From computer algebra to gravitational waves

The first finite length differential sequence has been introduced by Janet (1920). Thanks to the first book of Pommaret (1978), the Janet algorithm has been extended by Blinkov, Gerdt, Quadrat, Robertz, Seiler and others who introduced Janet and Pommaret bases in computer algebra. Also, new intrinsic tools have been developed by Spencer in the study of Lie pseudogroups or by Kashiwara in differential homological algebra. The achievement has been to define differential extension modules through the systematic use of differential double duality. Roughly, if D = K[d] is the non-commutative ring of differential operators with coefficients in a differential field K, let D be a linear differential operator with coefficients in K. A direct problem is to find the generating compatibility conditions (CC) in the form of a differential operator D1 such that Dξ = η implies D1η = 0 and so on. Taking the adjoint operators, we have ad(D) ◦ ad(D1) = ad(D1 ◦ D) = 0 but ad(D) may not generate all the CC of ad(D1). If M is the D-module defined by D and N is the D-module defined by ad(D) with torsion submodule t(N), then t(N) = ext 1 (M) “measures” this gap that only depends on M and not on the way to define it. Also, R = homK(M, K) is a differential module for the Spencer operator d : R → T ∗ ⊗ R, first introduced by Macaulay with his  inverse systems (1916). When D : T → S2T ∗ : ξ → L(ξ)ω = Ω is the Killing operator for the Minkowski metric ω with perturbation Ω, then N is the differential module defined by the Cauchy = ad(Killing) operator and t(N) = ext 1 (M) = 0 because the Spencer sequence is isomorphic to the tensor product of the Poincaré sequence by a Lie algebra. The Cauchy operator can be thus parametrized by stress functions having nothing to do with Ω, like the Airy function for plane elasticity. This result is thus pointing out the terrible confusion done by Einstein (1915) while ”adapting” to space-time the work done by Beltrami (1892) for space only. both of them using the same Einstein operator but ignoring it was self-adjoint in the framework of differential double duality (1995). Though unpleasant it is, we shall prove that the mathematical foundations of General Relativity are not coherent with these new results which are also illustrated by many other explicit examples.