Posted by: T. Boyd | September 21, 2009

Relativity and Einstein’s Clocks

He has made everything beautiful in its time. Also, he has put eternity into man’s heart. (Ecclesiastes 3:11)

It is time for another lively discussion with my bride about physics and astronomy.

She declares, “I read today that the atomic clock in Denver gains time over its sister clock in Europe.  And that gravity is responsible.   See, I told you no one really knows what time it really is!”

“Oh, that’s right,” I reply.  “Before we got married every clock in your house was set to different times, all fast, so you could get somewhere early or at least not too late, right?”

“Yes, it worked very well, and then I married you, and I have been late ever since,”  she answered.  “You went around the house and set every clock and watch so that they read the exact time to the nearest second.  It was crazy!”

I patiently added, “But I left your wrist watch alone, so you could have your own standard time. Anyway,  do you know why gravity affects a clock?”

“No,”  she replied cautiously, not sure she wanted to endure one of my long-winded expositions.  “But, go ahead.”

“Well,  Einstein said to think of a rocket ship that is accelerating in gravity-free space with the same acceleration that earth’s gravity would cause a body to fall.  All experiments would give the same result on the moving rocket or on a rocket at rest on the earth.”

I took a sip of my tea and continued, “Now, suppose that you are on the rocket ship that is accelerating, lying on the floor looking ‘upward’ toward the ceiling.  If there were a clock on the ceiling, sending a flash of light every second toward the floor, the flashes of light would be seen by you as arriving at a faster rate than one each second.  This is because you are accelerating upward.  To you, the ceiling clock would be running faster than one on the floor.”

“Wait a minute (no pun intended),” she interrupted.  “The clock would also be accelerating, would it not?  And so the effect should cancel.”

“No, ” I argued, “the propagation of light is independent of the motion of the source (the clock).  So the blips of light would arrive somewhat bunched together in time.”

“Now, because of the equivalence between an accelerating rocket ship, and one resting with the equivalent gravity on the earth, a clock on the ceiling of the rocket ship, would be running faster than one on the floor by the same amount in both situations,” I continued.

I concluded, triumphantly, “Therefore, because Denver, the mile-high city, is further away from the center of the earth than the city in Europe where the other clock is, the signals from the higher altitude clock will be received at a rate slightly faster than the one at the lower altitude!”

Unruffled by my logic, she smiles, “Precisely.  So, in other words, no one really knows what time it is.”


  1. Now, I still have this question about GR and clocks. By the equivalence between an accelerating laboratory (rocket) and one sitting at rest in a gravitational field, I would conclude the upper clock in the rocket at rest would run faster just because it is “uphill” in the gravitational field, not because the upper clock is in weaker gravity.

    I would like this to be confirmed by someone. If you don’t want to submit a comment, please write me at


  2. From my friend Russ via e-mail.

    You may like this article. The first of two paragraphs am cutting and pasting has the GPS clock offset numbers. I do not like the description of how they do the offset much but thats just me. In the second paragraph, he talks about accumulated difference between GPS and ground clocks. This, I think, at best (if he is correct on the point) is only theoretical because the GPS clocks are corrected daily by ground controllers.


    “For GPS satellites, GR predicts that the atomic clocks at GPS orbital altitudes will tick faster by about 45,900 ns/day because they are in a weaker gravitational field than atomic clocks on Earth’s surface. Special Relativity (SR) predicts that atomic clocks moving at GPS orbital speeds will tick slower by about 7,200 ns/day than stationary ground clocks. Rather than have clocks with such large rate differences, the satellite clocks are reset in rate before launch to compensate for these predicted effects. In practice, simply changing the international definition of the number of atomic transitions that constitute a one-second interval accomplishes this goal. Therefore, we observe the clocks running at their offset rates before launch. Then we observe the clocks running after launch and compare their rates with the predictions of relativity, both GR and SR combined. If the predictions are right, we should see the clocks run again at nearly the same rates as ground clocks, despite using an offset definition for the length of one second.

    We note that this post-launch rate comparison is independent of frame or observer considerations. Since the ground tracks repeat day after day, the distance from satellite to ground remains essentially unchanged. Yet, any rate difference between satellite and ground clocks continues to build a larger and larger time reading difference as the days go by. Therefore, no confusion can arise due to the satellite clock being located some distance away from the ground clock when we compare their time readings. One only needs to wait long enough and the time difference due to a rate discrepancy will eventually exceed any imaginable error source or ambiguity in such comparisons.”


    • Very good, Russ! I needed those numbers to compare the effects of GR and SR (general and special relativity, respectively). Another friend, John, argued that a clock in Denver would run slower than one at sea level because it was moving faster – because its angular velocity around the center of the earth would be greater (since it is further from the center).

      And I knew that should cause it to slow down by SR. But I also knew from the data that it ran faster, explained by GR – the weaker gravity. So, I figured, the GR effect must be greater than the SR effect. And sure enough, the numbers you give clear up how large the two effects are, and why GR “wins.”

      Thanks again.


  3. I have a question. Using this “gedanken”experiment, it seems to me that the ceiling clock would run faster whether or not the gravitational force were weaker. In other words, if the gravitational field were uniform, would not the “upper” clock run faster than the “lower” one?

    I throw out this question to those who know more about general relativity than I do.


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