# Linear Vs Sagnac

#### TheAntiRelative

Okay I've got a question about the difference between the sagnac effect in an inertial and non-inertial environment. Let me give you an example to base the info off of.

I want to make a really long linear sagnac-like arrangement to factor out any cross propagation. In the following example all I want is toward and away.

Put a set of 4 mirrors in space that will reflect light in a square. Bring two mirrors close together on each side to create a rectangle. Separate these two sets of close mirrors from eachother by 10 light seconds. Place a spaceship that is capable of travelling .25C in the plane of one of the 2 long optical paths (imagining that it doesn't block the other path)

Ignore the small travel time in which the beams propagate perpendicularly to the ships motion so - if you're ticky - when comparing to a ring that has a circumference the same as the linear optical path, just reduce the circumference by that tiny cross propagation amount. (close enough)

The purpose is to determine how the arrival time difference compares to a sagnac device. So for your non-inertial example imagine the same ship with 1 huge fiber optic cable attached to it's tail and nose. The length of the fiber cable is the same as the optical path in the first thought experiment and the ship is flying in a circle of that circumference.

For the Linear experiment:
Examination 1) Have the ship start at the 1/4 mark and emit a pulse forward and back simultaneously. Examine results with SR.
Exam 2) Have the ship start in the middle and examine results.

Questions:
1) Do both examinations arrive at the same arrival time difference?
2) Do both experiments result in the same arrival time difference in all situations of examination?

Anyone got Mathematica so they can show the linear experiment visually?

Related Special and General Relativity News on Phys.org

#### pervect

Staff Emeritus
Sorry - I've tried to read this twice, but I'm not convinced I understand the proposed arrangement without some sort of diagram.

In any event, the Sagnac effect applies only to rotating systems - there is no "linear" sagnac effect. I don't understand your diagram enough to know if your so-called linear system really rotates or not.

#### TheAntiRelative

Direction of spaceship travel is right to left.
No acceleration considerations.
Ignoring up/down propagation time.
Counter propagating beams emitted at specified starting point.

Thought Experiment 1:
Examination 1
Code:
/----{ }------------------------<[spaceship at 25%]< --{ }---\
|                                                            |
\--------------------{really long distance}------------------/
Examination 2
Code:
/----{ }------------<[spaceship at halfway]< ------------{ }--\
|                                                             |
\--------------------{really long distance}-------------------/

Non-inertial comparison thought experiment is a fiber-optic circle of a circumference equal to the total optical path above (20ls) attached to the front and back of the ship and travelling along the circumference at a speed that would make it all the way around in 80 seconds.

Sorry about the lack of clarity. Sometimes you can see it so clearly in your own mind that you don't realize how inadequate your words are. I hope this makes it a little better.

Edit: Oh yeah, no length to the ship either. I guess just call it a beamsplitter going out the bottom... You get the idea, really simplified and omitting insignificant details. Consider the reception point the same place as the emission point...

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#### pervect

Staff Emeritus
It's obvious from the geometry that pulses emitted from the spaceship to the right in your diagram will return to it before pulses emitted from the spaceship to the left.

In the lab frame, it will take light a time 2L/c to make a round trip, L being your "really long distance". During that time, the spaceship will move to the left .25*c*(2L/c) = .5L

At time 2L/c in the lab frame, the pulse emitted to the right will have arrived some time ago, while the pulse emitted to the left will have just reached the starting point.

#### yogi

The sagnac effect was originally thought to be a disproof of SR - but it really is not the same type of experiment as MMx - in MMx the source and the receiver are in the same frame - in a linear Sagnac experiment, the receiver moves relative to the source - For example the GPS correction that is introduced to compensate for the distance the receiver travels due to the earths rotation after the signal is sent from the satellite is typically referred to as a "one way Sagnac" correction. The additional distance - perhaps on the order of a few meters typically have nothing to do with the fact that the receiver is fixed on the curved surface of the earth - the same correction would be required if the receiver moved in a straight line after the signal is in transit (the curvature component is only a small fraction of a mm. whereas the length of the path to the receiver changes by a few meters)

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#### TheAntiRelative

I'm sorry, I once again left something out... :uhh: I was wanting to know what the results were according to the spaceship which is the inertial observer.

Thinking about it, I left out the most critical component of my question! *smack head* :grumpy:

#### pervect

Staff Emeritus
You can work the problem out in any frame that's convenient.

If you use an idealized model of the fiber optic cable where the velocity of propagation is the constant 'c', you'll get the same results with a fiber optic cable as mirrors.

If you start look at the index of refracation of the fiber optic cable so that the velocity of propagation is not 'c', you'll have a messy calcuation on your hands that I don't think you're interested in. Since the cable is under rather severe stress at the point where it changes direction you might have to take that into account, too - there's no such thing as a rigid body in relativity, so you'd want to assume something physially reasonable for the elasticity of the cable, and figure out how much it stretches.

To go through the calculation for the mirrors briefly in the space-ship frame (even if it's not necessarily the most convenient frame to use):

1) The distance between the mirrors will be Lorentz contracted to L' = L/gamma

2) We introduce a coordinate system centered on the space-ship, with positive x to the left of the space-ship, and the speed of light equal to c

Then:

The equation of light moving to the left is x-c*t = constant, the equation of light moving to the right is x+c*t = constant.

For the initial pulse sent at x=t=0, the equations will be

x-c*t=0, x+c*t=0

The equation of motion for the left mirror will be

x-v*t = constant, we can call this constant d0L, so x-v*t = d0L. d0L will be the inital distance to the left mirror at t=0 in the space-ship's frame

Similarly the equation of motion for the right mirror will be

x+vt = constant, we can callt his constant doR, the distance to the right mirror at t=0.

We can solve for the time when the light hits the mirror. For the left mirror we solve the simultaneous system of equations

x-c*t=0, x-v*t=d0L

to get t = d0L/(c-v) (and x=d0L/(c-v))

We can do the same for the right mirror.

The time taken to go from the left mirror to the right mirror, or vica-versa, will be L'/c , where L' is the length contracted distance between the mirrors. Note that L' = d0L+d0R.

Now all we have to do is figure out the last leg of the journey. At this point I hope the principles are clear, I will leave the final steps to someone else. We already know the equations of motion of the mirror, we need to calculate what the constant values of

x-c*t, x+c*t

are for the reflected light beams.

We should expect, of course, that the results will be just the Lorentz transform of the results we would get by working the problem out in the mirror frame.

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#### TheAntiRelative

pervect said:
You can work the problem out in any frame that's convenient.
There is a method to my madness. I'm trying to find out if something I've been told is true. Basically I've been told that the Sagnac Effect is completely geometric in nature and acceleration does not need to be considered.

I've been told that these two different experiments should come up with exactly the same phase shift from the ships' perspectives and something tells me that isn't correct.

pervect said:
If you use an idealized model of the fiber optic cable where the velocity of propagation is the constant 'c', you'll get the same results with a fiber optic cable as mirrors.
Evacuated air core fiber

pervect said:
Since the cable is under rather severe stress at the point where it changes direction you might have to take that into account, too - there's no such thing as a rigid body in relativity, so you'd want to assume something physially reasonable for the elasticity of the cable, and figure out how much it stretches.
Some really good lab assistant already figured that out for us so that the super rigid fiber we picked actually stretches out to be exactly the circumference that matches the path length of the mirror experiment when the spaceship/fiber combo are rotating.

What I was hoping to find out is if there is a vast difference between the two experiments or if a small approximation works just fine or if they are exactly the same but I don't trust my own ability to come up with the right answer.

#### pervect

Staff Emeritus
The mirror experiment is well defined. The cable experiment isn't so well defined, unless you define it to be the same as the mirror experiment :-)

The rotating relativistic disk has confused a whole lot of people, including some professional physicists who write confused papers :-(. The basic problem with the rotating disk is that many people naievly assume it's possible to synchronize all the clocks on a rotating disk via Einstein synchronization, while this turns out not to be the case.

It's a rather technical, but this is discussed for instance by Tartaglia

http://lanl.arxiv.org/abs/gr-qc/9805089

As a consequence, in any reference frame for which the vortex tensor differs
from zero,
i.e. any rotating "frame"

the concept of ”the whole physical space at a given instant” turns out
to be conventional, in the sense that it is lacking an operational meaning because
of the impossibility of a symmetrical and transitive synchronization procedure
I.e. if a,b,c, and d are all clocks at different places, you can synchronize a to b, and b to c, and c to d, but then you find that d isn't synchronized with a!

Your linear experiment as defined with mirrors doesn't suffer from this problem, but it's not really an example of the sagnac effect either. Attempting to re-think the mirror experiment as a rotating cable will run head-on into this problem.

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#### TheAntiRelative

pervect said:
Your linear experiment as defined with mirrors doesn't suffer from this problem, but it's not really an example of the sagnac effect either. Attempting to re-think the mirror experiment as a rotating cable will run head-on into this problem.
Hmm I was actually trying to work it the other direction from the cable to the mirrors but I suppose the problem still applies. I read some paper (sorry forgot where) that claimed these two experiments will get the same time difference in signals but that didn't sound right.

It sounds like you're saying the two experiments are thoroughly non-equivelent. Should I assume the two will not come to the same result either?

What further definition do I need to add to the rotating one? The circumference is 20 light seconds. The ship will complete the circle in 80 seconds. The emission event for both beams is at the same spacetime coord in both of the examples.

In essence my question is a yes/no. Do the reception events have the same interval between them by the ships frame in both experiments? Then after that question I can go off into looking at the "Why" of it.

#### pervect

Staff Emeritus
I'm getting the feeling there's some sort of communications problem from your questions. Possibly I haven't been clear enough.

Here's the way I see things.

The spaceship + mirrors problems is perfectly well defined. It has a perfectly well defined answer. The "cable" problem is not very well defined, however (unless you say that it is the same as the mirror problem).

Going back to the mirror problem, which is well defined. This problem is stated in a non-rotating coordinate system. The Sagnac effect is stated in terms of a rotating coordinate system. Therfore, the connection of the problem to the Sagnac experiment is tenuous. You won't, for instance, be able to directly apply the Sagnac formula that the total phase shift is proportional to the area of the loop, because you don't actually have a rotating loop.

i.e.

If you transform your system into rotating coordinates, or envision how it looks from an observer who is rotating, you don't have a loop that mantains its shape in the rotating coordinate system.

To really have a rotating loop that you could apply the textbook Sagnac formulas to, you'd need a loop or mirror arrangement that made the beam path appear to be static in rotating coordinates.

#### TheAntiRelative

pervect said:
To really have a rotating loop that you could apply the textbook Sagnac formulas to, you'd need a loop or mirror arrangement that made the beam path appear to be static in rotating coordinates.
I'm sorry. Either I'm not explaining myself well or I'm not understanding you correctly.

Would it help to say that the fiber optic ring is attached to a rotating disk such that in the co-rotating frame, the beam path appears to be static? The object is for the non-inertial to be the same(as possible) as the mirror problem at least in starting conditions but still comparing the results of an experiment in rotating non-inertial system to an experiment in inertial one. Additionally I want the non-inertial experiment to only be textbook Sagnac effect considerations alone.

Say for instance that I build the mirrors to be attached to a huge platform and then I measure the distance by the rest frame of the platform. Then I build a disc with a fiber cable around the perimeter. Before setting this disc in rotation I measure the optical path length to be the same as the mirrors.

I realize there is an issue with elasticity of the disk but for this comparison I'd like to factor that out and compare results in more of a "What if perfectly regid existed" situation. I know there are some interesting considerations with a disc which is why I orginally chose to not include it in the experiment. I only want to include GR effects on time/length in the final outcome of the comparison. That is why I was hoping to simply say that the elasticity of the fiber is such that when it rotates at the speed specified, it will stretch to a diameter estimated by a center point stationary observer as exactly necessary to create a circumference equal to the path length of the mirror setup.

I've been trying to avoid corrupting the conversaton by bringing up the paper I am trying to examine. It seems as though the experimental results published in the paper cannot be right, but I'm not sure about that.

If you decide to read them you'll see that I'm trying to set up an analogy that shows whether or not well proven precepts of GR and SR as they apply to Sagnac Effect contradict this paper.

Since we know that GR and SR are mathematically consistant, we can assume that if these two analagous experiments do not come up with same answer then there is something wrong with the interpretation in the papers below. Because by this paper, my two analogies should record the same time delay.

Generalized Sagnac Effect Ruyong Wang, Yi Zheng, and Aiping Yao - Physical Review Letters, 2004
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000093000014143901000001&idtype=cvips&gifs=yes [Broken]

And the earlier paper:
Modified Sagnac experiment for measuring travel-time
difference between counter-propagating light beams in a uniformly
moving fiber
R Wang, Y Zheng, A Yao, D Langley - Physics Letters A, 2003

In short, the paper claims a linear sagnac effect that is effected by linear cable length in then same way that additional circumference affects time delay between signals.

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#### pervect

Staff Emeritus
TheAntiRelative said:
I'm sorry. Either I'm not explaining myself well or I'm not understanding you correctly.

Would it help to say that the fiber optic ring is attached to a rotating disk such that in the co-rotating frame, the beam path appears to be static?
OK, I'm confused. Ascii diagrams suck.

Let's consider a long loop, which I will call A

-----------
-----------

If the loop is stationary in a rotating frame, after a 90 degree rotation, we get a long loop, which I will call B, that's oriented at a right angle to A.

||
||
||
||
||
||

The issue I'm stuck on is that your spaceship case never looks like "B". The loop axis always points in the same direction. So it's not rotating, IMO, it's just translating.

I've been trying to avoid corrupting the conversaton by bringing up the paper I am trying to examine. It seems as though the experimental results published in the paper cannot be right, but I'm not sure about that.

If you decide to read them you'll see that I'm trying to set up an analogy that shows whether or not well proven precepts of GR and SR as they apply to Sagnac Effect contradict this paper.

Since we know that GR and SR are mathematically consistant, we can assume that if these two analagous experiments do not come up with same answer then there is something wrong with the interpretation in the papers below. Because by this paper, my two analogies should record the same time delay.

Generalized Sagnac Effect Ruyong Wang, Yi Zheng, and Aiping Yao - Physical Review Letters, 2004
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000093000014143901000001&idtype=cvips&gifs=yes [Broken]

And the earlier paper:
Modified Sagnac experiment for measuring travel-time
difference between counter-propagating light beams in a uniformly
moving fiber
R Wang, Y Zheng, A Yao, D Langley - Physics Letters A, 2003
I haven't seen the whole paper, only the abstract, as I would have to pay. However, I must say that I'm skeptical too. It looks to me like the author is claiming to be able to detect absolute motion, I'm extremely skeptical.

Google finds an interesting link, it looks like the author has a paper of some sort on-line, which I *can* study.

http://www.wbabin.net/yuri/keilman5.htm [Broken]

I'll take a closer look at it and get back to you. My knee-jerk reaction is negative at this point, however.

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#### pervect

Staff Emeritus
The webpage I read looked like total nonsense, but when I double-checked the authors, they were not the same authors as were in the original paper.

I think it would be really interesting to read the Physical Review letters for responses to the original letter, I still have major doubts about the ideas being proposed.

Maybe someone with the appropriate on-line subscriptions can comment more.

#### TheAntiRelative

I don't think it's absolute motion that's being detected. I've had someone tell me that basically SR kinda predicts a phase shift in Sagnac because if you examine the rotating object from an inertial frame and look at the reception events. They're not at the same spacetime coordinates in one frame so they're not the same in any frame.

But as far as the mirror thing, yeah I'm talking about no rotation at all. Only translational movement. Then trying to compare that to the results of the rotational motion.

Supposedly it should come out with the same answer regardless of the process required to get answer. That's why I've been trying to put it into a yes/no format. If they cannot come out the same then something is odd about that paper and it'd be interesting to find out what is wrong with it.

#### TheAntiRelative

Anyone care to give a yes/no?

Should the two come up the same or different?

#### pervect

Staff Emeritus
I think I've already given my answer, that only one problem is well-defined. I'll repost a summary of my opinion: of the two problems/systems presented, one of them demands rigid, rotating objects, a notion that is ill-definined. (See the discusssion of Born rigidity in the sci.physics.faq, for instance).

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

"Linear Vs Sagnac"

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