A Unified Field Theory by N.F.J. Matthews - Related Research, Citations, and Researchers

"Involutive systems of differential equations: Einstein’s strength versus Cartan’s degré d’arbitraire" by Michael Sué
J. Math. Phys. 32, 392 (1991); http://dx.doi.org/10.1063/1.529424 (8 pages)


Three new theorems relating Einstein’s notions of ‘‘strength’’ and ‘‘compatibility’’ to the field of the initial‐value problem are presented. These theorems result (i) in a first proof of Matthews’ conjectures concerning this relation for a wider class of systems of partial‐differential equations, (ii) in a new interpretation of Einstein’s compatibility condition, and (iii) in the exact relation between Einstein’s strength and Cartan’s degré d’arbitraire.

"Theorem 1: Equality of the Cartan and Taylor numbers of arbitrariness."

"We will use these results to illustrate (Matthews') far-reaching conjectures especially where they surpass Theorem 1."

"Note: Matthews, contrary to Schutz and us, used nonvanishing and deviating differentiation orders... for the unknown functions in each case. Matthews allows even for a “splitting” of the expansion coefficients;..."

"This is the same result as ours in Sec. V, and Matthews showed it to be correct. But the interpretation of (7.8) already needs Matthews’ generalizations concerning the differentiation orders as well as the negative coefficients. For system (P) we need, as Matthews proved as well,..."

"As Matthews ideas allow us to get information about the numbers of derivatives which we are free to choose, they are far more general than Theorem 1. They are even more general than the Cauchy-Kowalewska Theorem, as they yield information even when the numbers of differential equations exceed the numbers of unknown functions. But they are, so far, only proven for the special expansions (7.7) and (7.8). Theorem 1, however, ensures now that Matthews ideas are correct as far as the systems fulfil Eq. (7.3). As such, Theorem 1 forms a stronghold to start from for further improvement."

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