D2 Derivation of the Lorentz transformations from Epstein Diagrams
If black and red want to compare their measured values for place and time which they assign to an event, then it presupposes that they have already crossed paths and synchronized their clocks. Synchronization means that both set their “master clocks” at this meeting at x = 0 = x' to t = 0 = t' and synchronized any other clocks within their system with the master clock. To compare the coordinates of an event is always to talk about a second contact of red and black. Otherwise the measured values of red and black would be arbitrary! This remark by the way applies to all space-time diagrams, not only to those of Epstein.
Consider two coordinate systems meeting at O as described in the preceding section. Now a red clock moves by a particular location of black, and the coordinates (t', x', y', z') are recorded. Which coordinates (t, x, y, z) will black attribute to this meeting? How can we, in general, convert such event coordinates, given that we account for the relativistic effects of time dilation, length contraction and desynchronization derived in section B from Einstein's basic postulates?
This question is answered by the Lorentz transformations already mentioned in A3 and A4. In this section we derive the Lorentz transformations from Epstein diagrams. For skeptics a second derivation follows in the next section based only on the three basic phenomena and their quantitative description in section B.
First we note that the convenient equations y = y' and z = z' still apply. As described in B3 distances perpendicular to the relative velocity of the two systems, that is, perpendicular to x and x', are the same for both black and red. We need only worry about the time coordinates and the spatial coordinates in the direction of relative motion v. And it is exactly these values which are perfectly represented in the Epstein diagram. We start with a nearly ‘empty’ Epstein diagram:
We mark an arbitrary point E in space-time. No doubts exist about where E lies in black and red: We need only project E onto the x-axis, respectively x'-axis. We obtain the points C and D (in the following figure), and
(1) x = OC and x’ = OD
Besides it is still true that
(2) y = y’ and z = z’
These projections additionally yield the auxiliary point Q, which we will later make use of. We now have the following picture:
E defines the event “the red clock of x' is at location x of black”. The time t' of this event can only be measured with the red clock which is locally present! Now we draw the projections on the two time axes and obtain the points F and B:
For red the clocks at O and D (or, somewhat later, at B and E) are synchronized. Black sees things differently. The time indicated by the local red clock for event E, is for black simply
(3) t’ = OF = CE
Thus we have accounted for the two effects of time dilation and desynchronization!
The time t of black is still missing in the diagram. Which clock does black use to measure the flyby of the red clock at location x? Clearly it uses the one at location x, that is, the one which was at point C when the meeting of the master clocks took place at point O. However, where is this clock, given that the red clock has moved from D to E? According to the dogma of Epstein everything moves equally far through space-time between two events. The black clock at location x is therefore at point G, and therefore we have by necessity CG = OA = OB = DE.
The two points E and G in the Epstein diagram both belong to the event “the red clock in its system at location x', flies past location x of black” !
This is often confusing for those, who are well versed with space-time diagrams for which an event always corresponds to a single point in the diagram.
Black attributes the following time coordinate to this event:
(4) t = CG = OA = OB = DE
We still have to find the values (t, x, y, z) for the associated values (t', x', y', z') and vice-versa. We already know how it is with y and z. We now investigate how one obtains the values (t, x) from (t', x'). The reverse transformations will be left as a small algebra exercise. We use lengths for all space-time distances and must therefore multiply time values by the speed of light c. Thus:
t • c = OA = OB = DE = DQ + QE = OD • tan(φ) + CE / cos(φ) = x’ • sin(φ) / cos(φ) + t’ • c / cos(φ)
Recalling the meaning of sin(φ) and cos(φ), we are already finished:
t • c = x’ • (v / c) / √ + t’ • c / √ = ( t’ • c + x’ • v / c ) / √
Dividing by c and writing the whole somewhat more conventionally, yields
The two terms in the numerator beautifully represent (together with the denominator) the effects of time dilation and desynchronization. For x' = 0 we have only the first effect whereas t' = 0 underscores the second effect.
Just as easily we can derive how one obtains x from (t', x'):
x = OC = OQ + QC = OD / cos(φ) + EC • tan(φ) = x’ / cos(φ) + t’ • c • sin(φ) / cos(φ) thus
x = x’ / √ + (t’ • c • v / c) / √ = ( x’ + v • t’ ) / √
Here the difference to the corresponding Galileo transformation is less significant, showing as it were only length contraction.
Here are the resulting transformations:
The transformations (t, x, y, z) ↔ (t', x', y', z') presented here must mutually cancel, if they are executed consecutively. This calculation is recommended to the reader as an exercise.
As author of this book I would like to sing the highest praise for the Epstein diagram. In the USA one would call me an ‘evangelist’. I am indeed quite proud to be the first Epstein evangelist to show the derivation of the Lorentz transformations from an Epstein diagram …
For the skeptics or otherwise incorrigible non-believers a derivation of the transformations follows in the next section which is completely devoid of Epstein diagrams and based only on the results of section B.