Jan 20 2024

## Conical Angle

On page 9 of the April 2007 booklet, it is proposed that the magnetic field of the electron guides the rectification of incoming gravitons “within some inclusive conical angle that is greater than zero”. As the electron grows in an atomic or molecular orbital, the magnetic field of the electron becomes stronger.

The Cauchy integral formula is used in many applications. Here we are applying it to gravity.

Since z is the direction straight out of the earth, let us call f’’(z) the gravitational force. “The Cauchy integral formula in the theorem in Sec. 50 can be extended so as to provide an integral representation of derivatives of f at z_{0}” ([1], pg. 165). Then the formula becomes:

f(z) = (1/2πi) ∫_{c} [(f(s) / (s – z)] ds ([1], pg. 166, formula (1))

Let us say z is a point at the center of the 3-dimensional electron, and (s – z) is the spin radius that starts off at 6.6676 x 10^{-16} m immediately after a spin flip at the end of τ_{i}. Let f(s) be the function that grows (s – z) as the orbital electron absorbs gravitons. Keep in mind that the conical angle grows also.

f’(z) is the rate at which the orbital electron absorbs gravitons, since an arc τ is often not directed at the center of the earth:

f’(z) = (1/2πi) ∫_{c} [(f(s) / (s – z)^{2}] ds ([1], pg. 166, formula (2))

f’’(z) is Newton’s second law of motion, F = ma:

f’’(z) = (1/πi) ∫_{c} [(f(s) / (s – z)^{3}] ds ([1], pg. 167, formula (4))

Since the electron is perfectly round, (s – z) still starts off at the radius of 6.6676 x 10^{-16} m at the beginning of an arc. As the electron grows in size in an orbital, at some point it is able to produce Pontrjagin classes, or higher k-planes, as long as f(s) is strong enough.

[1] Brown, James Ward and Churchill, Ruel V., “Complex Variables and Applications”, McGraw-Hill Higher Education, c. 2009