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In an earlier thread, I asserted that a rod has one true length, its rest length. If so, then the shorter coordinate length which is measured in some other frame must be somehow untrue. In this thread I argue that the coordinate length is a distorted view of the true length.

In the graphic below (fig. 1), there are two rods, each with a rest length of four units. The horizontal rod is at rest in frame S; the other rod is at rest in frame M. Frame M is moving at 0.6c relative to frame S. In this discussion, the focus will be on the rod in frame M [red], and its coordinate length in frame S [purple].

http://www.how-do-i-know-its-true.net/media/wpmu/uploads/blogs.dir/1/files/2011/02/RodMovementNumbered.png

[1] The rod which is at rest in frame M will always be parallel to the XM axis, no matter at what time the rod is drawn. Thus each instance of the rod in the graphic is parallel to the XM axis.

[2] The view of the rod from frame S (its coordinate length) is horizontal. Therefore, the view is not one view, but a composite of many views. The coordinate length is composed of many snapshots, each snapshot showing a specific point on the rod at a specific time in frame M. In the graphic, four instances of the rod are marked as they cross time TS = 6.68. The four marked points of the rod are at times TM = 5.35, 6.01, 6.67, and 7.33, respectively. The fifth mark, at TM = 7.75, is where the trailing end of the rod crosses TS = 6.68; no instance of the rod is drawn at that point.

[3] As one moves along the view of the rod in frame S, from leading end to trailing end, the time in frame M increases. This means that, in the view as seen from frame S, the trailing end of the rod has traveled farther than the leading end of the rod. This explains, qualitatively, the contracted coordinate length in frame S.

[4] When the velocity is reversed, the effect is the same, as shown in this graphic (fig. 2):

http://www.how-do-i-know-its-true.net/media/wpmu/uploads/blogs.dir/1/files/2011/02/RodMovementReverse.png

[5] The time differential from the leading end to the trailing end is equal to the relative velocity of the two frames multiplied by the rest length of the rod. In this example, (7.75 - 5.35) = 2.4 = 0.6 * 4, where T = ct and V = v/c. This can be better seen in figure 3:

http://www.how-do-i-know-its-true.net/media/wpmu/uploads/blogs.dir/1/files/2011/02/LengthContraction.png

Thus, the apparent contraction of the rod is directly related to the relative velocity of the frames. Taylor and Wheeler show that the contracted length can be developed by integrating from relative velocity 0 to V. (See exercise L-14 in Spacetime Physics.) Their interpretation is that the trailing end of the rod, as seen in frame S, begins to move before the leading end, thus contracting the rod in frame S. (The same is true for each differential segment of the rod. The differential time at each segment is smaller than at the trailing end, thus leading to a contraction proportional to the length.)

The interpretation proposed here is that the integration describes the compressive shifting of the individual snapshots in frame S. The coordinate length of the rod in frame S is thus a distorted view of the rod, while the rod itself is completely unaffected. The rest length of the rod is therefore its one true length.

Of course, the measured coordinate length is the same regardless of the interpretation of the result.

In the graphic below (fig. 1), there are two rods, each with a rest length of four units. The horizontal rod is at rest in frame S; the other rod is at rest in frame M. Frame M is moving at 0.6c relative to frame S. In this discussion, the focus will be on the rod in frame M [red], and its coordinate length in frame S [purple].

http://www.how-do-i-know-its-true.net/media/wpmu/uploads/blogs.dir/1/files/2011/02/RodMovementNumbered.png

[1] The rod which is at rest in frame M will always be parallel to the XM axis, no matter at what time the rod is drawn. Thus each instance of the rod in the graphic is parallel to the XM axis.

[2] The view of the rod from frame S (its coordinate length) is horizontal. Therefore, the view is not one view, but a composite of many views. The coordinate length is composed of many snapshots, each snapshot showing a specific point on the rod at a specific time in frame M. In the graphic, four instances of the rod are marked as they cross time TS = 6.68. The four marked points of the rod are at times TM = 5.35, 6.01, 6.67, and 7.33, respectively. The fifth mark, at TM = 7.75, is where the trailing end of the rod crosses TS = 6.68; no instance of the rod is drawn at that point.

[3] As one moves along the view of the rod in frame S, from leading end to trailing end, the time in frame M increases. This means that, in the view as seen from frame S, the trailing end of the rod has traveled farther than the leading end of the rod. This explains, qualitatively, the contracted coordinate length in frame S.

[4] When the velocity is reversed, the effect is the same, as shown in this graphic (fig. 2):

http://www.how-do-i-know-its-true.net/media/wpmu/uploads/blogs.dir/1/files/2011/02/RodMovementReverse.png

[5] The time differential from the leading end to the trailing end is equal to the relative velocity of the two frames multiplied by the rest length of the rod. In this example, (7.75 - 5.35) = 2.4 = 0.6 * 4, where T = ct and V = v/c. This can be better seen in figure 3:

http://www.how-do-i-know-its-true.net/media/wpmu/uploads/blogs.dir/1/files/2011/02/LengthContraction.png

Thus, the apparent contraction of the rod is directly related to the relative velocity of the frames. Taylor and Wheeler show that the contracted length can be developed by integrating from relative velocity 0 to V. (See exercise L-14 in Spacetime Physics.) Their interpretation is that the trailing end of the rod, as seen in frame S, begins to move before the leading end, thus contracting the rod in frame S. (The same is true for each differential segment of the rod. The differential time at each segment is smaller than at the trailing end, thus leading to a contraction proportional to the length.)

The interpretation proposed here is that the integration describes the compressive shifting of the individual snapshots in frame S. The coordinate length of the rod in frame S is thus a distorted view of the rod, while the rod itself is completely unaffected. The rest length of the rod is therefore its one true length.

Of course, the measured coordinate length is the same regardless of the interpretation of the result.

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