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

[JesseM]

I agree only that SR provides for reciprocal time dilation between 2 frames and that is all I was talking about.

Is this not true?

Yes. So, how can the GPS calculations, which are based only on a single frame, possibly contradict reciprocal time dilation? They simply don't address the issue of other frames one way or another--why should they, when one frame is sufficient for the purpose the satellites are designed for, namely pinpointing the location of transmitters on Earth? The GPS system was not designed as an exercise for relativity students to help teach them about comparing different frames.

Well, the satellite needs to be programmed for the relative motion.

It is concluded that moving clocks run slower as concluded from the article.

They only "conclude" anything about the rate of clocks in the single Earth-centered frame used in GPS calculations. Do you agree or disagree?

Since the time dilation is only one way, I guess that means the satellite is in absolute motion around the earth.

Is this correct?

Of course not. They don't say "the time dilation is only one way" in all possible frames you could use, only in the one actual frame they do use.

A hundred times?

Exaggeration is sometimes used to convey exasperation. I have asked certain questions, and made certain points, quite a number of times without getting any sort of substantive response from you. It would help if you would quote my posts section by section (paragraph by paragraph, sentence by sentence, whatever) and give your response to the points/questions in each section, rather than just quote the whole post and giving a two or three sentence response that doesn't address most of what I said.

I am OK with the Earth center relative motion.

So are you OK with the fact that there can be no "reciprocal time dilation" if we just use this one coordinate system, but that this in no way contradicts the claim that if you did use a different coordinate system you could get different answers to questions about the rate different clocks are ticking?

 

[Phrak]

Correct me, if I'm wrong, though something seems to be missing in your analysis. At this point you have t of an orbiting object compared to t at asymtotic infinity.

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

Shouldn't we want t_e on the surface of the Earth (with it's own ~24 hr orbit) compared to t_o in a freely falling orbit?

 

[cfrogue]

Yes. So, how can the GPS calculations, which are based only on a single frame, possibly contradict reciprocal time dilation? They simply don't address the issue of other frames one way or another--why should they, when one frame is sufficient for the purpose the satellites are designed for, namely pinpointing the location of transmitters on Earth? The GPS system was not designed as an exercise for relativity students to help teach them about comparing different frames.

Let A and B be two inertial frames in relative motion.

Now introduce a 3rd frame C.

Does this 3rd frame mean the reciprocal time dilation disappears between A and B?

 

[JesseM]

Let A and B be two inertial frames in relative motion.

Now introduce a 3rd frame C.

Does this 3rd frame mean the reciprocal time dilation disappears between A and B?

No. Now, suppose we do all our calculations from the perspective of frame C, and don't comment one way or another about how things might look in another frame. Does this mean we are contradicting the idea that there can be reciprocal time dilation between other frames, or claiming the existence of absolute time dilation?

 

[cfrogue]

No. Now, suppose we do all our calculations from the perspective of frame C, and don't comment one way or another about how things might look in another frame. Does this mean we are contradicting the idea that there can be reciprocal time dilation between other frames, or claiming the existence of absolute time dilation?

No, it means we are ignoring all the predictions of the theory.

 

[yuiop]

Shouldn't we want t_e on the surface of the Earth (with it's own ~24 hr orbit) compared to t_o in a freely falling orbit?

Yes, you would have to calculate the clock rate of the orbital clock and then calculate the clock rate of a clock on the surface and then compare the two. Pervect was was not specifically addressing the issue of comparing surface clocks to orbital clocks, but was focusing on whether or not the total time dilation of a clock can be broken down into simple gravitational and velocity terms. In post #8 of https://www.physicsforums.com/showthread.php?t=355378" I think I may have demonstrated that maybe you can.

(I am however, bothered that my result is the product of the gravitational and velocity time dilation terms and other sources are using the the sum of the gravitational and velocity terms. I need to look further into that :/)

 

[cfrogue]

Yes, you would have to calculate the clock rate of the orbital clock and then calculate the clock rate of a clock on the surface and then compare the two. Pervect was was not specifically addressing the issue of comparing surface clocks to orbital clocks, but was focusing on whether or not the total time dilation of a clock can be broken down into simple gravitational and velocity terms. In post #8 of https://www.physicsforums.com/showthread.php?t=355378" I think I may have demonstrated that maybe you can.

Is this an incomplete solution?

Should the calculations operate from a surface clock to the satellite and then from the satellite to the surface clock. After all, a theory should calculate the same in all directions for the same problem.

I wonder if the time dilation portion is absolute for both cases such that the satellite and the Earth based clocks all agree the satellite clock will beat slower for a space shuttle orbit.

 

[Jorrie]

Well, thank you. Is it correct that SR exhibits reciprocal time dilation?

As to number 2, the experimental evidence is asserting an affirmative.

But, let us simplify it even more.
Let O and O' be in collinear relative motion. Will each observer see the other's clock as running slower from SR?

I think JesseM has answered you already, but since he is concentrating more on the single frame calculations, it seems that you still have problems reconciling the coordinate dependent reciprocal time dilation assertion ("each observer see the other's clock as running slower?") with the coordinate independent fact that the satellite clock will be running slower over one orbit than the tower clock (gravitational time dilation equalized). You can make the latter calculation from either frame and the result remains the same.

At the instant of flyby, 'my' two clocks are very closely equivalent to your "Let O and O' be in collinear relative motion" and then they will observe this reciprocal effect. This is because you cannot favor one of the two (instantaneously) inertial frames. It essentially comes from their different definitions of simultaneity. However, over a longer period, neither of 'my' two clocks are 'purely inertial', but there is a big difference between the tower- and the satellite clock in terms of inertial status.

IMO, the best way of looking at it is that during the flyby, the two clocks momentarily follow equivalent spacetime paths and one cannot tell which one is physically 'running slower'. Over time however, the satellite clock follows a different spacetime path than the tower clock, because it does not stay in the same inertial frame (Pervect has explained that earlier). It is roughly the same as in the classical 'twin paradox' where the twin that is accelerated (changes inertial frames to turn around for the return flight) always records a lesser elapsed time.

Hence, no matter which frame you use as reference (for the calculations), the on-board GPS corrections for velocity time dilation are the same and there is no paradox...

 

[yuiop]

Is this an incomplete solution?

Should the calculations operate from a surface clock to the satellite and then from the satellite to the surface clock. After all, a theory should calculate the same in all directions for the same problem.

I wonder if the time dilation portion is absolute for both cases such that the satellite and the Earth based clocks all agree the satellite clock will beat slower for a space shuttle orbit.

Not quite sure what you are getting at here. What does " the satellite clock will beat slower for a space shuttle orbit" mean? All the calculations can tell you is how the proper times of various clocks evolve relative to a hypothetical clock at asymptotic infinity and predict what they will be reading when they come alongside each other and are directly compared.

 

[JesseM]

No, it means we are ignoring all the predictions of the theory.

What do you mean "ignoring"? Do you think they are denying any predictions of the theory, or do you agree that they're just not addressing predictions about comparisons between frame because this is not relevant to what they are interested in calculating? (in the case of the GPS system, what they are interested in is pinpointing the location on Earth of signals from GPS transmitters, a local question that all frames would agree on anyway)

 

[cfrogue]

I think JesseM has answered you already, but since he is concentrating more on the single frame calculations, it seems that you still have problems reconciling the coordinate dependent reciprocal time dilation assertion ("each observer see the other's clock as running slower?") with the coordinate independent fact that the satellite clock will be running slower over one orbit than the tower clock (gravitational time dilation equalized). You can make the latter calculation from either frame and the result remains the same.

At the instant of flyby, 'my' two clocks are very closely equivalent to your "Let O and O' be in collinear relative motion" and then they will observe this reciprocal effect. This is because you cannot favor one of the two (instantaneously) inertial frames. It essentially comes from their different definitions of simultaneity. However, over a longer period, neither of the two clocks are 'purely inertial', but there is a big difference between the tower- and the satellite clock in terms of inertial status.

IMO, the best way of looking at it is that during the flyby, the two clocks momentarily follow equivalent spacetime paths and one cannot tell which one is physically 'running slower'. Over time however, the satellite clock follows a different spacetime path than the tower clock, because it does not stay in the same inertial frame (Pervect has explained that earlier). It is roughly the same as in the classical 'twin paradox' where the twin that is accelerated (changes inertial frames to turn around for the return flight) always records a lesser elapsed time.

Hence, no matter which frame you use as reference (for the calculations), the on-board GPS corrections for velocity time dilation are the same and there is no paradox...

The twins issue is due to acceleration.

Now, the article is clear,

It is obvious that Eq. (24) contains within it the well-known effects of time dilation (the apparent slowing of moving clocks) and frequency shifts due to gravitation.
http://relativity.livingreviews.org/Articles/lrr-2003-1/

Can you explain this?

 

[cfrogue]

What do you mean "ignoring"? Do you think they are denying any predictions of the theory, or do you agree that they're just not addressing predictions about comparisons between frame because this is not relevant to what they are interested in calculating? (in the case of the GPS system, what they are interested in is pinpointing the location on Earth of signals from GPS transmitters, a local question that all frames would agree on anyway)

I agree GPS works.

But, I am wondering if you did calculate the integral from both the ground clock and the satellite clock?

This notion of absolute time is becoming interesting to me.

 

[cfrogue]

Not quite sure what you are getting at here. What does " the satellite clock will beat slower for a space shuttle orbit" mean? All the calculations can tell you is how the proper times of various clocks evolve relative to a hypothetical clock at asymptotic infinity and predict what they will be reading when they come alongside each other and are directly compared.

I was wondering if you did the integral from both an Earth based clock and a satellite clock?

What do you predict?

 

[Phrak]

Come to think of it Pervect, the best approach might be to begin with defining the metric in the weak field limit for an Earth sized planet, in Riemann normal coordinates.

g_tt=1+h_tt, g_XX=1, where h_tt=h_tt(r) is a perturbation proportional to the gravitational potential, then change to spherical coordinates.

 

[cfrogue]

Come to think of it Pervect, the best approach might be to begin with defining the metric in the weak field limit for an Earth sized planet, in Riemann normal coordinates.

g_tt=1+h_tt, g_XX=1, where h_tt=h_tt(r) is a perturbation proportional to the gravitational potential, then change to spherical coordinates.

Have you read the GPS mainstream on how to do the integral?

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

Chapter 4 eq 28

 

[JesseM]

I agree GPS works.

And so would you agree that the GPS calculations don't contradict the idea of reciprocal time dilation in different frames, they just doesn't address it one way or another?

But, I am wondering if you did calculate the integral from both the ground clock and the satellite clock?

You would get the same answers to questions that can be defined in a purely local manner, like what two clocks read at the moment they pass next to each other, but you could get different answers to questions that are frame-dependent, like the rate a clock is ticking at any given moment. For example, consider the SR example of a clock orbiting in a circle around a massless sphere, with another clock sitting on a tower attached to the sphere which is just the right height for the orbiting clock to pass right next to it. In this case, if you analyze things from the perspective of inertial frame A in which the tower clock is at rest, then at the moment the orbiting clock passes the tower clock, the orbiting clock is ticking slower in frame A; but if you analyze things from the perspective of inertial frame B in which the orbiting clock is instantaneously at rest when it passes the tower clock, then at the moment they pass the tower clock is ticking slower in frame B. However, both frames will agree on the times on each clock at the moment they pass since this is a purely local question, and they'll both make the same prediction about how much time elapses on each clock over the course of a full orbit, so they'll both predict that the orbiting clock will have elapsed less time than the tower clock the next time they pass each other. This is exactly like the twins paradox, since in this example the tower clock is moving inertially between meetings, while the orbiting clock is constantly accelerating.

 

[yuiop]

I was wondering if you did the integral from both an Earth based clock and a satellite clock?

What do you predict?

I am not sure my PC math unit has enough precision to cope with the microscopic differences we are talking about here. Have you got figures for the orbital radius and velocities?

 

[cfrogue]

I am not sure my PC math unit has enough precision to cope with the microscopic differences we are talking about here. Have you got figures for the orbital radius and velocities?

'

LOL, the space shuttle orbit for time dilation is not small.

The equations you request are in the article I posted. I imagine I could do them but who knows.

Anyway, did you do the integral from both perspectives?

How do they turn out?

 

[Jorrie]

The twins issue is due to acceleration.

It is obvious that Eq. (24) contains within it the well-known effects of time dilation (the apparent slowing of moving clocks) and frequency shifts due to gravitation.
http://relativity.livingreviews.org/Articles/lrr-2003-1/

Can you explain this?

Firstly, the 'twins issue' is due to a coordinate acceleration (i.e., a change of inertial frames); proper acceleration is not a requirement, I think. The satellite clock undergoes a continuous coordinate acceleration and is hence similar to the twins scenario.

Secondly, I thought your referenced Eq. (24) has been fully explained in this thread. Be that as it may, Eq. (28) and what follows directly below it explains it. It calculates the Earth time/proper time of the orbiting clock ratio, which by definition, is coordinate choice independent. What more is there to say?

 

[cfrogue]

And so would you agree that the GPS calculations don't contradict the idea of reciprocal time dilation in different frames, they just doesn't address it one way or another?

You would get the same answers to questions that can be defined in a purely local manner, like what two clocks read at the moment they pass next to each other, but you could get different answers to questions that are frame-dependent, like the rate a clock is ticking at any given moment. For example, consider the SR example of a clock orbiting in a circle around a massless sphere, with another clock sitting on a tower attached to the sphere which is just the right height for the orbiting clock to pass right next to it. In this case, if you analyze things from the perspective of inertial frame A in which the tower clock is at rest, then at the moment the orbiting clock passes the tower clock, the orbiting clock is ticking slower in frame A; but if you analyze things from the perspective of inertial frame B in which the orbiting clock is instantaneously at rest when it passes the tower clock, then at the moment they pass the tower clock is ticking slower in frame B. However, both frames will agree on the times on each clock at the moment they pass since this is a purely local question, and they'll both make the same prediction about how much time elapses on each clock over the course of a full orbit, so they'll both predict that the orbiting clock will have elapsed less time than the tower clock the next time they pass each other. This is exactly like the twins paradox, since in this example the tower clock is moving inertially between meetings, while the orbiting clock is constantly accelerating.

Now place an infinite number of clocks on towers as you specify and make the orbit of the satellite follow this path.

What does the integral tell you?

 

[JesseM]

Firstly, the 'twins issue' is due to a coordinate acceleration (i.e., a change of inertial frames); proper acceleration is not a requirement, I think. The satellite clock undergoes a continuous coordinate acceleration and is hence similar to the twins scenario.

You may know this already, but just to avoid confusion, coordinate acceleration in an inertial SR frame is always associated with proper acceleration and vice versa, in inertial frames you can't have one without the other (this is no longer true in non-inertial frames of course). A clock moving in a circle in flat SR spacetime (as opposed to one orbiting in GR due to spacetime curvature) would be experiencing proper acceleration, it would measure a nonzero reading on its accelerometer (the 'centrifugal force').

 

[cfrogue]

Firstly, the 'twins issue' is due to a coordinate acceleration (i.e., a change of inertial frames); proper acceleration is not a requirement, I think. The satellite clock undergoes a continuous coordinate acceleration and is hence similar to the twins scenario.

Secondly, I thought your referenced Eq. (24) has been fully explained in this thread. Be that as it may, Eq. (28) and what follows directly below it explains it. It calculates the proper time of the orbiting clock, which by definition, is coordinate choice independent. What more is there to say?

Einstein solved the twins by considering both twins and proving they come up with the same result. So, he showed both directions are necessary.


No, I have not seen this integral done from both directions.

Is only one preferred frame necessary under SR and GR?

Under this context, I could sit inside one frame and predict all events in the universe.

Is this your claim?

 

[JesseM]

Now place an infinite number of clocks on towers as you specify and make the orbit of the satellite follow this path.

What does the integral tell you?

Same thing, that the orbiting clock elapses less time over the course of an entire orbit. It would also be true in all frames that for two clocks on nearby towers, if the first read a time t1 when the orbiting clock passed it and the second read a time t2 (and the two tower clocks were synchronized in their mutual rest frame), then the time T elapsed on the orbiting clock between passing these two tower clocks would be less than (t2 - t1). But keep in mind, this is not inconsistent with the idea that there might be some inertial frame where both of these tower clocks were ticking slower than the orbiting clock during the time between the two passings...in this frame, the explanation for the fact that T < (t2 - t1) would be that the two tower clocks were out-of-sync (the relativity of simultaneity), with the second tower clock ahead of the first tower clock at the moment the orbiting clock was passing the first one, so even though the second tower clock ticked forward by less than the orbiting clock during the time it took for the orbiting clock to get from the first to the second, it could still be true that T < (t2 - t1).

Can you please answer the question I asked in my last post?

And so would you agree that the GPS calculations don't contradict the idea of reciprocal time dilation in different frames, they just doesn't address it one way or another?

 

[JesseM]

Einstein solved the twins by considering both twins and proving they come up with the same result. So, he showed both directions are necessary.

No, such a calculation is just an exercise to show that different frames give the same predictions about local events (something that is already guaranteed if you assume Lorentz-symmetric laws); once you accept this, if predicting local events is all you are interested in, then only one frame is necessary.

Is only one preferred frame necessary under SR and GR?

If you just want to make predictions about local events, only one frame is necessary. But any frame will give the same predictions about local events, so no frame is "preferred".

Under this context, I could sit inside one frame and predict all events in the universe.

Yup, in relativity you only need one coordinate system to predict all local events in the universe.

 

[Jorrie]

You may know this already, but just to avoid confusion, coordinate acceleration in an inertial SR frame is always associated with proper acceleration and vice versa, in inertial frames you can't have one without the other (this is no longer true in non-inertial frames of course). A clock moving in a circle in flat SR spacetime (as opposed to one orbiting in GR due to spacetime curvature) would be experiencing proper acceleration, it would measure a nonzero reading on its accelerometer (the 'centrifugal force').

Yup, I agree.

IMO, using three purely inertial clocks, one can demonstrate coordinate independent relativistic time dilation without invoking acceleration as part of the test. I do not wish to dilute this thread by debating it here, but unless already beaten to death in this forum, maybe we can devote another thread to it.

 

[Ich]

Come to think of it Pervect, the best approach might be to begin with defining the metric in the weak field limit for an Earth sized planet, in Riemann normal coordinates.

Definitely not, as long as you're not at the Earth's core.
FWIW, https://www.physicsforums.com/showthread.php?p=1600272#post1600272"'s another version of pervect's calculation. It's easiest to use the complete Schwarzschild solution (it's not that difficult) and then approximate from flat space, not the center.

 

[cfrogue]

Yup, I agree.

IMO, using three purely inertial clocks, one can demonstrate coordinate independent relativistic time dilation without invoking acceleration as part of the test. I do not wish to dilute this thread by debating it here, but unless already beaten to death in this forum, maybe we can devote another thread to it.

Can you do this please?

I do not think another thread is necessary.

 

[cfrogue]

Yup, in relativity you only need one coordinate system to predict all local events in the universe.

I am OK with this, but I would assume switching to another frame should produce a similar pattern and thus reciprocal time dilation even though there exists gravity and orbital considerations.

However, can both directions conclude the satellite clock will follow the adjustments consistent with the experimental evidence?

More specifically, when only the satellite is considered compared to Earth based clocks, will it conclude time dilation is absolute for relative v to the Earth and will the Earth when calculating the satellite conclude exactly the same thing?

 

[yuiop]

More specifically, when only the satellite is considered compared to Earth based clocks, will it conclude time dilation is absolute for relative v to the Earth and will the Earth when calculating the satellite conclude exactly the same thing?

The time dilation of the satellite clock is not a function of v relative to the Earth. If the satellite was in a high geosynchronous orbit it would have no velocity relative to the surface of the Earth but it would still have a velocity based time dilation component due to its velocity relative to the space that the Earth is rotating with respect to. Even though the geosynchronous satellite appears motionless from the point on the surface of the Earth immediately below the satellite, the satellite obviously has orbital velocity otherwise it would not remain in orbit.

 

[cfrogue]

The time dilation of the satellite clock is not a function of v relative to the Earth. If the satellite was in a high geosynchronous orbit it would have no velocity relative to the surface of the Earth but it would still have a velocity based time dilation component due to its velocity relative to the space that the Earth is rotating with respect to. Even though the geosynchronous satellite appears motionless from the point on the surface of the Earth immediately below the satellite, the satellite obviously has orbital velocity otherwise it would not remain in orbit.

So what is it a function of?

 

[JesseM]

cfrogue, did you read post #93? Can you please answer the question I asked at the end of that post, and also tell me if you understand the reasoning about why different frames can disagree about whether the orbiting clock or the two tower clocks are ticking slower, but still agree on the time T that elapses between passing the two tower clocks, and the times t1 and t2 that each tower clock reads when the orbiting clock passes it?

 

[cfrogue]

cfrogue, did you read post #93? Can you please answer the question I asked at the end of that post, and also tell me if you understand the reasoning about why different frames can disagree about whether the orbiting clock or the two tower clocks are ticking slower, but still agree on the time T that elapses between passing the two tower clocks, and the times t1 and t2 that each tower clock reads when the orbiting clock passes it?

Well, let me ask this question. Can you repost #93 in terms of orbital distance please?

This means, use the logic of #93 with a low orbit and a high orbit.

I think I would understand it better that way.

 

[JesseM]

Well, let me ask this question. Can you repost #93 in terms of orbital distance please?

This means, use the logic of #93 with a low orbit and a high orbit.

I think I would understand it better that way.

I don't understand. What scenario do you want to analyze? In that post I talked about an orbiting clock passing two clocks on towers attached to the ground--do you still want to have tower clocks? And the logic of post 93 was specifically based on the fact that the orbiting clock passed right next to two tower clocks in succession so the times could be compared locally--if you have two clocks at different heights, how can you make a local comparison of their readings? One would need to change heights to meet the other at some point, or else they could send radio signals with each tick and each clock could compare its own rate of ticking with the rate it was receiving signals from the other.

Also, the question at the end of post 93 had nothing to do with this particular scenario anyway (it was just a repost of a question I asked you earlier which you didn't answer), so can you please answer that?

 

[cfrogue]

I don't understand. What scenario do you want to analyze? In that post I talked about an orbiting clock passing two clocks on towers attached to the ground--do you still want to have tower clocks? And the logic of post 93 was specifically based on the fact that the orbiting clock passed right next to two tower clocks in succession so the times could be compared locally--if you have two clocks at different heights, how can you make a local comparison of their readings? One would need to change heights to meet the other at some point, or else they could send radio signals with each tick and each clock could compare its own rate of ticking with the rate it was receiving signals from the other.

Also, the question at the end of post 93 had nothing to do with this particular scenario anyway (it was just a repost of a question I asked you earlier which you didn't answer), so can you please answer that?

I will answer your question. But I must understand it first.

Please put #93 in the context of several orbits and distance to Earth to see if the logic works.
Then please explain the logic to me.

 

[JesseM]

I will answer your question. But I must understand it first.

But the question at the end of post 93 had nothing to do with the the scenario I was talking about in the earlier part of the post, so you don't need to understand anything about that scenario to answer it. The question at the end of post 93 was just asking whether, since you already agreed "reciprocal time dilation" involves comparing multiple frames, then since the GPS system does all its calculation in one frame, it in no way contradicts the idea of reciprocal time dilation, it just doesn't address comparisons between frames in the first place. Do you agree or disagree that a calculation that's confined to just one frame cannot possibly in itself contradict a claim about what happens when you compare multiple frames?

Please put #93 in the context of several orbits and distance to Earth to see if the logic works.

You already asked me to consider different orbits, and I said I didn't know what scenario you wanted me to consider, which is why I asked these questions:

I don't understand. What scenario do you want to analyze? In that post I talked about an orbiting clock passing two clocks on towers attached to the ground--do you still want to have tower clocks? And the logic of post 93 was specifically based on the fact that the orbiting clock passed right next to two tower clocks in succession so the times could be compared locally--if you have two clocks at different heights, how can you make a local comparison of their readings? One would need to change heights to meet the other at some point, or else they could send radio signals with each tick and each clock could compare its own rate of ticking with the rate it was receiving signals from the other.

If you give me a specific well-defined scenario to analyze, I can "see if the logic works", but I can't if you won't even answer my questions about what scenario you're imagining!