# Special relativity considered as an average

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The equations of special relativity can be derived by assuming that the speed of light needs to appear to be fixed in any valid inertial frame, and by eliminating the disagreements that arise between calculations performed using different reference frames, by using a "root product" or "geometric mean" averaging process.

## Longitudinal calculations

The Doppler predictions for signals given off by an approaching or receding object, if we assume that the speed of light is fixed globally with respect to (a) the observer, and (b), the emitter, are:

(a) freq'/freq = c/(c+v), and
(b) freq'/freq= (c-v)/c


, where v is recession velocity (for approaching bodies, we simply make v negative).

While both equations predict redshifts for recession and blueshifts for approach, the sizes of the effect are different, so if there really was an absolute global reference for the propagation of light, we should be able to tell how fast we were going wrt that absolute reference by measuring the real Doppler relationship reported by our equipment. But this would lead to a violation of Galileo's and Newton's idea that the physics of stationary and simply-moving systems should be identical.

Special relativity (1905) and Lorentz Ether Theory before it (1904) reconciled absolute global lightspeed constancy for everyone, by using a third, intermediate prediction:

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


, which is obviously the geometric mean or root product average of the previous two prediction (it's what you get by multiplying them together and then square rooting). This sort of averaging produces an intermediate result that lies on an agreed ratio between the two originals

Under SR, instead of saying that one or other of the equations should be correct, we produce agreement by declaring that neither are correct, and say that both miss their target by exactly this ratio. Since the discrepancy between A and B is the ratio 1:[1-vv/cc], the intermediate prediction disagrees with both (a) and b by the square root of this ratio, SQRT[1 – vv/cc]. This special ratio is referred ot as the Lorentz factor, and given the Greek letter γ, pronounced “gamma”.

(a) × (1 - v^2/c^2) = (b)
(a) × gamma × gamma = (b)
(a) ×gamma = (SR), (b) ÷ gamma = (SR),


If we believe that a body moving wrt to the global reference frame for light-propagation ages more slowly by gamma, then if lightspeed is fixed in our frame, we get (a) × (a Lorentz redshift), giving special relativity prediction. On the other hand, if we suppose that lightspeed is fixed for the emitter, and that we are time-dilated, we get (b) multiplied by a Lorentz blueshift , again giving the same intermediate SR prediction.

No matter which hypothetical frame we use for the supposed global speed of light, by applying Lorentz redshifts to every object moving with respect to this frame, we get precisely the same final physical prediction.

## Transverse calculations

Similarly, the Doppler predictions for signals given off by an object as it passes us, at the exact moment when it is neither approachigm or receding, assuming that lightspeed is fixed w.r.t (a) the observer, and (b), the emitter, are:

(a) freq'/freq = 1 (no effect due to the emitter's motion), and
(b) freq'/freq= 1-v^2/c^2 (the aberration redshift)


, where v is now arguably a speed rather than a velocity.

Once again SR take the two conflicting predictions and averages them together to produce an intermediate prediction that can be interpreted as the result of either (a) or (b), coupled with a Lorentz slowdown of whoever is thought to be “moving” w.r.t. the arbitrarily-selected reference frame.

Once again, the upshot is that if the equations are those of SR, we can believe that spacetime is flat and lightspeed is globally fixed for anyone and everyone without anyone being able to isolate any frame as having priority over any other.

## Lorentz length-contraction

since the Doppler length-change effect makes the apparent photographable lengths of moving bodies appear to stretch or contract by exactly the same ratio as the Doppler increase or decrease int their signals' energies, we can repeat exactly the same working as before, writing the apparent length-change (len'/len instead of apparent frequency-change freq'/freq.

Again we can say that the assumption of a fixed lightspeed gives conflicting predictions for photographable length depending on the selected frame for light-propagation (la) and (lb), but that if we instead use their geometrical mean, $\sqrt{ (la) × (lb) }$ , we can get the same physical prediction for all observers, as long as we also say that any body moving with respect to our arbitrarily-chosen frame also contracts by the same Lorentz ratio as before, $\sqrt{ 1 - v^2/c^2 }$.