John Nordberg's geometrical interpretation of Relativity's correction factor

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The Correction Factor

Einstein's Special Theory of Relativity contains a correction factor. It is:

Two expanding concentric spheres

This is a correction factor between our current sublight definition of time and a Speed-of-Light-Definition-of-Time. This can be shown geometrically with two concentric expanding spheres: one expanding at a sublight speed "V," and one expanding at the speed of light "c."

The equation for the two expanding spheres

The equation for a sphere is:

The equation for the velocity of the two expanding spheres

The SI definition of velocity -- in units -- is Length divided by Time. Thus, to obtain the velocity of an expanding sphere, first, divide both sides of the equation by time squared:

This is effectively the squared velocity of an expanding sphere.

The difference in velocities between the two spheres

Second, subtract the squared velocity of the outer sphere from the inner sphere -- taking the square root to obtain just the velocity:

This equation contains a "delta V" on the left. This is the difference in velocity between a sphere expanding at the speed of light, and one expanding at an arbitrary sublight speed. However, this equation expresses the difference in 3 directions: x, y and z -- not just one.

The difference in velocities between the two spheres in just the x direction

To obtain the difference in just one direction -- say, the x direction -- drop the y and z subscripts on the left, and drop the y and z terms on the right. The result is:

There are three velocities in this equation. Each velocity is a fraction. The denominator in each uses the traditional definition of time -- using the definition of a second. This can lead to a mathematical error, since moving the timepieces involved would change the definition of motion that a second represents. However, the "seconds" in the denominators of these three velocity fractions can be "eliminated," so-to-speak, by dividing all three fractions by the traditional definition of the speed of light. This step is illustrated by:

How to eliminate the error caused by using traditional seconds

The Speed-of-Light-Definition-of-Time explicitly treats "velocity" and "speed" fractions as the ratios of two lengths -- the distance an object travels divided by the distance light travels.

Correcting by dividing by "c" squared

Thus, dividing a traditional velocity by the traditional definition of the speed of light -- or, in this case, "c" squared -- eliminates the traditional seconds in the denominators and gives:

On the left of this equation is: delta V in the x direction divided the speed of light. This is effectively the method The Speed-of-Light-Definition-of-Time uses to measure velocities and speeds: a change in distance of an object, moving less than or equal to the speed of light, divided by a change in distance of light. Simplifying:

Relativity is a correction factor for using a sublight definition of time

Thus the difference in velocity between a sphere expanding at the speed of light and one expanding at a velocity less than the speed of light can be used to geometrically derive the correction factor in Relativity:

(This also highlights one of the problems with Relativity: it is only a solution in one dimension instead of 3 dimensions.)