Length contraction and the speed of light

[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?

 

[Mentz114]

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?

The light rays come from the middle of the train towards each end of the train. They are thin yellow lines inclined at 45^o.

 

[Stephanus]

The light rays come from the middle of the train towards each end of the train. They are thin yellow lines inclined at 45^o.

Thanks!

 

[CycoFin]

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.

If we measure length of something that is moving, there is always simultaneity convention. Length contraction is cause of this simultaneity convention, there is no "real" contraction.

One can measure length of moving object only using one statonary clock and measure endpoints as they flyby. But even then measurement is less because his clock is running slower (yes, slower and the reason is simultaneity convention)

 

[PAllen]

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 ?

You could just keep repeating the experiment with randomly varied firings until you get an image reasonably centered on the film (make it a CCD for reusability). That was the point of my comment: "if you get the picture". As long as you get a picture, you have your measurement. So, imagine a 'ruler gun' that keeps identical shooting rulers, and a flash bulb that fires randomly. Continue till you get a picture.

 

[PAllen]

If we measure length of something that is moving, there is always simultaneity convention. Length contraction is cause of this simultaneity convention, there is no "real" contraction.

One can measure length of moving object only using one statonary clock and measure endpoints as they flyby. But even then measurement is less because his clock is running slower (yes, slower and the reason is simultaneity convention)

Your clock never runs slow. You conclude clocks moving relative to you run slow. The one clock measurement you describe depends on simultaneity only in that you have to know the speed of the rod. To measure speed you either need two synchronized clocks or rely on SR Doppler formula which is based in Einstein clock synch.

In my proposed experiment, the simultaneity requirement is hidden in the geometry of the experiment. The set up that the line from film center to flash bulb is orthogonal to the motion of the ruler builds in simultaneity of the bulb/film frame. To me, though, it is hard not to accept this as comparably 'physically real' as an measurement we make about a moving body.

 

[Dale]

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.

For this to be a valid length you do have to adopt the convention that the one way speed of light is isotropic. That convention is equivalent to Einstein's synchronization convention in that frame.

 

[PAllen]

For this to be a valid length you do have to adopt the convention that the one way speed of light is isotropic. That convention is equivalent to Einstein's synchronization convention in that frame.

There is a more basic way simultaneity is built in - the geometry of the line from film center to bulb being orthogonal to ruler motion. In a different (e.g. ruler) frame, they won't be orthogonal, and the image would be considered affected by motion (of film) during the arrival time difference at one end of the image versus the other.

I'm only posing this as an example of something that feels like it should be considered 'a real length measurement'. It is as much a measurement of length of a moving object as transverse doppler is of time dilation (if you don't assume isotropy of light speed, you get different Doppler formula).

 

[1977ub]

And of course, travelers on the ruler will decide that photons from the bulb flash hit different parts of the ruler at different times, and different parts of the image on the film are created at different times.

 

[CycoFin]

Your clock never runs slow. You conclude clocks moving relative to you run slow. The one clock measurement you describe depends on simultaneity only in that you have to know the speed of the rod. To measure speed you either need two synchronized clocks or rely on SR Doppler formula which is based in Einstein clock synch.
.

My clock is running slow if I check the clock on one end of the rod, wait, and then check the time on the other end of the rod as they pass by and calculate difference and compare that to my stationary clock. These two rod clocks are running slow if I measure them with two different spots, but time elapse they show on my one spot is running faster than mine. This is because rod have different concept of simultaneously.

 

[harrylin]

If we measure length of something that is moving, there is always simultaneity convention. [..]

MMX is totally indifferent about simultaneity conventions. However, it is dependent on the light postulate, and the simultaneity convention is also based on that postulate. Imagine observations using the solar frame, as you may do according to the relativity postulate. On Earth is a slowly rotating MMX interferometer near the Earth's equator, so that it circles the Sun in one year. Assuming that the light postulate is correct, how do you explain the null result without length contraction?

PS in fact you can pick any inertial frame you like for the physical analysis, as long as you stick to that one (it's faulty, or at least messy, to change calibration of measurement equipment during a measurement).

 

[CycoFin]

There is length contraction but IMO it has very bad name. Length illusion would be better name for the fact that we measure moving objects shorter.
Actual physical length is not contracted. We measure shorter lengths because we "see" objects future/past points depending what point of the object we choose to be our reference point.

Back to train example. Let moving train be 100m measured by people on train. Let moving train measured by station people to be 50m. During this measurement point 0m station observer looks train's clock on that end. Same time (defined by station people) 50m point station observer looks train's clock on that point and if these two compare results they find out that train's clocks have different readings (so that 50m clock have more time, we can "see" the future of the train).

Let there be station observer on 200m mark. His recorded time from the clock of the end of the train as the end was bypassing him equals to 0m point observer for the start of the train. This is same time for train observers so train people measures that 200m station equals to their 100m train.

 

[harrylin]

You are missing the point. I have an extended object that lies along its local x-axis. Its length is the integral I gave.

Not sure who is missing the point. You probably mean its proper length. Proper length is the length of an object as measured with respect to a reference system that is in the same state of motion and which is pretended to be in rest. It's similar to such things as magnetic fields and kinetic energy. The proper kinetic energy and magnetic field of an electron in uniform motion are zero. Are kinetic energy and magnetic field therefore "non-physical"? Or are the kinetic energy and magnetic field of that electron "non-physical", but the kinetic energy of a power drill and the magnetic field of an electromagnetic coil "physical"?

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.

There is no ambiguity, but there is full symmetry between inertial frames. However, your example is unnecessarily complicated because of relativity of simultaneity; it's a bit of a red herring. Please correct me if I'm wrong, but I get the impression that you think that length contraction is caused by RoS. I just gave the example of MMX which in the general case as originally considered is indifferent about RoS, and in a parallel thread I demonstrated the need of length contraction if the two postulates are true (in combination with time dilation), before looking at simultaneity conventions. - #48
You can choose any simultaneity convention you like, it changes nothing for the physics.

[..] will $L/\gamma$ be proven to be an unmeasurable quantity ?

Once more, I still wonder if you think:
- is $t/\gamma$ an unmeasurable quantity ?
- is E_kin an unmeasurable quantity ?

 

[rede96]

I'm only posing this as an example of something that feels like it should be considered 'a real length measurement'. It is as much a measurement of length of a moving object as transverse doppler is of time dilation (if you don't assume isotropy of light speed, you get different Doppler formula).

Forgive me if this is off point or just semantics, but isn't any measurement we make of any system 'real' in that it tells us something about the system we are measuring? But it is how we interpret the measurement that is important. I would of thought that in the example of the ruler given, the 'photo' taken is measuring the effects of relativistic speeds on the observations of length of an object, as seen from a different frame. It is not measuring the physical or 'real' properties of the object or in this case the ruler. As that isn't possible for all properties unless one is at rest wrt to the ruler and you can physically examine it. So in that respect it isn't a 'real' measurement of the length of the ruler.

Anyway, what I was really interested in is how the ruler would look if we could take a high resolution photo of it at that speed. The meter ruler would appear shorter of course but would the mm increments all be equidistant or would they appear closer together at the ends or in the middle etc. The reason for my asking is I was curious how natural light (so coming from all directions) would reflect off the surface of the ruler and into the camera lens with the ruler traveling at such high speeds.

 

[CycoFin]

Anyway, what I was really interested in is how the ruler would look if we could take a high resolution photo of it at that speed. The meter ruler would appear shorter of course but would the mm increments all be equidistant or would they appear closer together at the ends or in the middle etc. The reason for my asking is I was curious how natural light (so coming from all directions) would reflect off the surface of the ruler and into the camera lens with the ruler traveling at such high speeds.

Check this https://en.wikipedia.org/wiki/Terrell_rotation

 

[harrylin]

[..] Length illusion would be better name for the fact that we measure moving objects shorter. [..] Same time (defined by station people) 50m point station observer looks train's clock on that point and if these two compare results they find out that train's clocks have different readings (so that 50m clock have more time, we can "see" the future of the train). [..]

In this thread other examples were given in which no clocks are used, but just two postulates from which such "illusions" follow. Of course, you are free to call the postulates of SR illusions, as they refer to physical measurements and not to physical reality, which may be somewhat hidden to us. We cannot do better than that.

 

[harrylin]

Forgive me if this is off point or just semantics, but isn't any measurement we make of any system 'real' in that it tells us something about the system we are measuring? But it is how we interpret the measurement that is important. I would of thought that in the example of the ruler given, the 'photo' taken is measuring the effects of relativistic speeds on the observations of length of an object, as seen from a different frame. It is not measuring the physical or 'real' properties of the object or in this case the ruler. As that isn't possible for all properties unless one is at rest wrt to the ruler and you can physically examine it. So in that respect it isn't a 'real' measurement of the length of the ruler. [...]

I'm afraid that the kinetic energy and magnetic field of electrons in an electron beam are always "unreal" or zero when examined with co-moving measurement equipment ...

According to SR, the effect of very high speed on an object is that it will be length contracted according to measurements with the "rest" system, while the proper length will be the same as the length in rest. All verified experiments up to this day are consistent with that prediction.

 

[Dale]

It seems that all of the physics content has finished and now we are just discussing semantics and philosophy. Thread closed.

 

[Mentz114]

There is nothing controversial in this post - I just want to finish off what I started in the the earlier thread which descended into argument.

I made the calculation based on the definition of rest length $L=\int_0^L dx$ which means we are adding up the notches on a ruler laid alongside the object we are measuring. Because there is no time in that definition, any time gaps between measuring the ends do not affect the result ( the object is not moving wrt the ruler).

When we define a similar quantity but from the frame of a moving ruler, we get $L'=\int_0^{L/\gamma} dx'$. The change in the upper limit is because the projected length of $L$ onto the $x'$-axis is $L/\gamma$ ( this can be shown).

From the LT joining the rod frame ( unprimed) and the moving ruler frame (primed) we have $dt'=\gamma(dx+\beta dt)$. So the integral $L'=\int_0^{L/\gamma} dx'$ now has a time part, and to complete it we need two times $t_0,t_1$ for the limits of the integration. It seems obvious that these correspond to the times of the measurements of the ends of the rod.

Suppose we want the result of the measurement to be $L$. This means that the integrand should integrate to $\gamma L$, to cancel the $\gamma$ in the upper limit. The only solution of ${dx'}=\gamma\beta dt + \gamma dx = \gamma dx$ is $dt=0$. To find the operational delay ( the delay in the ruler frame) we put $dt=0$ into $dt'=\gamma dt + \gamma\beta dx$ to get $dt'=\gamma\beta L$. In the rod frame the measurements appear simultaneous ($dt=0$).

I think this agrees with other similar calculations.

Now I repeat the with requirement $L'=L/\gamma$. This means the integrand in $L'$ must evalute to 1 (unity).
The rest is algebra and gives

$dt'=(1-\frac{L}{\gamma})/{\beta}$

This is fine if $\beta^2>0$ and is positive for $L<c\gamma$ (putting $c$ back into $L$) which includes most laboratories. If it goes negative it means we waited too long and the rod is gone.

So according to this $L/\gamma$ can be the result of a suitable timed measurement.

There's nothing here that isn't known but I haven't seen it shown convincingly in a coordinate independent way.

What the experimenter chooses here is a time delay. I'm not sure how that relates to clock-synchronisation. Defining everything as integrals along curves removes the need for synchronisation, I think.

(I'd like to thank @harrylin, @A.T., @PAllen, WannaBeNewton and others who responded usefully in the previous thread)

 

[PeterDonis]

Defining everything as integrals along curves removes the need for synchronisation, I think.

Sort of. Instead of picking different clock synchronizations for the different frames, you're picking different spacelike curves along which to integrate. (Drawing a spacetime diagram makes this obvious.)

You could say that the rulers in relative motion pick out the different spacelike curves for you, but that's not really true, because the rulers are not spacelike curves; they are world tubes, and the spacelike curves in question are particular curves picked out of those world tubes. What picks them out? A definition of simultaneity--which events in the different rulers' world tubes are considered as being "at the same time", so that the spacelike curve they form realizes a "proper length".

It's true that, for the special case of a ruler in free fall in flat spacetime, the spacelike curves that define its "proper length" at each instant of time can be picked out by an independent criterion, namely, being orthogonal to the worldlines of all the points of the ruler. But as soon as you introduce either non-inertial (or more precisely, non-uniform) motion, or spacetime curvature, that uniqueness goes away.

None of this invalidates what you're saying; I just think it's important to make clear the limitations of any definition of "length".

 

[Dale]

Please do not try to reopen this topic.