## E2    Epstein Diagrams for Mass and Momentum

Thus we obtain ‘dynamic mass’ mv by dividing ‘rest mass’ m0 by our well-known radical, which corresponds in the Epstein diagram to the cos(φ). Therefore we can represent mv and m0 in a simple diagram (Note: German 'Ruhemasse' means 'rest mass'):

Does the projection of mv on the horizontal axis have a meaning and if so what is it? We obtain this projection with mv • sin(φ), and if we remember what the sine value means in the Epstein diagram, then we immediately have

< ? > = mv • sin(φ) = mv • v / c = p / c

Thus we can completely label the Epstein diagram for mass and momentum (Note: German 'Impuls' means 'momentum'):

The faster an object is the larger the angle φ becomes, and the more the dynamic mass and the momentum of the object grow (if the rest mass remains constant). sin(φ) and cos(φ) keep their former meaning. Here one sees again, what a beautiful representation we would have if c simply had the unit-less value 1: one space-time unit per space-time unit. The transition from the usual technical units to these more ‘natural’ ones is, however, easy to make and is open to the willing reader. We nevertheless permit ourselves to speak of this diagram as the mass-momentum diagram.

Let us solve a standard task using such a mass-momentum diagram: How fast does an object have to move in order to double its mass? If we give m0 5 little squares then mv amounts to 10 little squares:

A compass arc with radius 10 squares gives us mv and φ. We read from the diagram:
v / c = sin (φ) ≈ (8.6 or 8.7 squares) / (10 squares) ≈ 0.86 or 0.87

The object must therefore move with approximately 87% of the speed of light. The calculation of v / c yields the exact value √ (3) / 2 with the approximate value 0.8660.

The opposite question (‘How large is mv / m0 for an object, which moves with 90% of c?') can be answered just as easily with an Epstein mass-momentum diagram. If one wants to completely avoid using a pocket calculator then one should make certain that mv in the denominator corresponds to a simple number of squares (best 10 or 20). For three or more digits of accuracy then millimeter scaled graph paper must be used ...

David Eckstein's copy of the German edition of [15]                                      Samuel Edelstein's copy of [15]

Although recommended by many authors, both the English and the German editions of the book are out of print. A few used copies are available at rather high prices via Amazon.