axiom of choice - variation or curvature

AntonioLao · **axiom of choice - variation or curvature -** 08-18-2005, 03:16 PM

Calculus of variations hinges on the existence of a local infinitesimal change. Moreover, this change is a virtual change, that is to say it is an outcome depending on a choice and implying that the general continuity conditions are intact. However, if spacetime is already quantized then this choice is unnecessary. Nevertheless, a connection between a derivative and a local infinitesimal curvature must be found. The present conjecture is that they are equivalent. To differentiate a function is the same as multiplying it by a local infinitesimal curvature. Finding the second derivative is the same as the product of two curvatures and the function. The advantage of curvature multiplication is that it involved no choices and it is also integrable into finding a mean curvature. The products of curvatures also give the intrinsic Gaussian curvatures.

When the radii of curvatures( $R_1$ and $R_2$ ) replaced the principal curvatures ( $\kappa_1$ and $\kappa_2$ ) the mean curvature (H) becomes

$H=\frac{1}{2}(\kappa_1+\kappa_2)=\frac{R_1+R_2}{2R_1 R_2}$

Except for the factor of ½, the mean curvature is inversely general covariant to the reduced mass. Furthermore, it will be used to derive mass from first principle basing on the products of curvatures.