Can Rotating Clocks on a Rim Be Synchronized Using Simultaneity Principles?

[ghwellsjr]

Can you describe an inertial reference frame in which the scenario of the three trains and finish line is not a tie, using Einstein's convention? I agree that there is no point going on with the other examples if I do not understand this, and I appreciate your taking the time to help me with it.

Yes, an IRF moving along the finish line.

 

[PeterDonis]

I think you still don't understand that the tie issue is not something that is intrinsic to nature. It is something that is determined by human definition (or convention).

Exactly.

 

[JVNY]

Exactly.

Agreed. I am coming from having only studied inertial motion, and only knowing the basics of the Einstein convention. So this is an area of not knowing what I don't know, and I appreciate all your patience.

 

[WannabeNewton]

Agreed. I am coming from having only studied inertial motion, and only knowing the basics of the Einstein convention.

If I recall from previous discussions, you have access to MTW right? Read chapter 6 and work through all the problems in said chapter; it's a brilliant chapter.

 

[JVNY]

If I recall from previous discussions, you have access to MTW right? Read chapter 6 and work through all the problems in said chapter; it's a brilliant chapter.

Excellent, thanks.

 

[JVNY]

Yes, an IRF moving along the finish line.

Yes, agreed; I was not thinking broadly enough (I was just focused on frames in inertial motion along the same axis). L-15 may be the wrong citation for what I was proposing. Here is a good excerpt from Spacetime Physics, page 63, on both the line of simultaneity that I described (perpendicular to the direction of relative motion) and the IRF in the motion that you describe (in which the train finishing events are not simultaneous):

Only in the special case of two or more events that occur at the same point (or in a plane perpendicular to the line of relative motion at that point -- see Section 3.6) does simultaneity in the laboratory frame mean simultaneity in the rocket frame. When the events occur at different locations along the direction of relative motion, they cannot be simultaneous in both frames.​

Section 3.6 goes into greater detail, showing that in the plane transverse to relative motion, (a) events simultaneous in one frame are simultaneous in the other, and (b) there is no length contraction; both of these results are required by symmetry (on pages 66-67).

 

[JVNY]

. . . Next, a rim observer can locate the 180 degree point on the opposite end of the diameter as follows. First, place a two-way mirror anywhere on the rim. Simultaneously send co- and counter-rotating signals from the clock. The signals will reflect off of the mirror and return (in opposite rotations) to the clock. Record the order that the signals arrive back at the clock. Repeat the process, moving the mirror until the two signals arrive back at the clock simultaneously. The mirror is now located at the opposite end of the diameter to the clock, half the distance of a one-way trip around the rim. . .

. . . This won't work if the ring is rotating. The degree marker at which a mounted two-way mirror would reflect the counter-propagating light beams such that they arrive back at the clock (placed at the 0 degree marker) simultaneously would not read what we normally think of as 180 degrees i.e. as the degree marker diametrically opposed to the 0 degree marker. If we in fact placed the two-way mirror at what we normally think of as the 180 degree marker then the prograde beam would arrive at the clock before the retrograde beam. Using your method we would in fact place the two-way mirror at a degree marker that is shifted, from what we normally think of as the 180 degree marker, in the direction favoring the retrograde beam in order to compensate for the Sagnac time delay (which is what results in the associated phase shift). So your method would only work if the ring is non-rotating. If you really wanted to place the two-way mirror at the "normal" 180 degree mark then you would need to take the Sagnac time delay into account when recording the arrival times of the counter-propagating signals, upon reflection, as opposed to looking for simultaneous arrival times, upon reflection. . .

I have researched this further and found that the method should properly locate the 180 degree mark. The Sagnac effect depends on the area enclosed by the signal path. Where the area is zero, there is no Sagnac effect. In particular, when a signal travels outbound in one direction around the rotating rim, reflects, and then retraces its path in the opposite direction, the area is zero and the Sagnac effect is zero. Ashby and Allan describe the underlying theory in "Practical implications of relativity for a global coordinate time scale," available at: http://tf.nist.gov/timefreq/general/pdf/133.pdf.

In particular, on page 657 they describe an experiment by Besson in which the portable clock turned around and retraced its path, creating a zero area, and no effect from Earth rotation (see first attachment).


On pages 658-59 they describe an experiment taking a portable clock ("PC") from the US Naval Observatory west to the National Bureau of Standards, then back east again, retracing its path, and note that the Sagnac correction cancels out (-9.6 ns effect westward offset by +9.6 ns effect eastward) (see second attachment, from page 659).

Therefore two signals sent simultaneously in opposite directions from 0 degrees on the rotating disk that reflect off of a mirror at 180 degrees on the disk and retrace their paths back should arrive at 0 degrees simultaneously; by retracing its path, each signal covers a zero area and the Sagnac effects for each from the outbound and retrace legs perfectly offset each other.

 

[yuiop]

This won't work if the ring is rotating. The degree marker at which a mounted two-way mirror would reflect the counter-propagating light beams such that they arrive back at the clock (placed at the 0 degree marker) simultaneously would not read what we normally think of as 180 degrees i.e. as the degree marker diametrically opposed to the 0 degree marker. If we in fact placed the two-way mirror at what we normally think of as the 180 degree marker then the prograde beam would arrive at the clock before the retrograde beam.

I am afraid I have to agree with JVNY here and disagree with your conclusion. If we have a non rotating ring with a mirror fixed at the 180 degree mark, such that light signals sent out in opposite direction from the 0 degree mark, return simultaneously to the the 0 degree mark after reflection, then when the ring is spun up, the signals will return simultaneously to the 0 degree mark, whatever the angular velocity of the ring. Basically, this is due to the fact that it makes no difference whether the light follows the prograde path followed by the retrograde path, or vice versa. The round trip time is the same.

It is however true, that the retrograde beam would arrive at the double sided mirror at 180 degree mark before the prograde beam, but that is a separate issue.

Here is slightly different scenario that might clarify things. Imagine we have a double sided mirror at the 0 degree mark and a double sided source also at the 0 degree mark. If signals are simultaneously sent in opposite directions around the ring and allowed to travel all the way around and reflect off the mirrors at the 0 degree mark they will both arrive back at the 0 degree mark simultaneously, with each signal having completed 2 trips around the ring. One signal will have completed a prograde circumnavigation followed by a retrograde circumnavigation, while the other will done the same but in reverse order, but the total times will be the same.

 

[WannabeNewton]

Therefore two signals sent simultaneously in opposite directions from 0 degrees on the rotating disk that reflect off of a mirror at 180 degrees on the disk and retrace their paths back should arrive at 0 degrees simultaneously; by retracing its path, each signal covers a zero area and the Sagnac effects for each from the outbound and retrace legs perfectly offset each other.

It is however true, that the retrograde beam would arrive at the double sided mirror at 180 degree mark before the prograde beam, but that is a separate issue.

You guys are right, I was confusing the reflection events at the two-way mirror with the entire round-trip circuits.

Sorry JVNY!

 

[JVNY]

No problem, and thanks to both of you for picking this back up.

[Note: I have edited this post from last night to change the discussion of the clock initialization method]. The article also gives a method for initializing the clocks by way of signals around the rim. I used a different method:

. . . Finally, the master clock can synchronize a dependent clock located on the mirror as follows. The master clock sends a signal in the co-rotating direction that instructs the dependent clock to set itself to a specified time equal to the proper time of the master clock when it sends the signal plus half the time that a signal takes to make a complete co-rotating one-way trip (previously determined by the timed one way co-rotating trip at the beginning of this process). . .

The article, although referring to sending signals around earth, says that it is possible to synchronize clocks by a radar method. I believe that it says that the master clock signals the dependent clock to set itself at (a) the master clock's time when it sends the signal, plus (b) half the master clock's proper time that a signal takes to go out to the dependent clock and back again, plus or minus (c) the appropriate amount of the Sagnac effect for the distance from the master to dependent clock (plus or minus depending upon the signal direction). I think that you get the same result for the dependent clock at 180 degrees following the method in the quotation above. In the original of this post from last night I suggested that you needed to use instead half of the average of one complete co-rotating and one complete counter-rotating trip (since using the average of the two cancels out the Sagnac effect, as yuiop points out). But I think in hindsight that using half of the average reintroduces a Sagnac error. Using half of the master clock's proper time that a single complete circuit takes in the same direction should work as long as the signal to the dependent clock is sent in the same direction.

So it is possible to synchronize the clocks using a convention consisting only of signals sent around the rim. There is no need to send signals from the axis to the rim or otherwise have any signal travel off of the rim. The paper analyzes synchronization of clocks around the Earth (not a rim), so I am analogizing to the rim.

Interestingly, the paper states that using this convention on Earth creates an Earth centered inertial coordinate system. So by analogy one should probably conclude that using the convention on the rim creates an axis centered inertial coordinate system -- even though no signal comes from the axis, rather everything takes place on the rim.