
![](https://www.scirp.org/html/22-7501431\57c6a4bd-264a-46f5-8e16-b5da4fb6df90.jpg)
class
if
![](https://www.scirp.org/html/22-7501431\222baf85-85ba-4b24-835c-2b0540420cf2.jpg)
and
.
We set
![](https://www.scirp.org/html/22-7501431\11b29db4-5378-4cc4-97d5-af1f0f1d0231.jpg)
with
when
. If
is a vector bundle over
with local coordinates
and
is the
-jet bundle of
with local coordinates
the Spencer operator just allows to distinguish a section
from a section
by introducing a kind of “difference” through the operator
![](https://www.scirp.org/html/22-7501431\3255f343-b568-4cbe-b706-22378f0326be.jpg)
with local components
![](https://www.scirp.org/html/22-7501431\8fab51b9-c721-4860-8054-0eadebd799fd.jpg)
and more generally
.
Minus the restriction of
to the kernel
of the canonical projection
![](https://www.scirp.org/html/22-7501431\61176061-2a78-4728-9ad2-11799c71ade7.jpg)
can be extended to the Spencer map
![](https://www.scirp.org/html/22-7501431\95314c1b-5186-4988-a8bb-23cb079a9d01.jpg)
defined by
.
The kernel of
is made by sections such that
.
Finally, if
is a system of order
on
locally defined by linear equations
the
-prolongation
![](https://www.scirp.org/html/22-7501431\23ab9c46-4c93-4702-a804-f88ad5d1e0e1.jpg)
is locally defined when
by the linear equations
![](https://www.scirp.org/html/22-7501431\25769058-94b5-4a15-b6e5-6e35d1f71a76.jpg)
and has symbol
![](https://www.scirp.org/html/22-7501431\c3836667-af8b-41f1-8181-2948f2074783.jpg)
locally defined by
![](https://www.scirp.org/html/22-7501431\24ea1f81-f0af-4a8c-902f-27aeff74e570.jpg)
if one looks at the top order terms. If
is over
, differentiating the identity
![](https://www.scirp.org/html/22-7501431\c432e29a-c69e-4601-b045-85bccca0b349.jpg)
with respect to
and substracting the identity
we obtain the identity
![](https://www.scirp.org/html/22-7501431\a6852f8f-a465-48c3-9ae1-4306ba29872a.jpg)
and thus the restriction
. This first order operator induces, up to sign, the purely algebraic monomorphism
on the symbol level
[8,14]. The Spencer operator has never been used in GR though an accelerometer in a rocket merely measures one of the components of the Spencer operator involving second order jets.
DEFINITION 2.1:
is said to be formally integrable (FI) when the restriction
![](https://www.scirp.org/html/22-7501431\a1966354-ccc6-4739-81e7-66cabf919ab8.jpg)
is an epimorphism
. In that case, the Spencer form
is a canonical equivalent formally integrable first order system on
with no zero order equations.
DEFINITION 2.2:
is said to be involutive when it is formally integrable and the symbol
is involutive, that is all the sequences
![](https://www.scirp.org/html/22-7501431\dbcb0b3f-e337-42c2-b5bb-36f5e95a706c.jpg)
are exact
. Equivalently, using a linear change of local coordinates if necessary, we may successively solve the maximum number
of equations with respect to the leading or principal jet coordinates of strict order
and class
. Then
is involutive if
is obtained by only prolonging the
equations of class
with respect to
for
. In that case, such a prolongation procedure allows to compute in a unique way the principal jets from the parametric other ones and may also be applied to nonlinear systems as well [8,15].
When
is involutive, the linear differential operator
![](https://www.scirp.org/html/22-7501431\666d7722-d553-4fe4-8bf6-74a4f35556ab.jpg)
of order
with space of solutions
is said to be involutive and one has the canonical linear Janet sequence [8]:
![](https://www.scirp.org/html/22-7501431\c2008153-313d-41ae-a69b-a311c00ec8f2.jpg)
with Janet bundles
![](https://www.scirp.org/html/22-7501431\8bce4964-99e2-4c1a-9865-cf6eea8e1c0d.jpg)
Each operator
is first order involutive as it is induced by
![](https://www.scirp.org/html/22-7501431\915df1d0-f9b0-4c00-bc05-58fe25c5beb4.jpg)
and generates the compatibility conditions (CC) of the preceding one. As the Janet sequence can be cut at any place, the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence, contrary to what many people believe in GR.
Similarly, we have the involutive first Spencer operator
![](https://www.scirp.org/html/22-7501431\adfcba3b-b065-4179-8f17-49ac5eac25a6.jpg)
of order one induced by
. Introducing the Spencer bundles
the first order involutive
Spencer operator
is induced by
![](https://www.scirp.org/html/22-7501431\85017589-87f4-4acb-9edc-63ad9d249861.jpg)
and we obtain the canonical linear Spencer sequence [8,14]:
![](https://www.scirp.org/html/22-7501431\36e69eef-ce2b-469c-9282-c0c5a3a3edaf.jpg)
as the Janet sequence for the first order involutive system
. Introducing the other Spencer bundles
![](https://www.scirp.org/html/22-7501431\1d7c46a1-35c8-4740-b275-544ff0c71b2e.jpg)
with
, the linear Spencer sequence is induced by the linear hybrid sequence:
![](https://www.scirp.org/html/22-7501431\4e881ae2-767d-4693-b214-4648c7f58abb.jpg)
which is at the same time the Janet sequence for
and the Spencer sequence for
![](https://www.scirp.org/html/22-7501431\f1ecc9a8-d53d-4c95-af2b-8fad9faf9fd9.jpg)
[8,14]. Such a sequence projects onto the Janet sequence and we have the following commutative diagram with exact columns:
![](https://www.scirp.org/html/22-7501431\8af2b673-1baa-4922-a8b7-66b0bb84a2f5.jpg)
In this diagram, only depending on the linear differential operator
, the epimorhisms
for ![](https://www.scirp.org/html/22-7501431\0a2af3c4-f4e6-4193-8bdc-eabec93ad905.jpg)
are induced by the canonical projection
![](https://www.scirp.org/html/22-7501431\90b8463f-24a2-4780-ba90-7673fc39b392.jpg)
if we start with the knowledge of
or from the knowledge of an epimorphism
![](https://www.scirp.org/html/22-7501431\d217144a-0764-4d03-9984-dcab54d7087c.jpg)
if we set
. In the theory of Lie equations considered,
,
is a transitive involutive system of infinitesimal Lie equations of order
and the corresponding operator
is a Lie operator. As an exercise, we invite the reader to draw this diagram in the affine and projective 1-dimensional cases.
EXAMPLE 2.3: If we restrict our study to the group of isometries of the euclidean metric
in dimension
, exhibiting the Janet and the Spencer sequences is not easy at all, even when
, because the corresponding Killing operator
involving the Lie derivative
and providing twice the so-called infinitesimal deformation tensor
of continuum mechanics, is not involutive. In order to overcome this problem, one must differentiate once by considering also the Christoffel symbols
and add the operator
.
Now, one can prove that the Spencer sequence for Lie groups of transformations is locally isomorphic to the tensor product of the Poincaré sequence by the Lie algebra of the underlying Lie group [7,8]. Hence, if two Lie groups
act on
, it follows from the definition of the Janet and Spencer bundles that the Spencer sequence for
is embedded into the Spencer sequence for
while the Janet sequence for
projects onto the Janet sequence for
but the common differences are isomorphic to
. This rather philosophical comment, namely to replace the Janet sequence by the Spencer sequence, must be considered as the crucial key for understanding the work of the brothers E. and F. Cosserat in 1909 [7,17-19] or H. Weyl in 1918 [7,16], the best picture being that of Janet and Spencer playing at see-saw. Indeed, when
, one has 3 parameters (2 translations + 1 rotation) and the following commutative diagram which only depends on the left commutative square:
![](https://www.scirp.org/html/22-7501431\8474ac09-8556-4a71-9780-4543cb6047b1.jpg)
In this diagram, there is no way to compare
(curvature alone as in Vessiot) with
(curvature + torsion as in Cartan).
For proving that the adjoint of
provides the Cosserat equations which can be parametrized by the adjoint of
, we may lower the upper indices by means of the constant euclidean metric and look for the factors of
and
in the integration by parts of the sum:
![](https://www.scirp.org/html/22-7501431\14fbec01-5d45-4852-a9b1-cab26c1c2b48.jpg)
in order to obtain:
![](https://www.scirp.org/html/22-7501431\8058f53b-a9b4-4516-ad66-fb304ff20b3f.jpg)
Finally, we get the nontrivial first order parametrization
![](https://www.scirp.org/html/22-7501431\7a71855c-905c-4cf9-ba85-563c3439672b.jpg)
by means of the three arbitrary functions
, in a coherent way with the Airy second order parametrization obtained if we set
![](https://www.scirp.org/html/22-7501431\28c4f7e2-7653-427b-804a-2d87a908cd3c.jpg)
when
as we shall see in the third part.
The link between the FI of
and the CC of
is expressed by the following diagram that may be used inductively:
![](https://www.scirp.org/html/22-7501431\cb9085a4-2448-4879-99ee-0a96c5366c4c.jpg)
The “snake theorem” [8,20] then provides the long exact connecting sequence:
.
If we apply such a diagram to first order Lie equations with no zero or first order CC, we have
and we may apply the Spencer
-map to the top row obtained with r = 2 in order to get the commutative diagram:
![](https://www.scirp.org/html/22-7501431\295dafc9-9bb2-4710-8f0c-b6b339e764a8.jpg)
with exact rows and exact columns but the first that may not be exact at
. We shall denote by
the coboundary as the image of the central
, by
the cocycle as the kernel of the lower
and by
the Spencer
-cohomology at
as the quotient.
In the classical Killing system,
is defined by
![](https://www.scirp.org/html/22-7501431\75749efe-bb5e-4980-817b-7ba3123a3679.jpg)
Applying the previous diagram, we discover that the Riemann tensor is a section of the bundle
![](https://www.scirp.org/html/22-7501431\3aa0cb55-bfc1-4763-8e3a-d747f9ba55d4.jpg)
with
![](https://www.scirp.org/html/22-7501431\c11f55a2-3915-4515-a8f3-90453bd357f0.jpg)
by using the top row or the left column. Though we discover the two properties of the Riemann tensor through the chase involved, we have no indices and cannot therefore exhibit the Ricci tensor of GR by means of the usual contraction or trace.
Let us proceed the same way with the conformal Killing system
![](https://www.scirp.org/html/22-7501431\39ec0210-cabc-44e4-8e9b-dd02f1b90807.jpg)
obtained by introducing
![](https://www.scirp.org/html/22-7501431\1aea6cb2-0131-4afd-9a5c-2a57ac6f4f04.jpg)
or, equivalently, by eliminating
in
.
Now
is defined by
![](https://www.scirp.org/html/22-7501431\2a2b67fd-8502-4303-b34a-8c2d7db520d1.jpg)
but we have
with
and the Weyl tensor is a section of the bundle
![](https://www.scirp.org/html/22-7501431\d513bc11-86ca-4f4b-9de3-14382dc4433d.jpg)
with
.
Similarly, we have no indices and cannot therefore exhibit the Ricci tensor. However, when
, among the components of the Spencer operator we have
![](https://www.scirp.org/html/22-7501431\29e7c4a9-3ca4-4cf9-97be-5679c8e16343.jpg)
and thus
.
Such a result allows to recover the electromagnetic field in the image of the Spencer operator
and Maxwell equations by duality along the way proposed by Weyl in [16] but the use of the Spencer operator provides the only possibility to exhibit a link with Cosserat equations.
Comparing the classical and conformal Killing systems by using the inclusions
we finally obtain the following commutative and exact diagram where a diagonal chase allows to identify
with
![](https://www.scirp.org/html/22-7501431\c26f238b-7ff3-443f-a54e-4da71de16779.jpg)
and to split the right column [7,12,20]:
![](https://www.scirp.org/html/22-7501431\f53dfc88-52ad-442e-b13c-ab36f532d8e2.jpg)
SECOND FUNDAMENTAL RESULT: The Ricci tensor only depends on the “difference” existing between the clasical Killing system and the conformal Killing system, namely the
second order jets (elations once more). The Ricci tensor, thus obtained without contracting the indices as usual, may be embedded in the image of the Spencer operator made by 1-forms with value in 1-forms that we have already exhibited for describing EM. It follows that the foundations of both EM and GR are not coherent with jet theory and must therefore be revisited within this new framework.
4. Third Part: Algebraic Analysis
EXAMPLE 3.1: Let a rigid bar be able to slide along an horizontal axis with reference position
and attach two pendula, one at each end, with lengths
and
, having small angles
and
with respect to the vertical. If we project Newton law with gravity
on the perpendicular to each pendulum in order to eliminate the tension of the threads and denote the time derivative with a dot, we get the two equations:
![](https://www.scirp.org/html/22-7501431\235fb91f-b29c-4241-b7e6-c415e5439ba4.jpg)
As an experimental fact, starting from an arbitrary movement of the pendula, we can stop them by moving the bar if and only if
and we say that the system is controllable.
More generally, we can bring the OD equations describing the behaviour of a mechanical or electrical system to the Kalman form
with input
and output
. We say that the system is controllable if, for any given
one can find
such that a coherent trajectory
may be found. In 1963 [21], R. E. Kalman discovered that the system is controllable if and only if
.
Surprisingly, such a functional definition admits a formal test which is only valid for Kalman type systems with constant coefficients and is thus far from being intrinsic. In the PD case, the Spencer form will replace the Kalman form.
EXAMPLE 3.2:
![](https://www.scirp.org/html/22-7501431\06b1e2ea-4705-4a41-a251-a658d73fb4c5.jpg)
can always be achieved and the system is thus controllable in the sense of the definition but
is not controllable in the sense of the test.
EXAMPLE 3.3:
.
Any way to bring this system to Kalman form provides the controllability condition
if
but nothing can be said if
. Also, getting
from the second equation and substituting in the first, we get the second order OD equation
![](https://www.scirp.org/html/22-7501431\31872795-f4d9-498e-ab39-e0722cca9de3.jpg)
for which nothing can be said at first sight.
PROBLEM 1: Is a SYSTEM of OD or PD equations “controllable” (answer must be YES or NO) and how can we define controllability?
Now, if a differential operator
is given, a direct problem is to find (generating) compatibility conditions (CC) as an operator
such that
.
Conversely, the inverse problem will be to find ![](https://www.scirp.org/html/22-7501431\17179e09-865a-498f-9239-57d8a6251806.jpg)
such that
generates the CC of
and we shall say that
is parametrized by
. Of course, solving the direct problem (Janet, Spencer) is necessary for solving the inverse problem.
EXAMPLE 3.4: When
, the Cauchy equations for the stress in continuum mechanics are
![](https://www.scirp.org/html/22-7501431\6cf77a37-91ea-4799-a473-59317fd4f483.jpg)
with
.
Their parametrization
![](https://www.scirp.org/html/22-7501431\9511fadf-db6f-45b6-b048-ac7f2b87e9f3.jpg)
has been discovered by Airy in 1862 and
is called the Airy function. When
, Maxwell and Morera discovered a similar parametrization with 3 potentials (exercise).
EXAMPLE 3.5: When
, the Maxwell equations
where
is the EM field are parametrized by
where
is the 4-potential. The second set of Maxwell equations can also be parametrized by the so-called pseudopotential which is a pseudovector density (exercise).
EXAMPLE 3.5: If
,
is the Minkowski metric and
is the gravitational potential, then
and a perturbation
of
may satisfy in vacuum the 10 second order Einstein equations for the 10
:
![](https://www.scirp.org/html/22-7501431\4df31b6d-26d1-43c0-beec-4de69078f2e4.jpg)
The parametrizing challenge has been proposed in 1970 by J. Wheeler for 1000 $ and solved negatively in 1995 by the author who only received 1 $.
PROBLEM 2: Is an OPERATOR parametrizable (answer must be YES or NO) and how can we find a parametrization?
Let
be a unitary ring, that is
![](https://www.scirp.org/html/22-7501431\bbda4ab1-779b-46ca-9fb3-2019d3437859.jpg)
and even an integral domain, that is
or
.
We say that
is a left module over
if
![](https://www.scirp.org/html/22-7501431\11cf738a-6d04-4504-a020-f2656c175e86.jpg)
and we denote by
the set of morphisms
such that
.
DEFINITION 3.6: We define the torsion submodule
.
There is a sequence
![](https://www.scirp.org/html/22-7501431\81f60bdf-a94e-439e-aa0f-5831e4187b2a.jpg)
where the morphism
is defined by
![](https://www.scirp.org/html/22-7501431\dd72f6f0-0462-4756-be21-db2a280909f3.jpg)
because we have at once
.
PROBLEM 3: Is a MODULE
torsion-free, that is
(answer must be YES or NO) and how can we test such a property?
In the remaining of this paper we shall prove that the three problems are indeed identical and that only the solution of the third will provide the solution of the two others [1,22-24].
Let
be a differential field, that is a field
![](https://www.scirp.org/html/22-7501431\cdd3d395-af42-428e-a794-c8956713b8ea.jpg)
with
commuting derivations
with
![](https://www.scirp.org/html/22-7501431\e1d7055a-ab9c-40ec-95d5-b123c35b791b.jpg)
such that
![](https://www.scirp.org/html/22-7501431\097b0ab8-555b-4e36-8361-eb4b21874443.jpg)
and
.
Using an implicit summation on multiindices, we may introduce the (noncommutative) ring of differential operators
![](https://www.scirp.org/html/22-7501431\6304bc52-3b19-4350-bf9f-ea06d0b35b8e.jpg)
with elements
such that
and
. Now, if we introduce differential indeterminates
, we may extend
to
![](https://www.scirp.org/html/22-7501431\f7b51ff7-de33-48e8-9258-b35295b3271a.jpg)
for
. Therefore, setting
we obtain by residue the differential module or
- module
. Introducing the two free differential modules
, we obtain equivalently the free presentation
. More generally, introducing the successive CC as in the preceding section, we may finally obtain the free resolution of
, namely the exact sequence
.
The “trick” is to let
act on the left on column vectors in the operator case and on the right on row vectors in the module case. Homological algebra has been created for finding intrinsic properties of modules not depending on their presentation or even on their resolution.
EXAMPLE 3.7: In order to understand that different presentations may nevertheless provide isomorphic modules, let us consider the linear inhomogeneous system
with
. Differentiating twice, we get
and the two fourth order CC:
![](https://www.scirp.org/html/22-7501431\f0f1fbd1-3ae0-46fc-945b-4759da277fe7.jpg)
However, as
, we also get the CC
and the two resolutions:
![](https://www.scirp.org/html/22-7501431\140c3663-5a83-45b8-8ff3-6b6e6680a23b.jpg)
where we can identify the two differential modules involved on the right with
because:
![](https://www.scirp.org/html/22-7501431\47a98ac6-a614-4685-9014-1279b19087ca.jpg)
We now exhibit another approach by defining the formal adjoint of an operartor
and an operator matrix
:
DEFINITION 3.8:
![](https://www.scirp.org/html/22-7501431\eb94fc04-e9e8-4f14-97e9-c20f805f5128.jpg)
![](https://www.scirp.org/html/22-7501431\0250b959-161e-4312-a9c2-dd8eafd43bd9.jpg)
from integration by part, where
is a row vector of test functions and
the usual contraction.
PROPOSITION 3.9: If we have an operator
, we obtain by duality an operator
![](https://www.scirp.org/html/22-7501431\a2743232-2cf4-436e-991e-de456d8b36f3.jpg)
where
is obtained from
by inverting the transition matrix and EM provides a fine example of such a procedure [10].
Now, with operational notations, let us consider the two differential sequences:
![](https://www.scirp.org/html/22-7501431\31fea7cd-0b65-4258-b6ca-2971719130a0.jpg)
![](https://www.scirp.org/html/22-7501431\91c35480-21fe-4fa9-9b23-027edb314195.jpg)
where
generates all the CC of
. Then
![](https://www.scirp.org/html/22-7501431\7c345bf9-ff1c-4041-b751-efa834023171.jpg)
but
may not generate all the CC of
.
EXAMPLE 3.10: With
for
, we get
for
. Then
is defined by
while
is defined by
but the CC of
are generated by
. Passing to the module framework, we obtain the sequences:
![](https://www.scirp.org/html/22-7501431\0aad30d5-8c3c-4eba-8d82-d1c5672dff53.jpg)
THEOREM 3.11: The cohomology
at
of the lower sequence does not depend on the resolution of
and is a torsion module called the first extension module of
.
Exactly like we defined the differential module
from
, let us define the differential module
from
. The proof of the next theorem is quite tricky and out of the scope of this paper [1,22-24]:
MAIN THEOREM 3.12:
.
FORMAL TEST 3.13: The double duality test needed in order to check whether
or not and to find out a parametrization if
has 5 steps which are drawn in the following diagram where
generates the CC of
and
generates the CC of
:
![](https://www.scirp.org/html/22-7501431\67810297-004e-41ef-8d67-ce73142f3f06.jpg)
THEOREM 3.14:
parametrized by
.
COROLLARY 3.15: If
and
is surjective, then
if and only if
is injective [24,25].
EXAMPLE 3.16: (Kalman test revisited) If we multiply the Kalman system
on the left by a test row vector
, we obtain:
![](https://www.scirp.org/html/22-7501431\4ff9e0f9-5523-4f4f-a0b3-9c6cc75c921f.jpg)
Differentiating the zero order equations and using the first order ones, we get
and so on. Using the Cayley-Hamilton theorem, we stop at
and find back exactly the Kalman test but in a completely different intrinsic framework.
EXAMPLE 3.17: (Double pendulum revisited) Using two test functions
and
, we get:
![](https://www.scirp.org/html/22-7501431\66a7a174-2e5a-4f57-bf0a-d9ce1370d5ca.jpg)
and obtain at once the zero order equation
.
Differentiating twice and substituting, we also get
and
is injective if and only if
.
EXAMPLE 3.18: (Airy parametrization revisited) When
, we may study the infinitesimal deformation
by means of the Killing operator
when
is the euclidean metric. Then
provides (up to sign and factor
) the Cauchy equations
for the stress tensor density [12,16,26]. The following diagram describes the Poincaré scheme:
.
Accordingly, the second order Airy parametrization is nothing else than the adjoint of the only Riemann CC involved, namely
which is the linearization of the Riemann tensor of Example 1.9.
EXAMPLE 3.19: (Einstein equations revisited) Contrary to the Ricci operator, the Einstein operator is selfadjoint because it comes from a variational procedure, the sixth terms being exchanged between themselves under
. For example, we have:
![](https://www.scirp.org/html/22-7501431\05faa292-6df0-437c-a5f1-e56412db3c2f.jpg)
and the adjoint of the first operator is the sixth. Accordingly, one has the following diagram where
:
![](https://www.scirp.org/html/22-7501431\63e6a04e-c779-439e-bdfe-4e72f4cd7575.jpg)
THIRD FUNDAMENTAL RESULT: Comparing this diagram to the previous one proves that Einstein equations are not coherent with Janet and Spencer sequences as conformal geometry has not been introduced in this last part.
EXERCISE 3.20: Prove that
![](https://www.scirp.org/html/22-7501431\4d1fe015-32c4-4aac-8c7f-078aeff385fd.jpg)
is controllable if and only if
(Riccati) and find a parametrization.
EXERCISE 3.21: Prove that the infinitesimal contact transformations of Example 1.10 admit the injective parametrization
![](https://www.scirp.org/html/22-7501431\0a7d3ab2-10d0-4e44-afa1-34536b36fef8.jpg)
5. Conclusion
The mathematical foundations of General Relativity leading to Einstein equations are always presented in textbooks or papers without any reference to conformal geometry. However, comparing the classical Killing equations to the conformal Killing equations while constructing corresponding differential sequences, the Ricci tensor appears as the kernel of the canonical projection of the Riemann tensor onto the Weyl tensor. After obtaining such a result in a purely intrinsic way, that is without using indices, we have been able to introduce “diagram chasing” in order to relate for the first time electromagnetism and gravitation to the Spencer
-cohomology of the classical and conformal Killing symbols. Accordingly, the mathematical foundations of general relativity are not coherent with jet theory and must therefore be revisited within this new framework along the lines we have sketched. Finally, the fact that Einstein equations cannot be parametrized, contrary to most other equations of physics or engineering, also brings a deep structural question on these equations that will have to be solved in the future by means of algebraic analysis.