Measuring the One Way Speed of Light

[RandyD123]

TL;DR Summary

Referencing a YouTube Video

Does this video even make sense? And if so, is it right or wrong?

 

[Paul Colby]

yes, it’s correct.

with that said, you can’t muck with the isotropy of the speed of light arbitrarily without modifying Maxwells equation. The type or functional form of any isotropy is limited. Within these limits, it’s a matter of conversion.

 

[phinds]

Does this video even make sense? And if so, is it right or wrong?

I suggest a forum search if you want more discussion. The topic has beaten to death numerous time here on PF.

There are numerous links at the bottom of this page.

 

[Dale]

Summary:: Referencing a YouTube Video

Does this video even make sense? And if so, is it right or wrong?

It is correct. The one way speed of light is indeed a convention.

I would disagree a bit with him about some of his statements to the effect that we cannot know the one way speed of light. Because it is a convention, not only can we know but we do know with certainty simply by choosing our convention.

 

[Ibix]

Short answer: to measure one way speed you need clocks at opposite ends of a straight track. How do you synchronise those clocks? There is no answer to that which doesn't depend on assuming a one-way speed of light.

 

[Dale]

I wonder what the FLRW spacetime would look like under an anisotropic c synchronization.

 

[pervect]

I wonder what the FLRW spacetime would look like under an anisotropic c synchronization.

If you're really curious, choose your favorite line element for the FLRW metric, pick an approrpirate diffeomorphism to remap t (isotropic) to t' (non-isotropic), and compute the new line element.

 

[lerus]

As far as I understand, historicaly the very first measurement of speed of light - in 1676 by Olaus Roemer using Jupiter's satellites was a one-way measurement.
The second measurement of speed of light by James Bradley in 1726 using Stellar Aberration was also a one way measurement.
Another possible approach is to use Doppler effect - we can use source that emits light with known frequency and receiver that is moving with the known speed.
We can measure frequency of received light and calculate speed of light using Doppler formula.

 

[Ibix]

As far as I understand, historicaly the very first measurement of speed of light - in 1676 by Olaus Roemer using Jupiter's satellites was a one-way measurement.
The second measurement of speed of light by James Bradley in 1726 using Stellar Aberration was also a one way measurement.

The problem with any one-way measure, including the ones you cite, is that they assume that the speed of light is the same in both directions. Romer, for example, effectively looks at a distant clock (the Jovian moons) and attributes apparent rate variation solely to changing light travel time due to its changing distance. In a relativistic analysis, this turns out to mean that he assumed the Einstein clock synchronisation convention, which is to say that he assumed that the speed of light was isotropic. One could re-analyse the results using a non-isotropic synchronisation convention and get a different result.

 

[vanhees71]

I wonder what the FLRW spacetime would look like under an anisotropic c synchronization.

Just too ugly compared to the standard coordinates with the proper time of the comoving observers (comoving with the "cosmic substrate" or the rest frame of the cosmic microwave radiation).

 

[Dale]

the very first measurement of speed of light - in 1676 by Olaus Roemer using Jupiter's satellites was a one-way measurement.

It is easy to set up experiments where the light path is one way. The issue is that all such experiments depend on some method of clock synchronization. Your assumption about clock synchronization determines the speed you get. In the case of Romer’s measurement he was using slow clock transport and assumed the isotropy of slow clock transport. This is equivalent to assuming the Einstein synchronization convention.

This analysis is described here:
https://openlibrary.org/books/OL689312M/Special_relativity_and_its_experimental_foundations

 

[Paul Colby]

It is easy to set up experiments where the light path is one way. The issue is that all such experiments depend on some method of clock synchronization.

I’m skeptical. One could use a pulsed light source and two partially silvered mirrors separated by some distance. A single distant observer with a single clock could reside equally distant from each mirror. The distant observer would see two pulses separated by the time of flight of the pulse between the mirrors. The source is moved and the pulse sent along the reverse direction.

Now this experiment doesn’t solve the problem as usually discussed because the equal length paths to the distant observer each contain a lateral component in opposite directions. However, the magnitude of these contributions depends on the functional form of the speed anisotropy.

 

[Ibix]

I’m skeptical. One could use a pulsed light source and two partially silvered mirrors separated by some distance. A single distant observer with a single clock could reside equally distant from each mirror. The distant observer would see two pulses separated by the time of flight of the pulse between the mirrors. The source is moved and the pulse sent along the reverse direction.

Are you basically suggesting sending light pulses along one edge and the other two edges of a closed triangular path? That's a two-way measurement.

More generally, choosing an anisotropic speed of light just leads to a non-orthogonal coordinate system on spacetime. That doesn't have any measurable consequences beyond making the maths nastier.

 

[Paul Colby]

sending light pulses along one edge and the other two edges of a closed triangular path? That's a two-way measurement.

How so? Only one observer and only one clock.

 

[vanhees71]

The point is that you need a local observer with one clock at a defined rate (e.g., using the definition of the second in the SI via the Cs standard). So what you can measure concerning the speed of light are local observables at the place of this one clock. The standard example is "radar", i.e., a signal that is sent to a distant object, being reflected and then detected again, measuring the time it takes to detect the signal again. That's measuring the two-way speed of light of course.

To measure a one-way speed of light you need two clocks, one at the place of the emission and one at the place of detection of the light signal. To make sense of the clock readings as a "one-way speed of light" the clocks must be somehow synchronized, and it depends on the synchronization procedure you use, which "one-way speed of light" you measure.

 

[Dale]

I’m skeptical. One could use a pulsed light source and two partially silvered mirrors separated by some distance. A single distant observer with a single clock could reside equally distant from each mirror. The distant observer would see two pulses separated by the time of flight of the pulse between the mirrors. The source is moved and the pulse sent along the reverse direction.

This is assumes already that the one way speed of light is isotropic.

How so? Only one observer and only one clock.

That is actually the identifying feature of a two way measurement. Actual one way measurements require two clocks so they require an assumption about simultaneity.

Two way measurements don’t assume simultaneity since they use a single clock, but to infer a one way speed they have to assume isotropy. What you are describing assumes isotropy, so it is a two way measurement, as also evidenced by the use of a single clock.

 

[Paul Colby]

Well, I knew this was a lost cause. I’m just describing a standard time of flight measurement. The time difference measured by my single observers clock is the time of flight between mirrors plus the time of flight difference between the much longer equal length paths to the observer. Now, this second time difference depends on the details of the anisotropy assumed as a function of direction. In the limit the observer is infinitely far away the path directions become identical. So it’s a limiting procedure. One must show that the time difference times the total distance remains significant. This is indeed the case for the anisotropy forms assumed. It’s not clear to me this holds in general.

 

[Paul Colby]

That is actually the identifying feature of a two way measurement. Actual one way measurements require two clocks.

Only if one defines a partiality reflective mirror as a clock in this case.

 

[Dale]

In the limit the observer is infinitely far away the path directions become identical.

This doesn’t work. In the limit the directions become arbitrarily close but the distance becomes arbitrarily long. The time difference from any anisotropy in the speed of light decreases as the directions become close, but it increases as the distance increases. The two effects together mean that even in the limit of a distant observer the anisotropy assumption is still non-negligible.

Only if one defines a partiality reflective mirror as a clock in this case.

It has nothing to do with that. The mirror isn’t a clock. The experiment is a two way experiment because the direction of the light is changed, a single clock is used, and the calculation of the speed of light depends on an assumption about the isotropy of the speed of light. All of those are characteristics of two way measurements.

I recommend that you actually work through the math of your proposed experiment. Either you will see where the isotropy assumption comes in or I can point it out.

 

[Paul Colby]

Okay, let me write this out with some care. We have a very large optical bench fresh from the manufacture with an x and a y coordinate system. We're going to measure the time of flight for light pulses using a detector and a standard issue time of flight box. First off our assumption is that the speed of light is dependent on direction which for our table I can write as, ##c(\theta)##, where ##\theta## is the angle between a light ray and the x axis. We make the following further assumptions that ##c(\theta)## is a real single valued analytic function of ##\theta##.

We place two beam splitters (1,2) of negligible dimension, 1 at ##x=-W, y=0## and 2 at ##x=W, y=0##. The detector, D, is placed at ##x=0,y=L##. The beam splitters are adjusted so pulses from each will be directed to the detector and to the other beam splitter. Now, the time of flight depends on distance and direction. Now, we fire a pulse through 1 to 2. The pulse is split at 1 and then at 2. The time of flight between 1 to 2 is

##\Delta_{12} = \frac{2W}{c(0)}##

##\Delta_{21} = \frac{2W}{c(\pi)}##

The split pulses travel along different legs of the triangle to the detector. Their time of flights are,

##\Delta_{1D} = \frac{\sqrt{(L^2+W^2)}}{c(\frac{\pi}{2}-\alpha)}##
##\Delta_{2D} = \frac{\sqrt{(L^2+W^2)}}{c(\frac{\pi}{2}+\alpha)}##

##\alpha = \arctan{\frac{W}{L}}##

The time between received pulses is,

##\Delta_A = \Delta_{12} + \Delta_{2D} - \Delta_{1D}##

Reversing the direction of the pulse (sending through 2 then 1)

##\Delta_B = \Delta_{21} + \Delta_{1D} - \Delta_{2D}##

So, my question / observation is; will ##\Delta_A = \Delta_B## for all ##c(\theta)##? Clearly not since there are choices which make ##\pm(\Delta_{2D} - \Delta_{1D})## negligible while ##\Delta_{12} - \Delta_{21}## is not.

 

[Dale]

We make the following further assumptions that c(θ) is a real single valued analytic function of θ.

There is a further condition that you have neglected. That is that the function ##c(\theta)## must have the two way speed of light equal to ##c##. In other words, for any constant path (in an inertial frame) of length ##s##, the time for light to traverse that path forward plus the time to traverse the same path backward is ##2s/c##. This is required because the two-way speed of light is measurable and is ##c##.

So, my question / observation is; will ##\Delta_A = \Delta_B## for all ##c(\theta)##? Clearly not since there are choices which make ##\pm(\Delta_{2D} - \Delta_{1D})## negligible while ##\Delta_{12} - \Delta_{21}## is not.

Any such choices are ruled out by the two way speed of light condition.

 

[Paul Colby]

There is a further condition that you have neglected.

What is the origin of this requirement?

Ah, MM. So, If I produce a function ##c(\theta)+c(\theta+\pi) = 2c## which yield ##\Delta_A \ne \Delta_B##. then what?

 

[Paul Colby]

Any such choices are ruled out by the two way speed of light condition.

Okay, try

##c(\theta) = c + \epsilon\cos^7\theta##

This function meets the requirement, ##c(\theta)+c(\theta+\pi)=2c##, yet yields ##\Delta_A \ne \Delta_B##.

 

[Sagittarius A-Star]

This function meets the requirement, ##c(\theta)+c(\theta+\pi)=2c##, yet yields ##\Delta_A \ne \Delta_B##.

That's not the requirement. For example, one requirement is:

##\Delta_{12} + \Delta_{21} = \frac{4W}{c}##

 

[Paul Colby]

That's not the requirement. For example, one requirement is:

Good catch.

Try ##c(\theta) = \frac{c}{1+\epsilon\cos^7\theta}##

also, we’re discussing the actual measurements which are ##\Delta_A## and ##\Delta_B##.

 

[DrGreg]

So, If I produce a function ##c(\theta)+c(\theta+\pi) = 2c## which yield ##\Delta_A \ne \Delta_B##. then what?

No, you want$$
\frac{1}{c(\theta)} + \frac{1}{c(\theta+\pi)} = \frac{2}{c}
$$

 

[Paul Colby]

No, you want$$
\frac{1}{c(\theta)} + \frac{1}{c(\theta+\pi)} = \frac{2}{c}
$$

Yes, that holds exactly for ##c(\theta)## given in #25.

 

[Dale]

What is the origin of this requirement?

The origin of the requirement is experiment. Whatever function we choose for ##c(\theta)## must be compatible with experiment. Experimentally, if we bounce light around any closed path we measure ##\Delta t= s/c## where ##s## is the path length and ##\Delta t## is the time between emission and reception.

Okay, try

##c(\theta) = c + \epsilon\cos^7\theta##

This function meets the requirement, ##c(\theta)+c(\theta+\pi)=2c##, yet yields ##\Delta_A \ne \Delta_B##.

You are correct. I gave a weaker statement than the actual experimental constraint. I thought that the weaker constraint I stated implied the experimental constraint, but as you have shown it does not. My apologies.

With the correct constraint it is clear that ##\Delta_A=\Delta_B## by direct calculation.

##\Delta_A=\Delta_{12}+\Delta_{2D}-\Delta_{1D}##
##\Delta_A=\Delta_{12}+\Delta_{2D}-\Delta_{1D}+\Delta_{D1}-\Delta_{D1}##
##\Delta_A=\Delta_{12}+\Delta_{2D}+ \Delta_{D1}-(\Delta_{1D}+ \Delta_{D1})##
##\Delta_A=(2R+2W-4W)/c##

And similarly with ##\Delta_B##

 

[Paul Colby]

The origin of the requirement is experiment. Whatever function we choose for ##c(\theta)## must be compatible with experiment. Experimentally, if we bounce light around any closed path we measure ##\Delta t= s c##.

You are correct. I gave a weaker statement than the actual experimental constraint. I thought that the weaker constraint I stated implied the experimental constraint, but as you have shown it does not. My apologies.

With the correct constraint it is clear that ##\Delta_A=\Delta_B## by direct calculation.

##\Delta_A=\Delta_{12}+\Delta_{2D}-\Delta_{1D}##
##\Delta_A=\Delta_{12}+\Delta_{2D}-\Delta_{1D}+\Delta_{D1}-\Delta_{D1}##
##\Delta_A=\Delta_{12}+\Delta_{2D}+ \Delta_{D1}-(\Delta_{1D}+ \Delta_{D1})##
##\Delta_A=(2R+2W-4W)/c##

And similarly with ##\Delta_B##

Well, we need to pick a set of requirements and stick with them. I follow that ##\Delta_{1D}+\Delta_{D1}## must average but lose it on ##\Delta_{2D}+\Delta_{D1}##. I can’t reproduce your result.

also, ##\Delta_{D1}## and the like don’t appear in the measurement.

 

[Dale]

I follow that ##\Delta_{1D}+\Delta_{D1}## must average but lose it on ##\Delta_{2D}+\Delta_{D1}##. I can’t reproduce your result.

Since the time around any closed loop equals the length (perimeter) of the loop divided by ##c## we have ##\Delta_{12}+\Delta_{2D}+ \Delta_{D1}=(2R+2W)/c## and ##\Delta_{1D}+ \Delta_{D1}=4W/c##

also, ##\Delta_{D1}## and the like don’t appear in the measurement.

##\Delta_{D1}-\Delta_{D1}=0## so we are free to add it to your original expression for ##\Delta_A##

 

[Paul Colby]

Since the time around any closed loop equals the length (perimeter) of the loop divided by c we have

Yeah, I don’t accept this as a requirement. It may well be a valid result given the usual velocity anisotropy but I believe it need not hold for those being considered here.

 

[Dale]

Yeah, I don’t accept this as a requirement. It may well be a valid result given the usual velocity anisotropy but I believe it need not hold for those being considered here.

You are not free to reject it. It is required by experiment. Any convention that does not satisfy this requirement is experimentally falsified.

 

[Paul Colby]

You are not free to reject it. It is required by experiment. Any convention that does not satisfy this requirement is experimentally falsified.

Fine, which experiment.

 

[Dale]

Fine, which experiment.

Any non-rotating ring interferometer experiment

 

[Paul Colby]

Any non-rotating ring interferometer experiment

Like all things these are done to finite precision. So, we’ve established the point I was trying to make which is the class of acceptable ##c(\theta)## is really rather narrow. One is restricted to ones that are not measurable. The ones I proposed are measurable and therefore disallowed for some value of the paramete ##\epsilon##.