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On the interpretation of a spacetime diagram 
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#1
Jan1711, 07:37 PM

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Below is a spacetime diagram which depicts the movement of a rod, four units long, relative to another rod of equal length. The red lines are the "moving" rod; the purple lines are the "stationary" rod. The gray lines are the world lines of the unit marks (0, 1, 2, 3, 4) on each rod.
As I attempt to interpret the diagram, I am moved to ask a question which has been asked before on this forum, including by me. There is nothing new in the question itselfonly my level of understanding has (hopefully) progressed. To set up the question, I stipulate that the stationary rod and the moving rod are identical, having been fabricated on the same highprecision equipment. The stationary rod was then fastened to a bench in the lab, and the moving rod was set in motion. As the moving rod passes over the resting rod, it is moving at a constant velocity of 0.6c. As can be seen in the diagram, instruments mounted on the moving rod will measure the resting rod to be 3.2 units long. Likewise, instruments mounted on the resting rod will measure the moving rod to be 3.2 units long. My question focuses on the stationary rod, because it has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. Knowing that the stationary rod is four units long, would we not have to conclude that the instruments on the moving rod are incorrect when they report its length to be 3.2 units? 


#2
Jan1711, 08:12 PM

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#3
Jan1711, 09:46 PM

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In the case of a relative speed of 0.6c, which each rod correctly measures, gamma is 1.25, which means that each one sees the other one's times as lengthened by a factor of 1.25 and sees the other's rod as being shortened by a factor 1/1.25 which equals 0.8. Let's say that light takes 1 unit of time to travel 1 unit of length (whatever you want that to be) so c equals 1 unit of length per 1 unit of time.
There are many ways to illustrate how this is happening but one of the easiest, in the case where the two rods pass directly adjacent to each other is to time how long it takes for each rod to pass by the other one. So let's look at it from the standpoint of the "stationary" rod. It's clock/timer is running at the normal rate and so when the two rod's meet, it starts its timer. How long will it take for the other end of the moving rod to get to the leading edge of the stationary rod? Well, since the rod is 3.2 units long and it is traveling at 0.6 units of length per 1 unit of time, and since time is equal to distance divided by speed, the time will be 3.2 divided by 0.6 which equals 5.3333 units of time. That is what the stationary rod measures of how long it takes for the moving rod to pass a certain point. Now since length equals speed multiplied by time we get 0.6 times 5.3333 which equals 3.2. (Image that!) Now let's see how the moving rod measures the stationary rod's length. We're still examining this from the lab frame. When the two rods meet, the moving rod starts it's timer. How long will it take for the leading edge of the moving rod to get to the trailing edge of the stationary rod? Well, since the stationary rod is 4.0 units long and the moving rod is traveling at 0.6 units of length per 1 unit of time, and since time is equal to distance divided by speed, the time will be 4.0 divided by 0.6 which equals 6.6666 units of time (according to the stationary rod). Now to see what the moving rod measures for how long it takes for a point on the moving rod to pass the length of the stationary rod we have to divide by gamma so the result is 5.3333. Now since length equals speed multiplied by time, we get 0.6 multiplied by 5.3333 which equals 3.2. Please note that this is all done in the lab frame. It is also possible to transform the entire scenario into another frame, such as one in which the "moving" rod is at rest but this will result in the exact same calculation as was done for the "stationary" rod so it's rather unexciting. 


#4
Jan1711, 09:57 PM

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On the interpretation of a spacetime diagram
On this thread I drew up some illustrations of two ruler/clock systems moving past each other, showing how each system measures the other rulers to be shrunk and the other clocks to be sloweddown and outofsync. You can see from the diagrams how the situation is completely symmetrical, and if you add to that the fact that all laws of physics work the same way in the coordinate systems defined by each ruler/clock system, you may get a better idea of why there is no basis for preferring one point of view over the other (including the fact that there's no basis for labeling either 'at rest' in an absolute sense).



#5
Jan1811, 06:47 AM

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Suppose we construct two identical parallel rods, each of unit length, with devices attached to measure projected lengths (e.g. A sliding T). Now we rotate one rod 45 deg and each rod measures the projected length of the other rod. Each rod finds the projection of the other rod to be .7. The stationary rod has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. Knowing that the stationary rod is 1 unit long, would we not have to conclude that the instruments on the rotated rod are incorrect when they report the projection of its length to be .7 units? 


#6
Jan1811, 06:31 PM

P: 221

Perhaps this is the experiment which can (in principle) be conducted to determine whether length contraction is physically real. 


#7
Jan1811, 06:53 PM

P: 848

What about considering the situation with a focus on the 4dimensional geometry. The situation described below is very similar to what you've set up with the two poles. However, to give the situation a definite physical character to it we use the wellknown poleinthebarn example. My Special Relativity physics prof gave this one to us as a homework problem. The challenge was to use spacetime diagrams to illustrate how a pole that was too long to fit in a barn in one coordinate system could be observed to fit easily in the barn in the view from another system.
I think is helpful to first view the pole and the barn as two 4dimensional objects. The pole flies toward the barn at relativistic speed. The spacetime diagram is sketched as the pole (red 4D object) moving to the left and the barn (blue 4D object) moving to the right at the same relativistic speed with respect to a rest system (black coordinates). Again, we assert something more than just coordinate systems: We regard the two objects as literally 4D objectsyou construct the 4D objects by extruding the easily visualized 3D objects along their respective 4th dimensions. The impression of speed is a manifestation of the relative rotational orientations of the 4D objects with respect to each other (and with respect to a photon world line). Please don't get into the side bar philosophical issue of whether I'm trying to float a literal interpretation of a static 4D universe here. This is a pedagogical choice. Thus, for the sake of understanding the spacetime diagrams, we at least temporarily abandon caution for the sake of getting the hang of crosssection views of 4D objects. Once we accept this 4D object view, we must then adapt another curious aspect of our 4D world experience. Observers experience the laws of physics only by living in a continuous sequence of instantaneous 3D crosssections of the 4D universe in which the following results: For any rotational angle of a given 4D object's world line (angle of the world line of the 3D extrusion along the 4th dimension world line), any instantaneous 3D crosssection view of that 4D object (representing a normal spatial axis) will be symmetrically oriented about a world line representing the world line of a photon of light. That is, the angle between the spatial dimension (axis, i.e., X1') and the photon world line is (for any observer) always the same as the angle between the observer's 4th dimension (i.e., world line directionX4'). Thus, for two observers moving with respect to each other, their 4th dimension world lines will have different angular orientations with respect to a photon world lineand consequently their spatial axes will have different rotational axes as well. This accounts for the different crosssection views of 4D objects. In the sketch below we have a red coordinate observer (view from the vantage point of the pole) and a blue observer (view from the viewpoint of the barn). The front and back doors are initially open. When the pole reaches the barn (blue guy's view) the blue guy waits until the pole is inside the barn, then he quickly closes both doors, then opens the back door just in time to let the pole fly on through. But, just for an instant the pole is INSIDE THE BARN WITH BOTH DOORS CLOSED. However, the red guy (moving with the pole) observes the front and back doors opening at different times, so that, in his crosssection view of the 4D objects, the pole is never inside the barn with both doors closed (it better not be, since in his world the pole is too long to fit into the barn). Notice, for the red guy, the back door opens before the pole is completely inside the barn. This of course illustrates the length contraction as well as time dilation and differing impressions of simultanaeity. The point emphasized here is that the blue and red guys are each viewing dimensions of crosssections views of objects that are in reality 4dimensional. The 4D objects are the reality. Our 3D images are just particular crosssection views that depend on the realtive orientations with respect to photon world lines. It was important to use a SYMMETRIC spacetime diagram in order to sketch the 4D objects to the same scale on the computer screen. 


#9
Jan1811, 07:02 PM

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#10
Jan1811, 07:27 PM

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#11
Jan1911, 12:57 AM

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#12
Jan1911, 01:32 AM

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#13
Jan1911, 01:50 AM

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#14
Jan1911, 03:01 AM

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P.S. Edited to correct the number of extra carriages from 4 to 6.6666 


#15
Jan1911, 05:18 AM

P: 221

On first look, I think this is the same as the poleinthebarn scenario. If so, then the answer is that you can't fit more cars in the gap as the speed increases, because (roughly speaking) the gap isn't on the track all at once (that is, at the same time). Suppose we accept your premise. There are two observers of the train, moving at different speeds relative to the train. The one observer will gauge that he can put two extra cars in the gap; the other will see room for three extra cars. Which one is right? 


#16
Jan1911, 05:31 AM

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#17
Jan1911, 05:43 AM

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#18
Jan1911, 06:06 AM

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