1 The rigid rod paradox with 8 clocks

The first experiment is called clock synchronisation. Halfway in between clock #4 and #5, there is a light source which emits a reset signal. The setup is such that the length lightpath to each clock is the same. Using rule 1 the light signal will reach all the clocks simultaneous. This is important because all the clocks at any moment will all show the same count.

The second experiment starts with making an exact copy of rod #1. Also attached to each clock there is an engine which can be fired with a standard burst in either the forward or backward direction. Each clock also has an observer. The second experiment consists that each observer on rod #2 fires his engine with a standard burst in the same direction when his clock is reset. This burst will give the rod a certain speed v. The now moving observers will perform the next tasks when they reach the next clock at rest: They will write down the reading of the clock at rest and the reading of their own moving clock. This is the result: They are the same for all observers. The number of counts of the moving clocks is less than the number of counts of the clocks at rest. This is not so strange because it means that the physical forces which influence the behaviour of each clock are identical. Specific what this means is that all the moving clocks stay synchronised. This is rule 2. You can repeat this experiment, but still, rule 2 applies.

Experiment 3 is almost identical to experiment 2. That means all the engines are fired after the reset signal is received. This defines the starting condition of experiment 3. The starting condition of experiment 3 is a moving rod with the speed v. Experiment 3 involves that a certain moment the light signal between clock #4 and #5 of the moving rod issues a reset signal. Like before the moving observers write down the results when they reach the next clock at rest. This is the result: All the observers write down the same number of counts for the clocks at rest. For the moving clocks, the results are different. The clock in front will have the lowest count. The clock at the back the highest count. Physical the clock in the back is reset the first.

Experiment 4 is the same as Experiment 3 with the difference that we again make an exact copy of rod #2 before the reset signal is issued. This is rod #3. The extra complication is that in experiment 4 both the moving rod #2 and #3 receive the same reset signal. The next complication is that when each of the clocks of rod #3 receives a reset signal also the engine is fired in the same direction as experiment #2. However, this will also give a physical complication, because the engine in the back will start first and in front of the latest. As such the physical forces will try to compress the rod. The opposite case is also possible. That means physical forces will try to expand the length of the physical rod.

Experiment 2 belongs to what you can call a symmetrical experiment. This is the case if you start from a state at rest than in either direction the results are the same i.e. how higher the speed how slower the moving clock ticks. What is also the case, after reaching a certain speed and the speed is decreased the clock starts to run faster until the speed reaches zero.

Experiment 4 belongs to what you can call an asymmetrical experiment. This is the case when the starting condition involves a moving rod. In that case when a clock receives a burst in the same direction as the original speed the clock will start to run slower. In the opposite direction, it is first faster and then slower.

These results are maybe different as what is expected. They challenge the concept of what means at rest.

2 The rigid rod paradox with 8 clocks

On Wednesday, November 13, 2019 at 10:26:30 AM UTC-6, Nicolaas Vroom wrote:

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Consider a rod with 8 clocks, equally spaced, a distance l apart. The rod is considered at rest. This implies that the speed of light c in all directions is the same. We call this rule 1. The clocks are numbered from #1 to #8. The strategy is to perform a certain number of experiments. The first experiment is called clock synchronisation. Halfway in between clock #4 and #5, there is a light source which emits a reset signal. The setup is such that the length lightpath to each clock is the same. Using rule 1 the light signal will reach all the clocks simultaneous. This is important because all the clocks at any moment will all show the same count.

As stated this implies the clocks are in a circle. If they were in a straight line they could not all be the same distance from the source of the light pulse. However I accept that the clocks can be synchronized with the result you state by using a slightly more elaborate procedure.

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The second experiment starts with making an exact copy of rod #1. Also attached to each clock there is an engine which can be fired with a standard burst in either the forward or backward direction. Each clock also has an observer. The second experiment consists that each observer on rod #2 fires his engine with a standard burst in the same direction when his clock is reset. This burst will give the rod a certain speed v. The now moving observers will perform the next tasks when they reach the next clock at rest: They will write down the reading of the clock at rest and the reading of their own moving clock. This is the result: They are the same for all observers. The number of counts of the moving clocks is less than the number of counts of the clocks at rest. This is not so strange because it means that the physical forces which influence the behaviour of each clock are identical. Specific what this means is that all the moving clocks stay synchronised. This is rule 2. You can repeat this experiment, but still, rule 2 applies.

This is where you are misunderstanding SR. You are applying an impulse to each clock/observer on rod #2 and the result is that each clock/observer is instantaneously accelerated to some speed, and they all start at the same time in this frame. Also, in this frame, they remain the same distance apart and synchronized as you say, BUT ONLY IN THIS FRAME. In the frame of the observers on the accelerated clocks the other clocks are no longer synchronized, and the spacing between the clocks on rod #2 is no longer the same as on rod #1.

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Experiment 3 is almost identical to experiment 2. That means all the engines are fired after the reset signal is received. This defines the starting condition of experiment 3. The starting condition of experiment 3 is a moving rod with the speed v. Experiment 3 involves that a certain moment the light signal between clock #4 and #5 of the moving rod issues a reset signal. Like before the moving observers write down the results when they reach the next clock at rest. This is the result: All the observers write down the same number of counts for the clocks at rest. For the moving clocks, the results are different. The clock in front will have the lowest count. The clock at the back the highest count. Physical the clock in the back is reset the first.

This experiment is not so clear what you intend. I believe you are saying that rod #2 is initially moving as speed v and that when alongside rod #1 that the engines are fired "simultaneously" to cause rod #2 to stop alongside rod #1. It is important to be clear what you mean by simultaneous, i.e. in what frame are they simultaneous? If in the rod #1 frame, and if rod #2 was accelerated as in experiment #1, then yes, all the clocks will show the same time differences. But be aware that the engines are not being fired simultaneously in the moving rod #2 frame. Nor is the initial spacing between the clocks on rod #2 in the moving frame the same as on rod #1 measured in its rest frame.

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Experiment 4 is the same as Experiment 3 with the difference that we again make an exact copy of rod #2 before the reset signal is issued. This is rod #3. The extra complication is that in experiment 4 both the moving rod #2 and #3 receive the same reset signal. The next complication is that when each of the clocks of rod #3 receives a reset signal also the engine is fired in the same direction as experiment #2. However, this will also give a physical complication, because the engine in the back will start first and in front of the latest. As such the physical forces will try to compress the rod.

You don't mention, and maybe are not aware, that in experiment 1 that rod #2 will be stretched, under tension, immediately after firing the engines that accelerated the rod.

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The opposite case is also possible. That means physical forces will try to expand the length of the physical rod. Experiment 2 belongs to what you can call a symmetrical experiment. This is the case if you start from a state at rest than in either direction the results are the same i.e. how higher the speed how slower the moving clock ticks. What is also the case, after reaching a certain speed and the speed is decreased the clock starts to run faster until the speed reaches zero.

I'm not following what you are intending to say here. If you are seeing some inconsistency, that is because you are not appreciating all the effects of the accelerations and SR. Rod #2 will be stretched immediately after the impulse and, in the frame moving with rod #2, the clocks will no longer be synchronized IN THAT FRAME. They will remain synchronized in the rest frame of rod #1 however, and will continue to be spaced the same distance, until the stiffness of rod #2 pulls the clocks back to their "normal" spacing. There are three effects: time dilation, change in synchonization and change in spacing. If you correctly understand all of these there is no paradox or inconsistency.

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Experiment 4 belongs to what you can call an asymmetrical experiment. This is the case when the starting condition involves a moving rod. In that case when a clock receives a burst in the same direction as the original speed the clock will start to run slower. In the opposite direction, it is first faster and then slower. These results are maybe different as what is expected. They challenge the concept of what means at rest. For much more detail read this: https://www.nicvroom.be/Article_Review_Moving%20Bodies_Appendix2.htm Nicolaas Vroom

3 The rigid rod paradox with 8 clocks