Light Clock Thought Experiment: Parallel Moving Mirrors

[yogi]

I have raised this issue in a different manner before - but never got a satisfactory analysis. In the traditional parallel moving mirrors (separated by distance d) light clock thought experiment, the photon is considered to travel back and forth over the same path in the moving train (both emission and reception occur at the same place in the train) and the external observer in the Lab frame sees the light path as a zig-zag sawtooth which prompts him to conclude that it requires more time for the over and back event as measured in the lab. But if the photon enters from the lab frame perpendicular to the parallel moving mirrors, (through a hole in one mirror) the lab frame will see the over and back distance as d and the observer in the train will see the sawtooth. Since both measure the velocity of light as c, more actual time will accumulate on the trainman's clock during one sawtooth than on the lab clock. On the other hand, if the photon initiated from the lab frame enters at an angle such that it appears to the trainman as though the photon travels over and back along the same path inside the train, the trainman's clock will record less time, and the lab clock will record more time during one sawtooth.

Any comments

 

[LURCH]

My only comment would be that your evaluation of the nature of this thoght experiment is right on the money. I would add that in the case of the second scenario you propose, wherein the photon enters the train from the lab in a direction that appears vertical from the lab's frame of reference, it will not strike the other mirror. For example, if the photon enters from the top and travels "strait down", from the lab-bound observer's point of view, the lower mirror will have already passed by when the photon arrives. The observer inside the train will see the photon enter and travel from the upper mirror at an angle towards the rear of the train, missing the lower mirror.

Of course, this means that the observer in the train will measure a longer amount of time for the photon to travel (from the upper mirror to the floor) than the observer in the lab.

 

[yogi]

Lurch - If the measurements are as postulated - then it is a misleading no-no to state, as is commonly done, "that each observer sees the other guys clock as running slow" - The clock that accumulates the most time will depend upon the experiment being performed and how the spacetime interval involved in that experiment transforms to the other system.

PS - I had assumed the mirrors had a long length so the photon would not miss the bottom mirror during the time it transversed the distance d.

 

[LURCH]

Originally posted by yogi
Lurch - If the measurements are as postulated - then it is a misleading no-no to state, as is commonly done, "that each observer sees the other guys clock as running slow" - The clock that accumulates the most time will depend upon the experiment being performed and how the spacetime interval involved in that experiment transforms to the other system.

PS - I had assumed the mirrors had a long length so the photon would not miss the bottom mirror during the time it transversed the distance d.

Well, it could be considered a "no-no", but not a big one. It simply assumes a definition of light-clock in which the clock is stationary relative to at least one observer.

For instance, in this situation:

...if the photon enters from the lab frame perpendicular to the parallel moving mirrors, (through a hole in one mirror) the lab frame will see the over and back distance as d and the observer in the train will see the sawtooth. Since both measure the velocity of light as c, more actual time will accumulate on the trainman's clock during one sawtooth than on the lab clock.

you have constructed a light-clock that is stationary from the frame of reference of the observer in the lab. So the situation becomes opposite to the one you described at the beginning of your post, and all the same observations hold true, but in reverse. The photon's points of emission and reflection are now stationary from the lab's frame of reference, and the measurements will be the same as if the clock were outside the train, in the lab. Whether the observer in the lab is watching a light-clock that is in his hand or inside the train makes no difference, so long as the points of emission and reflection are stationary relative to him.

So the assumed definition of "light-clock" includes the stipulation that the photon's lateral progress is zero, as measured by some observer. If the photon's lateral progress is not zero, then you do not have a light-clock, but rather a light-source; a point from which photons are radiated away.

(Oh yea, I get it now about the mirrors in the train being long)

 

[yogi]

So - having gotten your attention - we consider a very long train (5 light years long as measured in the lab system) that travels at 0.707c toward a planet Alpha 5 light years distance from our Earth based lab. The photon in our Imaginary light clock enters perpendicular to a hole near the beginning of the train (as proposed above) so that it appears as a sawtooth inside the long train as it lumbers through space on its journey. In the lab the propertime recorded by each over and back reflection will be 2d/c but inside the train the trainman records each over and back reflection as
2(1.4)(2d/c). When the front of the train reaches Alpha, The rear will just be passing our lab and the last sawtooth is completed - By this scenaro the trainman will have recorded more time than the labman. Something is obviously wrong - but what?

 

[yogi]

Opps - correction - the trainman watching the sawtooth will record the cycle time as 4d/c - and the time recorded in the lab for the trip will be 1.4 times 5 light years. The number of cycles will be (1.4 x 5 light years)/(2d/c). If both the labman and trainman record the same number of cycles, the trainman's clock will accumulate more time during the trip - but the measurement of the total trip time on the train should always be less than the proper time measured in the lab!

A more difficult problem is how to build a train track from Earth to Alpha.

 

[Creator]

Originally posted by yogi
In the lab the propertime recorded by each over and back reflection will be 2d/c but inside the train the trainman records each over and back reflection as 2(1.4)(2d/c);.. (corrected to 4d/c) ...By this scenaro the trainman will have recorded more time than the labman. Something is obviously wrong - but what?

Yogi;
'Time dilation' is NOT a comparison of time rates as measured from EACH person's frame as you indicate above.
Rather, it compares the clock rates in each frame as measured BY ONE person's frame

See bold print in your previous statement. Correcting that idea may help you resolve your difficulty.

Creator

 

[yogi]

Creator - I would take issue with your statement -- in the lab frame (earth) the lapsed time is on average approximately 10 usec for the high speed u Meson - but for the clock in the muon's frame, the proper time for the decay process is on average 2 usec. The muon only lives 2 usec whether it is moving or not - but it gets the benefit of covering more space in that 2 usec interval when it moves relative to the lab frame. If the muon could travel at c like a photon, it would circumscribe the universe in that 2 second period as we and all else in the lab frame expired during the eons.