1. Introduction
Let us start this paper with a personal but meaningful story that has oriented my research during the last forty years or so, since the French “Grandes Ecoles” created their own research laboratories. Being a fresh permanent researcher of Ecole Nationale des Ponts et Chaussées in Paris, the author of this paper has been asked to become the scientific adviser of a young student in order to introduce him to research. As General Relativity was far too much difficult for somebody without any specific mathematical knowledge while remembering his own experience at the same age, he asked the student to collect about 50 books of Special Relativity and classify them along the way each writer was avoiding the use of the conformal group of space-time implied by the Michelson and Morley experiment, only caring about the Poincaré or Lorentz subgroups. After six months, the student (like any reader) arrived at the fact that most books were almost copying each other and could be nevertheless classified into three categories:
• 30 books, including the original 1905 paper ( [1] ) by Einstein, were at once, as a working assumption, deciding to restrict their study to a linear group reducing to the Galilée group when the speed of light was going to infinity. It must be noticed that people did believe that Einstein had not been influenced in 1905 by the Michelson and Morley experiment of 1887 till the discovery of hand written notes taken during lectures given by Einstein in Chicago (1921) and Kyoto (1922).
• 15 books were trying to “prove” that the conformal factor was indeed reduced to a constant equal to 1 when space-time was supposed to be homogeneous and isotropic.
• 5 books only were claiming that the conformal factor could eventually depend on the property of space-time, adding however that, if there was no surrounding electromagnetism or gravitation, the situation should be reduced to the preceding one but nothing was said otherwise.
The student was so disgusted by such a state of affair that he decided to give up on research and to become a normal civil engineer. As a byproduct, if group theory must be used, the underlying group of transformations of space-time must be related to the propagation of light by itself rather than by considering tricky signals between observers, thus must have to do with the biggest group of invariance of Maxwell Equations ( [2] [3] ). However, at the time we got the solution of this problem with the publication of ( [4] ) in 1988 (See [5] for recent results), a deep confusion was going on which is still not acknowledged though it can be explained in a few lines ( [6] ). Using standard notations of differential geometry, if the 2-form
describing the EM field is satisfying the first set of Maxwell equations, it amounts to say that it is closed, that is killed by the exterior derivative
. The EM field can be thus (locally) parametrized by the EM potential 1-form
with
where
is again the exterior derivative, because
. Now, if E is a vector bundle over a manifold X of dimension n, then we may define its adjoint vector bundle
where
is obtained from E by inverting the transition rules, like
is obtained from
and such a construction can be extended to linear partial differential operators between (sections of) vector bundles. When
, it follows that the second set of Maxwell equations for the EM induction is just described by
, independently of any Minkowski constitutive relation between field and induction. Using Hodge duality with respect to the volume form
, this operator is isomorphic to
. It follows that both the first set and second set of Maxwell equations are invariant by any diffeomorphism and that the conformal group of space-time is the biggest group of transformations preserving the Minkowski constitutive relations in vacuum where the speed of light is truly c as a universal constant. It was thus natural to believe that the mathematical structure of electromagnetism and gravitation had only to do with such a group having:
the main difficulty being to deal with these later non-linear transformations. Of course, such a challenge could not be solved without the help of the non-linear theory of partial differential equations and Lie pseudogroups combined with homological algebra, that is before 1995 at least ( [7] ).
From a purely physical point of view, these new nonlinear methods have been introduced for the first time in 1909 by the brothers E. and F. Cosserat for studying the mathematical foundations of EL ( [8] - [14] ). We have presented their link with the nonlinear Spencer differential sequences existing in the formal theory of Lie pseudogroups at the end of our book “Differential Galois Theory” published in 1983 ( [15] ). Similarly, the conformal methods have been introduced by H. Weyl in 1918 for revisiting the mathematical foundations of EM ( [3] ). We have presented their link with the above approach through a unique differential sequence only depending on the structure of the conformal group in our book “Lie Pseudogroups and Mechanics” published in 1988 ( [4] ). However, the Cosserat brothers were only dealing with translations and rotations while Weyl was only dealing with dilatation and elations. Also, as an additional condition not fulfilled by the classical Einstein-Fokker-Nordström theory ( [16] ), if the conformal factor has to do with gravitation, it must be defined everywhere but at the central attractive mass as we already said.
From a purely mathematical point of view, the concept of a finite length differential sequence, now called Janet sequence, has been first described as a footnote by M. Janet in 1920 ( [17] ). Then, the work of D. C. Spencer in 1970 has been the first attempt to use the formal theory of systems of partial differential equations that he developed himself in order to study the formal theory of Lie pseudogroups ( [18] [19] [20] ). However, the nonlinear Spencer sequences for Lie pseudogroups, though never used in physics, largely supersede the “Cartan structure equations” introduced by E.Cartan in 1905 ( [21] [22] ) and are different from the “Vessiot structure equations” introduced by E. Vessiot in 1903 ( [23] ) or 1904 ( [24] ) for the same purpose but still not known today after more than a century because they have never been acknowledged by Cartan himself or even by his successors.
The purpose of the present difficult paper is to apply these new methods for studying the common nonlinear conformal origin of electromagnetism and gravitation, in a purely mathematical way, by constructing explicitly the corresponding nonlinear Spencer sequence. All the physical consequences will be presented in another paper.
2. Groupoids and Algebroids
Let us now turn to the clever way proposed by Vessiot in 1903 ( [23] ) and 1904 ( [24] ). Our purpose is only to sketch the main results that we have obtained in many books ( [4] [7] [13] [15], we do not know other references) and to illustrate them by a series of specific examples, asking the reader to imagine any link with what has been said. We break the study into 8 successive steps.
1) If
, we shall denote by
the open sub-fibered manifold of
defined independently of the coordinate system by
with source projection
and target projection
. We shall sometimes introduce a copy Y of X with local coordinates (y) in order to avoid any confusion between the source and the target manifolds. In order to construct another nonlinear sequence, we need a few basic definitions on Lie groupoids and Lie algebroids that will become substitutes for Lie groups and Lie algebras. The first idea is to use the chain rule for derivatives
whenever
can be composed and to replace both
and
respectively by
and
in order to obtain the new section
. This kind of “composition” law can be written in a symbolic way by introducing another copy Z of X with local coordinates (z) as follows:
We may also define
and obtain similarly an “inversion” law.
DEFINITION 2.1: A fibered submanifold
is called a system of finite Lie equations or a Lie groupoid of order q if we have an induced source projection
, target projection
, composition
, inversion
and identity
. In the sequel we shall only consider transitive Lie groupoids such that the map
is an epimorphism and we shall denote by
the isotropy Lie group bundle of
. Also, one can prove that the new system
obtained by differentiating r times all the defining equations of
is a Lie groupoid of order
.
Let us start with a Lie pseudogroup
defined by a system
of order q. Roughly speaking, if
but such a definition is totally meaningless in actual practice as it cannot be checked most of the time. In all the sequel we shall suppose that the system is involutive ( [4] [7] [13] [15] [25] ) and that
is transitive that is
or, equivalently, the map
is surjective.
2) The Lie algebra
of infinitesimal transformations is then obtained by linearization, setting
and passing to the limit
in order to obtain the linear involutive system
by reciprocal image with
. We define the isotropy Lie algebra bundle
by the short exact sequence
.
3) Passing from source to target, we may prolong the vertical infinitesimal transformations
to the jet coordinates up to order q in order to obtain:
where we have replaced
by
, each component being the “formal” derivative of the previous one .
4) As
, we may use the Frobenius theorem in order to find a generating fundamental set of differential invariants
up to order q which are such that
by using the chain rule for derivatives whenever
acting now on Y. Specializing the
at
we obtain the Lie form
of
.
Of course, in actual practice one must use sections of
instead of solutions and we now prove why the use of the Spencer operator becomes crucial for such a purpose. Indeed, using the algebraic bracket
, we may obtain by bilinearity a differential bracket on
extending the bracket on T:
which does not depend on the respective lifts
and
of
and
in
. This bracket on sections satisfies the Jacobi identity and we set ( [4] [7] [13] [25] ):
DEFINITION 2.2: We say that a vector subbundle
is a system of infinitesimal Lie equations or a Lie algebroid if
, that is to say
. Such a definition can be tested by means of computer algebra. We shall also say that
is transitive if we have the short exact sequence
. In that case, a splitting of this sequence, namely a map
such that
or equivalently a section
over
, is called a
-connection and its curvature
is defined by
.
PROPOSITION 2.3: If
, then
.
Proof: When
, we have
and we just need to use the following formulas showing how D acts on the various brackets (See [7] and [25] for more details):
because the right member of the second formula is a section of
whenever
. The first formula may be used when
is formally integrable.
口
EXAMPLE 2.4: With
and evident notations, the components of
at order zero, one, two and three are defined by the totally unusual successive formulas:
For affine transformations,
and thus
.
For projective transformations,
and thus
.
THEOREM 2.5: (prolongation/projection (PP) procedure) If an arbitrary system
is given, one can effectively find two integers
such that the system
is formally integrable or even involutive.
COROLLARY 2.6: The bracket is compatible with the PP procedure:
EXAMPLE 2.7: With
and
, let us consider the Lie pseudodogroup
of finite transformations
such that
. Setting
and linearizing, we get the Lie operator
where
is the Lie derivative because it is well known that
in the operator sense. The system
of linear infinitesimal Lie equations is:
Replacing
by a section
, we have:
Let us choose the two sections:
We let the reader check that
. However, we have the strict inclusion
defined by the additional equation
and thus
though we have indeed
, a result not evident at all because the sections
and
have nothing to do with solutions. The reader may proceed in the same way with
and compare.
5) The main discovery of Vessiot, as early as in 1903 and thus fifty years in advance, has been to notice that the prolongation at order q of any horizontal vector field
commutes with the prolongation at order q of any vertical vector field
, exchanging therefore the differential invariants. Keeping in mind the well known property of the Jacobian determinant while passing to the finite point of view, any (local) transformation
can be lifted to a (local) transformation of the differential invariants between themselves of the form
allowing to introduce a natural bundle
over X by patching changes of coordinates
. A section
of
is called a geometric object or structure on X and transforms like
or simply
. This is a way to generalize vectors and tensors (
) or even connections (
). As a byproduct we have
as a new way to write out the Lie form and we may say that
preserves
. We also obtain
. Coming back to the infinitesimal point of view and setting
, we may define the ordinary Lie derivative with value in
by introducing the vertical bundle of
as a vector bundle over
and the formula:
while we have
where
is a multi-index as a way to write down the system
of infinitesimal Lie equations in the Medolaghi form:
EXAMPLE 2.8: With
, let us consider the Lie group of projective transformations
as a lie pseudogroup. Differentiating three times in order to eliminate the parameters, we obtain the third order Schwarzian OD equation and its linearization over
:
Accordingly, the prolongation
of any
over Y such that
becomes:
It follows that
is a generating third order differential invariant and
is in Lie form.
Now, we have:
and the natural bundle
is thus defined by the transition rules:
The general Lie form of
is:
We obtain
in Medolaghi form as follows:
Using a section
, we finally get the formal Lie derivative:
and let the reader ckeck directly that
,
, a result absolutely not evident at first sight.
6) By analogy with “special” and “general” relativity, we shall call the given section special and any other arbitrary section general. The problem is now to study the formal properties of the linear system just obtained with coefficients only depending on
, exactly like L.P. Eisenhart did for
when finding the constant Riemann curvature condition for a metric
with
( [7], Example 10, p 246 to 256, [26] ). Indeed, if any expression involving
and its derivatives is a scalar object, it must reduce to a constant because
is assumed to be transitive and thus cannot be defined by any zero order equation. Now one can prove that the CC for
, thus for
too, only depend on the
and take the quasi-linear symbolic form
, allowing to define an affine subfibered manifold
over
. Now, if one has two sections
and
of
, the equivalence problem is to look for
such that
. When the two sections satisfy the same CC, the problem is sometimes locally possible (Lie groups of transformations, Darboux problem in analytical mechanics, …) but sometimes not ( [25], p. 333).
7) Instead of the CC for the equivalence problem, let us look for the integrability conditions (IC) for the system of infinitesimal Lie equations and suppose that, for the given section, all the equations of order
are obtained by differentiating r times only the equations of order q, then it was claimed by Vessiot ( [23] with no proof, see [7], pp. 207-211) that such a property is held if and only if there is an equivariant section
where
is a natural vector bundle over
with local coordinates
. Moreover, any such equivariant section depends on a finite number of constants c called structure constants and the IC for the Vessiot structure equations
are of a polynomial form
.
EXAMPLE 2.9: Comig back to Example 2.7 first considered by Vessiot as early as in 1903 ( [23] ), the geometric object
must satisfy the Vessiot structure equation
with a single Vessiot structure constant
in the situation considered where
and
(See ( [27] ) for other examples and applications). As a byproduct, there is no conceptual difference between such a constant and the constant appearing in the constant Riemannian curvature condition of Eisenhart ( [26] ).
8) Finally, when Y is no longer a copy of X, a system
is said to be an automorphic system for a Lie pseudogroup
if, whenever
and
are two solutions, then there exists one and only one transformation
such that
. Explicit tests for checking such a property formally have been given in ( [15] ) and can be implemented on computer in the differential algebraic framework.
3. Nonlinear Sequences
Contrary to what happens in the study of Lie pseudogroups and in particular in the study of the algebraic ones that can be found in mathematical physics, nonlinear operators do not in general admit CC, unless they are defined by differential polynomials, as can be seen by considering the two following examples with
. With standard notations from differential algebra, if we are dealing with a ground differential field K, like
in the next examples, we denote by
the ring (which is even an integral domain) of differential polynomials in y with coefficients in K and by
the corresponding quotient field of differential rational functions in y. Then, if
, we have the two towers
and
of extensions, thus the tower
. Accordingly, the differential extension
is a finitely generated differential extension. If we consider u and v as new indeterminates, then
and
are both differential transcendental extensions of K and the kernel of the canonical differential morphism
is a prime differential ideal in the differential integral domain
, a way to describe by residue the smallest differential field containing
and
in
. Of course, the true difficulty is to find out such a prime differential ideal.
EXAMPLE 3.1: First of all, let us consider the following nonlinear system in y with second member
:
The differential ideal
generated by P and Q in
is prime because
and thus
is an integral domain.
We may consider the following nonlinear involutive system with two equations:
We have also the linear inhomogeneous finite type second order system with three equations:
Though we have a priori two CC, we let the reader prove, as a delicate exercise, that there is only the single nonlinear second order CC obtained from the bottom dot:
EXAMPLE 3.2: On the contrary, if we consider the following new nonlinear system:
we obtain successively:
The symbol at order 3 is thus not a vector bundle and no direct study as above can be used because the differential ideal generated by
is not perfect as it contains
without containing
(See [15] and [28] for more details). The following nonlinear system is not involutive:
We have the following four generic nonlinear additional finite type third order equations:
Though we have now a priori three CC and thus three additional equations because the system is not involutive, setting
, there is only the single additional nonlinear second order equation:
Differentiating once and using the relation
, we get:
a result leading to a tricky resultant providing a third order differential polynomial in
.
However, the kernel of a linear operator
is always taken with respect to the zero section of F, while it must be taken with respect to a prescribed section by a double arrow for a nonlinear operator. Keeping in mind the linear Janet sequence and the examples of Vessiot structure equations already presented, one obtains:
THEOREM 3.3: There exists a nonlinear Janet sequence associated with the Lie form of an involutive system of finite Lie equations:
where the kernel of the first operator
is taken with respect to the section
of
while the kernel of the second operator
is taken with respect to the zero section of the vector bundle
over
.
COROLLARY 3.4: By linearization at the identity, one obtains the involutive Lie operator
with kernel
satisfying
and the corresponding linear Janet sequence:
where
and
.
Now we notice that T is a natural vector bundle of order 1 and
is thus a natural vector bundle of order
. Looking at the way a vector field and its derivatives are transformed under any
while replacing
by
, we obtain:
and so on, a result leading to:
LEMMA 3.5:
is associated with
that is we can obtain a new section
from any section
and any section
by the formula:
where the left member belongs to
. Similarly
is associated with
.
More generally, looking now for transformations “close” to the identity, that is setting
when
is a small constant parameter and passing to the limit
, we may linearize any (nonlinear) system of finite Lie equations in order to obtain a (linear) system of infinitesimal Lie equations
for vector fields. Such a system has the property that, if
are two solutions, then
is also a solution. Accordingly, the set
of its solutions satisfies
and can therefore be considered as the Lie algebra of
.
More generally, the next definition will extend the classical Lie derivative:
DEFINITION 3.6: We say that a vector bundle F is associated with
if there exists a first order differential operator
called formal Lie derivative and such that:
1)
.
2)
.
3)
.
4)
.
LEMMA 3.7: If E and F are associated with
, we may set on
:
If
denotes the solutions of
, then we may set
but no explicit computation can be done when
is infinite dimensional. However, we have:
PROPOSITION 3.8:
is associated with
if we define:
and thus
is associated with
.
Proof: It is easy to check the properties 1, 2, 4 and it only remains to prove property 3 as follows.
by using successively the Jacobi identity for the algebraic bracket and the last proposition.
口
EXAMPLE 3.9: T and
both with any tensor bundle are associated with
. For T we may define
. We have
and the four properties of the formal Lie derivative can be checked directly. Of course, we find back
. We let the reader treat similarly the case of
.
PROPOSITION 3.10: There is a first nonlinear Spencer sequence:
with
. Moreover, setting
, this sequence is locally exact if
.
Proof: There is a canonical inclusion
defined by
and the composition
is a well defined section of
over the section
of
like
. The difference
is thus a section of
over
and we have already noticed that
. For
we get with
:
We also obtain from Lemma 3.5 the useful formula
allowing to determine
inductively.
We refer to ( [7], p 215-216) for the inductive proof of the local exactness, providing the only formulas that will be used later on and can be checked directly by the reader:
(1)
(2)
(3)
There is no need for double-arrows in this framework as the kernels are taken with respect to the zero section of the vector bundles involved. We finally notice that the main difference with the gauge sequence is that all the indices range from 1 to n and that the condition
amounts to
because
by assumption.
口
COROLLARY 3.11: There is a restricted first nonlinear Spencer sequence:
DEFINITION 3.12: A splitting of the short exact sequence
is a map
such that
or equivalently a section of
over
and is called a
-connection. Its curvature
is defined by
. We notice that
is a connection with
if and only if
. In particular
is the only existing symmetric connection for the Killing system.
REMARK 3.13: Rewriting the previous local formulas with A instead of
we get:
(1*)
(2*)
(3*)
When
and though surprising it may look like, we find back exactly all the formulas presented by E. and F. Cosserat in ( [10], p 123). Even more strikingly, in the case of a Riemann structure, the last two terms disappear but the quadratic terms are left while, in the case of screw and complex structures, the quadratic terms disappear but the last two terms are left. We finally notice that
is a
-connection if and only if
, a result contradicting the use of connections in physics. However, when
, we have
and thus:
does not depend on the lift of
.
COROLLARY 3.14: When
there is a second nonlinear Spencer sequence stabilized at order q:
where
and
are involutive and a restricted second nonlinear Spencer sequence:
such that
and
are involutive whenever
is involutive.
Proof: With
we have
. Setting
, we obtain
and
restricts to
.
Finally, setting
, we obtain successively:
We obtain therefore
and
restricts to
.
In the case of Lie groups of transformations, the symbol of the involutive system
must be
providing an isomorphism
and we have therefore
for
like in the linear Spencer sequence.
口
REMARK 3.15: In the case of the (local) action of a Lie group G on X, we may consider the graph of this action, that is the morphism
. If q is large enough, then there is an isomorphism
obtained by eliminating the parameters and
. If
with
is a basis of infinitesimal generators of this action, there is a morphism of Lie algebroids over X, namely
when q is large enough and the linear Spencer sequence
is locally exact because it is locally isomoprphic to the tensor product by
of the Poincaré sequence
where d is the exterior derivative ( [7] ).
We may also consider similarly
and
, depending on the choice of the independent variable among the source x or the target y.
Surprisingly, in the case of Lie pseudogroups or Lie groupoids, the situation is quite different. We recall the way to introduce a groupoid structure on
from the groupoid structure on
when
, that is how to define
. We get successively with
:
and so on with more and more involved formulas.
Now, if we want to obtain objects over the source x according to the non-linear Spencer sequence, we have only two possibilities in actual practice, namely:
As we have already considered the first, we have now only to study the second. In
, we have:
LEMMA 3.16:
is a quasi-linear rational function of
,
. With more details, when
, we have
and
with
and when
, we have
, that is to say
.
Proof: In the groupoid framework, we have:
Doing the substitutions:
while using the fact that
and
, we obtain at once:
Proceeding by induction, we finally obtain:
that is to say
because
, thus
or, equivalently,
.
口
REMARK 3.17: The passage from
to
is exactly the one done by E. and F. Cosserat in ( [10], p 190), even though it is based on a subtle misunderstanding that we shall correct later on.
REMARK 3.18: According to the previous results, the “field” must be a section of the natural bundle
of geometric objects if we use the nonlinear Janet sequence or a section of the first Spencer bundle
if we use the nonlinear Spencer sequence. The aim of this paper is to prove that the second choice is by far more convenient for mathematical physics.
4. Variational Calculus
It remains to graft a variational procedure adapted to the previous results. Contrary to what happens in analytical mechanics or elasticity for example, the main idea is to vary sections but not points. Hence, we may introduce the variation
and set
along the “vertical machinery” but notations like
or
have no meaning at all.
As a major result first discovered in specific cases by the brothers Cosserat in 1909 and by Weyl in 1916, we shall prove and apply the following key result:
THE PROCEDURE ONLY DEPENDS ON THE LINEAR SPENCER OPERATOR AND ITS FORMAL ADJOINT.
In order to prove this result, if
can be composed in such a way that
, we get:
Using the local exactness of the first nonlinear Spencer sequence or ( [25], p 219), we may state:
LEMMA 4.1: For any section
, the finite gauge transformation:
exchanges the solutions of the field equations
.
Introducing the formal Lie derivative on
by the formulas:
LEMMA 4.2: Passing to the limit over the source with
for
, we get an infinitesimal gauge transformation leading to the infinitesimal variation:
(3)
which does not depend on the parametrization of
. Setting
, we get:
(3*)
LEMMA 4.3: Passing to the limit over the target with
and
, we get the other infinitesimal variation where
is over the target:
(4)
which depends on the parametrization of
.
EXAMPLE 4.4: We obtain for
:
Introducing the inverse matrix
, we obtain therefore equivalently:
both with:
For the Killing system
with
, these variations are exactly the ones that can be found in ( [10], (50) + (49), p 124 with a printing mistake corrected on p 128) when replacing a 3 × 3 skew-symmetric matrix by the corresponding vector. The last unavoidable Proposition is thus essential in order to bring back the nonlinear framework of finite elasticity to the linear framework of infinitesimal elasticity that only depends on the linear Spencer operator.
For the conformal Killing system
(see next section) we obtain:
but
is far from being a 1-form. However,
and thus
is a pure 1-form if we replace
by
. Hence,
is a scalar for any
and we have
. As we shall see in section V.A, we have
for any section
and we obtain therefore successively:
These are exactly the variations obtained by Weyl ( [3], (76), p. 289) who was assuming implicitly
when setting
by introducing a connection. Accordingly,
is the variation of the EM potential itself, that is the
of engineers used in order to exhibit the Maxwell equations from a variational principle ( [3], p. 26) but the introduction of the Spencer operator is new in this framework.
The explicit general formulas of the two lemma cannot be found somewhere else (The reader may compare them to the ones obtained in [19] by means of the so-called “diagonal” method that cannot be applied to the study of explicit examples). The following unusual difficult proposition generalizes well known variational techniques used in continuum mechanics and will be crucially used for applications:
PROPOSITION 4.5: The same variation is obtained whenever
with
, a transformation only depending on
and invertible if and only if
.
Proof: First of all, setting
, we get
for
, a transformation which is invertible if and only if
. In the nonlinear framework, we have to keep in mind that there is no need to vary the object
which is given but only the need to vary the section
as we already saw, using
over the target or
over the source. With
, we obtain for example:
and so on. Introducing the formal derivatives
for
, we have:
We shall denote by
with
the corresponding vertical vector field, namely:
However, the standard prolongation of an infinitesimal change of source coordinates described by the horizontal vector field
, obtained by replacing all the derivatives of
by a section
over
, is the vector field:
It can be proved that
over the source, with a similar property for
over the target ( [25] ). However,
is not a vertical vector field and cannot therefore be compared to
. The solution of this problem explains a strange comment made by Weyl in ( [3], p 289 + (78), p 290) and which became a founding stone of classical gauge theory. Indeed,
is not a scalar because
is not a 2-tensor. However, when
, then
is a
-connection and
is a true scalar that may be set equal to zero in order to obtain
, a fact explaining why the EM-potential is considered as a connection in quantum mechanics instead of using the second order jets
of the conformal system, with a shift by one step in the physical interpretation of the Spencer sequence (See [4] for more historical details).
The main idea is to consider the vertical vector field
whenever
. Passing to the limit
in the formula
, we first get
. Using the chain rule for derivatives and substituting jets, we get successively:
and so on, replacing
by
in
in order to obtain:
where the right member only depends on
when
.
Finally, we may write the symbolic formula
in the explicit form:
Substituting in the previous formula provides
and we just need to replace q by
in order to achieve the proof.
Checking directly the proposition is not evident even when
as we have:
but cannot be done by hand when
.
口
For an arbitrary vector bundle E and involutive system
, we may define the r-prolongations
and their respective symbols
defined from
where
is the vector bundle of q-symmetric covariant tensors. Using the Spencer δ-map, we now recall the definition of the Spencer bundles:
and of the Janet bundles:
When
, we may obtain by induction on r the following fundamental diagram I relating the second linear Spencer sequence to the linear Janet sequence with epimorphisms
:
Chasing in the above diagram, the Spencer sequence is locally exact at
if and only if the Janet sequence is locally exact at
because the central sequence is locally exact (See [7] [13] [25] for more details). In the present situation, we shall always have
. The situation is much more complicate in the nonlinear framework and we provide details for a later use.
Let
be a section of
satisfying the same CC as
, namely
. As
is a quotient of
, we may find a section
such that:
Similarly, as
is a natural bundle of order q, then
is a natural bundle of order
and we can find a section
such that:
and we are facing two possible but quite different situations:
• Eliminating
, we obtain:
and thus
over the target if we set
over the source, even if
may not be a section of
. As
is killed by
, we have related cocycles at
in the Janet sequence over the source with cocycles at
or
over the target.
• Eliminating
, we obtain successively:
where we have over the source:
However, we know that
is associated with
and is thus not affected by
which projects onto
. Hence, only
is affected by
in a covariant way and we obtain therefore over the source:
where
. It follows that
with
in the first non-linear Spencer sequence for
.
We invite the reader to follow all the formulas involved in these technical results on the next examples. Of course, whenever
is formally integrable and
is a lift of
, then we have
and
because
.
EXAMPLE 4.6: In the case of Riemannian structures, we have
because we deal with a non-degenerate metric
with
and may introduce
. We have by definition
that we shall simply write
and obtain therefore:
Our purpose is now to compute the expression:
In order to eliminate the derivatives of
over te target we may multiply the first equation by B and substract from the second while using the fact that
with
in order to get:
These results can be extended at once to any tensorial geometric object but the conformal case needs more work and we let the reader treat it as an exercise. He will discover that the standard elimination of a conformal factor is not the best way to use in order to understand the conformal structure which has to do with a tensor density and no longer with a tensor.
In the non-linear case, the non-linear CC of the system
defined by
only depend on the differential invariants and are exactly the ones satisfied by
in the sense that they have the same Vessiot structure constants whenever
is formally integrable, in particular involutive as shown in Example 2.7. Accordingly, we can always find
over
. In the linear case, the procedure is similar but slightly simpler. Indeed, if
is an involutive Lie operator, we may consider only the initial part of the fundamental diagram I:
and study the linear inhomogeneous involutive system
with
and
. If we pick up any lift
of
and chase, we notice that
is such that
.
EXAMPLE 4.7: In the Example 2.7, using the involutive system
, we have
and the fundamental diagram I:
with fiber dimensions:
It is important to point out the importance of formal integrability and involution in this case. For this, let us start with a 1-form
, denote its variation by
and consider only the linear inhomogeneous system
with no CC for A. If the ground differential field is
with commuting derivations
, let us choose
,
. As a lift
of A, we let the reader check that we may choose in K:
Using one prolongation, we have:
If
, we may denote its variation by B and get at once
. Such a result is contradicting our initial choice
and we cannot therefore find a lift
of
. Hence, we have to introduce the new geometric object
with
and CC
leading to
while using the previous diagrams. We can therefore lift
to
by choosing in K:
However, we have now to add:
and lift
to
over
by choosing in K:
The image of the Spencer operator is
that is to say:
and we check that
, namely:
a result which is not evident at first sight and has no meaning in any classical approach because we use sections and not solutions.
Now, if
are such that
, it follows that
such that
and the new
differs from the initial
by a gauge transformation.
Conversely, let
be such that
. It follows that
and one can find
such that
providing
.
PROPOSITION 4.8: Natural transformations of
over the source in the nonlinear Janet sequence correspond to gauge transformations of
or
over the target in the nonlinear Spencer sequence. Similarly, the Lie derivative
in the linear Janet sequence corresponds to the Spencer operator
or
in the linear Spencer sequence.
With a slight abuse of language
when
and we get
that is
and
.
Passing to the infinitesimal point of view, we obtain the following generalization of Remark 3.12 which is important for applications.
COROLLARY 4.9:
.
Recapitulating the results so far obtained concerning the links existing between the source and the target points of view, we may set in a symbolic way:
In order to help the reader maturing the corresponding nontrivial formulas, we compute explicitly the case
and let the case n arbitrary left to the reader as a difficult exercise that cannot be achieved by hand when
:
EXAMPLE 4.10: Using the previous formulas, we have
,
and:
The delicate point is that we have successively:
When
, we obtain therefore the simple groupoid composition formulas
and thus:
Using indices in arbitrary dimension, we get successively:
As a very useful application, we obtain successively:
where sections of jet bundles are used in an essential way, and the important lemma:
LEMMA 4.11: When the transformation
is invertible with inverse
, we have the fundamental identity over the source or over the target:
EXAMPLE 4.12: We proceed the same way for studying the links existing between
over the source,
over the target and the nonlinear Spencer operator. First of all, we notice that:
and the components of
thus factor through linear combinations of the components of
. After tedious computations, we get successively when
:
These formulas agree with the successive constructive/inductive identities:
showing that
is linearly depending on
and we finally get:
while using successively the relations
,
,
and so on when
is the inverse of
, in a coherent way with the action of
on
which is described as follows:
Restricting these formulas to the affine case defined by
, we have thus
. It follows that
on one side and
in a coherent way. It is finally important to notice that these results are not evident, even when
, as soon as second order jets are involved.
We shall use all the preceding formulas in the next example showing that, contrary to what happens in elasticity theory where the source is usually identified with the Lagrange variables, in both the Vessiot/Janet and the Cartan/Spencer approaches, the source must be identified with the Euler variables without any possible doubt.
EXAMPLE 4.13: With
and the finite OD Lie equation
with
and corresponding Lie operator
over the source, we have:
Differentiating once the first equation and substracting the second, we obtain therefore:
whenever
. Finally, setting
, we get over the target:
Differentiating
in order to obtain
, we get over the source:
We may summarize these results as follows:
We invite the reader to extend this result to an arbitrary dimension
.
EXAMPLE 4.14: The case of an affine stucture needs more work with
. Indeed, let us consider the action of the affine Lie group of transformations
with
acting on the target
considered as a copy of the real line X. We obtain the prolongations up to order 2 of the 2 infinitesimal generators of the action:
There cannot be any differential invariant of order 1 and the only generating one of order 2 can be
. When
we get successively
,
and
transforms like
a result providing the bundle of geometric objects
with local coordinates
and corresponding transition rules. For any section
, we get the Vessiot general system
of second order finite Lie equations
which is already in Lie form and relates the jet coordinates
of order 2. The special section is
and we may consider the automorphic system
obtained by introducing any second order section
, for example
providing
. It is not at all evident, even on such an elementary example, to compute the variation
induced by the previous formulas and to prove that, like any field quantity, it only depends on
on the condition to use only source quantities. For this, setting
, varying and substituting, we obtain:
Now, linearizing the preceding Lie equation over the identity transformation
, we get the Medolaghi equation:
and the striking formula
over the source for an arbitrary
. We finally point out the fact that, as we have just shown above and contrary to what the brothers Cosserat had in mind, the first order operators involved in the nonlinear Spencer sequence have strictly nothing to do with the operators involved in the nonlinear Janet sequence whenever
. For example, in the present situation,
has nothing to do with
. Similarly, using the comment before example 4.7 in the linear framework, we have the first order Spencer operator
on one side and the second order Lie operator
on the other side.
The next delicate example proves nevertheless that target quantities may also be used.
EXAMPLE 4.15: In the last example, depending on the way we use
on the source or
on the target, we may consider the two (very different) Medolaghi equations:
Now, starting from the single OD equation
in sectional notations, we may successively differentiate and prolongate once in order to get:
Substracting the second from the first as a way to eliminate
, we obtain a linear relation involving only the components of the nonlinear Spencer operator in a coherent way with the theory of nonlinear systems, namely:
At first sight it does not seem possible to know whether we have a linear combination of the components of
or of the components of
. However, if we come back to the original situation
, we have eliminated
over the source and we are thus only left with
over the target. Hence it can only depend on
and we find indeed the striking relation:
provided by the simple second order Medolaghi equation
over the target. It is essential to notice that no classical technique can provide these results which are essentially depending on the nonlinear Spencer operator and are thus not known today.
EXAMPLE 4.16: The above methods can be applied to any explicit example. The reader may treat as an exercise the case of the pseudogroup of isometries of a non degenerate metric which can be found in any textbook of continuum mechanics or elasticity theory, though in a very different framework with methods only valid for tensors. With the previous notations, let
with
and consider the following nonlinear system
with
. One obtains therefore:
and thus the same recapitulating formulas linking the source and target variations:
It is also difficult to compute or compare the variational formulas over the source and target in the nonlinear Spencer sequence, even when
and
( [29] ).
EXAMPLE 4.17: Let us prove that the explicit computation of the gauge transformation is at the limit of what can be done with the hand, even when
. We have successively:
and thus:
Setting
and passing to the limit when
, we finally obtain:
If we use the standard euclidean metric
, we may thus introduce the pure 1-form
. We should consider the defining formula
and have to introduce the additional term
which is only leading to the additional infinitesimal term
because
. We finally obtain:
and this result can be easily extended to an arbitrary dimension with the formula:
Comparing this procedure with the one we have adopted in the previous exampes, we have:
However, taking into account the formulas
and
, we also get:
Working over the target is more difficult. Indeed, we have successively ( care to the first step):
More generaly, we let the reader prove that the variation of
over the target (respectively the source) is described by “minus” the same formula as the variation of
over the source (respectively the target). In any case, the reader must not forget that the word “variation” just means that the section
is changed, not that the source is moved. Accordingly, getting in mind this example and for simplicity, we shall always prefer to work with vertical bundles over the source, closely following the purely mathematical definitions, contrary to Weyl ( [3], §28, formulas (17) to (27), p 233-236). The reader must be now ready for comparing the variations of
and
.
In order to conclude this section, we provide without any proof two results and refer the reader to ( [7] ) for details.
PROPOSITION 4.18: Changing slightly the notation while setting
, we have:
where
acts on
and
acts on
. It follows that gauge transformations exchange the solutions of
among themselves.
COROLLARY 4.19: Denoting by
the cyclic sum, we have the so-called Bianchi identity:
5. Applications
Before studying in a specific way electromagnetism and gravitation, we shall come back to Example 4.10 and provide a technical result which, though looking like evident at first sight, is at the origin of a deep misunderstanding done by the brothers Cosserat and Weyl on the variational procedure used in the study of physical problems (Compare to [14] ).
Setting
for simplicity while using Lemma 4.11 and the fact that the standard Lie derivative is commuting with any diffeomorphism, we obtain at once:
The interest of such a presentation is to provide the right correspondence between the source/target and the Euler/Lagrange choices. Indeed, if we use the way followed by most authors up to now in continuum mechanics, we should have source = Lagrange, target = Euler, a result leading to the conservation of mass
when
is the original initial mass per unit volume. We may set
and obtain therefore
, a choice leading to:
but the concept of “variation” is not mathematically well defined, though this result is coherent with the classical formulas that can be found for example in ( [4] [9] ) or ( [3], (17) and (18) p 233, (20) to (21) p 234, (76) p 289, (78) p 290) where “points are moved”.
On the contrary, if we adopt the unusual choice source = Euler, target = Lagrange, we should get
, a choice leading to
and thus:
which is the right choice agreeing, up to the sign, with classical formulas but with the important improvement that this result becomes a purely mathematical one, obtained from a well defined variational procedure involving only the so-called “vertical” machinery. This result fully explains why we had doubts about the sign involved in the variational formulas of ( [4], p. 383) but without being able to correct them at that time. We may finally revisit Lemma 4.11 in order to obtain the fundamental identity over the source:
which becomes the conservation of mass when
and
.
In addition, as many chases will be used through many diagrams in the sequel, we invite the reader not familiar with these technical tools to consult the books ( [30] [31] ) that we consider as the best references for learning about homological algebra. A more elementary approach can be found in ( [32] ) that has been used during many intensive courses on the applications of homological algebra to control theory. As for differential homological algebra, one of the most difficult tools existing in mathematics today, and its link with applications, we refer the reader to the various references provided in ( [33] ).
Finally, for the reader interested by a survey on more explicit applications, we particularly refer to ( [2] [34] [35] [36] ) for analytical mechanics and hydrodynamics, ( [5] [37] [38] ) for coupling phenomenas, ( [36] [39] [40] ) for the foundations of Gauge Theory, ( [36] [41] ) for the foundations of General Relativity.
A) POINCARE, WEYL AND CONFORMAL GROUPS
When constructing inductively the Janet and Spencer sequences for an involutive system
, we have to use the following commutative and exact diagrams where we have set
and used a diagonal chase:
It follows that the short exact sequences
are allowing to define the Janet and Spencer bundles inductively. If we consider two involutive systems
, it follows that the kernels of the induced canonical epimorphisms
are isomorphic to the cokernels of the canonical monomorphisms
and we may say that Janet and Spencer play at see-saw because we have the formula
.
When dealing with applications, we have set
and considered systems of finite type Lie equations determined by Lie groups of transformations. Accordingly, we have obtained in particular
when comparing the classical and conformal Killing systems, but these bundles have never been used in physics. However, instead of the classical Killing system
defined by the infinitesimal first order PD Lie equations
and its first prolongations
defined by the infinitesimal additional second order PD Lie equations
or the conformal Killing system
defined by
and
but we may also consider the formal Lie derivatives for geometric objects:
We may now introduce the intermediate differential system
defined by
and
, for the Weyl group obtained by adding the only dilatation with infinitesimal generator
to the Poincaré group. We have the relations
and the strict inclusions
when
,
,
but we have to notice that we must have
for the conformal system and thus
if we do want to deal again with an involutive second order system
. However, we must not forget that the comparison between the Spencer and the Janet sequences can only be done for involutive operators, that is we can indeed use the involutive systems
but we have to use
even if it is isomorphic to
. Finally, as
and
, the first Spencer operator
is induced by the usual Spencer operator
and thus projects by cokernel onto the induced operator
. Composing with
, it projects therefore onto
as in EM and so on by using the fact that
and d are both involutive or the composite epimorphisms
. The main result we have obtained is thus to be able to increase the order and dimension of the underlying jet bundles and groups as we have ( [29] ):
that is
when
and our aim is now to prove that the mathematical structures of electromagnetism and gravitation only depend on the second order jets.
With more details, the Killing system
is defined by the infinitesimal Lie equations in Medolaghi form with the well known Levi-Civita isomorphism
for geometric objects:
We notice that
and refer the reader to ( [27] ) for more details about the link between this result and the deformation theory of algebraic structures. We also notice that
is formally integrable and thus
is involutive if and only if
has constant Riemannian curvature along the results of L. P. Eisenhart ( [26] ). The only structure constant c appearing in the corresponding Vessiot structure equations is such that
and the normalizer of
is
if and only if
. Otherwise
is of codimension 1 in its normalizer
as we shall see below by adding the only dilatation. In all what follows,
is assumed to be flat with
and vanishing Weyl tensor.
The Weyl system
is defined by the infinitesimal Lie equations:
and is involutive if and only if
. Introducing for convenience the metric density
, we obtain the Medolaghi form for
with
:
Finally, the conformal system
is defined by the following infinitesimal Lie equations:
and is involutive if and only if
or, equivalently, if
has vanishing Weyl tensor.
However, introducing again the metric density
while substituting, we obtain after prolongation and division by
the second order system
in Medolaghi form and the Levi-Civita isomorphim
restricts to an isomorphism
if we set:
Contracting the first equations by
we notice that
is no longer vanishing while, contracting in k and j the second equations, we now notice that
is no longer vanishing. It is also essential to notice that the symbols
and
only depend on
and not on any conformal factor.
The following Proposition does not seem to be known:
PROPOSITION 5.A.1:
is the only symmetric
-connection with vanishing trace.
Proof: Using a direct substitution, we have to study:
Multiplying by
, we have to study:
or equivalently:
that is to say:
Now, we have:
Finally, taking into account that
is a
-connection, we have:
Hence, collecting all the remaining terms, we are left with
.
As for the unicity, it comes from a chase in the commutative and exact diagram:
obtained by counting the respective dimensions with
and
while checking that