C4 Length Contraction in the Epstein Diagram
Also the length contraction can be easily and quantitatively correctly read from the Epstein diagram. Consider the following diagram:
What does red say?
The segment OC is at rest in my system and it is only getting older. It moves through time by OB = CD. The segment has the length OC = BD which is its proper length. Black moves in space-time an equivalent distance, i.e. from O to A. Purely spatially speaking it moves with
v = - c • sin(φ) from O to F. Therefore less time elapses for black, only OE = OA • cos(φ) = OB • cos(φ)
What does black say?
The segment OC moves in space-time toward BD while I simply grow older by OA. Meanwhile the segment OC ages only by
OJ = OB • cos(φ) = OA • cos(φ). I measure a length of OG = HI in the x-direction for this segment. It is OG = OC • cos(φ). Purely spatially I measure that a segment OG moves toward HI with speed v = c • sin(φ) .
All statements are in complete agreement with the results of B2 and B3!
In its own system each object has its proper length and moves only through time. The tilt of angle φ causes the lengths of fast objects to be measured as shortened ‘shadows’. In addition, the progression of the time is slowed for these fast objects. The beauty of this is that both effects are quantitatively correctly shown. Also: The two principles of the maximum proper time and the maximum proper length become evident in the Epstein diagram!
From length contraction, as described in B3, only the direction of movement in space is affected, which agrees with the direction of the relative velocity. Epstein illustrates this with two diagrams [15-91]:
The USA is at rest. It moves only through time:
The USA is observed from above by a spacecraft racing from east to west. For passengers in the spacecraft the USA is therefore racing from west to east. The time axis of the USA tilts through an angle φ and the spatial projection of the USA shrinks, but only in the direction of the relative motion:
Space and time actually shrink down to mere shadows (see Minkowski quote of C1), shadows of the movement through space-time, trapped in the speed of light!