Where the standard Minkowski spacetime diagram has a decidedly square "look and feel" to it, the Epstein diagram looks somewhat circular, or rather like a conversion from rectangular Cartesian coordinates to polar coordinates. The circle has a radius of cT, i.e. the speed of light multiplied by the time read on a clock that moves radially from the center. It is usual to make the radius 1 spacetime unit, as shown on the diagram. Further, we only show two dimensions (the radius is a two-vector, one of space and one of time), but it can be done for more dimensions. With two dimensions of space and one of time, the circle will be a sphere and for full 3-D space plus time, it will become what is known as a 4-D hyper-sphere.

The blue and red arrows from the origin are rotated relative to each other by an angle determined by the relative speed between two inertial observers. In the diagram above it is V=0.6c, so let us use this in a worked example. The grid shown is arbitrarily chosen to coincide with the blue axes. In one second of blue coordinate time, the blue observer has moved 1 light-second straight up, just like in any spacetime diagram. In the same time, the red observer has moved 0.6 light-seconds to the right and 0.8 light-seconds up, so the red unit vector points to 0.6,0.8 on the grid.

Using Pythagoras, we know that 0.6

^{2}+0.8

^{2}= 1

^{2}, so what Dave wanted to happen now actually makes sense: add a space displacement vector to a time displacement vector and you get an invariant spacetime vector. What is more, it can be shown that if we take the origin and the two arrow-head positions each as a separate event in spacetime, the values conform to the Lorentz transformations (LTs) for converting space and time intervals observed in one inertial frame to another inertial frame. If the red frame's clock reads 1s when reaching the position (0.6, 0.8), a clock synchronized in the blue frame will observe the same event at 0.8 seconds. More about this later.

If we flash a light at the origin (when the two observers were collocated there), the light-pulse will go along the the blue +x axis, i.e. 90 degrees. The present red frame will however not observe that light flash to progress along the blue +x axis, but rather along its own red +x axis. This is a peculiar feature of the Epstein diagram - the normal common lightcone is replaced by a light-sphere that "looks the same" for all inertial observers, but they observe different portions of it. Lewis Carroll Epstein has cautioned against taking this concept too literally - he actually called it "a myth" in his book, albeit a very useful one. It helps to visualize special relativity more naturally.

Before I leave you with the above to sink in a little, just one more tantalizing bit of Epstein. Look in the bottom part of the diagram at the dotted line from the red +x, drawn normal to the blue +x axis. Can you see the Lorentz contraction there? It is just the projection of the length of 1 light-second on the red x-axis onto the blue x-axis. Can it really be that simple?

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Regards

Jorrie