Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Differential Homological Algebra and General Relativity (1907.12387v1)

Published 25 Jul 2019 in math.GM

Abstract: In 1916, F.S. Macaulay developed specific localization techniques for dealing with "unmixed polynomial ideals" in commutative algebra, transforming them into what he called "inverse systems" of partial differential equations. In 1970, D.C. Spencer and coworkers studied the formal theory of such systems, using methods of homological algebra that were giving rise to "differential homological algebra", replacing unmixed polynomial ideals by "pure differential modules". The use of "extension modules" and "differential double duality" is essential for such a purpose. In particular, 0-pure differential modules are torsion-free and admit an "absolute parametrization" by means of arbitrary potential like functions. In 2012, we have been able to extend this result to arbitrary pure modules, introducing a "relative parametrization" where the potentials should satisfy compatible "differential constraints". We recently discovered that General Relativity is just a way to parametrize the Cauchy stress equations by means of the formal adjoint of the Ricci operator in order to obtain a "minimum parametrization" by adding sufficiently many compatible differential constraints, exactly like the Lorenz condition in electromagnetism. These unusual purely mathematical results are illustrated by many explicit examples and even strengthen the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory.

Summary

We haven't generated a summary for this paper yet.