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Old Aug 17th 2017, 03:55 PM   #1
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How real is the fitzgerald contraction of SR

Here is a simple question presented without a lot of maths but just a diagram I made in Inkscape paint software (attached).

It shows a couple of box cars going through a tunnel which ar attached together (not shown). However when the wheels are placed over two sensors simultaneously (shown as yellow triangles) a light bulb turns on. If only one of the sensors is activated the light bulb will not turn on. Now if the box cars are stationary and placed as in the diagram the light bulb will turn on.

However, now for the tricky part. If the box cars are move close to the speed of light and travelling through the tunnel and they are length contracted the wheels will never simultaneously touch the sensors. Will the light bulb ever turn on in this scenario, for the briefest of time, or does the fitzgerald contraction of SR prevent this from happening?

What makes this even more fun is that a passenger in the box car sees the railway lines contracted instead so he sees the sensors as closer together. At just the right speed the sensor will be close enough together so that they will be both under the wheels of the first box car. (3rd picture). In this scenario the light bulb would turn on ... or would it?
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How real is the fitzgerald contraction of SR-train-tunnel.png  
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Last edited by kiwiheretic; Aug 17th 2017 at 08:27 PM. Reason: readability
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Old Aug 17th 2017, 05:56 PM   #2
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Originally Posted by kiwiheretic View Post
Will the light bulb ever turn on in this scenario, for the briefest of time, or does the fitzgerald contraction of SR prevent this from happening?
No, the lightbulb will not turn on. The Lorentz contraction is quite physically real phenomena.

Originally Posted by kiwiheretic View Post
At just the right speed the sensor will be close enough together so that they will be both under the wheels of the first box car. (3rd picture). In this scenario the light bulb would turn on ... or would it?
It still won't turn on because you argue that when the box cars are basically at rest their wheels contact the sensors simultaneously. In the scenario where the box cars are moving, the distance between the sensors is also Lorentz contracted.

There are many scenarios such as yours in the relativity literature. I can find more of them if you'd like?
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Old Aug 17th 2017, 08:19 PM   #3
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Originally Posted by Pmb View Post
It still won't turn on because you argue that when the box cars are basically at rest their wheels contact the sensors simultaneously. In the scenario where the box cars are moving, the distance between the sensors is also Lorentz contracted.
Why not? Why is the box car's frame of reference less real than a frame of reference at rest with respect to the train tracks? When the box car's observer sees the two wheels of the first box car contacting the sensor's simultaneously (as in 3rd picture) why is this frame of reference less valid if Einstein said all laws of physics apply in all frames? Why then would the light bulb not come on?

I realise that there are many paradoxical scenarios but most of them aren't paradoxes under GR, only SR so this is why I cast this problem so it would fall under the domain of SR.

Last edited by kiwiheretic; Aug 17th 2017 at 08:24 PM.
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Old Aug 17th 2017, 08:51 PM   #4
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Originally Posted by kiwiheretic View Post
Why not? Why is the box car's frame of reference less real than a frame of reference at rest with respect to the train tracks?
I may have misunderstood your question so I'll start from scratch. You wrote
Originally Posted by kiwiheretic View Post
However, now for the tricky part. If the box cars are move close to the speed of light and travelling through the tunnel and they are length contracted the wheels will never simultaneously touch the sensors. Will the light bulb ever turn on in this scenario, for the briefest of time, or does the fitzgerald contraction of SR prevent this from happening?
Let S represent the rest frame of the tracks. Let L be the distance between the wheels of the outer set of the two box cars wheels. Then as measured in frame S L is also the distance between the two sensors.

Let L' be the distance between those same two sets of wheels when the box cars are moving with speed v. Then L' < L. Therefore the sensors will not trigger the light bulb.

Then you wrote
Originally Posted by kiwiheretic View Post
What makes this even more fun is that a passenger in the box car sees the railway lines contracted also so he sees the sensors are closer together. At just the right speed the sensor will be close enough together so that they will be both under the wheels of the first box car. (3rd picture). In this scenario the light bulb would turn on ... or would it?
The easiest way to answer this is to know that the light bulb lighting up is an invariant event. That is to say that if it occurs in one frame of reference it must occur in all frames of reference, much like a mouse being alive or dead.

The difficult way to answer this requires a detailed analysis of why the light bulb lights up. In the top photo all you show is two sensors and a light bulb but no detailed explanation of how the light bulb is lit up. Once that's done then you can see why it isn't it up. By this I mean something like the following: Let each sensor send an electric signal along what looks to be wires. The switch on the left sends a signal to the right at a finite speed down the wire to the lightbulb. The switch on the left sends a signal to the left at a finite speed down the wide to the bulb. Now here's the catch: In the rest frame of the tracks the speed of each signal is the same. However in the rest frame of the box cars the speeds of the signals are not the same and therefore the light bulb doesn't receive the signals at the same time so that the light bulb will not light up - No paradox! I.e. in both frames the light bulb doesn't light up.

People often neglect the differing speeds of those two signals and perhaps think of them as traveling instantaneously, which cannot happen.

If you'd like to follow the gory details of the derivation of the velocity transformation rules then I invite you to visit my website at:
Velocity Transformation Rules

To convince yourself of what I just explained I recommend using an actual, but still hypothetical, signaling system, i.e. one that actually sends a real signal down the wires at a finite speed.

Want to take a guess as to what would happen if the speed could actually be instantaneous in the rest frame of the track?

Originally Posted by kiwiheretic View Post
I realise that there are many paradoxical scenarios but most of them aren't paradoxes under GR, only SR so this is why I cast this problem so it would fall under the domain of SR.
The thought never entered my mind about such things in GR. I had something in mind from a physics journal.

Last edited by Pmb; Aug 17th 2017 at 08:53 PM.
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Old Aug 17th 2017, 09:26 PM   #5
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The paradox I had in mind is from the article

Length Contraction Paradox by Wolfgang Rindler, American Journal of Physics, 29 (6): 365–366

Go to: https://en.wikipedia.org/wiki/Ladder_paradox

and scroll down to where it says Man falling into grate variation.

Last edited by Pmb; Aug 17th 2017 at 10:58 PM.
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Old Aug 18th 2017, 02:09 AM   #6
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I take it if we put sensors in the wheels of the box car (rather than just the track) and ran it thru a circuit contained in the box on the box car which tests for simultaneous sensors on the wheels it would not show identical results as the circuit on the ground for speeds near that of c.
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Old Aug 18th 2017, 02:36 AM   #7
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Originally Posted by kiwiheretic View Post
I take it if we put sensors in the wheels of the box car (rather than just the track) and ran it thru a circuit contained in the box on the box car which tests for simultaneous sensors on the wheels it would not show identical results as the circuit on the ground for speeds near that of c.
Yes. It'd be the same as if the box car was now playing the same role as the track used to and the track with sensors now playing the role of the sets of wheels on the box cars. It'd be a symmetrical setup.

Last edited by Pmb; Aug 18th 2017 at 02:39 AM.
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Old Aug 18th 2017, 06:46 AM   #8
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From the train's frame of reference, the tracks, and so the distance between the sensors, is contracted so is smaller than the distance between wheels. The wheels do not contact the sensors at the same time so the light does not turn on.

From the ground's (track's) frame of reference, the distance between the wheels is contracted so is smaller than the distance between sensors. The wheels do not contact the sensors at the same time so the light does not turn on.
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Old Aug 18th 2017, 08:14 AM   #9
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Originally Posted by HallsofIvy View Post
The wheels do not contact the sensors at the same time so the light does not turn on.
The distance between sensors contracts. However I think you made the same mistake I did in my first response in this thread, i.e. not paying close enough attention to one of the illustrations kiwiheretic posted.

Take another look at the bottom illustration in diagram in kiwiheretic's diagram noting the reason why he uses two boxcars. He is choosing a speed such that the distance between one boxcar's set of wheel's is the same distance as the contracted distance between the two sensors. At one point a single boxcar's set of wheel's will simultaneously contact the two sensors. The reason why the bulb doesn't light up during contact is due to the differing rates of signal flow along the wires leading from the sensor to the light bulb.

Last edited by Pmb; Aug 18th 2017 at 11:38 AM.
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