Length contraction and the speed of light

[Mentz114]

I hope to lay to rest two of the misconceptions about special relativity that are evident in the many questions asked here.

1) Why is the speed of light a constant ?

Everybody believes Pythagoras's theorem that the length of the hypotenuse of a right angle triangle is $\sqrt{s_1^2+s_2^2}$ where $s_1,\ s_2$ are the lengths of the other two sides. It a geometrical fact.

In SR the length of the hypotenuse of a right angled triangle in the $t,x$ plane is $\sqrt{s_1^2-s_2^2}$ where $s_1$ is the longer of the two sides ( unless they are equal in which case the order is irrelevant).

It is a postulate of SR that the Lorentzian length of the path of light is zero. So any two events that lie on a light path will have proper length of zero. Furthermore if these points are Lorentz transformed they will still lie on the light path.

The constancy of light speed thus follows from a postulate and a geometrical fact that is just as geometric as Pythagoras's theorem.

No physical reason for this is known. One might as well ask 'why is Pythagoras's theorem true'. Any reasons given will be purely geometric.

2) Where is 'length contraction' ?

A lot of effort has gone into 'proving the constancy of c' by using cross-frame calculations involving contracted length and dilated time. These are redundant as I hope the above demonstrates. But there is an opportunity to show what is happening in frame-independent or operational terms.

We have train and platform, with a light source in the middle of the train sending a beam to two receivers one at each end.

The diagram 'train-frame' shows this in coordinates in which the train is at rest (train coords). The blue worldline is someone on the platform receding at $0.5c$ ($\gamma=1.1547$). In the train frame the diagram is symmetric and the light beams hit the receivers at the same time on the three train clocks. The distance covered by the light is equal to the clock time that elapsed so $c=1$.

The second diagram shows the scenario in the platform coordinates. It is no longer symmetric and the light beams do not hit the receivers at the same clock time. The distance traveled by the light has shrunk for the back receiver and grown for the front receiver. The corresponding clock times have shrunk/increased by the same factor so $c=1$ in these coordinates.

It is straightforward to show that the distances in the new coordinates ( L_1, L_2) are the Doppler shrunk/stretched lengths as measured in the train frame (these are acquired by radar distance measurement).

So - where is the contracted length or distance? All we need to balance the books is proper times and radar distances. The first is an invariant and the second is an operationally defined distance - not a fudged definition of distance.

'Contracted distance/length' exists only as a factor in a Lorentz transformation. It is not required in frame independent calculations. It is used an imaginary fudge-factor that helps get the right answer to a pointless calculation.

Persisting with this useless exercise suggests that one may be looking for a counter example which will *disprove* the postulates. There is no more hope of doing that than finding a counter example to Pythagoras. It is impossible without abandoning the geometry in which case we no longer talking about SR but some other theory.

 

[harrylin]

I hope to lay to rest two of the misconceptions about special relativity that are evident in the many questions asked here.

1) Why is the speed of light a constant ?[..]

Probably you mean: "Why is the speed of light invariant?".

The invariance of the speed of light relates to the first postulate
The constancy of the speed of light relates to the second postulate.
These two are often confounded and even intermingled.

Furthermore, different people interpret SR differently, and even attach different meanings to "why".

2) Where is 'length contraction' ?
[..] 'Contracted distance/length' exists only as a factor in a Lorentz transformation. [..] It is used an imaginary fudge-factor that helps get the right answer to a pointless calculation.

I don't agree with the idea that a physical phenomenon is a "fudge-factor". Here's another point of view that makes more sense to me:
http://scitation.aip.org/content/aapt/journal/ajp/72/10/10.1119/1.1778390
"The result emphasizes the reality of Lorentz contraction by showing that the contraction is a direct consequence of the first and second postulates of special relativity"

 

[Mentz114]

Probably you mean: "Why is the speed of light invariant?".

The invariance of the speed of light relates to the first postulate
The constancy of the speed of light relates to the second postulate.
These two are often confounded and even intermingled.

Furthermore, different people interpret SR differently, and even attach different meanings to "why".

I don't agree with the idea that a physical phenomenon is a "fudge-factor". Here's another point of view that makes more sense to me:
http://scitation.aip.org/content/aapt/journal/ajp/72/10/10.1119/1.1778390
"The result emphasizes the reality of Lorentz contraction by showing that the contraction is a direct consequence of the first and second postulates of special relativity"

Yes, I didn't even express myself well. This thread comes from frustration.

I see you still think that 'length contraction' is a physical phenomemon.

 

[harrylin]

[...] I see you still think that 'length contraction' is a physical phenomemon.

Special relativity describes physical phenomena...

PS compare https://www.physicsforums.com/threads/time-dilation-in-relativity-theory.818279/

 

[Mentz114]

Special relativity describes physical phenomena...

PS compare https://www.physicsforums.com/threads/time-dilation-in-relativity-theory.818279/

I've seen it. Thanks for the input. I have no problem with differential ageing. As Rindler says 'time dilation is cumulative' so it is easily verified.

I didn't want this thread to be a discussion about the physicality of 'length contraction' but ...

In the train scenario I used, if the train physically contracts from the POV of the platform won't the train clocks have been in motion relative to each other for a short time ? This would desynchronise those clocks wouldn't it ?

 

[harrylin]

I've seen it. Thanks for the input. I have no problem with differential ageing. As Rindler says 'time dilation is cumulative' so it is easily verified.

Assuming differential aging as a fact, length contraction was indirectly verified by such experiments as https://en.wikipedia.org/wiki/Kennedy–Thorndike_experiment as well as, in principle, by experiments with moving mirrors such as described in post #2.

[..] In the train scenario I used, if the train physically contracts from the POV of the platform won't the train clocks have been in motion relative to each other for a short time ? This would desynchronise those clocks wouldn't it ?

Yes, the effect is extremely small but that is exactly what SR predicts: assuming that the train was slowly accelerated without oscillations, and that it is uncompressed and un-stretched just as before the acceleration, then those clocks should be slightly out of tune relatively to each other from the POV of the platform system. As during such acceleration the rear clock has a slightly higher speed than the front clock, the rear clock is predicted to be slightly behind on the front clock.

 

[Mentz114]

Assuming differential aging as a fact, length contraction was indirectly verified by such experiments as https://en.wikipedia.org/wiki/Kennedy–Thorndike_experiment as well as, in principle, by experiments with moving mirrors such as described in post #2.

Yes, the effect is extremely small but that is exactly what SR predicts: assuming that the train was slowly accelerated without oscillations, and that it is uncompressed and un-stretched just as before the acceleration, then those clocks should be slightly out of tune relatively to each other from the POV of the platform system. As during such acceleration the rear clock has a slightly higher speed than the front clock, the rear clock is predicted to be slightly behind on the front clock.

Thanks but I don't think we are talking about the same acceleration. The inertial LT does not accelerate anything.

Bringing in the acceleration that got the train up to speed has confused things. The kinematic 'length contraction' is attibuted to relative velocity and it cannot be attributed to the speed-up acceleration.

 

[harrylin]

[..] Bringing in the acceleration that got the train up to speed has confused things. The kinematic 'length contraction' is attibuted to relative velocity and it cannot be attributed to the speed-up acceleration.

Acceleration is a means to arrive at a different velocity. The paper that I referred to in post #2 first considers constant velocity.

Often a distinction is made between dynamic and kinematic length contraction. However, that distinction is a bit artificial in practice, in view of the general kind of situations that SR is meant to deal with, such as MMX&KTX, particle accelerators, etc. also trains commonly change velocity. SR has no problem with such cases, it was designed for them; and considering the dynamics increases our physical insight.

Consider also if you follow Einstein's argumentation in §3 and 4 of http://fourmilab.ch/etexts/einstein/specrel/www/:
"Let us in “stationary” space take two systems of co-ordinates [..] Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K) [..]
If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize [emphasis mine]

Length contraction and time dilation basically mean that a clock that from our perspective is moving, is according to us Lorentz contracted and ticking at a reduced rate compared with the same clock in rest. As long as we may assume (or if we verify) that acceleration doesn't cause permanent deformation or other damage, it's quite irrelevant if we do those measurements in the same system by means of a change of state of motion of the clock or if we compare measurements by means of two independent inertial systems that are in different states of motion.

Now, if you did not mean that any physical change took place, then I cannot understand your earlier question. Please rephrase your sentence so that it is free from physical action and I will be able to understand its meaning:
"In the train scenario I used, if the train physically contracts from the POV of the platform won't the train clocks have been in motion relative to each other for a short time ? This would desynchronise those clocks wouldn't it ?"

 

[Mentz114]

Often a distinction is made between dynamic and kinematic length contraction.
...
...

Nice to hear from you so soon. I chopped your text to avaoid repetition. I did read it.

There is no dynamic contraction/expansion the in inertial system comprising the train and platform setup.

In the covariant calculation I did in the first post there is no quantity identifiable as length contraction as defined by Rindler and others.
Covariant calculations are based on $\gamma$ and integrals taken along the worldlines. This makes them independent of clock-synchronisation.
I postulate that anything that de[ends on clock-synchronisation is not covariant and cannot be physical.

If one defines rest length correctly, and integrates it along a time-like curve it remains the same. It is an invariant. The length of the train does not change.

The LT is a change of the map used to describe a certain segment of reality. It cannot change that reality.

 

[1977ub]

The LT is a change of the map used to describe a certain segment of reality. It cannot change that reality.

Would you say that there is a "reality" to momentum? The train's momentum is measured to be different in different frames.

 

[Mentz114]

@1977ub How does the LT increase any momentum ? The train is already moving, we use the LT to change a map that describes the train and platfrom.

 

[WannabeNewton]

A lot of this is just an argument over semantics because what is "physical" or "real" is not related to physics whatsoever. Instead of just repeating the same old song and dance I'll just refer to older posts of mine:

https://www.physicsforums.com/threads/is-length-contraction-an-illusion.756797/#post-4766303
https://www.physicsforums.com/threa...-length-contracted-space.749963/#post-4726643
https://www.physicsforums.com/threa...ts-in-special-relativity.734244/#post-4639014

I want to stress the last point that length contraction due to acceleration and that due to a Lorentz boost are really two aspects of the same phenomenon. This is easily seen simply by noting that in both cases all we are doing is taking simultaneity surfaces of an inertial frame and intersecting them with the worldline (or congruence of worldlines) of the object, with the result that the object has a different length in the inertial frame compared to its proper length in its instantaneous rest frame, which is itself from simultaneity surfaces of the rest frame (assuming they can be defined).

 

[Mentz114]

Thanks for the input.

A lot of this is just an argument over semantics because what is "physical" or "real" is not related to physics whatsoever.

No it's not. I have challenged anyone to show me the 'contracted length' when we transform between frames in the scenario above. That is unambiguous.

I want to stress the last point that length contraction due to acceleration and that due to a Lorentz boost are really two aspects of the same phenomenon.

Huh ? There is no acceleration in Rindlers formula for kinetic length contraction $L'=L/\gamma$.

 

[1977ub]

@1977ub How does the LT increase any momentum ? The train is already moving, we use the LT to change a map that describes the train and platfrom.

Momentum is measured to change between frames, whether you used classic or relativistic momentum.

Why can a length not change when you shift frames?

 

[harrylin]

Nice to hear from you so soon. I chopped your text to avaoid repetition. I did read it. There is no dynamic contraction/expansion the in inertial system comprising the train and platform setup.
[..].

Yes that's clear; however, it remains a mystery for me what you tried to ask me in post # 5, if it wasn't concerning dynamics. And obviously, fixed lengths cannot change in an inertial system. Nobody would think otherwise.

I chopped the part about your metaphysics vs that of the paper that I cited. The fact that you can hide something cannot prove that it doesn't exist. My metaphysics is somewhat in-between and perhaps more nuanced than either, but I prefer to stick to discussing solid physics.

 

[harrylin]

[..]
Why can a length not change when you shift frames?

Perhaps I'm here only bashing sloppy semantics, but when you shift frames nothing physical happens - only your perspective (your definition of simultaneity) changes. As a result of your change of measurement system, your measure of the same length will be different.

 

[1977ub]

Perhaps I'm here only bashing sloppy semantics, but when you shift frames nothing physical happens - only your perspective (your definition of simultaneity) changes. As a result of your change of measurement system, your measure of the same length will be different.

I was responding to this:

If one defines rest length correctly, and integrates it along a time-like curve it remains the same. It is an invariant. The length of the train does not change.

The LT is a change of the map used to describe a certain segment of reality. It cannot change that reality.

Also, now I see that "rest length" was mentioned initially. The phrase "length of the train" seems a bit more ambiguous.

 

[A.T.]

Perhaps I'm here only bashing sloppy semantics,

You could be less sloppy and explicitly distinguish between "length" and "proper length", to avoid the confusion.

 

[Mentz114]

Momentum is measured to change between frames, whether you used classic or relativistic momentum.

Why can a length not change when you shift frames?

A length measurement can change between frames. This does not affect the thing being measured in any way.

 

[Mentz114]

Yes that's clear; however, it remains a mystery for me what you tried to ask me in post # 5, if it wasn't concerning dynamics. And obviously, fixed lengths cannot change in an inertial system. Nobody would think otherwise.

I chopped the part about your metaphysics vs that of the paper that I cited. The fact that you can hide something cannot prove that it doesn't exist. My metaphysics is somewhat in-between and perhaps more nuanced than either, but I prefer to stick to discussing solid physics.

Yes, the first thing was a bit subtle and slippery. I have already disproved one of my postulates and found the so called 'contracted length' of the train. It is the rest length of the train projected onto the $t'=0$ line in the platform frame and is equal to $L/\gamma$.

In the rest frame we can write $L=\int_0^L\ dx$ and in the train frame $L'=\int_0^{L/\gamma}\ dx'$. Also ${dx'}^2=\gamma^2(dx^2+\beta^2dt^2)$. Now by choosing a measurement which is simultaneous at both ends in the rest frame ($dt=0$) but not in the platform frame ($dt'\ne 0$) we can get $L=L'$.

So we can get any remote measurement to give any value by choosing a number. The only measurement that assumes no simultaneity convention is the one in the rest frame. This little calculation also shows that all measurements on a moving target must involve time in the length integral because $dx$ and $dt$ get mixed by the LT.

I haven't worked out how this affects my broader assertion about the ontological status of LC but I'm working on it.

 

[harrylin]

You could be less sloppy and explicitly distinguish between "length" and "proper length", to avoid the confusion.

No, that is a different possible misunderstanding.

 

[harrylin]

[..] I have already disproved one of my postulates and found the so called 'contracted length' of the train.

Very good!

The only measurement that assumes no simultaneity convention is the one in the rest frame. [..]

Any measurement involving two separated clocks uses a simultaneity convention...

 

[1977ub]

A length measurement can change between frames. This does not affect the thing being measured in any way.

But which measurement is "true" - ( see ladder / barn paradox ). Which frame has the "right" to claim the "actual" measurement?

 

[Mentz114]

But which measurement is "true" - ( see ladder / barn paradox ). Which frame has the "right" to claim the "actual" measurement?

I think this is a misunderstanding. Where did I say that one measurement is more true than another ? In fact I just showed in a coordinate independent way that all length measurements on a moving object require a choice of simultaneity. This puts them all on the same footing.

The other thing is this. When I say 'boost the rest frame of the train', I do not mean put rockets on the train and blast in into space. What I mean is that I will calculate what the worldmap ( Rindlers phrase) looks like in the coordinates of the platform guy. So I do a Lorentz transformation of the train rest-frame.

 

[harrylin]

But which measurement is "true" - ( see ladder / barn paradox ). Which frame has the "right" to claim the "actual" measurement?

According to SR we cannot establish by means of such measurements a "true frame". My comment in post #22 is related to that.

I think this is a misunderstanding. Where did I say that one measurement is more true than another [..]

Once more, you seemed to suggest something like that in an indirect way as I remarked in my post #22.

 

[Mentz114]

Very good!

Thank you. Can I have a biscuit as well ?

Any measurement involving two separated clocks uses a simultaneity convention...

If you mean that all the train clocks must be showing the same time, then this is not true. We don't look at any clocks when we measure.

If you mean all the lab clocks must be running at the same rate, that is automatically true.

The integral $\int_0^L dx$ has no time in it, so we can measure the near end, have tea and then measure the far end and subtract.

No special tinkering with timings is required.The 'contracted length' is a geometrical construct made by projecting the train spatial slice into the platform spatial slice. We can work this out using the kinetic decomposition described for instance in Malaments book Topics in the Foundations of General Relativity and Newtonian Gravitation Theory (chaper 2.4). The spatial projection metric is

$h_{ab}= g_{ab}-u_a u_b$ which has $h_{xx}=\gamma^2$ which means that lengths in the x-direction differ by a factor $\gamma$.

But we cannot call this 'contracted length' without a choice of simultaneity, which I hope will be revealed in the temporal projections.

Stay tuned.

 

[CycoFin]

People on train can measure one end of train, have a tea, and then measure other end. Let say they get result 100 meters.

We station people see that their result is 100m but we measure less because we must use concept of simultaneously because train is moving.
Of course we can also measure one end, have a tea, and then measure other end and then calculate (how much train moved when having tea) correct train length.
Or if we don't want to calculate, we just measure train length with simultaneous (which train people disagree) measurements. Both techniques gives us less than 100 meters.

Length contraction is not physical

 

[harrylin]

If you mean that all the train clocks must be showing the same time, then this is not true. We don't look at any clocks when we measure. If you mean all the lab clocks must be running at the same rate, that is automatically true. [..]

Neither. As remarked by me and 1977ub in preceding posts, it seems as if you privilege the system that you designate "rest system" in comparison to the one that you call "moving system". However, those systems are arbitrary inertial systems, and according to SR they are on equal footing. Both systems use the simultaneity convention that is discussed in §1 of http://fourmilab.ch/etexts/einstein/specrel/www/ . If I correctly understand your train example, it's a simple variant of Einstein's train example, using two clocks at a distance in the train and also two such clocks on the platform, and you designate each as "inertial frame" for measurements. Therefore I disagreed with your comment that"The only measurement that assumes no simultaneity convention is the one in the rest frame.". To the contrary: both frames assume the Poincare-Einstein simultaneity convention.

The integral $\int_0^L dx$ has no time in it, so we can measure the near end, have tea and then measure the far end and subtract. [...]

Once more: that is only true according to the perspective of the platform observer, and the train observer makes the same claim about the train.

 

[harrylin]

People on train can measure one end of train, have a tea, and then measure other end. Let say they get result 100 meters.

We station people see that their result is 100m but we measure less because we must use concept of simultaneously because train is moving.
Of course we can also measure one end, have a tea, and then measure other end and then calculate (how much train moved when having tea) correct train length.
Or if we don't want to calculate, we just measure train length with simultaneous (which train people disagree) measurements. Both techniques gives us less than 100 meters.

Length contraction is not physical

Please explain the null result of a moving interferometer, assuming that SR's postulates are correct.
PS. note that no simultaneity convention is used in such measurements, which very much simplifies the analysis.

 

[PAllen]

Suppose I have 1 meter square sheet of film, and flash bulb placed e.g. 100 meters above it. A 2 meter (rest length) ruler, moving parallel to the film plane in the direction of its length, and e.g. 1 cm above this plane, is approaching. I time the flash to fire such its light will reach the film when the center of the ruler is above the center of the film. The ruler is moving at .968 c. What I get is an image on the film of a ruler shadow 1/2 meter long. What is your (Mentz114) philosophy for treating this as 'not physical'? No coordinates or conventions are involved. Just a few objects at mutual rest, and another object moving relative to those.

 

[Mentz114]

Neither. As remarked by me and 1977ub in preceding posts, it seems as if you privilege the system that you designate "rest system" in comparison to the one that you call "moving system". However, those systems are arbitrary inertial systems, and according to SR they are on equal footing. Both systems use the simultaneity convention that is discussed in §1 of http://fourmilab.ch/etexts/einstein/specrel/www/ . If I correctly understand your train example, it's a simple variant of Einstein's train example, using two clocks at a distance in the train and also two such clocks on the platform, and you designate each as "inertial frame" for measurements. Therefore I disagreed with your comment that"The only measurement that assumes no simultaneity convention is the one in the rest frame.". To the contrary: both frames assume the Poincare-Einstein simultaneity convention.

Once more: that is only true according to the perspective of the platform observer, and the train observer makes the same claim about the train.

You are missing the point. I have an extended object that lies along its local x-axis. Its length is the integral I gave. Then I have an apparatus which is moving inertially along the same axis wrt the object. What happens when the apparatus tries to measure the length of the moving object ? There is no symmetry or ambuguity. Or frame dependence.

Going back to this

$L=\int_0^Ldx$
$L'=\int_0^{L/\gamma}dx'$
${dx'}^2=\gamma^2(dx^2+\beta^2dt^2)$
So
$L'=\int_0^{L/\gamma}\sqrt{\gamma^2(dx^2+\beta^2dt^2)}$

The $dt$ refers to the rods rest frame and it is the time between measurements. If we set that to zero then $L'=L$. But now the in the apparatus frame we have ${dt'}^2=\gamma^2\beta^2dx^2$. So the time gap between the measurements in the apparatus frame is $dt'=\gamma\beta L$. No problems.

But what happens if I do this procedure to force $L'=L/\gamma$. Will there be real solution or will $L/\gamma$ be proven to be an unmeasurable quantity ?

 

[Mentz114]

Suppose I have 1 meter square sheet of film, and flash bulb placed e.g. 100 meters above it. A 2 meter (rest length) ruler, moving parallel to the film plane in the direction of its length, and e.g. 1 cm above this plane, is approaching. I time the flash to fire such its light will reach the film when the center of the ruler is above the center of the film. The ruler is moving at .968 c. What I get is an image on the film of a ruler shadow 1/2 meter long. What is your (Mentz114) philosophy for treating this as 'not physical'? No coordinates or conventions are involved. Just a few objects at mutual rest, and another object moving relative to those.

Missed the point. You have demonstrated what everybody knows - if one measures the length of a moving body the result is not necessarily the rest length.

Please check my calculations and tell me if there is a blunder.

 

[PAllen]

Missed the point. You have demonstrated what everybody knows - if one measures the length of a moving body the result is not necessarily the rest length.

Please check my calculations and tell me if there is a blunder.

Your calculation doesn't correspond directly to my experiment. You ask:

"Will there be real solution or will L/γ be proven to be an unmeasurable quantity ?"

and my experiment shows it is measurable, and the apparatus doing the measurement knows nothing about the rest length. It just measures a ruler as 'obviously 1/2 meter long'. There are not even any clocks needed. If a picture is captured, then it shows the (moving) ruler is 1/2 meter long.

 

[Mentz114]

Your calculation doesn't correspond directly to my experiment. You ask:

"Will there be real solution or will L/γ be proven to be an unmeasurable quantity ?"

and my experiment shows it is measurable, and the apparatus doing the measurement knows nothing about the rest length. It just measures a ruler as 'obviously 1/2 meter long'. There are not even any clocks needed. If a picture is captured, then it shows the (moving) ruler is 1/2 meter long.

You do need clocks. "I time the flash to fire such its light will reach the film when the center of the ruler is above the center of the film." What procedure is used to determine the correct time to fire the bulb ?

 

[Stephanus]

I hope to lay to rest two of the misconceptions about special relativity that are evident in the many questions asked here.

1) Why is the speed of light a constant ?

Everybody believes Pythagoras's theorem that the length of the hypotenuse of a right angle triangle is $\sqrt{s_1^2+s_2^2}$ where $s_1,\ s_2$ are the lengths of the other two sides. It a geometrical fact.

In SR the length of the hypotenuse of a right angled triangle in the $t,x$ plane is $\sqrt{s_1^2-s_2^2}$ where $s_1$ is the longer of the two sides ( unless they are equal in which case the order is irrelevant).

It is a postulate of SR that the Lorentzian length of the path of light is zero. So any two events that lie on a light path will have proper length of zero. Furthermore if these points are Lorentz transformed they will still lie on the light path.

The constancy of light speed thus follows from a postulate and a geometrical fact that is just as geometric as Pythagoras's theorem.

No physical reason for this is known. One might as well ask 'why is Pythagoras's theorem true'. Any reasons given will be purely geometric.

2) Where is 'length contraction' ?

A lot of effort has gone into 'proving the constancy of c' by using cross-frame calculations involving contracted length and dilated time. These are redundant as I hope the above demonstrates. But there is an opportunity to show what is happening in frame-independent or operational terms.

We have train and platform, with a light source in the middle of the train sending a beam to two receivers one at each end.

The diagram 'train-frame' shows this in coordinates in which the train is at rest (train coords). The blue worldline is someone on the platform receding at $0.5c$ ($\gamma=1.1547$). In the train frame the diagram is symmetric and the light beams hit the receivers at the same time on the three train clocks. The distance covered by the light is equal to the clock time that elapsed so $c=1$.

The second diagram shows the scenario in the platform coordinates. It is no longer symmetric and the light beams do not hit the receivers at the same clock time. The distance traveled by the light has shrunk for the back receiver and grown for the front receiver. The corresponding clock times have shrunk/increased by the same factor so $c=1$ in these coordinates.

It is straightforward to show that the distances in the new coordinates ( L_1, L_2) are the Doppler shrunk/stretched lengths as measured in the train frame (these are acquired by radar distance measurement).

So - where is the contracted length or distance? All we need to balance the books is proper times and radar distances. The first is an invariant and the second is an operationally defined distance - not a fudged definition of distance.

'Contracted distance/length' exists only as a factor in a Lorentz transformation. It is not required in frame independent calculations. It is used an imaginary fudge-factor that helps get the right answer to a pointless calculation.

Persisting with this useless exercise suggests that one may be looking for a counter example which will *disprove* the postulates. There is no more hope of doing that than finding a counter example to Pythagoras. It is impossible without abandoning the geometry in which case we no longer talking about SR but some other theory.

In 'Janus' train, the lights come from the train to the platform.
https://www.physicsforums.com/threads/length-contraction.817911/#post-5135255
Do in your pictures the lights come from the platform?