Measuring the Rate of a Moving Clock

“Moving clocks run slow” is a conclusion drawn from any attempt to evaluate the rate of a clock in a moving reference frame using Einstein’s Theory of Special Relativity. All books on Special Relativity give examples of how the rate of a moving clock can and should be measured. After reading several, I have concluded that they are all derived from the same source and all contain the same confusing wording. So, at the risk of adding still another, I am offering my version here. First I will describe a process for measuring the rate of a moving clock using Newtonian analysis. Then I will extend it to Special Relativity.

Suppose I am standing by some railroad tracks while a train goes by. On the train is a conductor with a watch whose rate I wish to measure. In the language of Physics, I am in one reference frame (the ground), while the conductor is in another (the train). Even Newtonian Physics has a concept of relativity. I consider the conductor and his watch to be moving with respect to me while the conductor considers me and my watch to be moving with respect to him.

Here is how I might measure the rate of the conductor’s watch. When he passes me I take a look at his watch and compare it to my own. Whether or not the times match is not important. I am after the rate of the conductor’s watch. But one reading is not enough to allow me to evaluate that rate. And now the conductor and his watch have passed. But I planned ahead. I stationed an assistant some ways down the tracks with a watch of her own. When the conductor passes her, she gets a second reading. Now I can evaluate the conductor’s watch. For instance, lets say I recorded the conductor’s watch as reading one minute after noon when my watch read exactly noon. My assistant then recorded the conductor’s watch as reading four and a half minutes after noon when her watch read four minutes after noon. Four minutes elapsed on our watches while only three and a half minutes elapsed on the conductor’s watch. I conclude that the conductor’s watch is running slow.

This is a good spot to emphasize the importance of synchronizing. When measuring the rate of the conductor’s watch I need to synchronize my and my assistant’s watches. I don’t need to synchronize with the conductor. Lets say that I failed to synchronize my and my assistant’s watches. Lets say that my assistant’s watch reads a full minute ahead of mine all the time. For the above example, that would mean that three minutes actually elapsed between our readings and I should have concluded that the conductor’s watch was running fast. Synchronizing watches using Newtonian physics is pretty intuitive. Bring them together. Set them to read the same time. Separate them again. In relativity there is a very specific methodology that is beyond the scope of this discussion.

You might be tempted to say that the conductor, confident in the accuracy of his watch, can conclude that his watch is not running slow, but that our watches are running fast. However, doing so would require the conductor to accept that I have synchronized my and my assistant’s watches properly. The arrangement described does not give the conductor the ability to make a direct and independent measurement of the rate of either my or my assistant’s watch. Of course, the conductor could recruit one of the passengers on the train to get a second reading from my or my assistant’s watch. In that case the conductor would be mirroring my measurement method.

This example contains the key elements required to evaluate the rate of a clock in a moving reference frame using either Newtonian Physics or Special Relativity.

· The evaluator must take two readings off the same moving clock.

· To do that the evaluator needs two clocks in his own reference frame that are separated by some distance and synchronized.

· The “evaluatee” cannot draw an inverse conclusion about the evaluator’s clock rate, at least not using readings from his own single clock.

Now many people who have had the patience to read to this point may be thinking that this discussion is like beating a dead horse. After all, everything I have written so far is intuitive. But that is the reason I wrote it. For Special Relativity requires that some things that are intuitive to us must not be true. Which intuitive parts are not true according to Special Relativity? Well, in the discussion above, two parts. You can guess by the first sentence of this discussion that Special Relativity predicts I will measure the rate of the conductor’s watch, even if it is in perfect operating order, to be running slow. But that non-intuitive reality cannot stand by itself. How can I conclude that the conductor’s watch is running slow and the conductor also conclude that my watch is running slow? That won’t happen unless some other non-intuitive thing happens. The other non-intuitive thing is that the conductor believes my and my assistant’s watches are not synchronized. So he is not surprised at my conclusion when he is certain that his watch is running fine.

If you have followed what I have written so far, then there is only this summary to go to understand that two non-intuitive concepts about time can combine together to form a logically consistent alternative reality. Let’s say that the conductor and I are both believers in Special Relativity. I need not change anything I have done to measure the rate of the conductor’s watch. It’s just that I will expect the results I have described from a perfectly good watch. The conductor need not change his actions either. He can pick off the time from my and my assistant’s watches just as before. But instead of concluding that our watches are running fast, he knows that our watches are really running slow. As far as he is concerned, the contradictory readings are caused by my and my assistant’s watches being out of synch. He knows this even after I assure him that I synchronized the watches. For according to Special Relativity, what I consider synchronized in my reference frame is not synchronized from his reference frame.

What would happen if the conductor actually did recruit a passenger and tried to measure the rate of my watch using my method? The calculations would be a mirror of mine. The conductor would compare clock readings and conclude that my watch was running slow and I would conclude that the conductor and the passenger had not synchronized their watches.