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Originally Posted by CraigS
Ok. I think I get this with a slight variation.
(I may not have this quite right, yet so please bear with me for a bit)....
I thought that the contraction occurs in any component of an object's dimensions, which lies parallel to the direction of travel at the moment of measurement. (Ie: where this component intersects the other's 'hypersurface of simultaneity').
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That's correct. Consider a rod travelling along the direction of the length of the rod.
The rod undergoes length contraction. The diameter of the rod which is a dimension perpendicular to the direction of motion remains unaltered.
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I've attached a pdf showing two sticks moving off at the same speed in opposite directions. Check out the B stick. Where the ends of the B stick intersect A stick's hypersurface you can see the length has contracted from one unit, to less than one unit.
So, from this diagram, the length contraction happens if the direction of travel is in opposite directions, but happens to the component of the stick's dimension, which lies parallel to the direction of travel.
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The pdf is showing 2 Minkowski diagrams. The length contraction is based on the stationary observer's frame of reference.
In the stationary frame of reference the time and spatial axes are perpendicular or orthogonal. In the sticks frame of refernece which is moving relative to the stationary observer the axes are rotated and are now less than 90 degrees to each other. The faster the stick moves the greater the rotation of the axes. The lines intersecting the ends of the sticks are parallel to the rotated time axis. Since the lines run parallel to the time axes the ends have been simultaneously measured. The sticks are located over the rotated spatial axis.
The projection of the lines onto the rotated spatial axis gives the contracted length.
Regards
Steven