I On the physical meaning of Minkowski's spacetime model

[Dale]

Do you mean that for clocks at rest in a rotating frame a light beam traversing a closed path of length $L$ is not always $L/c$ (as measured by clocks at fixed points in rotating frame) ?

That is correct. A light beam traversing clockwise or counterclockwise will give different times.

 

[cianfa72]

That is correct. A light beam traversing clockwise or counterclockwise will give different times.

Which is the reason for this ?

 

[Dale]

Which is the reason for this ?

The second postulate.

 

[cianfa72]

The second postulate.

Sorry, not sure to understand. The second postulate is about the invariance of speed of light as measured in inertial frames.

 

[Dale]

Sorry, not sure to understand. The second postulate is about the invariance of speed of light as measured in inertial frame.

Yes, from that you can derive the fact that in the rotating reference frame a light beam traversing clockwise or counterclockwise will give different times.

 

[Ibix]

Sorry, not sure to understand. The second postulate is about the invariance of speed of light as measured in inertial frame.

So analyse what happens to two light pulses going in opposite directions in a rotating closed (circular is easiest) path. Assuming the emitter is attached to the rotating apparatus do simultaneously emitted pulses return to the emitter simultaneously?

 

[cianfa72]

So analyze what happens to two light pulses going in opposite directions in a rotating closed (circular is easiest) path. Assuming the emitter is attached to the rotating apparatus do simultaneously emitted pulses return to the emitter simultaneously?

Ah ok, one can analyze it from the point of view of the inertial frame where the second postulate holds. No, the two emitted pulses do not return to the emitter simultaneously as measured in the inertial frame by Einstein's synchronized clocks in it. This latter fact is frame invariant (i.e. there is no coincidence of the two reception events), hence a light beam traversing clockwise or counterclockwise will give different times.

 

[Ibix]

Yes.

 

[Nugatory]

No, the two emitted pulses do not return to the emitter simultaneously as measured in the inertial frame by Einstein's synchronized clocks in it.

No qualification about clocks or synchronization is needed.

 

[cianfa72]

No qualification about clocks or synchronization is needed.

This because we are looking at the coincidence of events (whether they are the same spacetime point or not).

 

[Sagittarius A-Star]

Ah ok, one can analyze it from the point of view of the inertial frame where the second postulate holds.

You can also analyze it in a rotating frame. The circumference with respect to this frame shall be called $U'$.

You can define an inertial reference frame with only one $x'$ axis in the range $-U'/2 < x' < +U'/2$, curled around the rim of the circular disk and rotating with it, and one $t'$ axis. This happens all in the same potential of the pseudo-gravitation caused by the centrifugal force.

But the standard Lorentz transformation to/from the non-rotating inertial frame is only permitted, if the coordinate time $t'$ is define by an Einstein-synchronization along the rim of the disk. That means you need 2 different clocks at the locations $x'\approx-U'/2$ and $x'\approx+U'/2$, even if the locations almost coincide. The Sagnac-effect uses only one clock, with measures as time-difference 2x the term for "relativity of simultaneity" in the LT (independent of the signal-velocity in an optical fiber).

Source:
http://www.physicsinsights.org/sagnac_1.html