K7   Deriving the Formula for Addition of Speeds from an Epstein Diagram

In D3, we promised a proof of the formula (red box in D4) for the relativistic addition of speeds based directly on Epstein diagrams and not using the Lorentz transformations. Basis for this is a drawing by Epstein himself in Appendix A of the second edition of [15].  We have redrawn Epstein's figure in a way to best support our proof:

On the left one sees the Epstein diagram for the following situation: Blue moves with velocity w' in the reference system of red, and thus sin(β) = w' / c. The proper length of the distance traveled is AB, and the trip takes time AF for blue, while red measures time AG = AK.

On the right you see that red moves relative to black, and thus, as usual, sin(α) = v / c. Also for black blue moves from the start to the end of the segment CD: If blue reaches I, the endpoint D of the segment is in O and therefore they coincide for black spatially in E. The time intervals that red and blue measure are unchanged (CM = AK and CH = AF, respectively), and also the proper length of blue's path from the viewpoint of red is unchanged (AB = CD = QO).

The formula for addition of speeds is proved if we can show

We do not need all of the following segments. Of course, their physical meaning is also irrelevant for the proof. However:

AK = AG = BL = CM = EO  mmmm
Time elapsed for red
AB = FG = KL = CD = QO length of blue's path for red, proper length of this path
AF = BG = CH = EI Time elapsed for blue
CE = HI = MO length of blue's path for black
CI = CQ = CP Time elapsed for black

The calculation is then quite simple:

The interpretation of this calculation is left to the reader: If one assumes that we have already proven the formula for addition of speeds, then these calculations should check the correctness of the above construction of the angle γ. If one accepts the correctness of the above drawing, then the calculation gives a proof for the addition of speeds, independent of the Lorentz transformations!

It requires some experience dealing with Epstein diagrams to recognize the correctness of the construction free of doubt. So it is perhaps not the didactic approach of first choice. Nevertheless, the drawing of Epstein shows clearly some of the most important aspects: Since the proper time of the sequence for blue is an invariant (AF = BG = CH = EI), then the addition of two velocities each smaller than c, will always yield a speed which is itself also smaller than c. If blue moves horizontally with respect to red (i.e., such as light with velocity c), then blue will also move horizontally with respect to black: v ⊕ c = c .

I would like to reiterate my thanks to Alfred Hepp for drawing my attention in the first place to appendix A in the second edition of [15]. He also persistently encouraged me to expand the construction of Epstein into a proof of the addition of speeds formula based solely on Epstein diagrams. Dissatisfied with my first proof, he also made significant contributions resulting in the present, much simpler derivation.

Another suggestion: The ⊕-addition makes the open interval (-1, 1) a commutative group. The identity element is zero and the inverse of v is -v. The addition is also monotonic in the sense that if a < b then it follows that a ⊕ d  <  b ⊕ d. Prove twice that ⊕-addition is associative: a) by formal algebraic calculation  and  b)  by physical interpretation. Why must the two endpoints 1 and -1 be excluded?