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{1}{c+v}\; \mathrm{and}\; \frac{1}{c-v}.

The arithmetic mean is

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

The harmonic mean is

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

The geometric mean is

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

The factor γ transforms a geometric mean into an arithmetic mean. The factor γ2 transforms a harmonic mean into an arithmetic mean.

Or consider the mean between these two with β = v/c:

\frac{1}{1+\beta}\; \mathrm{and}\; \frac{1}{1-\beta}.

The arithmetic mean is

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

The harmonic mean is

\left (\frac{1}{2}\left (\left (\frac{1}{1+\beta} \right )^{-1}+\left (\frac{1}{1-\beta} \right )^{-1} \right ) \right )^{-1}=1.

The geometric mean is

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

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.