Synchronization of Earth clocks

[psmitty]

If we have two clocks, one stationary at surface of Earth, and the other one very slowly moved at the surface of Earth, in the direction of Earth rotation, the clock that made the trip should be running ahead of the same stationary clock once it makes a complete circle and stops where it started from (if I got it right).

In an inertial frame, they should remain synchronized at low relative speeds, but on Earth, being a non-inertial frame, no matter how slowly you move a clock, it goes out of sync when you move it.

Also, if you move it half way in one direction, and return it back, it will get back to sync with the stationary one.

Should this be considered as time dilation, some sort of a or change in simultaneity? I mean, for a slowly moving clock, this difference in time seems to be a function of Earth's rotation speed and clock's traveled distance along the surface.

So how do I get this function for time difference?

 

[kamenjar]

I'm new to relativity, but this is my understanding.

It is the "time" traveled in an accelerated frame that causes times to differ, not the actual movement direction.

It is only the relative delta of time spent at larger speeds during movement of two clocks with respect to each other that counts

If you accelerate it in order take it around the earth, then decelerate it to stop it, The clock will be slower. If you never decelerated and just did a checkpoint at that time when the clock passed the time difference would be slightly greater because the clock was moving longer at high speed.

If you take off on a round trip to the nearest star and get yourself to relativistic speeds by spending a lot of time accelerating or a lot of energy to accelerate fast, then come back to the same place where you left that clock, the time difference would be much larger. If you wanted to stop over and visit that star (decelerate). The time difference would still be large, but not as large as in the previous case because the subject would spend less time near the speed of light.

In all four cases of the five clocks may have returned at the same time, but all clocks would show different times.

 

[JesseM]

Should this be considered as time dilation, some sort of a or change in simultaneity? I mean, for a slowly moving clock, this difference in time seems to be a function of Earth's rotation speed and clock's traveled distance along the surface.

So how do I get this function for time difference?

Are you looking for a full GR analysis or do you just want to treat the Earth as a rotating massless sphere in flat spacetime? If the latter you might look at analyses of the Sagnac effect which often use rotating frames...pages 102-106 of this book might also be helpful (gives the line element ds^2 in a rotating frame, which can be used to calculate elapsed time along any parametrized curve)...likewise p. 84 of this book gives a "time dilation" formula for $$d\tau /dt$$ in a rotating frame (rate at which clock time $$\tau$$ is increasing relative to coordinate time t)

 

[bcrowell]

Are you looking for a full GR analysis or do you just want to treat the Earth as a rotating massless sphere in flat spacetime?

I don't think the GR analysis gives a different result than the SR analysis, because the clock is moving along an equipotential. It's just the Sagnac effect.

 

[Passionflower]

Should this be considered as time dilation, some sort of a or change in simultaneity? I mean, for a slowly moving clock, this difference in time seems to be a function of Earth's rotation speed and clock's traveled distance along the surface.

So how do I get this function for time difference?

Assuming you are only interested in the special relativity approach we can ignore gravity completely.

Basically you want to compare the elapsed times of two accelerating observers making a loop from event A to B.

The simplest way to do this is to consider a third observer who is inertial, for instance an observer at the center of the Earth. He sees two clocks with two different accelerations going from event A to event B. Then it is simply a matter of calculation to verify that the clock with the least acceleration will age most. Which is the clock that goes against the rotation, because this will reduce the total acceleration.