decreasing curvature

[AntonioLao]

For every curvature K of the spacetime continuum there is a defined radius of curvature R. The simple relationship between K and R is that they are reciprocal of each other. That is the product of K and R is unity: KR=1. This suggests that K=1/R or R=1/K. If R is assigned as the expanding radius of the universe then clearly K decreases as R increases. As R approaches infinity then K approaches zero and the geometry of the universe becomes truly Euclidean and the force of gravity is zero everywhere.

However, from the general theory of conic sections, KR=1 is a special form of hyperbola which also indicate the inverse variation of K and R where 1 is the constant of variation. It is derived from the general equation of conics: Ax+Bxy+Cy+Dx+Ey+F=0 where A=C=D=E=0, B=1, and F=-1 then by direct substitution xy=1 and the term xy is now replaced by KR giving KR=1. Nonetheless, KR=1 is also a special form of rational functions such that for it to be a true equation both K and R must take on rational values.