Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials Revisited

The purpose of this paper is to revisit the well known potentials, also called stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) for 3-dimensional elasticity, finally by A. Einstein (1915) for 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949, 1962) in order to relate potentials to Lagrange multipliers. Using the methods of Algebraic Analysis, namely mixing differential geometry with homological algebra and combining the double duality test involved with the Spencer cohomology, we shall be able to extend these results to an arbitrary situation with an arbitrary dimension n. We shall also explain why double duality is perfectly adapted to variational calculus with differential constraints as a way to eliminate the corresponding Lagrange multipliers. For example, the canonical parametrization of the stress equations is just described by the formal adjoint of the components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed is equal to for any minimal parametrization, the Einstein parametrization being “in between” with potentials. We provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today, but it could be strictly impossible to obtain them without using the above methods. We also revisit the possibility (Maxwell equations of electromagnetism) or the impossibility (Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various equations of physics. It is nevertheless important to notice that, when n and the algorithms presented are known, most of the calculations can be achieved by using computers for the corresponding symbolic computations. Finally, though the paper is mathematically oriented as it aims providing new insights towards the mathematical foundations of general relativity, it is written in a rather self-contained way.

Homological Solution of the Lanczos Problems in Arbitrary Dimension

When D is a linear partial differential operator of any order, a direct problem is to look for an operator D1 generating the compatibility conditions (CC) D1η = 0 of Dξ = η. Conversely, when D1 is given, an inverse problem is to look for an operator D such that its CC are generated by D1 and we shall say that D1 is parametrized by D = D0. We may thus construct a differential sequence with successive operators D, D1, D2, ..., each operator parametrizing the next one. Introducing the formal adjoint ad() of an operator, we have but ad (Di-1) may not generate all the CC of ad (Di). When D = K [d1, ..., dn] = K [d] is the (non-commutative) ring of differential operators with coefficients in a differential field K, then D gives rise by residue to a differential module M over D while ad (D) gives rise to a differential module N =ad (M) over D. The differential extension modules with ext0(M) = homD (M, D) only depend on M and are measuring the above gaps, independently of the previous differential sequence, in such a way that ext1 (N) = t (M) is the torsion submodule of M. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension n and to prove for the first time that the extension modules highly depend on the Vessiot structure constants c. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint Beltrami = ad (Riemann) which can indeed be parametrized by the operator Lanczos = ad (Bianchi) in arbitrary dimension, “one step further on to the right” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when D is a Lie operator of finite type, we shall prove that . It will follow that the Riemann-Lanczos and Weyl-Lanczos problems just amount to prove such a result for i = 1,2 and arbitrary n when D is the classical or conformal Killing operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.