Abstract
The purpose of this short but difficult paper is to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT) in the light of a modern approach to nonlinear systems of ordinary or partial differential equations, using new methods from Differential Geometry (D.C. Spencer, 1970), Differential Algebra (J.F. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). The main idea is to identify the differential indeterminates of Ritt and Kolchin with the jet coordinates of Spencer, in order to study Differential Duality by using only linear differential operators with coefficients in a differential field K. In particular, the linearized second order Einstein operator and the formal adjoint of the Ricci operator are both parametrizing the 4 first order Cauchy stress equations but cannot themselves be parametrized. In the framework of Homological Algebra, this result is not coherent with the vanishing of a certain second extension module and leads to question the proper origin and existence of gravitational waves. As a byproduct, we also prove that gravitation and electromagnetism only depend on the second order jets (called elations by E. Cartan in 1922) of the system of conformal Killing equations because any 1-form with value in the bundle of elations can be decomposed uniquely into the direct sum (R, F) where R is a section of the Ricci bundle of symmetric covariant 2-tensors and the EM field F is a section of the vector bundle of skew-symmetric 2-tensors. No one of these purely mathematical results could have been obtained by any classical approach. Up to the knowledge of the author, it is also the first time that differential algebra in a modern setting is applied to study the specific algebraic feature of most equations to be found in mathematical physics, particularly in GR.
Highlights
The first motivation for studying the methods used in this paper has been a 1000$ challenge proposed in 1970 by J
The purpose of this short but difficult paper is to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT) in the light of a modern approach to nonlinear systems of ordinary or partial differential equations, using new methods from Differential Geometry
We have proved in many books [4] [5] and in [6] [13] [14] that the situation is similar for Maxwell equations, a result leading to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT), of Electromagnetism (EM)
Summary
Passing to the module framework, we just recognize the definition of ext ( M ) when M is determined by. ( ) Applying homD ( M , D) , we may define ND = coker d1* and exhibit the following free resolution of N by right D-modules:. A torsion-free ( injective)/reflexive ( bijective) module is described by an operator that admits respectively a single/double step parametrization. −∂3φ = ξ 2 , ∂2φ − x3∂1φ = ξ 3 ⇒ ξ1 − x3ξ 2 = φ It defines a free differential module M D which is reflexive and even projective. Any resolution of this module splits, like the short exact sequence 0 → D2 → D3 → D → 0 , and the corresponding differential sequence of operators is locally exact like the Poincaré sequence Any resolution of this module splits, like the short exact sequence 0 → D2 → D3 → D → 0 , and the corresponding differential sequence of operators is locally exact like the Poincaré sequence ([3], p. 684-691)
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