- #36

Sagittarius A-Star

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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.alexandrinushka said:@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 c^{2}- v^{2}as a denominator, while R&R take c^{2}as a denominator.

- 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##

That's a nicely formatted formulaalexandrinushka said:Wiki gives one that is the same with the link:

On Wiki they do indeed proceed by later removing the v^{2}(as it is negligible due to the high c^{2}), but I don't think R&R ever apply approximations in their derivation.

- 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.

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