Undergrad One way speed of light experiment proposal

[Nugatory]

Sorry, the experiment I suggested is not the one in the picture. It is just "somehow similar" to it. Read again.

Ah - OK I think I see what you mean then. If I do, then you may be misunderstanding the problem in the same way the original poster did. The challenge is not to produce a measurement that is accurate within experimental error if you ignore relativistic effects (that's easy, it's what slow clock synchronization does, and it's what Fizeau himself was doing). The challenge is to propose an experiment that could, in principle, measure the one-way speed of light when your instruments are precise enough that relativistic effects do matter. Original poster explicitly stated his assumption that the accuracy would by low enough that relativistic effects don't matter; in your proposal the same assumption is hiding in the assumption that the rod is rigid.

If you perform your experiment with sufficient accuracy, you will find that the measured "one-way speed of light" changes according to the speed of the moving rod, a very strong hint that that's not what it's really measuring. Much of the discussion in #10 of this thread applies here as well.

 

[Ibix]

I don't think I understood the @kkris1's problem last week. I think the point of his experiment (or the way I would phrase it if I were in his shoes) was that you could measure a value of the speed of light as a function of the velocity of the rod - call it $c'(v)$ - and then observe that this is asymptotic to some value as $v\rightarrow 0$. And this asymptote would indeed be $c$. One way speed of light, right?

Since he's getting c for the one-way speed, clocks synchronised by this method must agree with Einstein synchronised clocks (DanMP's addition of two clocks disguised as cogs makes this explicit). The un-expressed question is: what decision did kkris1 make that led to the Einstein convention and not any other?

The decision Einstein made is obvious: he chose to regard two clocks as synchronised if both agree that the other appears to lag by the same amount (so if we both look at the other's clock when our own reads 12:00:00 and we both see the other's reading 11:59:59 then we conclude that the difference is due to the light speed delay, not due to mis-setting). This is based on the assertion that the speed of light is the same in both directions. On the other hand, if we assert that the speed of light is not the same in both directions then we each need to apply different corrections for the light-speed delay, and both "other" clocks reading 11:59:59 is not synchronised.

But what decision did kkris1 make? It goes back to the Einstein's train thought experiment that I pointed out in #2: he chose to be at rest with respect to his apparatus, like the platform observer rather than the train observer.

Do Einstein's train thought experiment. But add two copies of kkris1's rig - one on the platform and one on the train. Since the rig is asymptotic to Einstein synchronised clocks in its own rest frame, the platform rig must (nearly) agree with the platform clocks and the train rig must (nearly) agree with the train clocks. But because we've embedded these copies in Einstein's train, it's obvious that they can't agree with each other.

So if kkris1 uses his rig at rest with respect to himself to measure the one-way speed of light, he'll get one value. If he uses it not at rest with respect to himself he'll get another. Deciding to use the value when the rig is at rest with respect to him is a choice, and it's equivalent to Einstein's more upfront decision.

 

[DanMP]

Ah - OK I think I see what you mean then. If I do, then you may be misunderstanding the problem in the same way the original poster did. The challenge is not to produce a measurement that is accurate within experimental error if you ignore relativistic effects (that's easy, it's what slow clock synchronization does, and it's what Fizeau himself was doing). The challenge is to propose an experiment that could, in principle, measure the one-way speed of light when your instruments are precise enough that relativistic effects do matter. Original poster explicitly stated his assumption that the accuracy would by low enough that relativistic effects don't matter; in your proposal the same assumption is hiding in the assumption that the rod is rigid.

Yes, the length of the rod would probably be to small for high enough accuracy, although we can use fiber optic, with big coils perpendicular to the axis/rod, in order to increase the "time of flight" and, as a consequence, reduce the wheels diameter and angular speed, but this would not increase the time difference between AB and BA measurements, so I'm not sure if the measurements can be accurate enough.

In order to reduce rod rigidity issue we can put the motor that rotates it right in the middle (halfway between A and B).

If you perform your experiment with sufficient accuracy, you will find that the measured "one-way speed of light" changes according to the speed of the moving rod ...

Why? And what movement? The rotation?

 

[Nugatory]

Why? And what movement? The rotation?

Yes. You can't spin the rotating apparatus up without it distorting as different parts are moving at different speeds. There's a good overview of this surprisingly complicated problem here: http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf.

You can find a way of correcting for all this stuff, but the correction process itself will end up with the assumption that the one-way speed is equal to the two-way speed hidden somewhere.

 

[Bartolomeo]

The idea is interesting, but not enough, as discussed until now.

Still, from it (the elements quoted above) the measurement can be done if the rod is rigid, equipped with toothed wheels at the ends (A and B) and rotates. The light sent from B through one notch gets to A through the next notch of the other wheel, somehow similar to Armand H. L. Fizeau experiment:
View attachment 204229
(the image is taken from http://www.chegg.com/homework-help/questions-and-answers/experiment-measure-speed-light-using-apparatus-armand-h-l-fizeau-see-figure-distance-light-q2735778).

In this scenario, length contraction is not a problem if we want to compare the one way speed of light from A to B with the one from B to A. The problem is the distance (too short) and the perfect "synchronization" of the wheels, but although the results would not be perfect, it may appear a difference ...

This is the well - known method, it is so - called Double Fizeau Toothed wheel. Quite interesting one. Stephan Marinov once conducted this experiment. Md. Farid Ahmet conducted it with higher precision but, according to his report he was unable to detect anisotropy of light. His article was in arXiv.org
The method was first proposed by R.W. Wood and had been mentioned or considered by many people (S.J. Prokovnik, F.E Hacket, E. Feenberg, H. Arzeliez, W. A. Rodriguez e.t.c.).
But, by all means the rod will relativistically twist in ether and due to this twist measured velocity of light will be c. Herbert Ives (who measured time dilation by means of transverse Doppler effect) explained existence of this twist in his 1938 paper - Herbert E. Ives - Theory of Double Fizeau Toothed Wheel (1939).
Papers by Prokovnik and Ives contain many links.
S.J. Prokovnik - The Empty Ghosts of Michelson and Morley A Critique of the Marinov Coupled-Mirrors Experiment

 

[DanMP]

Yes. You can't spin the rotating apparatus up without it distorting as different parts are moving at different speeds. ...
You can find a way of correcting for all this stuff, but the correction process itself will end up with the assumption that the one-way speed is equal to the two-way speed hidden somewhere.

Thank you. I reached a similar conclusion during the weekend :)

This is the well - known method, it is so - called Double Fizeau Toothed wheel
...

I knew it but forgot the name ... and didn't find it on Google at "speed of light experiment toothed wheels". Thank you.

 

[kkris1]

"I don't think I understood the @kkris1's problem last week. I think the point of his experiment (or the way I would phrase it if I were in his shoes) was that you could measure a value of the speed of light as a function of the velocity of the rod - call it c ′ ( v ) c′(v) - and then observe that this is asymptotic to some value as v → 0 v→0. And this asymptote would indeed be c c. One way speed of light, right? Since he's getting c for the one-way speed, clocks synchronised by this method must agree with Einstein synchronised clocks (DanMP's addition of two clocks disguised as cogs makes this explicit). The un-expressed question is: what decision did kkris1 make that led to the Einstein convention and not any other? "

Let me rephrase the experiment:
Let's have opaque rod with narrow slits close to both ends A and B. Let's position two lasers so the light is getting through both slits to the photo sensors at A' and B' respectively. Now we move the rod out, accelerate it to the speed v (around 10m/s, so the length contraction is negligible ~10^-14m ;besides, we can't really tell what is being contracted, the rod or the distance between the lasers, since we can assume that the rod is at rest and the lasers are moving with constant speed -v).
When the light from laser A will be transmitted through the leading slit, the light from laser B will pass through the trailing slit. The clocks at A' and B' will be synchronized, since if leading slit is at A, the trailing slit is simultaneously at B. Having both clocks synchronized, we can now measure one way speed of light.
There is no dependency of the speed of the rod and measured value of one way speed of light.

Using this experiment, we do not even need synchronized clocks. Instead we can reflect the light from A' and B' to the photo sensor somewhere in front and using digital oscilloscope measure the difference in arrival time of the two signals.

 

[Ibix]

There is no dependency of the speed of the rod and measured value of one way speed of light.

This is false, as I pointed out in the part of my post that you quoted. And several other posters pointed out.

Also - accelerate the whole apparatus to 0.6c (not just the rod) and repeat. Are your "simultaneous" events simultaneous?

And what value of c are you putting into your length contraction calculation? And, for that matter, how do you justify using the length contraction equation at all, since it is based on assuming Einstein's simultaneity convention?

 

[kkris1]

"This is false, as I pointed out in the part of my post that you quoted. And several other posters pointed out. Also - accelerate the whole apparatus to 0.6c (not just the rod) and repeat. Are your "simultaneous" events simultaneous? And what value of c are you putting into your length contraction calculation? And, for that matter, how do you justify using the length contraction equation at all, since it is based on assuming Einstein's simultaneity convention?"

How can you practically accelerate anything to 0.6c? I'm talking about real , not just "thought" experiment. But even if you did, it would have changed nothing, since everything in the experiment would be affected the same way.(e.g length contraction, time dilation)

We can talk only about relative length contraction; our 10m rod gliding 10m/s would be contracted according to formula:
L'=L(sqrt(1-v^2/c^2) which will yield ~(10^-13)m . I'm aiming to position the lasers with ~(10^-9)m (1nm ) accuracy.
Regarding dependency of the measured value of one way speed of light on v: there is no such dependency.
If the rod glides with lower speed (say 1m/s) past the lasers, the signals from photo sensors will be longer and less sharp, but still we should be able find their peaks, or middle points and their physical separation will be 10m.
Just imagine- when leading slit of the rod triggers the signal from A', at the same time 10m behind another signal is created at B'. This two signals separated by the distance of 10m are going with the speed of light to the sensor somewhere in front and you can measure the time between these two signals. Your distance is 10m, divided by the time between these two signals will give you one way speed of light.

 

[Dale]

@kkris1 we have already asked you multiple times to work out the math. What you are saying is simply false.

You assume that the length contraction is negligible at 10 m/s, but that assumption already assumes that the one way speed of light is c. If the one way speed of light is not c then such assumptions cannot be made and you need to work out the math, as has been requested multiple times.

There is simply no way to measure the one way speed of light which does not assume the very thing it purports to measure

 

[Ibix]

"This is false, as I pointed out in the part of my post that you quoted. And several other posters pointed out. Also - accelerate the whole apparatus to 0.6c (not just the rod) and repeat. Are your "simultaneous" events simultaneous? And what value of c are you putting into your length contraction calculation? And, for that matter, how do you justify using the length contraction equation at all, since it is based on assuming Einstein's simultaneity convention?"

How can you practically accelerate anything to 0.6c? I'm talking about real , not just "thought" experiment.

There are two obvious answers to this. First, impractical and impossible are not the same thing - so all you are doing here is saying that you refuse to think about your theory in a regime where it will fail. Fine if you admit that you're approximating something, but you claim you're not.

Second, back of the envelope, if your whole apparatus is moving at $u$ ($u <<c$) then the flight time of light (your rod is length l; Einstein synchronisation assumed) is $l/c$. The error from ignoring the relativity of simultaneity is approximately $\gamma ul/c^2$. That makes the fractional timing error approximately $u/c$, treating $\gamma\simeq 1$. Modern atomic clocks are accurate to one part in 10¹⁴, so can potentially detect the synchronisation errors when the apparatus moves at a speed of three microns per second. That's probably optimistic, but I have a lot of orders of magnitude in hand before generating velocities is even remotely challenging. 0.6c just makes it more dramatic.

But even if you did, it would have changed nothing, since everything in the experiment would be affected the same way.(e.g length contraction, time dilation)

Incorrect, as shown above.

We can talk only about relative length contraction; our 10m rod gliding 10m/s would be contracted according to formula:
L'=L(sqrt(1-v^2/c^2) which will yield ~(10^-13)m .

You are here explicitly assuming the Einstein synchronisation convention. You need to derive the length contraction equation without assuming the speed of light is the same in both direction. Or, indeed making any assumption about the speed of light. Good luck...

I'm aiming to position the lasers with ~(10^-9)m (1nm ) accuracy.
Regarding dependency of the measured value of one way speed of light on v: there is no such dependency.
If the rod glides with lower speed (say 1m/s) past the lasers, the signals from photo sensors will be longer and less sharp, but still we should be able find their peaks, or middle points and their physical separation will be 10m.

Nonsense. Your entire purpose in keeping $v$ low was to attempt to duck length contraction. If there's no effect from traveling quickly, why are you specifying low speed?

Just imagine- when leading slit of the rod triggers the signal from A', at the same time 10m behind another signal is created at B'. This two signals separated by the distance of 10m are going with the speed of light to the sensor somewhere in front and you can measure the time between these two signals. Your distance is 10m, divided by the time between these two signals will give you one way speed of light.

...except that you assumed the speed of light when you used the length contraction equation. So you will get the answer you put in.

 

[Herman Trivilino]

;besides, we can't really tell what is being contracted, the rod or the distance between the lasers, since we can assume that the rod is at rest and the lasers are moving with constant speed -v).

This claim alone reveals a misconception. Length contraction is real for both, not just one or the other. Perhaps if you can understand the role simultaneity takes in making this arrangement (symmetry of length contraction) possible you would be better able to understand the objections voiced by the other posters.

 

[Ibix]

You need to derive the length contraction equation without assuming the speed of light is the same in both direction.

Just for fun...

If the round trip speed of light is to be $c$, it is easy to see that the one-way speeds ($c_+$ in the +x direction and $c_-$ in the -x direction) must satisfy $2/c=1/c_++1/c_-$, where $c$ is the usual invariant two-way speed of light.

Imagine a radar set moving inertially. It emits a radar pulse in the +x direction at some time $t_e$ and receives the reflection at time $t_r$. The event the pulse reflects off is assigned coordinates $(x_a,t_a)$ (a for asymmetric):$$\begin{eqnarray*}
x_a&=&c(t_r-t_e)/2\\
t_a&=&t_e+(t_r-t_e)\frac c{2c_+}
\end{eqnarray*}$$If we assume the Einstein synchronisation convention and let $c_+=c$ then this reduces to:$$\begin{eqnarray*}
x&=&c(t_r-t_e)/2\\
t&=&(t_r+t_e)/2
\end{eqnarray*}$$These are the usual Einstein coordinates (for more discussion see Dolby and Gull - https://arxiv.org/abs/gr-qc/0104077). We can eliminate $t_e$ and $t_r$ to get a transformation between the asymmetric speed coordinates and the familiar Einstein coordinates. Writing this in matrix form:$$\begin{eqnarray*}
\left(\begin{array}{c}x_a\\t_a\end{array}\right)&=&
\left(\begin{array}{cc}1&0\\\frac{c-c_+}{cc_+}&1\end{array}\right)
\left(\begin{array}{c}x\\t\end{array}\right)\\
\vec{x_a}&=&\mathbf m \vec {x}
\end{eqnarray*}$$Writing in matrix form makes it easy to chain together coordinate transforms, so we can write a generalisation of the Lorentz transform that transforms between frames using asymmetric light speeds as $\vec{x'_a}=\mathbf {m\Lambda m}^{-1}\vec{x_a}$, where $\mathbf\Lambda$ is the usual Lorentz transform matrix:$$
\mathbf\Lambda=\left(\begin{array}{cc}\gamma&-v\gamma\\-\frac{v}{c^2}\gamma&\gamma\end{array}\right)$$This is basically switching from our asymmetric coordinates to the usual Einstein coordinates, switching to Einstein coordinates in another frame, then switching back to our asymmetric coordinates that correspond to that frame. The algebra is tedious, but we eventually find:$$\begin{eqnarray*}
x'_a&=&\gamma\left(\left(1-\left(1-\frac c{c_+}\right)\frac vc\right)x_a-vt_a\right)\\
t'_a&=&\gamma\left(\left(1-\frac v{c_+}+\frac vc\right)t_a-\left(\frac 2{c_+}-\frac c{c_+^2}\right)\frac vcx_a\right)
\end{eqnarray*}$$Note that $v$ is the velocity of the primed Einstein frame as measured by the unprimed Einstein frame; I should probably carry the calculation through to get the velocity as measured by the unprimed asymmetric frame (since it has a different notion of time, it has a different notion of velocity) but it's more convenient here not to. Note also that $\gamma$ has its usual relativistic meaning.

Now we can derive the correct length contraction formula for the case where the speed of light is asymmetric. Imagine a rod of length $l$ at rest in the unprimed frame. One end is at $x_a=0$ and the other at $x_a=l$. In the primed frame at $t'_a=0$ one end is also at the origin and the other is at $x'_a=l'$. Substituting $x_a=l$, $x'_a=l'$ and $t'_a=0$ into our transformation equations we get$$l'=\left(\frac {c_+}{c_+-(1-c_+/c)v}\right)\frac l\gamma$$Now we have everything we need to analyse kkris1's experiment. Assuming that the one-way speed of light is $c_+$ the expected flight time of the light along a rod of length $l$ is $T=l/c_+$. The systematic timing error from ignoring length contraction is $\delta T=(l-l')/v$ so the fractional error is $$\frac{\delta T}T=\frac{c_+}{\gamma v} \left(\gamma-\frac {c_+}{c_+-(1-c_+/c)v}\right)$$Substituting l=10m, v=1m/s we can plot the fractional error as a function of $c_+$ (note that the possible range of $c_+$ is $c/2\leq c_+$).

Clearly, length contraction is only negligible for $c\simeq c_+$ - so neglecting length contraction as kkris1 does is equivalent to assuming the Einstein synchronisation convention. Note that this isn't the same choice and error I mentioned in my earlier posts; there appear to be many choices implicit in the experimental design. In other words, simultaneity convention remains a convention.

I hope I typeset all the above correctly (Edit: I didn't - minor corrections above). A Maxima .mac file to do the algebra is:

/* Maxima batch file to derive length contraction in coordinates where */
/* the speed of light is not isotropic                                 */

/* Define Einstein x,t coordinates in terms of the emission and reception */
/* times (te and tr) of a radar set at rest.                              */
x:c*(tr-te)/2;
t:(tr+te)/2;

/* Ditto for the anisotropic coordinates */
xa:x;
ta:te+(tr-te)*(c/(2*cp));

/* Eliminate tr and te...*/
linsolve([xx=x, tt=t], [tr,te]);
/* ...and get the expression for ta in terms of x and t (xa is trivial) */
ratsimp(subst(rhs(%[2]),te,subst(rhs(%[1]),tr,ta)));

/* Transcribe the results into matrix form and define the Lorentz transform */
m:matrix([1,0],[(c-cp)/(c*cp),1]);
L:matrix([gamma,-v*gamma],[-v*gamma/c^2,gamma]);

/* Express the complete transformation matrix and apply it to a vector */
La:m.L.invert(m);
Laxa:ratsimp(La.matrix([xxa],[tta]));

/* Now get the length contraction formula by insisting that ta'=0... */
solve(Laxa[2][1],tta);
/* ...and xa=l. This expression is l', the length contraction formula. */
ratsimp(subst(l,xxa,subst(rhs(%[1]),tta,Laxa[1][1])));
/* Simplified it manually - just check we didn't mess up - this should be 0 */
ratsimp(%+cp*l*gamma*(1-v^2/c^2)/((1-cp/c)*v-cp));

 

[Dale]

Just for fun...

Excellent post. Very well written as well as a good analysis.

 

[Nugatory]

If the round trip speed of light is to be $c$, it is easy to see that the one-way speeds ($c_+$ in the +x direction and $c_-$ in the -x direction) must satisfy $2/c=1/c_++1/c_-$, where $c$ is the usual invariant two-way speed of light.

That constraint is one of the things that make the postulate so compelling; only a very strange and bizarre anistropy could satisfy that constraint everywhere. But we're still talking postulate here - if the alternatives to a postulate are strange and bizarre... that's probably why we chose the postulate in the first place.

 

[Ibix]

Excellent post. Very well written as well as a good analysis.

Thank you.

That constraint is one of the things that make the postulate so compelling; only a very strange and bizarre anistropy could satisfy that constraint everywhere. But we're still talking postulate here - if the alternatives to a postulate are strange and bizarre... that's probably why we chose the postulate in the first place.

It's also a choice to use orthogonal coordinates; the anisotropic ones share a time axis with Einstein coordinates but the x axes are skewed with respect to one another. That's why the algebra gets so much nastier.

 

[DrGreg]

One other thing to mention about using a non-standard synchronisation: it doesn't just affect the one-way speed of light, it affects all (one-way) coordinate speeds.

For example, using Einstein synchronisation, we know that if two equal masses collide with equal but opposite velocities, inelastically so that they stick together, the resulting mass will have zero velocity. Using a different synchronisation, those masses would not have equal coordinate speeds before collision, so in such a coordinate system $\gamma m\textbf{v}$ is not conserved. You'd need a different formula to get a conserved momentum.