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Happy New Year! May your 2007 be good.
In "Ladder Paradox (i)" we viewed the "long arrow in the short box" as pure relativistic length contraction. In this 'installment' we will view it from a different angle: space-time rotation on a so-called space-propertime diagram (to be defined soon).
The experimentally confirmed relativistic principle that light propagation is isotropic in every inertial frame makes it very simple to synchronize clocks. You send a time-stamped electromagnetic signal from one clock's location to the others. If the distance (d) between the clocks is known, the receiving observer adds the light travel time (tt=d/c, where c is the speed of light in free space) to the time stamp and sets the local clock accordingly. This, in principle, is the "Einstein method" of clock synchronization.
When two inertial frames are in motion relative to each other, clocks synchronized by the Einstein method in one inertial frame will not be synchronized in the other frame. Figure 4 illustrates the situation of our arrow (red) and our box (blue) on a two-dimensional space-propertime diagram.
Figure 4 
On a space-propertime diagram, the arrow, which is moving to the right at 0.8c, is rotated as shown - the rotation angle is actually φ = -asin(v/c). Note that the rotation is through more than the 45 degrees allowed in Minkowski space-time diagrams. Here the speed of light corresponds to a 90-degree rotation.
An important observation: the 15ft arrow is projected onto the box frame as a 9ft x-axis length and a 12ns t-axis synchronization offset of the arrow's tail clock. This means that observers using synchronized clocks in the box frame will, at the time of event A, observe the arrow's tail clock to be 12ns ahead of the arrowhead's clock. This is despite the fact that in the arrow frame, the head- and tail clocks are perfectly synchronized!
The time-wise movement of the box is vertical upwards, because it is stationary in the frame. The arrow moves along the dotted red lines, as shown in figure 5, where event B is illustrated. (Event B is when the arrow's tail enters the box.) The arrow's tail reaches the x-position of the box's front 11.25 seconds after the arrowhead entered the box, in accordance with the times given in "Ladder Paradox (i)".
Figure 5 
Note that in this frame, both the box and the arrow moved a space-propertime interval of 11.25 units This is an important observation: in a gravity-free environment, all inertial objects trace out the same space-propertime intervals, irrespective of their relative movements. This is independent of which observer's inertial frame is chosen as reference. It is equivalent to the invariance of the space-time interval of Minkowski space-time.
This leads to another easy observation: when an object's world line (space-time path) is not parallel to the coordinate time axis, it apparently trades spatial movement for temporal movement in a vector-subtraction fashion. This is obviously dependant on which observer's inertial frame is chosen as reference. It is the equivalent of relativistic time dilation.
After 12.5 seconds of box time, the arrowhead reaches the far side of the box, with the tail of the arrow well inside the box. Because we do not have "space-propertime vision", we observe only the projection of the arrow onto the space axis - hence the long arrow fits into the short box!
Figure 6 
Whether the physical arrow could momentarily fit into the box, is a matter of who's looking at it! In part 3C of this mini-series, we will analyze the situation from the arrow's point of view. This will make it clear that length contraction cannot be a physical phenomenon, but only an observed one.
Regards, Jorrie
PS: Space-propertime diagrams (Edit: also known as Loedel- or Brehme diagrams) are not well recognized in mainstream physics. This is not because they are false, but because of a certain lack of generality. One of the tools of the relativist, the 'light-cone', is not easily represented on a space-propertime diagram.
However, we are mostly interested in time-like and light-like intervals, so for many purposes, the space-propertime diagram works just fine and is somewhat more intuitive than the Minkowski space-time diagram. You can read more on space-like, light-like and time-like intervals on the website Relativity 4 Engineers.
-J
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