Geometric mean

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The geometric mean of two numbers or two equations is obtained by multiplying them together, and then taking the square root of the result.

This process is used to find an averaged number which lies on a ratio between the two original numbers, or an equation whose solutions all depart from those of the two original equations by matching ratios (which obey a divergence equation').


Simple numbers

The geometric mean of 5 and 10 is just over 7 (about ~7.071). five times ten is fifty, root(50) is ~7.071 .

If we multiply five by the "divergence", ~1.4142, we get ~7.071, and if we then multiply ~7.071 by ~1.4142 we get ten.

We can do this trick with any two numbers to obtain a third intermediate number that diverges from both by a common ratio.


We can also use the technique to average the predictions of two different equations, and generate a "divergence equation" that can map the central equation to the two original equations.

Under special relativity, the longitudinal Doppler shift obtained by assuming that light propagates at "simple c" with respect to the observer is:

(a)  freq'/freq = c/(c+v)

The corresponding prediction assuming that light propagates at "simple c" with respect to the emitter, is:

(b) freq'/freq = (c-v)/c

An averaged equation using the geometric mean of the two earlier conflicting predictions would give us

freq'/freq = SQRT[ c/(c+v) ×  (c-v)/c ], = SQRT[ c(c-v) / c(c+v) ] , = SQRT[ (c-v) / c(c+v) ] 

This intermediate equation is the one that appears in Einstein 1905 paper of special relativity, as a way of reconciling the two earlier predictions.

In the case of special relativity, the "divergence equation" is [math]\sqrt{ 1 - v^2/c^2 }[/math] , the "Lorentz factor", which maps between either of the two flanking equations and the central averaged equation.

Although the use of geometric mean approach was fairly obvious in Einstein's 1905 equation, later texts tended to make the "averaging" operation less obvious by stating the relationship as either (a) or (b), multiplied or divided by the divergence equation (gamma).