Geometry

Nonmetric Foundations for Differential Geometry and Complex and Bicomplex Reformulations in the 4D Case

Authors: Warren D. Smith
Comments: 49 Pages. [CLAUDE was used]

I — "six-foundations theorem." The metric isn't the only possible basis for Riemannian/Lorentzian geometry. The Riemann tensor, connection, Ricci-Weyl pair, and – in 4D – Lanczos tensor or superpotential paired with Ricci, each determine the geometry. The proof – an algorithm integrating the geodesic-deviation equation outward from a seed point – was implemented as a check. Both Lanczos-based foundations are strictly richer than the metric. Goal: new foundations for differential geometry, physics.

II — extra 4D-only beauty. The 4D real-valued Riemann tensor R_abcd admits two duals, one for each antisymmetric index-pair. The complex numbers ℂ extend the reals ℝ by adjoining i=√-1;bicomplexes ℂ⊗ℂ≅ℂ⊕ℂ extend ℂ with another √-1 symbol. Build ℂ⊗ℂ directly on ℝ by adjoining two commuting special symbols i⃖ and i⃗ obeying i⃖²=i⃗²=−1,   i⃖ i⃗=i⃗ i⃖=ⅱ⃡,   and ⅱ⃡²=+1. One bicomplex-valued tensor (1+i⃖*)R_abcd(1+i⃗*), packages R_abcd, its left-, right- and double-duals *R_abcd, R*_abcd, and *R*_abcd. A tower of complex (sometimes bicomplex) extensions of familiar 4D tensors then naturally arise. Since (bi)complex, each now packages (4×) 2× as much. We explain these (bi)complexified tensors and catalogue their identities. Various classical results become single statements: the Bel-Robinson tensor is a modulus square; Bianchi's identity complexifies; "Petrov type" means the Jordan structure of a complexified Weyl tensor matrix; "algebraic speciality" becomes a single equation; a standard Bel criterion condition wrongly classifies Schwarzschild, which our cleaner formulation repairs. Goals: concise natural (re)formulation of 4D differential geometry; useful identity-compendium.

III — combines both. Part I's reconstructions, since linear, accept II's complexified curvatures to produce 4D metrics with extra bicomplex imaginary parts. Reconstructing de Sitter universe from bicomplexified Riemann yields a bicomplex metric with anti-de Sitter as its ⅱ⃡ imaginary part. We call this mysterious component the "shadow metric" despite not always being a metric. Nontrivial Kottler black hole metric shadows are not altered-parameter Kottlers. This process automatically associates three extra bicomplex-imaginary parts with any 4-metric; Ricci, Lanczos, and Weyl tensors effectively simplify to ordinary-complex (just one imaginary part); Bel-Robinson and traceless-Ricci tensors remain pure-real. Riemann's other two (i⃖ and i⃗) components automatically carry the ADM "3+1 time slicing" Einstein field equation reformulation's "momentum constraint." (Confirmation: This lattermost correspondence was already in the 1+3 literature.)Goal: speculatively this provides the natural framework for unifying gravity with quantum field theories.

Also II.12 proves that C⁴ Lanczos superpotentials don't exist on a 2-parameter family of 5D manifolds. Apparently they generically exist exactly in 4D.
Category: Geometry