Why does time dilation only affect GPS satellites in one direction?

[cfrogue]

Can anyone answer this?
GPS satellites are preprogrammed before launch due to relativistic effects mainly time dilation and gravity.

Yet, once in orbit, the satellite should view the Earth frame as "moving" and thus apply time dilation to the Earth frame meaning the Earth clocks should beat slower. This does not happen.

Questions, why does time dilation only apply one way with GPS satellites.


The effects are emphasized for several different orbit radii of particular interest. For a low Earth orbiter such as the Space Shuttle, the velocity is so great that slowing due to time dilation is the dominant effect, while for a GPS satellite clock, the gravitational blueshift is greater. The effects cancel at . The Global Navigation Satellite System GALILEO, which is currently being designed under the auspices of the European Space Agency, will have orbital radii of approximately 30,000 km.

http://relativity.livingreviews.org/Articles/lrr-2003-1/
See chapter 5.

 

[JesseM]

Can anyone answer this?
GPS satellites are preprogrammed before launch due to relativistic effects mainly time dilation and gravity.

Yet, once in orbit, the satellite should view the Earth frame as "moving" and thus apply time dilation to the Earth frame meaning the Earth clocks should beat slower. This does not happen.

The satellites aren't programmed to calculate things with respect to a separate coordinate system where they (the satellites) are at rest, and in GR unlike in SR there wouldn't be a unique coordinate system where this was true anyway (so your statement that they should view Earth clocks as running slower isn't really correct, you could come up with a non-inertial coordinate system where a given satellite was at rest but where clocks on Earth were ticking faster). All the calculations on board the satellite are done with respect to the Earth-centered coordinate system where the satellites are in motion. Remember, there is no physical requirement that a given observer calculate things relative to a frame where they are at rest, any observer can use any coordinate system they please for making calculations.

 

[cfrogue]

The satellites aren't programmed to calculate things with respect to a separate coordinate system where they (the satellites) are at rest, and in GR unlike in SR there wouldn't be a unique coordinate system where this was true anyway (so your statement that they should view Earth clocks as running slower isn't really correct, you could come up with a non-inertial coordinate system where a given satellite was at rest but where clocks on Earth were ticking faster). All the calculations on board the satellite are done with respect to the Earth-centered coordinate system where the satellites are in motion. Remember, there is no physical requirement that a given observer calculate things relative to a frame where they are at rest, any observer can use any coordinate system they please for making calculations.

I am confused.
If you look at the diagram in the link I posted, it shows the speed relative to the Earth frame influences the clock beat rates in the satellites.

Thus, the satellite clocks are adjusted according to their orbit according to that chart.

There are two main adjustments according to the article, one for time dilation and one for gravity.

Do you agree this is correct or am I reading the article wrong.

 

[JesseM]

I am confused.
If you look at the diagram in the link I posted, it shows the speed relative to the Earth frame influences the clock beat rates in the satellites.

Thus, the satellite clocks are adjusted according to their orbit according to that chart.

There are two main adjustments according to the article, one for time dilation and one for gravity.

Do you agree this is correct or am I reading the article wrong.

You are correct, but all these statements refer to the Earth-centered coordinate system--when the satellites have a higher speed in this system, the rate of an unadjusted clock aboard the satellite will slow down relative to coordinate time in this system, so the rate has to be adjusted so they keep pace with coordinate time. No coordinate systems other than the Earth-centered one need to be used here.

 

[cfrogue]

You are correct, but all these statements refer to the Earth-centered coordinate system--when the satellites have a higher speed in this system, the rate of an unadjusted clock aboard the satellite will slow down relative to coordinate time in this system, so the rate has to be adjusted so they keep pace with coordinate time. No coordinate systems other than the Earth-centered one need to be used here.

OK, so it is correct that time dilation is a factor for the satellite clock.

Is this correct?

 

[JesseM]

OK, so it is correct that time dilation is a factor for the satellite clock.

Is this correct?

Yes.

 

[cfrogue]

Originally Posted by cfrogue
OK, so it is correct that time dilation is a factor for the satellite clock.

Is this correct?

Yes.

So, why does reciprocal time dilation not apply from the POV of the satellite since the Earth is moving relative to it and the satellite is at rest. Thus, the Earth clocks would seem to have to beat slower and thus the satellite would progressively get more out of sync over time.

 

[JesseM]

So, why does reciprocal time dilation not apply from the POV of the satellite since the Earth is moving relative to it and the satellite is at rest.

It seems like you are equating "the POV of the satellite" with "the POV of a coordinate system where the satellite is at rest". If so, this is incorrect--any observer is free to adopt any coordinate system they like to define a "point of view" for doing calculations, and in this case the satellites are programmed to calculate things from the perspective of the Earth-centered coordinate system. It might be true that if the satellite instead did calculations from the perspective of a satellite-centered coordinate system then they would conclude Earth clocks were running slower (though as I said, since we are talking about non-inertial coordinate systems this need not necessarily be true, time dilation is certainly not guaranteed to be 'reciprocal' in this way when dealing with two non-inertial systems), but in actual fact this is not the coordinate system the satellites are programmed to do their calculations in.

 

[cfrogue]

It seems like you are equating "the POV of the satellite" with "the POV of a coordinate system where the satellite is at rest". If so, this is incorrect--any observer is free to adopt any coordinate system they like to define a "point of view" for doing calculations, and in this case the satellites are programmed to calculate things from the perspective of the Earth-centered coordinate system. It might be true that if the satellite instead did calculations from the perspective of a satellite-centered coordinate system then they would conclude Earth clocks were running slower (though as I said, since we are talking about non-inertial coordinate systems this need not necessarily be true, time dilation is certainly not guaranteed to be 'reciprocal' in this way when dealing with two non-inertial systems), but in actual fact this is not the coordinate system the satellites are programmed to do their calculations in.

The article uses two terms, time dilation and gravity.

It makes the following statement.
the velocity is so great that slowing due to time dilation is the dominant effect

What do you think this means?

 

[JesseM]

The article uses two terms, time dilation and gravity.

It makes the following statement.
the velocity is so great that slowing due to time dilation is the dominant effect

What do you think this means?

I think they are talking about velocity-based time dilation vs. gravitational time dilation, the latter being a GR effect which isn't based on velocity.

 

[cfrogue]

I think they are talking about velocity-based time dilation vs. gravitational time dilation, the latter being a GR effect which isn't based on velocity.

OK, and is the velocity relative velocity between the Earth and satellite?

 

[mgb_phys]

Yes it's the relative velocity of the satelite and the ground station.
Just to add some extra complexity GPS clocks are actually deliberately slowed down - the General relativistic time going faster in lower gravity is bigger than the special relativistic time slowing down at high speed.

 

[cfrogue]

Yes it's the relative velocity of the satelite and the ground station.
Just to add some extra complexity GPS clocks are actually deliberately slowed down - the General relativistic time going faster in lower gravity is bigger than the special relativistic time slowing down at high speed.

That is what the article says. But, at some altitude they cancel. At low altitudes the relative v is the stronger influence according to the article.

So, if this is a relative v, why doesn't the Earth frame beat slower when the satellite is in orbit? It seems that the satellite would consider itself at rest and the Earth frame moving. Thus, the satellite would believe the Earth frame clocks should beat slower.

Why doesn't this happen?

 

[JesseM]

That is what the article says. But, at some altitude they cancel. At low altitudes the relative v is the stronger influence according to the article.

So, if this is a relative v, why doesn't the Earth frame beat slower when the satellite is in orbit? It seems that the satellite would consider itself at rest and the Earth frame moving. Thus, the satellite would believe the Earth frame clocks should beat slower.

Why doesn't this happen?

There is only one type of velocity used--the velocity of the satellite in the Earth-centered coordinate system. It's a "relative" velocity in the sense that you're measuring the satellite's velocity relative to this particular coordinate system, but "relative" does not imply that anyone is calculating the velocity of the Earth clocks in a satellite-centered frame. You could do this on your own if you wanted to, but it is not part of the GPS calculations, the only frame that any GPS computers are using is the Earth-centered one.

Do you understand that in relativity you are never required to use multiple frames, you can always get the answer to any physical question you're interested in using a single frame, even if the situation you are analyzing involves objects which are not at rest in that frame?

 

[pervect]

I think you'd be best off understanding the twin paradox in flat space-time first. The key problem here I think its that you are imagining that the satellite has some well-defined coordinate system that covers all of space. That is generally not true for an accelerating observer - one can construct various coordinate systems for them, but they are in general all local.

To avoid all these complexities and gain some insight, I'd again suggest that you consider the flat space-time non-rotating twin paradox, which is being beaten to death in a number of threads around here...

 

[cfrogue]

There is only one type of velocity used--the velocity of the satellite in the Earth-centered coordinate system. It's a "relative" velocity in the sense that you're measuring the satellite's velocity relative to this particular coordinate system, but "relative" does not imply that anyone is calculating the velocity of the Earth clocks in a satellite-centered frame. You could do this on your own if you wanted to, but it is not part of the GPS calculations, the only frame that any GPS computers are using is the Earth-centered one.

Do you understand that in relativity you are never required to use multiple frames, you can always get the answer to any physical question you're interested in using a single frame, even if the situation you are analyzing involves objects which are not at rest in that frame?

Yea, this is fine.

Maybe I should ask it this way.

Is the Earth moving relative to the satellite?

 

[cfrogue]

I think you'd be best off understanding the twin paradox in flat space-time first. The key problem here I think its that you are imagining that the satellite has some well-defined coordinate system that covers all of space. That is generally not true for an accelerating observer - one can construct various coordinate systems for them, but they are in general all local.

To avoid all these complexities and gain some insight, I'd again suggest that you consider the flat space-time non-rotating twin paradox, which is being beaten to death in a number of threads around here...

I am imagining the satellite is at virtual rest in its own system of coordinates. I say virtual because of the orbit.

Also I am imagining the Earth is moving relative to the satellite. I am not assuming some universal coordinate system. I don't have any idea what that would mean.

 

[JesseM]

Yea, this is fine.

Maybe I should ask it this way.

Is the Earth moving relative to the satellite?

When people talk about X moving relative to Y in SR, they really mean X is moving in the inertial rest frame of Y--there is no way to define "motion" except relative to a particular coordinate system. Since in GR there is no "standard" way to define the non-inertial rest frame of a given object, your question isn't really specific enough to have a definite answer, you need to define what type of coordinate system you want to use. There could be some non-inertial coordinate systems where the satellite was at rest and the Earth clocks were in motion, and others where both the satellite and the Earth clocks were at rest (at least for some period of time).

 

[cfrogue]

When people talk about X moving relative to Y in SR, they really mean X is moving in the inertial rest frame of Y--there is no way to define "motion" except relative to a particular coordinate system. Since in GR there is no "standard" way to define the non-inertial rest frame of a given object, your question isn't really specific enough to have a definite answer, you need to define what type of coordinate system you want to use. There could be some non-inertial coordinate systems where the satellite was at rest and the Earth clocks were in motion, and others where both the satellite and the Earth clocks were at rest (at least for some period of time).

I am ignoring all those complexities of GR and just focusing in on the relative v part as used in GPS.

Also, I would like to use the coords of the satellite since that is legal.

So, is the Earth moving relative to the satellite?

 

[JesseM]

I am ignoring all those complexities of GR and just focusing in on the relative v part as used in GPS.

But that's a v relative to a particular Earth-centered coordinate system. You can only talk about the v of anything relative to some coordinate system, velocity has no coordinate-independent meaning.

Also, I would like to use the coords of the satellite since that is legal.

But as I already told you, there is no single coordinate system that qualifies as "the" frame of the satellite. Since we are dealing with curved spacetime we can't use an inertial frame, and there are an infinite number of different possible non-inertial coordinate systems where the satellite is at rest, all of which are equally valid in GR (see the discussion in http://www.aei.mpg.de/einsteinOnline/en/spotlights/background_independence/index.html about the principle of 'diffeomorphism invariance' which says the laws of GR work in any kind of arbitrary coordinate system you can come up with).

So, is the Earth moving relative to the satellite?

In some satellite-centered coordinate systems it would be, in others it wouldn't.

 

[cfrogue]

But as I already told you, there is no single coordinate system that qualifies as "the" frame of the satellite. Since we are dealing with curved spacetime we can't use an inertial frame, and there are an infinite number of different possible non-inertial coordinate systems where the satellite is at rest, all of which are equally valid in GR (see the discussion in http://www.aei.mpg.de/einsteinOnline/en/spotlights/background_independence/index.html about the principle of 'diffeomorphism invariance' which says the laws of GR work in any kind of arbitrary coordinate system you can come up with).

Why can't I consider the satellite frame as the stationary frame?

Also, I am still not getting why the satellite will experience time dilation from the point of view of the Earth because of a relative v but the Earth does not beat slower than the satellite when the satellite is considered the at rest frame.

It would seem the satellite frame would claim the Earth clocks beat slower.

If you know the math, that would be easier for me.

 

[JesseM]

Why can't I consider the satellite frame as the stationary frame?

What "satellite frame"? This phrase simply does not have any unique meaning in curved spacetime--as I keep saying, you could define an infinite number of distinct non-inertial coordinate systems where the satellite is at rest, anyone of which might be called a "satellite frame".

Also, I am still not getting why the satellite will experience time dilation from the point of view of the Earth because of a relative v but the Earth does not beat slower than the satellite when the satellite is considered the at rest frame.

I never said anything of the sort. In at least some satellite-centered coordinate systems, it would be true that the Earth clock is running slower than the satellite clock, at least some of the time (it can't run slower eternally, since we know that if we synchronize the satellite clock with an Earth clock before launch, then later bring it back to Earth after a few orbits, all coordinate systems must agree that the satellite clock has elapsed less time in total, since different coordinate systems always agree about local facts like what two clocks read when they are next to each other).

It might be simpler to consider a satellite traveling in circles around a massless sphere in flat spacetime so we could ignore GR and just think about inertial frames. Then in the inertial rest frame of the sphere, the satellite is ticking slow by a constant amount (assuming its speed in this frame is constant), whereas in the instantaneous inertial rest frame of the satellite at any given moment, the Earth clock is ticking slower. However, since the satellite is moving non-inertially, there is no inertial frame where it remains at rest throughout an orbit, and although some frames may say the Earth clock is ticking slower at any given moment, all will agree that over the course of an entire orbit the satellite ticks forward by less than the Earth clock.

 

[cfrogue]

It might be simpler to consider a satellite traveling in circles around a massless sphere in flat spacetime so we could ignore GR and just think about inertial frames. Then in the inertial rest frame of the sphere, the satellite is ticking slow by a constant amount (assuming its speed in this frame is constant), whereas in the instantaneous inertial rest frame of the satellite at any given moment, the Earth clock is ticking slower. However, since the satellite is moving non-inertially, there is no inertial frame where it remains at rest throughout an orbit, and although some frames may say the Earth clock is ticking slower at any given moment, all will agree that over the course of an entire orbit the satellite ticks forward by less than the Earth clock.

So, where is the reciprocal time dilation?

 

[JesseM]

So, where is the reciprocal time dilation?

In this SR example, at any given instant it's reciprocal--if you pick the sphere's rest frame and find that at a given moment the satellite's clock is only ticking at 0.8 ticks/second, then if you look at that point on the satellite's worldline and pick the inertial frame where the satellite is instantaneously at rest, in that frame the Earth's clock is only ticking at 0.8 ticks/second. But the time dilation is not reciprocal over the course of an entire orbit, because the sphere is moving inertially while the satellite is moving non-inertially, so it's similar to the twin paradox where the aging of the two twins is also not reciprocal because one of them accelerated and the other didn't.

 

[cfrogue]

In this SR example, at any given instant it's reciprocal--if you pick the sphere's rest frame and find that at a given moment the satellite's clock is only ticking at 0.8 ticks/second, then if you look at that point on the satellite's worldline and pick the inertial frame where the satellite is instantaneously at rest, in that frame the Earth's clock is only ticking at 0.8 ticks/second. But the time dilation is not reciprocal over the course of an entire orbit, because the sphere is moving inertially while the satellite is moving non-inertially, so it's similar to the twin paradox where the aging of the two twins is also not reciprocal because one of them accelerated and the other didn't.

It seems like two concentric rings moving relative to each other.

Yes, gravity is decided and programmed correctly so that can be ignored.

How do you conclude one of the rings is moving inertially while the other is not?

Also, the Sagnac effect is not part of this either as that is also accounted for in the calculations and not part of this "v".

 

[pervect]

I am imagining the satellite is at virtual rest in its own system of coordinates. I say virtual because of the orbit.

Also I am imagining the Earth is moving relative to the satellite. I am not assuming some universal coordinate system. I don't have any idea what that would mean.

I'm not sure I understand your coordinate system. The simplest case is still very hard to analyze exactly. Suppose we consider a satellite in a powered orbit around some central point, so that space-time is flat and we can ignore GR.

The simplest case is probably to assume that the satellite is "not rotating" with respect to some internal gyroscopes. This means that there will be some amount of Thomas precession http://en.wikipedia.org/w/index.php?title=Thomas_precession&oldid=323066126, the effect of which will be that when the satellite makes one orbit around the Earth in the Earth frame, the angular position of the Earth relative to the satellite will not have changed by 360 degrees - so one orbit of the satellite around the Earth in the Earth frame will be physically different than one orbit of the Earth around the satellite in the satellite frame.

There should be other effects too - the acceleration that the satellite experiences can be interpreted in the non-inertial frame to act as a sort of gravity. The "Earth" will always be overhead, modulo some small offsets due to possible effects from the relativity of simultaneity that would need to be investigated more closely. This would tend make the Earth's clocks run faster due to the fictitious potential difference, just as the clocks in an accelerating elevator or rocket run at different rates in the nose and tail. This would tend to offset any effects due to its velocity.


It's really a messy calculation - and it won't give you much insight unless you're already very familiar with SR. Which is why I suggest that you attempt to analyze a simpler problem.

But even the simple analysis suggests that there are a couple of possible candidate explanations for the differences, those being the Thomas precession of the satellite, and the fact that in a non-inertial frame, clocks appear to run at different rates. Without a more detailed and lengthly analysis, I can't say offhand which effect does what.

 

[cfrogue]

I'm not sure I understand your coordinate system. The simplest case is still very hard to analyze exactly. Suppose we consider a satellite in a powered orbit around some central point, so that space-time is flat and we can ignore GR.

The simplest case is probably to assume that the satellite is "not rotating" with respect to some internal gyroscopes. This means that there will be some amount of Thomas precession http://en.wikipedia.org/w/index.php?title=Thomas_precession&oldid=323066126, the effect of which will be that when the satellite makes one orbit around the Earth in the Earth frame, the angular position of the Earth relative to the satellite will not have changed by 360 degrees - so one orbit of the satellite around the Earth in the Earth frame will be physically different than one orbit of the Earth around the satellite in the satellite frame.

There should be other effects too - the acceleration that the satellite experiences can be interpreted in the non-inertial frame to act as a sort of gravity. The "Earth" will always be overhead, modulo some small offsets due to possible effects from the relativity of simultaneity that would need to be investigated more closely. This would tend make the Earth's clocks run faster due to the fictitious potential difference, just as the clocks in an accelerating elevator or rocket run at different rates in the nose and tail. This would tend to offset any effects due to its velocity.


It's really a messy calculation - and it won't give you much insight unless you're already very familiar with SR. Which is why I suggest that you attempt to analyze a simpler problem.

But even the simple analysis suggests that there are a couple of possible candidate explanations for the differences, those being the Thomas precession of the satellite, and the fact that in a non-inertial frame, clocks appear to run at different rates. Without a more detailed and lengthly analysis, I can't say offhand which effect does what.

I think I can handle a messy calculation.

Why not present it and let's see.

The article I linked showed a relative v was an important factor in GPS pre-programming of clocks.

So, the coords I would like to use are that of the satellite and view the Earth as the moving frame.

Is this allowed?

 

[JesseM]

It seems like two concentric rings moving relative to each other.

I was assuming that the sphere is not rotating, so clocks on the surface are moving inertially. If you want to assume a rotating sphere, then neither clocks on the surface nor clocks in orbit would be moving inertially, but it would still be true that in any inertial frame, clocks in orbit have a greater average speed over the course of a full orbit than clocks on the surface, so clocks in orbit will tick less in total over one orbit.

Yes, gravity is decided and programmed correctly so that can be ignored.

Not in the actual GPS example it can't, since spacetime is curved there and it is impossible for any coordinate system to qualify as "inertial". That's why I specified that I was imagining a satellite rotating around a sphere of zero mass in flat spacetime.

How do you conclude one of the rings is moving inertially while the other is not?

If the sphere is rotating then neither set of clocks is moving inertially; again, I was thinking of a nonrotating sphere. Hopefully you agree that in SR there is always an objective truth about whether a clock is moving inertially or not, acceleration shows up as G-forces which can be measured with an accelerometer.

Also, the Sagnac effect is not part of this either as that is also accounted for in the calculations and not part of this "v".

The Sagnac effect doesn't need to be considered at all, since it's an optical phenomenon dealing with light, whereas we're talking about actual time dilation in a given frame.

 

[cfrogue]

I was assuming that the sphere is not rotating, so clocks on the surface are moving inertially. If you want to assume a rotating sphere, then neither clocks on the surface nor clocks in orbit would be moving inertially, but it would still be true that in any inertial frame, clocks in orbit have a greater average speed over the course of a full orbit than clocks on the surface, so clocks in orbit will tick less in total over one orbit.

Not in the actual GPS example it can't, since spacetime is curved there and it is impossible for any coordinate system to qualify as "inertial". That's why I specified that I was imagining a satellite rotating around a sphere of zero mass in flat spacetime.

If the sphere is rotating then neither set of clocks is moving inertially; again, I was thinking of a nonrotating sphere. Hopefully you agree that in SR there is always an objective truth about whether a clock is moving inertially or not, acceleration shows up as G-forces which can be measured with an accelerometer.

The Sagnac effect doesn't need to be considered at all, since it's an optical phenomenon dealing with light, whereas we're talking about actual time dilation in a given frame.

Yea, can you explain the v in the GPS pre-programming?

Is this a relative v or not.

If it is a relative v, then why does reciprocal time dilation not apply.

 

[cfrogue]

I was assuming that the sphere is not rotating, so clocks on the surface are moving inertially. If you want to assume a rotating sphere, then neither clocks on the surface nor clocks in orbit would be moving inertially, but it would still be true that in any inertial frame, clocks in orbit have a greater average speed over the course of a full orbit than clocks on the surface, so clocks in orbit will tick less in total over one orbit.

Not in the actual GPS example it can't, since spacetime is curved there and it is impossible for any coordinate system to qualify as "inertial". That's why I specified that I was imagining a satellite rotating around a sphere of zero mass in flat spacetime.

If the sphere is rotating then neither set of clocks is moving inertially; again, I was thinking of a nonrotating sphere. Hopefully you agree that in SR there is always an objective truth about whether a clock is moving inertially or not, acceleration shows up as G-forces which can be measured with an accelerometer.

The Sagnac effect doesn't need to be considered at all, since it's an optical phenomenon dealing with light, whereas we're talking about actual time dilation in a given frame.

The Sagnac effect on moving ground-based receivers must still be considered

http://relativity.livingreviews.org/Articles/lrr-2003-1/

Chapter 5

 

[JesseM]

The Sagnac effect on moving ground-based receivers must still be considered

http://relativity.livingreviews.org/Articles/lrr-2003-1/

Chapter 5

That's because they are using light signals to communicate. But if aren't worried about practical details of communication and just want to know how fast different clocks are ticking relative to a give frame of reference, we don't have to consider it.

 

[JesseM]

Yea, can you explain the v in the GPS pre-programming?

Is this a relative v or not.

If it is a relative v, then why does reciprocal time dilation not apply.

What do you mean by "relative"? It is a v defined in relation to a single coordinate Earth-centered system, so you can say it's defined "relative" to that coordinate system, but it's not "relative" in the sense that we are considering more than one frame of reference, so there are no issues of reciprocity. Similarly, in SR if you are traveling at 0.8c relative to me, and I am only using my inertial rest frame to define velocity and clock rates, then there is no issue of reciprocity here either--in this single frame, it is unambiguously true that I am at rest and you are moving at 0.8c, and that my clock is ticking at a normal rate while yours is slowed down by a factor of 0.6 (and if you also use my rest frame to do your calculations, you will agree--remember that any observer can calculate things from the perspective of any frame they like). Reciprocity would only enter into things if we wanted to also look at things from the perspective of your rest frame, but I have no obligation to do this, I can address any coordinate-invariant physical question using only my own rest frame to do the calculations.

Earlier I asked this question:

Do you understand that in relativity you are never required to use multiple frames, you can always get the answer to any physical question you're interested in using a single frame, even if the situation you are analyzing involves objects which are not at rest in that frame?

Your response was "Yea, this is fine", but the fact that you continue to talk as though we are somehow obligated to consider the issue of reciprocity in different frames (and it's not entirely clear you understand that 'reciprocity' only applies when we consider multiple frames as opposed to just one) suggests you aren't actually totally fine with this. So please tell me again, do you understand that there is never a requirement to use multiple frames in relativity, and that it only makes sense to talk about "reciprocity" in things like time dilation when we are comparing multiple frames?

Finally, note that even when we do deal with multiple frames, the idea that time dilation should be reciprocal only applies if both frames are inertial ones (the time dilation factor $$\sqrt{1 - v^2/c^2}$$ is only meant to apply in inertial frames). If you have an inertial observer A with a non-inertial observer B orbiting around him, and you consider both the inertial frame where A is at rest and the non-inertial rotating frame where B is at rest, then both frames will agree that B's clock is ticking slower than A's.

 

[yuiop]

In SR if two observers are moving away from each other with constant velocity, they each measure the clock of the other observer to be running slower than their own clock. This is the reciprocity that SR leads us to expect and under conditions of constant velocity there is no way to determine which observer's clock is "really" running slower. However, if one of the observers accelerates to turn around and head back towards the other observer, the reciprocity is broken and there is no argument about which clock was actually running slower when they get back together and compare clocks side by side. It is the same for orbiting clocks or particles in a magnetic storage ring. The circular motion is a form of acceleration and the reciprocity breaks down.

 

[JesseM]

In SR if two observers are moving away from each other with constant velocity, they each measure the clock of the other observer to be running slower than their own clock.

That's true if each observer uses their own rest frame to make calculations, which is what is normally assumed in SR problems. However, part of what seems to be confusing cfrogue is that there is nothing that forces one to use one's own rest frame to make calculations, this is purely a matter of convention--in SR both observers could agree to use a single inertial frame to make calculations, in which case they would both agree whose clock is running slower, so there is no issue of "reciprocity" here. Something just like this is going on with the GPS system, where all the satellites are using a single Earth-centered coordinate system to make calculations regardless of their own state of motion.

 

[pervect]

It might be instructive to work out the problem using GR in the Earth centered frame, using the Schwarzschild metric, for a circular orbit at the equator. This will also show that the usual SR velocity time dilation formula is only an approximation.

It might be helpful to recap how we get the equation for time dilation in SR, first.The time dilation that we wish to solve for is just
$$\frac{dt}{d\tau}$$

In SR, we know the flat space-time metric is

$$c^2 d\tau^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2$$

dividing both sides by dt^2, we get

$$c^2 \left( \frac{d\tau}{dt} \right) ^2 = c^2 - \left( \frac{dx}{dt} \right)^2 - \left( \frac{dy}{dt} \right)^2 - \left( \frac{dz}{dt} \right)^2$$

Solving for $d\tau / dt$ we get

$$\frac{d\tau}{dt} = \sqrt{1 - \frac{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 }{c^2} }$$

We recognize this as
$$\frac{d\tau}{dt} = \sqrt{1 - \frac{v^2}{c^2}}$$

but since we want the reciprocal, $dt / d\tau$, we get the usual relation

$$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Now, we just need to repeat this for the case of a gravitating body. We'll use Schwarzschild coordinates to get the curved space-time metric, rather than the flat metric we used in SR.

Note that in Schwarzschild coordinates, $\theta$ is Pi at the equator, and $\phi$ varies from 0 to 2Pi as it sweeps out the orbit. r is constant for a circular orbit.

So we write

$$c^2 d\tau^2 = g_{tt} dt^2 + g_{rr} dr^2 + g_{\theta\theta} d\theta^2 + g_{\phi\phi} d\phi^2$$

dividing both sides of the equation by dt^2, and dropping some terms that we know to be zero, such as $\frac{dr}{dt}$ and $\frac{d\theta}{dt}$, we get

$$c^2 \left( \frac{d\tau}{dt} \right)^2 = g_{tt} + g_{\phi\phi}\left(\frac{d\phi}{dt}\right)^2$$

We know that for the schwarzschild metric

http://en.wikipedia.org/w/index.php?title=Schwarzschild_metric&oldid=325313722

$$g_{tt} = \left(1-\frac{r_s}{r}\right)^2\,c^2$$

and we know that

$$g_{\phi\phi} = -r^2$$

as sin$\phi$ is one.

Putting this together we get
$$c^2 \left( \frac{d\tau}{dt} \right)^2 = c^2 \, \left(1-\frac{r_s}{r}\right)^2\ - r^2 \left(\frac{d\phi}{dt}\right)^2$$

This can be rewritten as

$$\frac{d\tau}{dt}= \sqrt{ g_{tt} } \sqrt{1 - \frac{1}{g_{tt}} \left( \frac{r \frac{d\theta}{dt} }{c} \right)^2$$This is almost in the form of the product of the GR and SR time dilation but not quite exactly.

Note that if we set $d\phi / dt$ to zero, we see that the time dilation is just the gravitational time dilation

$$\frac{dt}{d\tau}= \frac{1}{\sqrt{g_{tt}}}$$

But because the gravitational time dilation is so nearly unity, it provides only a tiny correction to the velocity in the SR formula to multiply it by g_tt so it's approximately correct to multiply the SR time dilation by the gravitational time dilation for a non-moving object to get the total time dilation.