Velocity-addition formula

The appearance of a velocity-addition formula under special relativity is sometimes presented as if it is somethign unique ot the theory. In fact, most C19th theories woudl have had a velocity-additiona formula, the difference between theirs and SR's (apart form the exact formulation used) beign how the result so fthe formula are interpreted.

Example: Newtonian theory v.a.f.=

Lets start with a Newtonian example. Under Newtonian theory, the frequency-shift seen in a recedign object is [math]\frac{frequency'}{frequency} = \frac{c-v}{c}[/math]. If a satellite-based radio station recedes at half lightspeed, the apprent frequency of its onboard clocks appears to halve, and if it recedes at lightspeed, the frequency of its signals drops all the way to zero, and it appears "frozen". Signals sent form this c-recedign body than cannot reach us.

However, if we are labelled A, and a radio station b is receding from us at half lightspeed, and another radio station C is a little further away and receding from A at another half lightspeed, and A, B and c always lie on a straight line, then a signal sent directly from A to C will have zero frequency and be inviewabe, the same signal sent form A to B, and then form B to C, will halve its frequency halved and halved again, so the signal stil lhas 25% of its orignal frequency

Since A is seen to behave differently dependign on wheteh rit is viewed directly or indirectly, i nth eindirect case, the original vlecoity-shift formula no longer holds we have v=c, but f'/f is noe one quarter instea dof zero.

To be able to calcualte the shift in a single step, we can use a vleoity-addition formula to say that o.5c=0.5c gives 0.75c, and use this effective vleocity vslue to calcualte the vsible shift.

Under NM, adding two veleocities that are each less than lightspeed will always give an "effective velocity" value that is less than c.

Under special relativity

We can do similar calcualtions under SR: with the SR shift formula, half-lightspeed recession gives a shift of 1:1/ROOT[3}, so the total shift over two stages is the sqre of this, or 1/3.

Using th rvleocity additon formula we find that 0.5c+o.5c = xxxxx, and then using xxxxx with v=cxxxx gives us ... 1/3

Differenc ein approach

Where these two exercises differ is that in the NM case, the geometry of the system is being modified by the inclusion fo transponder B. if both velcocities a frationally over half lightspeed, then a and B are unable to communicate directly, and are separated by a sinal horizon. However, B is within reach of both A and C, and can be used ot relays signals between them. The presence of B is changing the lightbeam geometry and geerating the concept of indirect radiation that we normally find under quntum mechanics. Under QM, A and C can ragard each other as birtual objects, that cannot be seen fdireclty, but whose prssence drs have viewable consequences.

Under SR

With SR, although the operatio nof the v.a.f. seems superficiallysimilar, the phenomenolgy isdifferent. Under SR the presenc eof an intemediate body is not allowed ot alter the signal-propagation properties of the region - if A and C are recedign at 0.999c, and each are moving away form each other at nearly half c, then we use the v.a.f. in reverse to redefine the more distant relative velocity AB so that the composite shift and the individual shifts all obey the same shoft law.

Under AR, these red]fined vleocities are not "effective" velcoities, but "real velcoities - the Lorentz transform changes our definitions of distances and times in such a way as to make the result of the Vaf real – we say, that simply how spacetime behaves under Minkowski spacetime.

Notes

As a consequwnce:physics bas don the reder NM equstions allows indoirect radiaiton through a gravitational horizon, phusics base don the Sr equations doesn;t. Thsi is why GR1960, which includes SR, cannot replicate or approximae Hawkign radiaiton.