Measuring Light Speed Anisotropy

[Paul Colby]

Sorry, me again. In another thread #47 I gave a experimental setup to detect an anisotropy in light speed of the form $c(\hat{n}) = c_o + \alpha (\hat{n}\cdot\hat{v})$. It was pointed out quite correctly that the error incurred in the long arm to the sensor would cancel the anisotropy making the time delay between pulses appear as $L/c+ = L/c- = L/c$. It was also pointed out that this is due to the existence of a coordinate transform which insures the isotropic result will always be measured. Cool.

It's fair to ask if this situation remains true if I were to consider another form for the anisotropy. If one considers $c(\hat{n}) = c_o + \alpha (\hat{n}\cdot\hat{v})^3$ then the error incurred in the long arm vanishes as the inverse cube of the distance to the detector when the long arm is in the plane $\hat{n}\cdot\hat{v} = 0$ while the anisotropy along the short arm would remain unaffected. I surmise that a coordinate change to introduce such an anisotropy can not exist otherwise the experiment wouldn't work for the reasons given before.

Is this correct or am I still confused?

 

[Ibix]

The condition an anisotropy has to satisfy to be consistent with relativity is that the two-way speed is c. By considering a simple straight line path bouncing straight back off a mirror you can see that this implies $$\frac {2}c=\frac 1 {c_+}+\frac 1 {c_-} $$where $c_\pm$ are the speeds in any pair of opposite directions. Your proposed anisotropy doesn't meet that criterion, since subbing in the particular case $c_\pm=c\pm\alpha$ makes the RHS $2/(c-\alpha^2/c)\neq 2/c$.

So I believe, contrary to what @PAllen said in the thread you linked, that the kind of anisotropy you were proposing there would be detectable, as would your revised version. Anything that satisfies my equation above should be undetectable by any means.

 

[PAllen]

The condition an anisotropy has to satisfy to be consistent with relativity is that the two-way speed is c. By considering a simple straight line path bouncing straight back off a mirror you can see that this implies $$\frac {2}c=\frac 1 {c_+}+\frac 1 {c_-} $$where $c_\pm$ are the speeds in any pair of opposite directions. Your proposed anisotropy doesn't meet that criterion, since subbing in the particular case $c_\pm=c\pm\alpha$ makes the RHS $2/(c-\alpha^2/c)\neq 2/c$.

So I believe, contrary to what @PAllen said in the thread you linked, that the kind of anisotropy you were proposing there would be detectable, as would your revised version. Anything that satisfies my equation above should be undetectable by any means.

I wasn’t intending to address his specific anisotropy equation. I was addressing the claim that geometry of his set up would inherently detect anisotropy because of purported error cancelling along the legs. Instead, I demonstrated the errors work in the direction of reducing the impact of anisotropy. Whether they fully cancel it, does, indeed, depend on the specific form. Elsewhere I described the tightness of the requirement as conspiratorial. Sorry if I didn’t clarify that I was not addressing his proposed equation, specifically.

 

[Paul Colby]

I think the first (linear) form of the anisotropy actually does vanish for all the reasons I've been given. Mixed in with all the discussion in the link thread are clock synchronization definitions etc that I find confusing and in some measure uninteresting. The experiment uses a single observer and a single clock so it's unclear how ones synchronization convention enters in. It was sounding like any anisotropy of any form would be swallowed up in a discussion of clocks, rulers, multiple observers on multiple train platforms. This seems to not be the case universally which is good because this would violate my intuition about what is or is not measurable/testable in principle. While I'm often wrong, this can be really irritating. Thanks for the clarification and help.

 

[Dale]

The experiment uses a single observer and a single clock so it's unclear how ones synchronization convention enters in.

The synchronization convention is inherent and unavoidable in anyone way speed measurement. A one way speed is how long it took to travel a certain distance starting at one point and stopping at a different point. To do that you have to subtract a start time from an end time, and since it is a one way measurement the start and the end are necessarily two different locations, and so a synchronization convention must necessarily be used. A one way speed implies a synchronization, by virtue of being one way.

It isn’t even a matter of how it is measured. It is simply part and parcel of what “one way speed” means. You cannot have a one way speed without a synchronization anymore than you can have an electron without a negative charge. It is a defining part of what makes it what it is.

It was sounding like any anisotropy of any form would be swallowed up in a discussion of clocks, rulers, multiple observers on multiple train platforms.

It is not any anisotropy which is swallowed up. It is anyone way speed which is swallowed up. One way speeds are inherently and unavoidably tied up in synchronization conventions.

Anisotropy can be physical and coordinate independent, but not the anisotropy of a one way speed.

 

[Paul Colby]

It isn’t even a matter of how it is measured. It is simply part and parcel of what “one way speed” means. You cannot have a one way speed without a synchronization anymore than you can have an electron without a negative charge. It is a defining part of what makes it what it is.

Well, within the measurement I described and the speed anisotropy I defined, the time between the pulses arriving at the detector is the time taken for a left going light pulse to traverse the distance L between beam splitters. The same can be said for a right going pulse. In a world obeying SR these are the same time because $\alpha = 0$ and light speed is isotropic. In a hypothetical world not conforming to SR where $\alpha \ne 0$ these time delays will be different under the constrains laid down.

 

[Dale]

and the speed anisotropy I defined, the time between the pulses

It would be a worthwhile exercise to calculate this. Start with the anisotropy you defined, calculate the synchronization convention as a transform from an inertial frame, describe the time between the pulses in the inertial frame, and transform to your defined frame.

 

[Paul Colby]

It would be a worthwhile exercise to calculate this. Start with the anisotropy you defined, calculate the synchronization convention as a transform from an inertial frame, describe the time between the pulses in the inertial frame, and transform to your defined frame.

In a hypothetical world which doesn't obey the relativity principle, what's an inertial frame? I would have to supply these definitions as well. There are no inertial frames in the world as we understand it today. Only good approximations exist.

 

[Dale]

In a hypothetical world which doesn't obey the relativity principle, what's an inertial frame?

I wouldn’t worry about such worlds. We don’t live in them. Again, I would recommend that you go through the exercise I suggested. It will be informative to you.

 

[Paul Colby]

I wouldn’t worry about such worlds. We don’t live in one. Again, I would recommend that you go through the exercise I suggested. It will be informative to you.

It's on my list. A world in which the anisotropy I gave exists likely doesn't work well, possibly even consistently. This really wasn't my point. There are papers I've seen in passing dealing with potential violations of Lorentz symmetry. In order to talk about such things one must have a complete handle of the proposed changes to the rules and their consequences. This is not one of those cases. These theories can't step on known facts. I'm very much in the Relativity is right camp. To be clear I can't claim to understand how this hypothetical speed anisotropy would transform between frames which is likely the intent of your suggested homework. It's enough for me that the fiction I defined is observable as defined.

I think the reason this warrants any comment on my part is that time of flight measurements are the bread and butter of many an experimentalist.

 

[PAllen]

Note, it is the requirement that all fundamental observables are isotropic that severely constrains the nature of anisotropy possibly assumed for non-observables like one way speed of light without syncrhonization. Thus, two way light speed, Doppler, a Roemer type experiment etc. all need to be isotropic. I prefer to think in terms of the universe being istotropic in its laws, but particular coordinate choices need not manifest this isotropy, just as there are many useful coordinates for the Schwarzschild geometry that don't manifest the timelike killing symmetry (outside the body or horizon).

 

[Dale]

There are papers I've seen in passing dealing with potential violations of Lorentz symmetry. In order to talk about such things one must have a complete handle of the proposed changes to the rules and their consequences.

Yes, with the research in string theory there has been a renewal of interest in possible Lorentz violating signals. The Standard Model Extension has been developed specifically for that. It is a test theory which includes the standard model, GR, and all possible Lorentz violations as parameters. It might be of interest to you, and it should be used my any of the more recent papers of the type you mentioned.