## K13 STR and Asano Diagrams

After ‘completing’ this book I discovered in the basement of the Central Library in Zurich a little book [45], in which the two brothers Seiichi and Shiro Asano present their "Space-Time Circular Diagrams". The first Japanese edition was published in 1983, thus coinciding with the first edition [15] by Epstein! The brothers Asano, as small boys, were impressed by all of the whoopla of Einstein's visit to Japan in 1922. They went on to have careers as an electrical engineer and a physician, respectively. After retiring, they decided to elucidate the STR for themselves and others.

Like Epstein they take as their starting point Minkowski’s equation Δτ^{2} = Δt^{2} – Δx^{2} – Δy^{2} – Δz^{2} (times and lengths are measured in the same units), suppress the y and z component and rearrange the remainder of the relationship Δτ^{2} = Δt^{2} – Δx^{2} so that it can be interpreted as the equation of a circle: Δt^{2} = Δτ^{2} + Δx^{2}. Also with the Asano brothers the straight line on which B moves with constant velocity v through space-time is tilted with respect to the time-axis of a stationary observer A by an angle φ, where sin(φ) = v / c. On [45-49] they consider right triangles that are congruent to those in the corresponding diagrams of Epstein.

A and B have met at O and both have set their clocks to zero. The sine of the angle AOB is v/c. At B_{1}, B_{2} and B_{3}, we have right angles according to the theorem of Thales.

When the time t_{3} has elapsed for A, which corresponds to the distance OA_{3}, then for B the time which corresponds to the distance OB_{3} has elapsed. At time t_{3}, B is the distance X_{3} = A_{3}B_{3} from A.

We obtain the corresponding congruent right triangles of the Epstein diagram by reflecting those of the Asano diagram through the angle bisector of AOB.

Also the Epstein diagrams characteristic semicircles around O occasionally appear in the Asano diagrams; however, they indicate only the elapsed time intervals for A. The diagrams show a dilation with center O and a stretching factor, which is proportional to time [45-50]:

But for the spatial axes the brothers have no good solution. One might say that they still tried to separate time, space-time and space. The dashed curves, which indicate at what speed a certain distance (in light-seconds) is reached, are quite complicated:

Do you notice the point which belong to the Pythagorean triple (6/8/10) and which also lies on the straight line for v = 0.6 • c?

Epstein diagrams are clearly preferable to those of Asano. They are based on a simple postulate and they are more easily drawn and read. But it is interesting to note that similar approaches appeared in different places simultaneously. The Asano brothers do not mention their ‘competitor’ Epstein in the first English edition of 1994.