Gamma factor between means

Consider the mean between two quantities:

$c+v\; \mathrm{and}\; c-v.$

The arithmetic mean is

$\frac{c+v+c-v}{2}=c.$

The harmonic mean is

$\left (\frac{(c+v)^{-1}+(c-v)^{-1}}{2} \right )^{-1}=\frac{c^2-v^2}{c}=\frac{c}{\gamma ^{2}}.$

where

$\gamma = \left ( 1-\frac{v^2}{c^2}\right )^{-1/2}$

which equals the gamma factor of the Lorentz transformation.

The geometric mean is

$((c+v)(c-v))^{1/2}=(c^{2}-v^{2})^{1/2}=\frac{c}{\gamma }$

Then the factor γ² transforms a harmonic mean into an arithmetic mean:

$\gamma^2 \left ( \frac{c^2-v^2}{c} \right )=c$

The inverse γ factor transforms an arithmetic mean into a geometric mean:

$\lambda^2 c = \left ( \frac{c^2-v^2}{c} \right )$

so that

$\lambda = \left ( 1-\frac{v^2}{c^2}\right )^{1/2}=\gamma^{-1}$

The inverse γ² factor transforms an arithmetic mean into a harmonic mean.

Or consider the mean between these two:

$\frac{c}{c+v}\; \mathrm{and}\; \frac{c}{c-v}.$

The arithmetic mean is

$\frac{c}{2}\left (\frac{c}{c+v}+\frac{c}{c-v} \right )= \frac{c}{c^2-v^2}=\gamma ^{2}.$

The harmonic mean is

$\left (\frac{1}{2}\left (\left (\frac{c}{c+v} \right )^{-1}+\left (\frac{c}{c-v} \right )^{-1} \right ) \right )^{-1}=1.$

The geometric mean is

$\left (\left (\frac{c}{c+v} \right )\left (\frac{c}{c-v} \right ) \right )^{1/2}=\gamma.$

The factor γ² transforms a harmonic mean into an arithmetic mean. The gamma factor relates the arithmetic mean and the harmonic mean, and the geometric mean combines them. An arithmetic mean and harmonic mean can be replaced by a geometric mean, in a sense.

Revised 1/2024.

Related