IJPAM: Volume 35, No. 3 (2007)

INTRODUCTION TO A PAIR OF MANIFOLDS CHARTED
ON ONE MINKOWSKI DOMAIN

Gregory L. Light
Providence College
Providence, Rhode Island, 02918, USA
e-mail: glight@providence.edu


Abstract.This paper introduces the idea of a ``combined manifold'', which essentially is the graph of a diffeomorphism from one manifold to another. Following a previously published work by the author, this present article addresses a combined space-time 4-manifold by introducing ``combined particles'' $\left\{ p=(x,y)\right\} $, where $\{x\}$ and $% \{y\}$ separately establish their own distinct 4-manifolds following General Relativity; this is made possible by assuming that $\{y\}$ only exert gravitational forces. We have proved that this combined 4-manifold is unique thanks to a distinct weighted average of the two metrics from a pair of sets of Einstein Field Equations, with the weights determined by two gravitational constants. As an illustration, we have decomposed the recognized gravitational constant $G$ and the Earth mass, and as a motivation, we tender the particle-wave duality as a theoretical support of our hypothetical combined 4-manifold.

Received: January 25, 2006

AMS Subject Classification: 83C05, 53C50, 53C15, 85A40

Key Words and Phrases: Einstein field equations, dark matter, 4-manifold, diagonal map, space-time geometry

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 35
Issue: 3