C2 Epstein Diagrams
The dogma reads: Everybody is always and everywhere moving at the speed of light c in 4D space-time. Everyone calls the direction in which they move their time and the directions orthogonal to it form their space. One second of time therefore corresponds to 299,792,458 m (≈ 300,000 km) of space (that one did not notice this earlier can be explained as a consequence of the ‘disproportionateness’…).
If we want to plot this movement in 4D space-time then we have the same problem as everyone else who attempts to draw a four-dimensional representation. One must be happy when a three-dimensional picture is clearly understood on a flat sheet of paper. In our case however these difficulties are easily eliminated: We represent only one direction in space, the one in which the two reference systems move relative to each other! Anyway, nothing exciting happens in the other two spatial directions according to our formulas of B3!
We will always use a black coordinate system with origin A and a red one with origin B, like the one shown above in B3. B moves with velocity v along the x-axis of black and A moves with velocity -v along the x'-axis of red. A and B met at O and at that point both set their clocks to zero. In addition, everyone tells time with the clocks of their respective system which have been synchronized respective to the master clocks of their system. A and B are each spatially at rest in their own system and move (in their own system) only through time. Our first attempt:
In order to avoid complications with causality we must forbid that red, which had an interaction at O with A, can ever influence the temporal past of black before point O. This means that the angle φ may not be larger than 90º. Otherwise red would be able to ignite its engine after some time, return to point O and arrive there at a time before the interaction between A and B, which already took place. We adhere emphatically to the following:
For systems, which can interact with one another, the angle φ between the two time axes (that is, the directions of the journey through 4D space-time) may not be larger than 90º.
It is important that the segments OA and OB are equal: In a given interval of time both cover the same distance in space-time! That is Epstein’s dogma. What is the meaning of the angle between the time axes? We depict the direction of the x-axis of black and mark the place, which B has reached in this x-direction, while black has just aged by the segment OA:
Thus OA = OB. OA is for black simply the time, which elapsed since the meeting with B at O. B has in this time, from the point of view of black, traversed the segment OX. This yields
since in the right triangle OXB angle φ occurs again at vertex B. We obtain B’s position in the coordinate system of A simply by projecting the space-time position of B perpendicularly onto the space-axis of A. In so doing we are being rather cavalier with our mix of units: OA, OB and OX are segments in space-time. When reducing to pure times and distances we must consider that 1 second of time corresponds to a distance of 1 light-second, or about 300,000 km! If we clean up the units of measurement in the above equation it looks as follows:
The unit-free number sin(φ) corresponds in the Epstein diagram to the ratio v to c! So far we have taken the point of view of black. That is unnecessary since Epstein diagrams have (unlike Minkowski diagrams!) the beautiful characteristic that they display symmetrical relationships symmetrically. Thus, we draw the above diagram again with the addition of a space-axis for B:
From the point of view of B: During the time OB, A traverses OX’.
The two triangles OXB and OX'A are congruent. The same absolute value for the relative velocity v arises in both coordinate systems. Because of the selected orientation of the axes we obtain however different signs for v: For red A moves in negative x'-direction, while for black B moves in positive x-direction. Thus we are in complete agreement with the presentation in B3.
Perhaps you are surprised that the time axes are not indicated with t or t’ (Note: in the diagrams the German word 'Zeit' stands for 'time', where as 'Raum' stands for 'space'). We reveal the reason for this in the next section. First we want to do another small calculation, which yields a very important result. For acute angles φ we have
The radical which appears in the calculation of time dilation and length contraction, has a simple geometrical meaning in the Epstein diagram! Spoken as black: sin(φ) projects OB on my space-axis, cos(φ) projects OB on my time-axis. We would like to exploit that immediately.