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

[BigFairy]

A valid question.

 

[cfrogue]

While you are on a 'timeout' and no doubt considering what JesseM and Pervect (amongst others) have told you, I suggest you also consider this ultra-simplistic satellite scenario.

Assume a hypothetical homogeneous, perfectly spherical, non-rotating Earth. Put an observer with a clock on top of a tower that reaches to the orbit of a manned satellite, which is in circular orbit. Gravitational time dilation is now equal for tower clock and satellite clock. Let the two observers record the times of their own and of each others clocks at every flyby. Now ask yourself:

1) At flyby, will each observer perceive the other observer's clock to (momentarily) run slower, because they are in relative motion? This is a frame dependent observation (your reciprocity issue).

2) When they compare clocks at each flyby, will they agree that the satellite clock recorded a shorter orbital period than the tower clock, i.e., that the satellite clock "lost time" relative to the tower clock in an absolute sense?

What would you answer?

While you are on a 'timeout' and no doubt considering what JesseM and Pervect (amongst others) have told you, I suggest you also consider this ultra-simplistic satellite scenario.

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?

 

[JesseM]

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

It "exhibits" this only if you compare rates of ticking in two different inertial frames. And note that in the example, the orbiting clock is moving non-inertially, so while it's true that in this clock's instantaneous inertial rest frame at any given moment the other clock is instantaneously ticking slower, all inertial frames will agree that over the course of an entire orbit, the non-inertial clock elapses less time, so in this sense there is no reciprocity in this example.

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?

By "see" do you mean what they see visually when they look at each other (which is influenced by the Doppler effect, so if they are moving towards each other they actually see the other clock running faster) or do you mean what they calculate in some frame? If they each do the calculations in their own rest frame, they'll each conclude that the other clock is running slower. But they are free to agree in advance to use the same frame to do their calculations rather than to each use the frame where they are at rest, in which case they will both agree about which clock is running slower.

 

[Phrak]

We're talking about the rate clocks are ticking relative to a given coordinate system, which is one of the ways of talking about time dilation in SR, rather than talking about the times on one clock as it passes next to two other clocks which share the same rest frame, which is a different way of talking about time dilation (though obviously they are related since any coordinate system's time can be defined in terms of a network of imaginary clocks which are at rest in that system).

This idea of combining SR time dilation from a clock whirling around in a circle with a contributing factor from gravitational potential doesn't seem to work at all.

 

[cfrogue]

It "exhibits" this only if you compare rates of ticking in two different inertial frames. And note that in the example, the orbiting clock is moving non-inertially, so while it's true that in this clock's instantaneous inertial rest frame at any given moment the other clock is instantaneously ticking slower, all inertial frames will agree that over the course of an entire orbit, the non-inertial clock elapses less time, so in this sense there is no reciprocity in this example.

By "see" do you mean what they see visually when they look at each other (which is influenced by the Doppler effect, so if they are moving towards each other they actually see the other clock running faster) or do you mean what they calculate in some frame? If they each do the calculations in their own rest frame, they'll each conclude that the other clock is running slower. But they are free to agree in advance to use the same frame to do their calculations rather than to each use the frame where they are at rest, in which case they will both agree about which clock is running slower.

"See", no, poor language on my part, calculate is what I meant.

But, let's ignore gravity for the moment. The paper on GPS does this many times to illustrate a point.

Given a sufficiently small path of the satellite does it make sense we will see time dilation from the sides.

Assume there exists an infinite number of clocks on the equator all in sync on the earth. Then a satellite proceeds in orbit along this path.


As to the other, the article says,

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/

See chapter 4.

How is your logic consistent with this?

Also, this v is relative in the equations and thus it would lead one to wonder where the reciprocal time dilation occurs.


Can you explain this?

 

[JesseM]

"See", no, poor language on my part, calculate is what I meant.

And do you agree that they'll only calculate different things about which clock is ticking slower if they use different frames to do their calculations? That they are free to agree to use the same frame for their calculations (even though one or both are not at rest in this frame), in which case they will naturally agree about which clock is ticking slower?

Given a sufficiently small path of the satellite does it make sense we will see time dilation from the sides.

Assume there exists an infinite number of clocks on the equator all in sync on the earth. Then a satellite proceeds in orbit along this path.

Sure, if the satellite uses its own local inertial rest frame to compare its rate of ticking with a clock on a tower that it's passing right next to, then the satellite will conclude that at that moment the tower clock is ticking slower. But the GPS calculations don't bother to calculate things from the perspective of any coordinate system but the Earth-centered one--that doesn't mean the GPS calculations somehow contradict the claim that if you used such an alternate coordinate system you might get a different answer to the question of which of two clocks was ticking slower at a given moment, they simply don't address the issue of alternate coordinate systems.

As to the other, the article says,

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/

See chapter 4.

How is your logic consistent with this?

When you say "as to the other", what part of my post are you referring to? I don't understand how this is supposed to contradict "my logic" in any way.

Also, this v is relative in the equations and thus it would lead one to wonder where the reciprocal time dilation occurs.

I've asked you a hundred times what you mean when you talk about velocities being "relative" and you never answer. The v in the equation is defined in terms of one particular coordinate system, and we can only talk about "reciprocal time dilation" when comparing multiple coordinate systems. Please tell me whether you agree or disagree with this (another question I keep asking over and over and you never give me a straight answer).

 

[S.Vasojevic]

In the GPS, the time variable becomes a coordinate time in the rotating frame of the earth, which is realized by applying appropriate corrections while performing synchronization processes. Synchronization is thus performed in the underlying inertial frame in which self-consistency can be achieved.

What is exactly your question?

 

[cfrogue]

And do you agree that they'll only calculate different things about which clock is ticking slower if they use different frames to do their calculations? That they are free to agree to use the same frame for their calculations (even though one or both are not at rest in this frame), in which case they will naturally agree about which clock is ticking slower?

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?

 

[cfrogue]

Sure, if the satellite uses its own local inertial rest frame to compare its rate of ticking with a clock on a tower that it's passing right next to, then the satellite will conclude that at that moment the tower clock is ticking slower. But the GPS calculations don't bother to calculate things from the perspective of any coordinate system but the Earth-centered one--that doesn't mean the GPS calculations somehow contradict the claim that if you used such an alternate coordinate system you might get a different answer to the question of which of two clocks was ticking slower at a given moment, they simply don't address the issue of alternate coordinate systems.

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

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

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

Is this correct?

Now, if I were in the space shuttle, would I conclude the Earth is moving and the shuttle is at rest?

 

[cfrogue]

i

I've asked you a hundred times what you mean when you talk about velocities being "relative" and you never answer. The v in the equation is defined in terms of one particular coordinate system, and we can only talk about "reciprocal time dilation" when comparing multiple coordinate systems. Please tell me whether you agree or disagree with this (another question I keep asking over and over and you never give me a straight answer).

A hundred times?

I am OK with the Earth center relative motion.

Are you?

 

[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
<br /> 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<br />

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?