Homological Solution of the Lanczos Problems in Arbitrary Dimension ()
1. Introduction
Introducing the Lanczos potential
as a 3-tensor satisfying the algebraic relations:
(1)
Lanczos claimed to have parametrized the Riemann tensor R through the relation:
(2)
where
is the covariant derivative. However, even if we can easily verify the algebraic conditions that must be satisfied by a Riemann candidate, namely:
(3)
the generating compatibility conditions (CC) of the underlying operator for the left member cannot be the (second) Bianchi identities:
(4)
which are produced by the well known parametrization described by the Riemann operator acting on a perturbation
of the background metric
, that is, when
is the Minkowski metric:
This contradiction can also be checked directly by substitution because we have:
Then Lanczos tried to parametrize the Weyl tensor C, only knowing the algebraic conditions that must be satisfied by a Weyl candidate, namely:
(5)
The last condition is reducing the number of linearly independent components from 20 to 10 for space-time, that is when the dimension is
, but the previous contradiction still holds.
50 years ago, while the author of this paper was “professional” in GR under the leadership of Prof. A. Lichnerowicz, he became familiar with the Lanczos problems. Since that time, he had no wish at all to enter this kind of “private domain” where a few persons were writing alternatively. Also, all the papers were covered with “computations” involving many technical formulas, one paper using computer algebra, another Gröbner bases, another Cartan exterior calculus, another Janet bases and so on during these 50 years. Finally, the author started to have doubts on the differential geometric conformal framework. A long time after, in 2001 and for quite different reasons, namely revisiting controllability in control theory on one side and the intrinsic proof of the impossibility to find potentials for Einstein equations in vacuum (contrary to the general dream of the GR community till now!) on the other side, the author wrote a big book, published by Kluwer (See Zbl 1079.93001). At this moment, being more familiar with differential homological algebra and the “parametrization problem”, the way towards the Lanczos problems became easier and we present it in four steps:
1) In the only dimension
considered indeed by Lanczos, the so-called Lanczos “potential”
has
components. As they must be related by the 4 additional relations
, we get 20 independent components, namely the number of (second) Bianchi identities. We claim that only the knowledge of the Spencer δ-cohomology allows to exhibit the proper identification with the Bianchi candidate vector bundle
in the short exact sequence of vector bundles where
is the symbol of the Killing equations:
and
when
. However, speaking about “potential” also means “parametrization”, ... but of what?. Here comes the main confusion of Lanczos, familiar with electromagnetism (EM) while using mainly quadratic lagrangians with the Riemann tensor in place of the EM field F such that
, with the Bianchi identities as differential constraint in the corresponding variational calculus. The operator that must be parametrized indeed by means of the formal adjoint of the Bianchi operator is thus the formal adjoint of the Riemann operator, going now backwards, that is from right to left in the adjoint sequence of the Killing resolution. Such a construction, using quite difficult results (side changing functor) of homological algebra, could not have been discovered by Lanczos and followers as such tools have only been available after 1995 through the works of pure mathematicians not interested by applications. As a byproduct, this new framework is allowing in particular to replace technical formulas by diagram chasing, without ANY formula.
2) As for the differential sequence involved, people use to refer to E. Calabi or H. Goldschmidt who effectively gave “ad hoc” results a long time ago, around 1965 but are not quoted because they cannot be used for our purpose. However, one should rather refer to the author’s many books (In particular the first one, published in 1978 and translated by MIR, Moscow, in 1983) in order to discover that such a differential sequence can be constructed for any Lie pseudogroup by using the Vessiot structure equations, still not known after 125 years!. This sequence is much more useful than the sequence constructed by Goldschmidt-Spencer who have never been aware of the work of Vessiot and has nothing to do with the work of Cartan who ignored this work.
3) Going from left to right in the differential sequence, the Riemann operator is generating the compatibility conditions (CC) of the Killing operator and the Bianchi operator is generating the CC of the Riemann operator. However, going backwards, that is to say from right to left, by taking the respective adjoint operators, it is not true in general that the successive operators have the same property. This remark has been the reason for introducing the differential extension modules in homological algebra, the aim being to study the possible “gaps” just described. By chance, in this case it works, contrary to what could happen in the example given of the infinite dimensional Lie pseudogroup of contact transformations. Indeed, in such a case, the differential sequence is existing because the Vessiot structure equation has only one Vessiot structure constant (totally unknown) like in the example of the constant Riemannian curvature. The case of unimodular contact transformations is even more difficult with two Vessiot structure constants but no link with any Maurer-Cartan equation.
4) Last but not least, the case of conformal isometries is even much more tricky:
First of all, it is clear that, when
, the first order Killing operator must be replaced by the first order conformal Killing operator while the second order Riemann operator must be replaced by the second order Weyl operator. However, ... what operator should be used in place of the first order Bianchi operator?. No chance, because we shall discover that, in dimension 4, it is a second order operator!. Such a result, neither known nor acknowledged up to now, has been checked with computer algebra by the author’s former PhD student A. Quadrat (INRIA) and appeared in book form ( [1] ). Acordingly, there does not exist a single reference on such a result. Needless to say that, in any other smaller or higher dimension, this material could not have been known by Lanczos himself or followers, a fact justifying the initial claim on the use of the Spencer δ-cohomology. For example, when
, the analogue of the Riemann operator is a third order operator with first order CC.
In order to recapitulate the above procedure, we have the following differential sequence, indicating below the fiber dimensions of the vector bundles involved with
:
(6)
where
is the sheaf of Killing vector fields for the Minkowski metric. Defining the operators:
we shall prove that Lanczos was in fact dreaming to construct the adjoint differential sequence:
(7)
where
for any vector bundle E where
is obtained from E by inverting the transition rules when changing local coordinates, exactly like T and
. Accordingly, all the problem will be to prove that each operator is indeed parametrized by the preceding one. As we shall see, the conformal situation could be treated similarly while starting with the conformal Killing operator followed by the Weyl operator and replacing each classical vector bundle F by the corresponding conformal bundle
. However, this will lead to a true nonsense because we shall discover that the analogue of the Weyl operator is of order 3 when
while the analogue of the Bianchi operator is of order 2, ... just when
. These striking results have been confirmed by computer algebra and the reader can even find the details in book form ( [1] ). It follows that both the Riemann and Weyl frameworks of the Lanczos potential theory must be entirely revisited. The aim of this paper is to overcome these problems by using differential homological algebra.
C. Lanczos (1893-1974) wrote three main papers (1939, 1949, 1962) on the search of potentials for parametrizing the Riemann and Weyl tensors ( [2] [3] [4] [5] ) and we refer the reader to the nice historical survey ( [6] ) for more details. However, Lanczos has been invited in 1962 by Prof. A. Lichnerowicz to lecture in France and this last lecture has been published in french. Getting inspiration from what happens in electromagnetism (EM) where the geometrical first set of Maxwell equations
when
is a closed 2-form can be parametrized by
for an arbitrary potential
with standard notations (See [7] for details), Lanczos created the concept of “candidate” while noticing that the Riemann and Weyl 4-tensors must “a priori” satisfy algebraic relations reducing the number of their components
and
respectively to 20 and 10 when
. Now, we have proved in many books ( [8] - [13] ) or papers ( [14] [15] [16] [17] [18] ) that it is not possible to understand the mathematical structure of the Riemann and Weyl tensors, both with their splitting link, without the following four important comments:
● The results discovered by E. Vessiot as early as in 1903 ( [19] ) are still neither known nor even acknowledged today, though they generalize the constant Riemaniann curvature condition discovered 25 years later by L. P. Eisenhart ( [20] ). They also allow to understand the direct link existing separately between the Riemann tensor and the Lie pseudogroup of isometries of a non-degenerate metric on one side or between the Weyl tensor and the group of conformal isometries of this metric on another side. Vessiot proved that, for any Lie pseudogroup
the pseudogroup of all local diffeomorphisms, there is a geometric object
, may be of a high order q large enough and not of a tensorial nature, which is characterizing
in the sense that one has:
where
is a fundamental set of differential invariants of order q and
must satisfy certain (non-linear in general) integrability conditions of the form:
called Vessiot structure equations, depending on a certain number of Vessiot structure constants c eventually satisfying algebraic Jacobi conditions
and we let the reader compare this situation to the Riemann or contact cases ( [1] ). We want to point out that these structure equations were perfectly known by E. Cartan (1869-1951) who never said that these results were at least competing with or even superseding the corresponding Cartan structure equations that he has developed about at the same time for similar purposes. The underlying reason is of a purely personal origin related to the differential Galois Theory within a kind of “mathematical affair” involving the best french mathematicians of that time. The original letters, given to the author of this paper by M. Janet, a friend of E. Vessiot, have ben published in ( [10] ) and have been put as a deposit in the main library of Ecole Normale Supérieure in Paris for future historical studies.
● A nonlinear operator with second member does not in general admit CC, ... unless it corresponds to the defining equations in Lie form of a Lie pseudogroup and the CC are the Vessiot structure equations in that case with structure constants determined by the chosen geometric object (compare again to the Riemannian geometry). We have shown in many books already quoted that, if
is a Lie operator and we set
, with bracket
induced by the ordinary bracket of vector fields, then the system
is the linearization of a non-linear version when
is a perturbation of
(twice the infinitesimal deformation tensor in elasticity) along the formula:
Similarly, we can choose for the generating CC
the linearization of a non-linear version described by the Vessiot structure equations:
that is exactly what we did for the flat Minkowski metric. However, Lanczos has been studying the CC
of
, ignoring that, contrary to the previous situation,
almost never comes from a linearization. It is therefore quite strange to discover that Lanczos never discovered that what he was doing with
and
while using quadratic Lagrangians in R, was exactly what is done in any textbook of elasticity or continum mechanics with
and
while using quadratic Lagrangians in
( [1] [7] [13] [16] ). We do believe that Lanczos was too much obsessed by comparing R in GR to F in EM. Like in any good crime story, the solution will be given in the last section and could not have been given before by any classical approach.
● The last invited lecture published in 1962 by Lanczos on his potential theory is never quoted because it is in French. Comparing it with a commutative diagram in a recently published paper on gravitational waves ( [18] ), we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality when he introduced his tentative 3-tensor potential. Our final purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment,
2. Mathematical Tools
2.1. Differential Sequences
In view of the many examples that will be presented in this paper, it becomes clear that there is a need for classifying the properties of systems of PD equations in a way that does not depend on their presentations and this is the purpose of differential homological algebra along the scheme:
in order to show that certain concepts, which are clear in one framework, may become quite obscure in the others and conversely, like the formal integrability and torsion concepts for example.
When E is a vector bundle over X and we have a system of order q on E, say
, we can introduce the canonical projection
and define a linear differential operator
. When
is given, the compatibility conditions for solving
can be described in operator form by
and so on. In general, if a system is not formally integrable, it is possible to obtain a formally integrable system, having the same solutions, by “saturating’’ conveniently the given PD equations through the adjunction of new PD equations obtained by various prolongations/projections (PP) and such a procedure must absolutely be done before looking for the compatibility conditions ( [21] [22] ).
In order to study differential modules, we shall simply forget about changes of coordinates and only consider trivial bundles. If K is a differential field with n commuting derivations
(Say
or
in the usual examples), we denote by
the subfield of constants of K, that is the set of elements killed by the n derivations (Say
in the usual examples). If
are formal derivatives (pure symbols in computer algebra packages!) which are only supposed to satisfy
in the operator sense for any
, we may consider the (non-commutative) ring
of differential operators with coefficients in K. If now
is a set of differential indeterminates, we let D act formally on y by setting
for any multi-index
and set
. We may also set
for
. Denoting by
the
subdifferential module generated by all the given OD or PD equations and all their formal derivatives, we may finally introduce the D-module
by residue. Here we recall that M is a module over a ring A or an A-module if
. We may introduce as usual the torsion submodule
and we say that M is a torsion module if
or that M is torsion-free if
.
It is not evident at all to exhibit the link existing between these two approaches and we proceed as follows. First of all, the ring D is filtred by the order of the operators and we have the filtration or inductive limit
. Moreover, it is clear that D, as an algebra, is generated by
and
with
if we identify an element
with the vector field
of differential geometry, but with
now. As a byproduct, the differential module
is also filtred by the order and we obtain an induced filtration or inductive limit
with
provided by the prolongations. Now, if we suppose that the system
is formally integrable (FI), that is all the OD or PD equations of order
are obtained by using only r prolongations, then we have the projective limit
obtained by successive jet projections. We have the following crucial technical proposition ( [13] [17] ):
Proposition 2.A.1:
is a differential module for the Spencer operator and we have a bijective correspondence
over K because K is a field.
Proof: for any
and
, we may set for any
:
and check that we have successively with
:
a result leading to
in the operator sense. Setting finally
with a slight abuse of notations when using the same notation
for the residue instead of the standard
. Setting
, it follows that R is a differential module for the law:
(8)
and we have
.
Q.E.D.
Through this paper, we shall only deal with linear differential operators. However, as explained in ( [9] [23] ), there is a nonlinear counterpart using the nonlinear Janet sequence coming from the Vessiot structure equations and a nonlinear Spencer sequence. The vertical machinery involved, that is a systematic use of fibered manifolds and vertical bundles, is much more difficult though we have chosen the notations of this paper in such a way that the interested reader may easily adapt them. As for the Vessiot structure equations first found in 1903 ( [19] ), they have been totally ignored during more than one century for reasons that are not scientific at all (See the original letters presented in [10] for explanations). Though we have written this paper in a rather self-contained way while using rather standard notations, the reader may refer to ( [24] [25] [26] ) for the differential geometric background, to ( [27] [28] ) for the elements of homological algebra needed through the various diagrames presented and to ( [29] [30] [31] ) for the main (difficult) concepts of differential homological algebra.
Collecting all the results so far obtained, if a differential operator
is given in the framework of differential geometry, we may keep the same operator matrix in the framework of differential modules which are left modules over the ring D of linear differential operators. We may also apply duality over D, that is apply
, provided we deal now with right differential modules or use the operator matrix of
and deal again with left differential modules obtained through the
conversion procedure. In actual practice, it is essential to notice that the new operator matrix may be quite different from the only transposed of the previous operator, even if we are dealing with constant coefficients.
Definition 2.A.2: If a differential operator
is given, a direct problem is to find (generating) compatibility conditions (CC) as an operator
such that
. Conversely, given
, the inverse problem will be to look for
such that
generates the CC of
and we shall say that
is parametrized by
... if such an operator
is existing!
Remark 2.A.3: Solving the direct problem (Janet, Spencer) is necessary for solving the inverse problem. However, though the direct problem always has a solution, the inverse problem may not have a solution at all and the case of the Einstein operator is one of the best non-trivial PD counterexamples (Compare [32] [33] [34] ). It is rather striking to discover that, in the case of OD operators, it took almost 50 years to understand that the possibility to solve the inverse problem was equivalent to the controllability of the corresponding control system ( [34] ) and the situation is similar in GR as the above result has been first found in 1995 ( [32] ).
As
, any operator is the adjoint of a certain operator and we recall that the double duality test needed in order to check whether
or not and to find out a parametrization if
when M is defined by
has 5 steps which are drawn in the following diagram where
generates the CC of
and
generates the CC of
:
(9)
Theorem 2.A.4: We have
parametrized by
in the differential module framework when N is defined by
. These results do not depend on the finite free presentations ofM or N (See [34] [35] ) for more details).
Corollary 2.A.5: In the differential module framework, if
is a finite free presentation of
and we already know that
by using the preceding test and Theorem, then we may obtain an exact
sequence
of free differential modules where
is the parametrizing operator, both with an inclusion
by chasing. However, there may exist other parametrizations
called minimal parametrizations
such that
is a torsion module and we have thus
(See [18] and [35] ).
As shown by the next examples, the main difficulty met in OD or PD applications is that
may not be formally integrable at all, even if
is involutive (See [12] for other examples).
Example 2.A.6: (Double pendulum) If a rigid bar is able to move horizontally with reference position x and we attach two pendula with respective length
and
making the (small) angles
and
with the vertical, the corresponding involutive control system is:
where g is the gravity. Multiplying these OD equations by two test functions
and integrating by parts, we get the adjoint system:
Multiplying the second equation by
, the third by
while using the first, we obtain the zero order OD equation
. Differentiating twice this time and substituting, we obtain the new zero order OD equation
. The determinant of the system of two zero order equations is then seen to be exactly
. It follows that the system is controllable if and only if
is different from
, a fact that the reader can check easily when moving the bar conveniently. If one length depends on time, the corresponding controllability condition cannot be obtained without computer algebra, even on such an elementary control system. The totally unexpected fourth order parametrization of the control system when it is controllable is:
●
:
●
,
.
It follows that the controllability of a control system is a “built in” property of this system that does not depend on the choice of the control variables, contrary to a tradition still existing in the control community (See Zbl 1079.93001 for a review). We invite the reader to use the Kalman approach that can be found in any control textbook today and to compare (See [12] or [34] for details).
Example 2.A.7: (Einstein equations) If
is the Einstein operator which is self-adjoint, then
is also the Einstein operator,
is the Cauchy operator and
is thus the Killing operator. It follows that
is the Riemann operator according to the Introduction. Using the previous theorem, any component of the Weyl tensor becomes a torsion element killed by the Dalembert operator as a “modern” description of the so-called Lichnerowicz waves (as they are called in France!) ( [16] ).
2.2. Variational Calculus
Having in mind “Optimal Control Theory” while using the notations of the previous Formal Test, let us assume that the two differential sequences:
(10)
are formally exact, that is
generates the CC of
and
generates the CC of
, namely
is a potential for
and
is a potential for
. We may consider a variational problem for a cost function or lagrangian
under the linear OD or PD constraint described by
.
● Introducing convenient Lagrange multipliers
while setting
for simplicity, we must vary the integral:
Integrating by parts, we obtain the Euler-Lagrange (EL) equations:
to which we have to add the constraint
obtained by varying
independently. If
is an injective operator, in particular if
is formally surjective (no CC) while
as inOD optimal control and M is torsion-free, thus free ( [12] ) or
and M is projective, then one can obtain
explicitly and eliminate it by substitution. Otherwise, using the CC
of
, we have to study the formal integrability of the combined system:
which may be a difficult task ( [12], Introduction and Chapter VI).
● However, we may also transform the given variational problem with constraint into a variational problem without any constraint if and only if the differential constraint can be parametrized. Using the parametrization of
by
, we may vary the integral:
whenever
and integrate by parts for arbitrary
in order to obtain the EL equations:
in a coherent way with the previous approach.
As a byproduct, if the field equations
can be parametrized by a potential
through the formula
, then the induction equations
can be obtained by duality in a coherent way with the double duality test, on the condition to know what sequence must be used. However, we have yet proved in ( [9] [10] [13] [36] ) that the Cauchy stress equations must be replaced by the Cosserat couple-stress equations and that the Janet sequence (only used in this paper) must be thus replaced by the Spencer sequence. Accordingly, the work of Lanczos ( [2] [3] [4] [5] ) and followers ( [37] [38] [39] [40] [41] ), using either exterior calculus, Janet and Gröbner bases or Pommaret bases, has been based on a confusion between fields and inductions on one side, but also between the Janet sequence and the Spencer sequence. By chance, as we always refer to intrinsic concepts like the extension modules that do not depend on the differential sequence used, all the results that will be presented can be adapted at once to the systematic use of the Spencer sequence in place of the Janet sequence.
3. Riemann/Lanczos Problem
The last invited lecture published in 1962 by Lanczos on his potential theory is never quoted because it is in French ( [5] ). Comparing it with a commutative diagram in a recently published paper on gravitational waves ( [16] ), we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the formal theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators.
When K is a differential field containing the field
of rational numbers and
are commuting formal derivatives, we may introduce the ring (which is even an integral domain)
of differential operators with coefficients in K. Accordingly, if a differential module M with torsion submodule
is defined by an operator
with coefficients in K, we may introduce the differential extension modules
and
for
. We have the long exact ker/coker long exact sequence of (left) differential modules (See [12] or [34] for details):
(11)
where the morphism
is defined by
and the adjoint differential module
is defined by the adjoint operator
. Then M is torsion-free if and only if
, that is
is a monomorphism or, equivalently,
can be parametrized by the operator
when
generates the compatibility conditions (CC) of
. Finally M is reflexive if and only if, in addition,
is an epimorphism, that is we have also
or, equivalently,
can be parametrized again by
when
generates the CC of
. As we have
and though this is not evident at first sight by exchanging M with N, we may also say that
if
generates the CC of
whenever
generates the CC of
and, similarly, that
when
generates the CC of
whenever
generates the CC of
. We shall provide an explicit description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their respective adjoint operators.
We now consider with details the Riemann/Lanczos problem which is at the same time the simplest of the two Lanczos problems as it can be solved in arbitrary dimension
but is also an example of the successive confusing works that have been done during the last fifty years as we already said. According to the last Section, the starting motivation seems absolutely natural at first. Indeed, considering the Killing operator
where
is the Lie derivative with respect to
and
is a nondegenerate metric with
. Accordingly, it is a Lie operator with
and we denote simply by
the set of solutions with
. Now, as we have explained many times, the main problem is to describe the CC of
in the form
by introducing the so-called Riemann operator
, using the standard notations that can be found at length in our many books ( [8] - [13] ) or papers ( [14] [42] ). We advise the reader to follow closely the next lines and to imagine why it will not be possible to repeat them for studying the Weyl/Lanczos problem. Introducing the Levi-Civita isomorphism
and the
Christoffel symbols
where
is the inverse matrix of
, we get
:
if we use jet coordinates with sections
transforming like
. The system
has a symbol
depending only on
with
and is finite type because its first prolongation is
. It cannot be thus involutive as can be seen directly on the following Janet board for finding a Pommaret basis when
and
is the euclidean metric:
Indeed, the only dot appearing in the board cannot provide any CC for the symbol
and we have therefore the short exact sequence:
by using the fact that
and counting the common dimension
, because an epimorphism between two spaces of the same dimension is also a monomorphism and thus an isomorphism. Accordingly, we need to use one additional prolongation and arrive to the:
● First comment: Using now one of the main results to be found in ( [8], ... [12] ), we know that, when
is formally integrable, then the CC of
are of order
where s is the number of prolongations needed in order to get an involutive symbol, that is
in the present situation, a result that should lead to CC of order 2 if
were formally integrable.
As
by counting the dimensions with
and
, we get
. Hence, we understand that the number of CC
of
is equal to the number of components of the Riemann tensor if and only if
is formally integrable, that is if and only if
has constant Riemannian curvature, a result first found by L.P. Eisenhart in 1926 ( [20] ) though in a different setting (See [8] for an explicit modern proof). Such a necessary condition for constructing an exact differential sequence could not have been used by Lanczos because the work of Spencer has only been known after 1970 ( [24] [26] ). Otherwise, if the metric does not satisfy this condition, CC may exist by using the Petrov classification but have no link with the Riemann tensor ( [22] ). We may therefore define the model vector bundle
with
in the sense of Lanczos by the short exact sequence:
A result leading to the operator
and the:
● Second comment: Applying the Spencer operator
to the top line of the preceding diagram, we get the commutative diagram:
Using a diagonal chase, we discover that
is just the Spencer δ-cohomology
at
along the following short exact sequence:
because
and we get the striking formula where the + signs have been replaced by signs:
This result, first found by the author in 1978 ( [8] ), clearly exhibit the two well known algebraic properties of the Riemann tensor. We now understand that Lanczos had in mind to linearize the Riemann tensor over the Minkowski metric, exactly like in GR, in order to construct a Lagrangian as a function of the corresponding linearization
of the Riemann tensor
, transforming the usual variational problem into a variational with a differential constraint described by the Bianchi identities leading to the operator
. As an equivalent alternative approach, his idea was to consider the curvature as a field by itself and construct the lagrangian on this field like in EM while adding the Bianchi identities as a differential constraint by using as many Lagrange multiplier as the number of Bianchi identities, a number not known by combinatorics at the time Lanczos was writing, a result leading to the:
● Third comment: Lanczos, who also knew continuum mechanics as an engineer, just copied the way used in elasticity (EL) and in electromagnetism (EM), for example introducing a Lagrangian as a function of the deformation
while adding a differential constraint described by the vanishing linearized Rieman tensor with therefore as many Lagrange multipliers as the number
of components of the Riemann tensor. It is crucial to notice that the same differential sequence is used one step before, that is with
and
while he was dealing with
and
previously, that is one step ahead in the sequence. We have proved recently that such a procedure is in total contradiction with the piezoelectricity and photoelasticity existing between EL and EM (See the picture in [7] ). It thus remains to exhibit the Bianchi operator exactly as we did for the Riemann operator, with the same historical comments already provided. However, now we know that
is formally integrable (otherwise nothing could be achieved and we should start with a smaller system!), the construction of the linearized Janet-type differential sequence as a strictly exact differential sequence but not an involutive differential sequence because the system
and thus the first order operator
are formally integrable though not involutive as
is finite type with
but not involutive. Doing one more prolongation only, we obtain the first order Bianchi CC from
in the following long exact symbol sequence (See the details below):
or from the short exact sequence:
showing that
( [8] [9] [13] ). We have in particular for
:
and thus
when
, a result leading to:
● Fourth comment: (Double Hodge duality) For an arbitrary n, it is not possible to recognize that one of the algebraic conditions for the Bianchi identity comes from the Spencer δ-map and is again an epimorphism as it was before for defining
, a result obtained by chasing in the commutative diagram obtained by applying
to the long exact symbol sequence finishing with
. It is not evident at all to discover that the modern description of the model vector bundle
is just equivalent to the one provided by Lanczos but only for
. For this, using local coordinates, we have the 4 linear equations with
:
to be compared with the 4 equations for the Lanczos tensor with
, namely:
Before reading the next lemma, we invite the reader to prove ... that they are identical!
Lemma 3.1: These two equations are identical only when
.
Proof: Using Hodge duality a first time, we may rewrite the first ones in the form:
Lowering the index i by means of the Eucldean metric for simplicity and setting
, we get:
Using again the Hodge duality but setting now
and so on, we get:
that is exactly the Lanczos formula, a result showing that, for
only, we discover that
are both killed by
.
We are thus able to exhibit the Lanczos potential
as a 3-tensor satisfying:
(1)
in the short exact sequence
but
this result does not provide any potential because ... the adjoint sequence is going backwards!.
Q.E.D.
Using adjoint operators and adjoint bundles while setting
when E is a vector bundle over X and using the Hodge duality, we obtain the short exact sequences with arrows reversed:
as a way to describe the Lagrange multiplier
in arbitrary dimension.
These results are leading to the:
● Fifth comment: The div-type operator induced (on the right) by the Bianchi operator has strictly nothing to do with the Cauchy operator (namely ad(Killing) on the left), contrary to what is still believed in GR. In addition, we have the:
● Sixth comment: We have proved in ( [38] [39] [40] ) that the usual Cauchy stress equations must be replaced by the Cosserat couple-stress equations or, equivalently, that the Janet sequence must be replaced by the Spencer sequence in a coherent way with the couplings existing between EL and EM. It is also important to notice that, in the non-linear framework, there is no analogue of
in the nonlinear Janet sequence or of D3 in the nonlinear Spencer sequence, a redhibitory reason leading to use only
and
or D1 and D2 both with their formal adjoints.
As a way to conclude this example, we may say that, for any
, the Riemann operator
is parametrizing the Bianchi operator
while the operator
is parametrizing the operator
. Nevertheless, according to ( [15] ), there may exist minimal parametrizations of
with a lower number of potentials equal to
, thus
when
because of the Euler-Poincaré characteristic
(See [18] ).
Remark 3.2: Lanczos has been trying in vain to do for the Bianchi operator what he did for the Riemann operator, a useless but possible “shift by one step to the right” and to do for the Weyl operator what he did for the Riemann operator. However, we shall discover that the dimension
, which is particularly “fine” for the classical Killing sequence, is particularly “bad” for the conformal Killing sequence, a result not known after one century because it cannot be understood without using the Spencer δ-cohomology in the following commutative diagram which is explaining therefore what we shall call the “Lanczos secret”. Following ( [21] ) and the fact that the two central vertical δ-sequences are exact, this diagram allows to construct the Bianchi operator
as generating CC for the Riemann operator
defined by a similar diagram and thus only depends on the symbol
. For the reader not familiar with homological algebra, we provide below the main diagram allowing to construct the Bianchi operator both with the corresponding fiber dimensions when
. In this commutative diagram, all the rows are exact and the columns are exact but eventually the left one:
Using the Spencer cohomology at
, the vector bundle
in this diagram or Riemann candidate in the language of Lanczos, is defined by the short exact sequence:
All the vertical down arrows are δ-maps of Spencer and all the vertical columns are exact but the first, which may not be exact only at
with cohomology equal to
because we have:
A snake-type chase similarly provides the identification
while using again the Spencer cohomology at
. The vector bundle
providing the Bianchi identities is thus defined by the exactness of the top row of the preceding diagram or, equivalently, using the left column, by the short exact sequence:
We conclude this Remark by saying that it is not even easy to discover that the bottom δ-map in the first column on the left is an epimorphism. In order to convince the reader of the powerfulness of these new methods, this result is left a an exercise (Hint: prove that:
by using a circular counterclockwise chase and that
).
Starting with the (classical) Killing operator
defined by
, we obtain successively the following differential sequences for various dimensions:
For example, we have the Euler-Poincaré characteristic:
when
or
when
.
Setting successively
and so on, it follows therefore from the previous study that each operator is parametrizing the following one. Applying double duality and introducing the respective adjoint operators, then
is parametrizing the Beltrami operator
with (canonical) potentials called Lanczos only when
while
is parametrizing the Cauchy operator
with (canonical) potentials called Airy, Beltrami, ... ( [15] [18] ). It must be finally noticed that
is also parametrizing the Cauchy operator ( [16] [18] ).
4. Weyl/Lanczos Problem
Starting now afresh with the conformal Killing operator CK such that
or, equivalently, introducing the metric density
, we have a new operator
defined by
and we obtain successively the following differential sequences for various dimensions
( [43] ):
For example, we have the Euler-Poincaré characteristic:
.
Proceeding exactly as before, we obtain for example when
the following commutative diagram where all the rows are exact and the columns are exact but eventually the left one:
Nevertheless, the same (but very tricky now!) chase as before allows to prove that the bottom δ-map in the first column on the left is again ... an epimorphism, a crucial result indeed, left again as a difficult exercise of diagram chasing (Hint: double step circular chase as before!).
Of course, in view of the dimensions of the matrices involved (up to
), we wish good luck to anybody trying to use computer algebra and refer to the computations done in ( [1] ) that have been done while knowing “a priori “ the dimensions that should be found.
Remark 4.1: Using the splitting of the Riemann tensor between the Ricci tensor and the Weyl tensor for the second column while taking into account the fact that the extension modules are torsion modules and thus that each component of the Weyl tensor is differentially dependent on the Ricci tensor, we obtain the following commutative and exact diagram:
It follows that the 10 components of the Weyl tensor must satisfy a first order linear system with 16 equations, having 6 generating first order CC. The differential rank of the corresponding operator is thus equal to
and such an operator defines a torsion module in which we have to look separately for each component of the Weyl tensor in order to prove that it is killed by the Dalembert operator ( [16] ). The situation is similar to that of the Cauchy-Riemann equations obtained when
by considering the conformal Killing operator CK. Indeed, any complex transformation
must be solution of the (linear) first order system
of finite Lie equations though we obtain
, that is
and
are separately killed by the second order Laplace operator
. We obtain the following striking technical lemma explaining the so-called gauging procedure of the Lanczos potential.
Lemma 4.2: When
, the central vertical arrow
is just described by the contraction formula
depending on the metric:
(12)
Proof: Let us write down the Bianchi operator in the form:
Contracting with
, we obtain:
Setting as usual
with
and contracting with
, we finally get :
as the way to use a contraction in order to exhibit Einstein equations.
With
, let us write down all the terms, using the Euclidean metric for simplicity instead of the Minkowski metric, recalling that only this later choice allows to find out both the Poincaré group and the differential sequence with successive operators
according to ( [22] ):
that is to say with all the terms:
where, in any case, we have
.
If we set
, the first line disappears because of the 3-form
and we are left with:
Using Hodge duality, we get with new indices:
arriving finally to the formula:
that is exactly twice the trace of the Lanczos tensor, namely:
This result explains why the Lanczos tensor
with 24 components is first reduced to 20 components through the condition
and finally to 16 components as in the diagram through the kernel of the above trace condition. It is thus impossible to understand this result, even for
, without the Spencer δ-cohomology and absolutely impossible to generalize this result in arbitrary dimension without the combination of the δ-cohomology and double duality in differential homological algebra.
Q.E.D.
Finally, using the previous definition
, such a result explains the confusion done by Lanczos and followers between the Riemann candidate
or the Weyl candidate
and their respective formal adjoint vector bundles having of course the same fiber dimension but quite different transition rules under changes of local coordinates.
We notice that the changes of the successive orders are totally unusual and refer to ( [22] ) for more details on the computer algebra methods. In particular, when
, the conformal analogue of the Bianchi operator is now of order 2, a result explaining why Lanczos and followers never succeeded adapting the Lanczos tensor potential L for the Weyl operator. We understand therefore that the solution of what we called “Lanczos secret” must be depending on a quite different homological framework. It is only after exhibiting it in the last section below that we will be able to say that we have thus solved the Riemann-Lanczos and Weyl-Lanczos parametrization problems in arbitrary dimension. In the meantime, we provide two examples that can be fully computed as a way to understand the use of adjoint differential sequences.
5. Motivating Examples
The two following examples will show how the differential extension modules may depend on the Vessiot structure constants.
Example 5.1: With
and
with
, let us consider the Lie operator
. The corresponding first order system:
is involutive whenever
and
where now d is the standard exterior derivative and
, exactly as in ( [19], p 438-440). We have the differential sequence:
with
or the resolution:
Multiplying
respectively by
, we obtain
in the form:
Then, multiplying
by
, we obtain
as:
We have therefore to consider the two cases:
●
: We have the new CC
and
. It follows that the torsion module
is generated by the residue of
because
and we may thus suppose that
. As for
, this torsion module is just defined by the system
for
and thus
.
●
: We must have the new CC:
It follows that
is now generated by the residue of
. Finally,
is defined by
and thus
.
Hence, both
and
highly depend on the Vessiot structure constant c.
Example 5.2: (Contact transformations)
With
or simply
, we may introduce the 1-form
and consider the system of finite Lie equations defined by
. Eliminating the factor
and linearizing at the q-jet of the identity, we obtain the first order involutive system of infinitesimal Lie equations:
with two equations of class 3, one equation of class 2 and thus one CC of order 1, namely
. Now, it is well known that this contact operator
can be parametrized by an operator
as follows:
and thus
. We have obtained the following formally exact sequence which is nevertheless not strictly exact because
is not formally integrable:
As M is therefore free and thus projective, it follows that the adjoint sequence is exact too.
Coming back to the Vessiot structure equations, we notice that
is not invariant by the contact Lie pseudogroup and cannot be considered as an associated geometric object. We have shown in ( [9], p 684-691) that the corresponding geometric object is a 1-form density
leading to the system of infinitesimal Lie equations in Medolaghi form:
and to the only Vessiot structure equation:
with the only structure constant c. In the present contact situation, we may choose
and get
but we may also choose
and get
, these two choices both bringing an involutive system. Let us prove that the situation becomes completely different with the new system:
having the only CC
.
Multiplying the three previous equations by the three test functions
, the only CC by the test function
and integrating by parts, we get the adjoint operators:
It follows that
with a strict inclusion and
. Similarly,
is defined by
and thus
.
Our problem will be now to construct and compare the differential sequences:
For this, linearizing the only Vessiot structure equation, we get the CC operator
and the corresponding system
in the form:
Multiplying on the left by the test function
and integrating by parts, we get the operator
in the form:
We obtain therefore the crucial formula
showing how the previous sequences are essentially depending on the Vessiot structure constant c. Indeed, if
, then
and the operator
is injective. This is the case when
. On the contrary, if
, then the operator
may not be injective as can be seen by choosing
. Indeed, in this case we get a kernel defined by
.
We invite the reader to treat similarly the case of unimodular contact transformations, namely transformations preserving the 1-form
, thus also the 2-form
and even the 3-form
that can be used as a volume form. The Vessiot structure equations for the ground geometric object
are now
with the only striking Jacobi condition
(See [1] for more details).
6. Generalized Lanczos Problem
In this last section, we prove that the following theorem allows to solve locally the Lanczos problem in a similar way for any Lie group of transformations:
Theorem 6.1: The Spencer sequence for any Lie operator
which is coming from a Lie group of transformations, with a Lie group G acting on X, is (locally) isomorphic to the tensor product of the Poincaré sequence for the exterior derivative by the Lie algebra
of G.
Proof: If M is the differential module defined by
, we want to prove that the extension modules
and
vanish, that is, if
generates the CC of
but also
generates the CC of
, then
generates the CC of
and
generates the CC of
. We also remind the reader that we have shown in ( [18] [22] ) that it is not easy to exhibit the CC of the Maxwell or Morera parametrizations when
and that a direct checking for
should be strictly impossible. It has been proved by L. P. Eisenhart in 1926 ( [20] ) that the solution space
of the Killing system has
infinitesimal generators
linearly independent over the constants if and only if
had constant Riemannian curvature, namely zero in our case. As we have a transitive Lie group of transformations preserving the metric considered as a transitive Lie pseudogroup, the three classical theorems of Sophus Lie assert than
where the structure constants c define a Lie algebra
. We have therefore
. Hence, we may replace locally the Killing system by the system
, getting therefore the differential sequence:
which is the tensor product of the Poincaré sequence by
. Finally, it follows from the above Theorem that the extension modules considered do not depend on the resolution used and thus vanish because the Poincaré sequence is self adjoint (up to sign), that is
generates the CC of
at any position, exactly like d generates the CC of d at any position. This (difficult) result explains why the adjoint differential modules we shall meet will be torsion-free or even reflexive. We invite the reader to compare with the situation of the Maxwell equations in electromagnetisme (See ( [12], p 492-494) for more details). However, we have explained in ( [11] [13] ) why neither the Janet sequence nor the Poincaré sequence can be used in physics and must be replaced by the Spencer sequence which is another resolution of
. Though this is out of the scope of this paper, we shall nevertheless shortly describe the relation existing between the above results and the Spencer operator, thus the Spencer sequence. For this, let us define for any
the section
. With the standard notations of ( [8] [11] [12] ) and
, the components of the Spencer operator become:
when q is large enough, that is
for the Killing system and
for the conformal Killing system in arbitrary dimension ( [43] ), we have involutive systems with vanishing symbols because both are finite-type. We obtain therefore the desired identification justifying our claim.
Q.E.D.
Corollary 6.2: When
is the Killing operator or the conformal Killing operator, then
and there is no gap. Moreover, if the differential module M defined by
is a torsion module as in the Theorem, then we have
in any case.
7. Conclusion
E. Vessiot discovered the so-called Vessiot structure equations as early as in 1903 and, only a few years later, E. Cartan discovered the so-called Maurer-Cartan structure equations. Both are depending on a certain number of constants like the single geometric structure constant of the constant Riemannian curvature for the first and the many algebraic structure constants of Lie algebra for the second. However, Cartan and followers never acknowledged the existence of another approach which is therefore still totally ignored today, in particular by physicists. Now, it is well known that the structure constants of a Lie algebra play a fundamental part in the Chevalley-Eilenberg cohomology of Lie algebras and their deformation theory. It was thus a challenge to associate the Vessiot structure constants with other homological properties related to systems of Lie equations, namely the extension modules determined by Lie operators. As a striking consequence, such a possibility opens a new way to understand and revisit the various contradictory works done during the last fifty years or so by different groups of researchers, using respectively Cartan, Gröbner or Janet bases while looking for a modern interpretation of the work done by C. Lanczos from 1938 to 1962. However, the reader must not forget that the Weyl tensor was not known by Lanczos, even as late as in 1967, and that it was not possible to discover any solution of the parametrization problem by potentials through double duality before 1990/1995, that is too late for the many people already engaged in this type of research. We finally hope that this paper will open a new domain for applying computer algebra while offering a collection of useful test examples.
NOTES
1We have only quoted the recent references on the Lanczos potential published after 2000.