Sagnac effect for matter beams

[Sagittarius A-Star]

@Sagittarius A-Star actually... hum...
The final formula for the time lag due to the Sagnac effect calculated by Rizzi and Ruggiero (their formula 25 with the proper Lorentz factor for the local rotating clock time) is different from the one in the link (formulae 9 and 13). The link takes c² - v² as a denominator, while R&R take c² as a denominator.

At Rizzi and Ruggiero they calculate in formula (25) the time-delta in the rotating frame, while in my link (formulae 9 and 13) they calculate the time-delta with reference to the non-rotating frame.

• Rizzi and Ruggiero formula 25 (clock on the disk at $\Sigma$, see also before formula 20 and text of figure 1):

$\Delta \tau = 2 \ v \ 2 \pi r \gamma / c^2$​

• The link - formulae 9 and 13 (clock in the stationary frame):

$\Delta t = 2 \gamma \ v \ 2 \pi r \gamma / c^2$​

Wiki gives one that is the same with the link:

On Wiki they do indeed proceed by later removing the v² (as it is negligible due to the high c²), but I don't think R&R ever apply approximations in their derivation.

That's a nicely formatted formula

• Wikipedia has exactly the same result as the others (but calculation only for light signals in vacuum, clock in the stationary frame):

$\Delta t = \frac {4v \pi r } {c^2(1-v^2/c^2)} = 2 \gamma \ v \ 2 \pi r \gamma / c^2$​

• You can also see from my posting #13: The formula for the Sagnac $\Delta t$ is nothing else than the term for "relativity of simultaneity" in the inverse Lorentz transformation, multiplied with $2$ (and with $\require {color} \color{orange} \gamma \color{black}$ for the stationary frame.

$\require {color} \Delta t = \color{orange} \gamma \color{black} (\Delta t' + \color{red} {\frac {v} {c^2} \Delta x'} \color{black})$​

You need only to plug-in:
$\Delta t' := 0$ (measured with 2 clocks at both end of the cable, that were synchronized on the rotating disk with light in both directions via the cable)
$\require {color} \color{red} \Delta x' \color{black} := \ 2 \pi r \gamma$
Multiply $\Delta t$ of the LT with $2$ because there are 2 signals in opposite direction involved.

 

[alexandrinushka]

That was only meant as a joke.

I am sorry silly of me

 

[Sagittarius A-Star]

I am sorry silly of me

That was not silly of you.

You are right regarding the twin paradox:

Hello @Nugatory we were discussing earlier if the Sagnac effect applies as well to non-circular paths. I was thinking about some clocks, that would move around a square shape (but it can also be a triangle, whatever, but not necessarily a circle). When back in front of the stationary clock in the lab, the clocks that have moved around the square will show a time lag.

 

[alexandrinushka]

@Sagittarius A-Star for the "nicely formatted" formula you'll be really disappointed: I copy pasted it from Wiki

Rizzi and Co say also that the space on the disk is not time-orthogonal (although I admit I don't really understand what they mean by that... I guess something like "impossible to define a common simultaneity relationship" or something) and that therefore:

"Moreover we stress that it is not possible to describe the relative space in terms of space-time foliation, i.e. in the form x0 = const, where x0 is an appropriate coordinate time, because the space of the disk, as we show in Appendix A.14, is not time-orthogonal." They go on by trying to link that weird space to physical space (with tangential projection techniques and then... I got lost. Although I remember from school what a tangent is and a projection as well).

Can you explain in ELI5 terms what is that supposed to mean? Why project stuff? Why can't we just "live" in this relative space (to use their terminology)?

If that is too much and crosses into metaphysics and you feel like it does not belong here, I'll completely understand. I am trying hard to link this stuff to our current experience and I fail



Also, thanks a lot for the previous comment. If I get it right, the time gap in the lab is just the time gap on the rotating disk multiplied by the Lorentz factor.
I have to admit I get confused when 1 is used instead of c²

You teach me humility really. When I was teaching Russian to a teenage girl and had to go 10 times through the same material I thought "she must be retarded" and lost my patience eventually.

I admire your ability to go over the same Lorentz factors, the switching between different frames and the nice formatting... all that for a complete stranger (and one who only has some high school math + some uni calculus). Thanks

 

[Sagittarius A-Star]

@Sagittarius A-Star for the "nicely formatted" formula you'll be really disappointed: I copy pasted it from Wiki

I overlooked this. I think it is better to create LaTeX formulas instead uploading pictures of formulas.

Rizzi and Co say also that the space on the disk is not time-orthogonal (although I admit I don't really understand what they mean by that... I guess something like "impossible to define a common simultaneity relationship" or something)

I am not familiar with the definition of orthogonality of coordinate systems, so maybe someone else can correct me, if I write something wrong. My understanding is:

• Consider first a 2-dimensional x-/y-coordinate system with Euclidean Geometry. The coordinate axes are orthogonal, if you can calculate every distance as $\sqrt{{\Delta x}^2+{\Delta y}^2}$ (Pythagorean theorem).
• In 1907, Hermann Minkowski found out, that according to SR we live in a 4-dimensional spacetime with a slightly different Geometry than the Euclidean (with different signs under the square-root). The 4 coordinate-axes are orthogonal, if you can calculate every spacetime interval as $\sqrt{|{{c^2\Delta t}^2-\Delta x}^2-{\Delta y}^2-{\Delta z}^2|}$. This is only the case for inertial frames.

If you transform to a Born chart of an Observer on the rim of a rotating disk ("Langevin observer"), limited to less than 360°, then the coordinate axes are not all orthogonal:

$ds^2 = - \left( 1 - \omega^2 r^2 \right) dt^2 + 2 \omega r^2 dt d\phi + dz^2 + dr^2 + r^2 d\phi^2$
...
Notice the "cross-terms" involving $dt\ d\phi$, which show that the Born chart is not an orthogonal coordinate chart.

Source:
https://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart

"Moreover we stress that it is not possible to describe the relative space in terms of space-time foliation, i.e. in the form x0 = const, where x0 is an appropriate coordinate time, because the space of the disk, as we show in Appendix A.14, is not time-orthogonal." They go on by trying to link that weird space to physical space (with tangential projection techniques and then... I got lost. Although I remember from school what a tangent is and a projection as well).

Can you explain in ELI5 terms what is that supposed to mean? Why project stuff? Why can't we just "live" in this relative space (to use their terminology)?

If that is too much and crosses into metaphysics and you feel like it does not belong here, I'll completely understand. I am trying hard to link this stuff to our current experience and I fail

Yes that is too much for me, sorry. Maybe the neightboring thread helps.

 

[PeterDonis]

I think it is better to create LaTeX formulas instead uploading pictures of formulas.

Not only is it better, it's in accordance with the forum rules.