# Time dilation, inertial frames and SR

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After reading several articles on time dilation and inertial frames I still have a question. The "Twin Paradox" at http://www.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm

Made it sound like the acceleration and deceleration is what caused time dilation. That clocks in two inertial frames would stay synchronized as long as they were at rest in their frame, even if they were separating (without power) from each other.

I must have missed something because "The Universe" Program pointed out that GPS satellites had to adjust their clocks because of their orbit speed using SR. Wouldn't the GPS unit on Earth and the satellites both be at rest in their own inertial frame? After acceleration is stopped it seems there would be no way for the clocks to know which one is fast and which one is slow.

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Identical, ideal clocks in two inertial frames will run at different rates, and accrue time (phase) differences. This difference, however, will be symmetrical — each frame will observer the other one as running slow. An acceleration breaks this symmetry, and one must accelerate in order to bring the clocks into the same frame. The accelerated clock is the one that will end up running at the "wrong" rate.

GPS clocks aren't in an inertial frame, even when adopting an earth-centered inertial frame of reference, where we assume an observer on the earth is at rest. The satellites are in orbit, and thus accelerating (centripetal acceleration). They actually run fast, because the gravitational dilation has a greater magnitude than the kinematic dilation, and they are in a smaller potential.

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Identical, ideal clocks in two inertial frames will run at different rates, and accrue time (phase) differences. This difference, however, will be symmetrical — each frame will observer the other one as running slow. An acceleration breaks this symmetry, and one must accelerate in order to bring the clocks into the same frame. The accelerated clock is the one that will end up running at the "wrong" rate.

Maybe my confusion is because of my understanding of inertial frames. For kinematic dilation, do you have to be experiencing a G-force from acceleration to be in a different frame? Or using GR, be experiencing more or less then 1 G-force from matter to be in a different frame? If this is not the case please give me an example or two of different inertial frames. Would a high speed propeller hub be in a different frame then the tip while rpm increased?

GPS clocks aren't in an inertial frame, even when adopting an earth-centered inertial frame of reference, where we assume an observer on the earth is at rest. The satellites are in orbit, and thus accelerating (centripetal acceleration). They actually run fast, because the gravitational dilation has a greater magnitude than the kinematic dilation, and they are in a smaller potential.

So, "GPS clocks aren't in an inertial frame". Do you mean different then Earth's frame? Is time dilation symmetrical (only considering kinematic dilation) between the satellite and earth? I have a few more questions but I will be able to word them better if my questions so far are answered.

Thank you for the reply. I did quite a bit of research before asking these questions but for some reason (especially SR time dilation) the clear picture eludes me.

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In SR inertial frames have no acceleration, i.e. Newton's laws describe the motion of the object. Any objects moving with respect to each other are in different inertial frames. We typically ignore gravity in looking at the twins paradox, so that the only acceleration is the twin that has to turn around to return to earth. That changes the inertial frame of the twin. In general relativity one finds that an object falling freely in a gravitational field is also in an inertial frame.

A propeller is not in an inertial frame because it is accelerating, owing to the rotation.

The earth and satellites are not in inertial frames, again owing to rotation. However, we "fake" an inertial frame (much like we do when we refer to the Coriolis force, which is an artifact of the rotating earth and our desire to treat the earth as an inertial frame). To an observer away from (and not rotating with) the earth, if one moves east at some speed, one is really moving faster than if one moves west at that same speed, so when using the earth's frame we add a Sagnac term, adding or subtracting time based on the motion in the east/west direction. One complete E/W circumnavigation will add or subtract 207 ns. Once we account for the Sagnac effect, our reference frame looks like an inertial frame, at some gravitational potential.

Clocks in orbit are moving relative to the earth (unless geostationary) and have a different gravitational potential, so both of these effects must be accounted for when comparing them to earth clocks.

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In SR inertial frames have no acceleration, i.e. Newton's laws describe the motion of the object. Any objects moving with respect to each other are in different inertial frames. ..In general relativity one finds that an object falling freely in a gravitational field is also in an inertial frame...The earth and satellites are not in inertial frames, again owing to rotation. However, we "fake" an inertial frame...

Excellent, excellent, excellent, I think. This is where the definition of inertial frames should start. Please bear with me, I think I see a light in the distance and it appears to be blue shifted.

I am in a inertial frame when in my car at constant speed and can accelerate or decelerate to a different frame (relative to any other object at a fixed location or moving at a constant speed).

The Earth is in a "fake" frame because of rotation but "the satellites are in orbit, and thus accelerating (centripetal acceleration)" and not a frame?

Not sure I see why the satellite is considered accelerating though, unless it has to do with the Earth rotation.

1. If a space ship takes a month to accelerate to .99c would it be wrong to look at it as a variable frame during acceleration? An entire trip could be made without a constant speed. How do we deal with time dilation while not at a constant speed?

2. Twin "a" takes a trip in space and comes back young. Twin "b" doesn't like being older then "a" so "b" takes a trip and comes back the same age as "a". Right?

3. I am still confused on what determines which twin will age faster unless it all happens during acceleration. During the time of constant speeds relative to each other, their clocks should be symmetrical, unless there is a space ether and not just a metric.

A propeller is not in an inertial frame because it is accelerating, owing to the rotation.

4. So at a constant rpm would the hub and tip experience time dilation because of the different speeds?

Edit-

5. Does the bolded text at the top contradict each other?

Edited by NowThatWeKnow
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Excellent, excellent, excellent, I think. This is where the definition of inertial frames should start. Please bear with me, I think I see a light in the distance and it appears to be blue shifted.

I am in a inertial frame when in my car at constant speed and can accelerate or decelerate to a different frame (relative to any other object at a fixed location or moving at a constant speed).

The Earth is in a "fake" frame because of rotation but "the satellites are in orbit, and thus accelerating (centripetal acceleration)" and not a frame?

Not sure I see why the satellite is considered accelerating though, unless it has to do with the Earth rotation.

1. If a space ship takes a month to accelerate to .99c would it be wrong to look at it as a variable frame during acceleration? An entire trip could be made without a constant speed. How do we deal with time dilation while not at a constant speed?

2. Twin "a" takes a trip in space and comes back young. Twin "b" doesn't like being older then "a" so "b" takes a trip and comes back the same age as "a". Right?

3. I am still confused on what determines which twin will age faster unless it all happens during acceleration. During the time of constant speeds relative to each other, their clocks should be symmetrical, unless there is a space ether and not just a metric.

4. So at a constant rpm would the hub and tip experience time dilation because of the different speeds?

Edit-

5. Does the bolded text at the top contradict each other?

The satellites experience a centripetal acceleration, because they are in a rotating frame. Even though the speed stays the same the velocity changes (direction of motion) so it's still an accelerating frame.

1. You can deal with accelerations in SR; you can take the time dilation equation and use the instantaneous speed at any time t, which you know from your acceleration, and integrate to get the accrued time dilation. (You are essentially creating an infinite number of inertial frames using this method)

2. Yes, if the trips are identical.

3. The breaking of the symmetry does all happen during the acceleration. But it's also independent of the amount of acceleration (i.e. the turnaround can be instantaneous or slow, and that won't affect the basic answer — the amount of dilation depends on v and the length of the trip)

4. Yes. This is quite similar to earth and satellites.

5. No. Constant velocity means no acceleration.

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The satellites experience a centripetal acceleration, because they are in a rotating frame. Even though the speed stays the same the velocity changes (direction of motion) so it's still an accelerating frame.

3. The breaking of the symmetry does all happen during the acceleration. But it's also independent of the amount of acceleration (i.e. the turnaround can be instantaneous or slow, and that won't affect the basic answer — the amount of dilation depends on v and the length of the trip)

I guess the stars in the Milky Way could be considered a rotating frame but on a scale that would have little impact on local motion.

3. I understand what you are saying but it does seem to create a paradoxical situation. Distant galaxies in opposite directions are separating from each other at speeds greater than light so has time stopped or is it running backwards in the other galaxy if they could see each other. An expanding ether of space independent of local motion would make sense to me.

You have answered some basic questions and I feel further reading will make much more sense to me. Thank you very much.

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I guess the stars in the Milky Way could be considered a rotating frame but on a scale that would have little impact on local motion.

Indeed. One needs to measure the amount of rotation to see the magnitude of the effect, but since we're all rotating like this it would only come into play for someone with different orbital parameters. Locally, assuming linear motion won't have much effect, and then we can assume we are at rest.

3. I understand what you are saying but it does seem to create a paradoxical situation. Distant galaxies in opposite directions are separating from each other at speeds greater than light so has time stopped or is it running backwards in the other galaxy if they could see each other. An expanding ether of space independent of local motion would make sense to me.

The expansion is an effect of General Relativity, and basically means that distant objects are in a different frame of reference than our own, and I think I'm right in saying that these are not simply different inertial frames. The space is expanding, (though calling it an ether can be problematic because of historical reasons) Locally, you have inertial frames and nothing exceeds c. But the remote frame is non-inertial, as viewed by us, and you lose the restriction on c unless you account for the expansion of space.

Or perhaps it's better to say that with regard to reference frames, the expansion of space is a non-inertial effect.

(If Martin happens along he may find some fault with my generalizations. Much of GR is outside my area of expertise)

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This gives me an idea for another thread. Maybe a catchy name will get more involved.

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