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

[cfrogue]

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:

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.

OK, perhaps you can explain this.

1) relative motion and reciprocal time dilation

2) Your absolute standard you are using that causes a v to occur but is not relative motion and time dilation is one way.

 

[cfrogue]

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.

So, if you assume there does not exist a gravity potential difference and there does not exist an orbital path, does this result in reciprocal time dilation?

 

[JesseM]

OK, perhaps you can explain this.

1) relative motion and reciprocal time dilation

2) Your absolute standard you are using that causes a v to occur but is not relative motion and time dilation is one way.

Huh? I don't understand what you want me to explain. Neither of these is a question, or even a complete sentence.

And can you please answer the question I asked you?

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?

Note that pervect's analysis only involved a single coordinate system too (Schwarzschild coordinates), so there can be no issue of reciprocity there.

 

[cfrogue]

Huh? I don't understand what you want me to explain. Neither of these is a question, or even a complete sentence.

And can you please answer the question I asked you?

OK, your question involved one frame.

I do not know how to answer this question since there is the Earth frame and the satellite frame.

How is there only one since all motion is relative? I am ignoring the orbit and the gravity difference.

Note that pervect's analysis only involved a single coordinate system too (Schwarzschild coordinates), so there can be no issue of reciprocity there.

How does this Schwarzschild metric reduce to SR with only inertial motion?

 

[cfrogue]

Huh? I don't understand what you want me to explain. Neither of these is a question, or even a complete sentence.

And can you please answer the question I asked you?

Note that pervect's analysis only involved a single coordinate system too (Schwarzschild coordinates), so there can be no issue of reciprocity there.

I think I know where we are not meeting the minds.

Gravity and acceleration are accepted as absolute motion.

Inertial motion is not.

Is this correct?

I am attempting to break the two into pieces as was done in the article.

Does this make sense?

 

[JesseM]

OK, your question involved one frame.

I do not know how to answer this question since there is the Earth frame and the satellite frame.

There is no "satellite frame" being used in the GPS system calculations. It seems like you think that because the satellites are moving relative to the Earth-centered frame that somehow obligates us to consider a separate "satellite frame", but that just isn't correct. Consider the analogy of the twin paradox in SR, where we are free to analyze the whole problem from the perspective of the Earth twin's frame, getting an answer to questions about how much each twin will have aged when they reunite without ever doing any calculations from the perspective of a frame where the traveling twin was at rest. You could do a separate analysis from the perspective of such a frame, but there is no obligation to if you are just interested in answering physical questions like how much each twin ages.

How is there only one since all motion is relative?

Frames are not real physical entities, it makes no sense to talk as though there were an objective truth about whether there is "only one" or there "are" multiple frames in any given situation. A frame is just a coordinate system, an imaginary spacetime grid which we have dreamed up for the purpose of making calculations. Frames don't come into existence because of the motion of objects--no matter what physical situation we are analyzing, no matter what number of objects are involved or what their motions are, we can analyze the situation entirely from the perspective of one frame, or we can analyze it from the perspective of a million distinct frames (including frames where none of the objects being analyzed are at rest), it's totally up to us. You are certainly free to imagine a "satellite frame" where the satellite is at rest if you really want to, but no such frame is actually used in the GPS computer calculations.

How does this Schwarzschild metric reduce to SR with only inertial motion?

It doesn't reduce to SR, the Schwarzschild metric is a curved spacetime while SR deals only with flat spacetime.

 

[JesseM]

I think I know where we are not meeting the minds.

I think the problem is that you are treating "frames" as real entities that either exist or don't exist depending on the motion of objects, when really they are just imaginary coordinate grids that we choose according to our own whims.

Gravity and acceleration are accepted as absolute motion.

Inertial motion is not.

Is this correct?

I'm not sure what you mean by this. In SR there is an absolute truth about whether something is moving inertially or accelerating, and in GR there is an absolute truth about whether an object is in freefall (which is what 'inertial' motion means in curved spacetime) or not, as well as an absolute truth about whether spacetime is curved (which is what 'gravity' means in GR) or flat. Even if this is what you were getting at, though, I don't really see what it has to do with our discussion about frames and "reciprocity".

I am attempting to break the two into pieces as was done in the article.

As pervect showed, you can only break it in two in an approximate way, the actual time dilation calculated in curved GR spacetime is not precisely equal to a product of velocity-based time dilation and gravitational time dilation.

 

[cfrogue]

There is no "satellite frame" being used in the GPS system calculations. It seems like you think that because the satellites are moving relative to the Earth-centered frame that somehow obligates us to consider a separate "satellite frame", but that just isn't correct. Consider the analogy of the twin paradox in SR, where we are free to analyze the whole problem from the perspective of the Earth twin's frame, getting an answer to questions about how much each twin will have aged when they reunite without ever doing any calculations from the perspective of a frame where the traveling twin was at rest. You could do a separate analysis from the perspective of such a frame, but there is no obligation to if you are just interested in answering physical questions like how much each twin ages.

Well, I assume a theory can answer questions from all perspectives within the theory and get the same answer for the same problem.

Do you agree?

For example, Einstein's solution to the twins paradox is that all parties agree the traveling twins ages less.

Frames are not real physical entities, it makes no sense to talk as though there were an objective truth about whether there is "only one" or there "are" multiple frames in any given situation. A frame is just a coordinate system, an imaginary spacetime grid which we have dreamed up for the purpose of making calculations. Frames don't come into existence because of the motion of objects--no matter what physical situation we are analyzing, no matter what number of objects are involved or what their motions are, we can analyze the situation entirely from the perspective of one frame, or we can analyze it from the perspective of a million distinct frames (including frames where none of the objects being analyzed are at rest), it's totally up to us.

If you look at the link I posted, there exists experiment evidence that v time dilation exists. Sure, it also accounts for gravity and orbit in the equations.

But, the link clearly demarcates this relative v.

Thus, one would assume a reciprocal time dilation relationship.

Why is this not the case?

It doesn't reduce to SR, the Schwarzschild metric is a curved spacetime while SR deals only with flat spacetime.

Then how does this apply to the relative v in the link?

 

[cfrogue]

I think the problem is that you are treating "frames" as real entities that either exist or don't exist depending on the motion of objects, when really they are just imaginary coordinate grids that we choose according to our own whims.

So, should we say reciprocal time dilation is false and that it is only one way as proven by GPS satellites?

As pervect showed, you can only break it in two in an approximate way, the actual time dilation calculated in curved GR spacetime is not precisely equal to a product of velocity-based time dilation and gravitational time dilation

I am OK with that. But, I want them to match the graph which is the experiment evidence and then I would like to know why reciprocal time dilation is false.

 

[JesseM]

Well, I assume a theory can answer questions from all perspectives within the theory and get the same answer for the same problem.

Do you agree?

As long as you are asking frame-independent questions about events which occur at a single local point in spacetime, like what ages the two twins are at the moments they meet. Different frames would have different answers to questions that are not "local" in this way, like what age the Earth twin is at the moment the traveling twin turns around (because of the relativity of simultaneity there will be different answers in different frames). I would say that only questions about local events are really "physical" questions.

If you look at the link I posted,

Which one?

there exists experiment evidence that v time dilation exists. Sure, it also accounts for gravity and orbit in the equations.

But, the link clearly demarcates this relative v.

Thus, one would assume a reciprocal time dilation relationship.

Why is this not the case?

Are you even reading what I write? "Reciprocal" time dilation relationships only make sense when you are comparing multiple frames, like how if you're moving away from me at 0.6c in SR, then in my inertial rest frame your clock is running slower than mine by a factor of 0.8, but in your inertial rest frame my clock is running slower than yours by a factor of 0.8. But frames are not real things, they are just imaginary coordinate grids which we can choose to use or not use as we please. If you choose to analyze this situation from both my rest frame and your rest frame, then you will see this sort of "reciprocal" time dilation, but if you choose to analyze the situation from only a single frame, then in this frame there is only going to be a single truth about which clock is running slower. Is this really so hard to understand?

 

[JesseM]

So, should we say reciprocal time dilation is false and that it is only one way as proven by GPS satellites?

Reciprocal time dilation is not "true" or "false" in any objective way, just like frames do not "exist" or "not exist" in any objective way. If you choose to analyze the same situation from the perspective of multiple frames, then different frames can have reciprocal perspectives about which clock is running slower. But if you choose to analyze this situation from the perspective of only one frame, then in this single frame there is only a single answer to the question of which clock is running slower.

 

[cfrogue]

As long as you are asking frame-independent questions about events which occur at a single local point in spacetime, like what ages the two twins are at the moments they meet. Different frames would have different answers to questions that are not "local" in this way, like what age the Earth twin is at the moment the traveling twin turns around (because of the relativity of simultaneity there will be different answers in different frames). I would say that only questions about local events are really "physical" questions.

We have to be very careful on expressing events do we not? For example, the light cone sets up an absolute standard on deciding whether events are in the absolute past. No observer in the universe can disagree on events as implemented by the light cone.

Thus, why do you think we can universally generalize events and put them under the context of R of S whch is a relative standard?

Which one?

The original link I posted.

Are you even reading what I write? "Reciprocal" time dilation relationships only make sense when you are comparing multiple frames, like how if you're moving away from me at 0.6c in SR, then in my inertial rest frame your clock is running slower than mine by a factor of 0.8, but in your inertial rest frame my clock is running slower than yours by a factor of 0.8. But frames are not real things, they are just imaginary coordinate grids which we can choose to use or not use as we please. If you choose to analyze this situation from both my rest frame and your rest frame, then you will see this sort of "reciprocal" time dilation, but if you choose to analyze the situation from only a single frame, then in this frame there is only going to be a single truth about which clock is running slower. Is this really so hard to understand?

I read what you wrote.

Perhaps you could specifically explain the experimental evidence of a relative v time dilation and then explain why a relative v does not imply reciprocal time dilation as demanded by SR.

That is the part I still do not get.

 

[cfrogue]

there is no mistake trout in the logic of this thread.

Oh, BTW,
you can integrate with the coords of Tau.

Here is the mainstream article.

http://arxiv.org/PS_cache/physics/pdf/0411/0411233v1.pdf

 

[JesseM]

We have to be very careful on expressing events do we not? For example, the light cone sets up an absolute standard on deciding whether events are in the absolute past. No observer in the universe can disagree on events as implemented by the light cone.

Thus, why do you think we can universally generalize events and put them under the context of R of S whch is a relative standard?

What does "universally generalize events" mean, and why do you think I am doing that?

I read what you wrote.

Perhaps you could specifically explain the experimental evidence of a relative v time dilation

What does "relative v time dilation" mean? I already noted the ambiguity of your use of the word "relative" in post #14:

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.

Then I noted it again in post #32 and asked what you meant by it, but you never answered this question:

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.

As I keep saying, all velocities and time dilations are defined relative to particular choices of coordinate system--do you disagree, and if so can you offer another way of defining velocity or time dilation which does not involve a particular choice of coordinate system?

and then explain why a relative v does not imply reciprocal time dilation as demanded by SR.

I keep telling you that "reciprocal time dilation" only makes sense when you are comparing two different frames, and that it is totally a matter of choice what frames you choose to use in your analysis. If you choose to analyze a pair clocks in relative motion using the two different frames where each clock is at rest, you will see reciprocal time dilation (i.e. in clock #1's rest frame clock #2 will be running slow, and in clock #2's rest frame clock #1 will be running slow). If you choose to analyze these two clocks from the perspective of a single frame, then there is only one answer to the question of which clock is running slower, so there is nothing "reciprocal" here. Do you think SR "demands" that there should be reciprocal time dilation in a single frame? Or do you think SR "demands" that we are forbidden to analyze a situation from the perspective of a single frame, that we are obligated to use multiple frames? If your answer to either of these questions is "yes", then you are just totally confused. And if your answer to both questions is "no", then I have no idea what you mean when you say reciprocal time dilation is "demanded by SR".

 

[cfrogue]

What does "relative v time dilation" mean? I already noted the ambiguity of your use of the word "relative" in post #14:

You said in #10
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.



So, you seem to mean there is relative motion and no time reciprocal time dilation.

How do you make this happen?

 

[cfrogue]

What does "universally generalize events" mean, and why do you think I am doing that?

You invoked R of S for describing events and that is not the only way to describe events. Thee is also the light cone.

 

[JesseM]

You said in #10
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.



So, you seem to mean there is relative motion and no time reciprocal time dilation.

How do you make this happen?

How can I answer this question if you won't answer my own question about what you mean by "relative motion"? Likewise you refuse to answer my question about whether you agree or disagree that "reciprocal time dilation" only makes sense when we are comparing multiple frames, that it is meaningless to talk about reciprocal time dilation in the context of a single frame. If you don't answer my questions about the meaning of your vague phrases, I can't answer yours.

 

[cfrogue]

As I keep saying, all velocities and time dilations are defined relative to particular choices of coordinate system--do you disagree, and if so can you offer another way of defining velocity or time dilation which does not involve a particular choice of coordinate system?

No I do not disagree.

So, I want the normal SR interpretation.

Why is this wrong?

 

[JesseM]

You invoked R of S for describing events and that is not the only way to describe events. Thee is also the light cone.

I didn't say anything about it being "the only way to describe events". In fact if you were paying attention, you'd have noticed I was saying that the only questions that have objective physical answers are questions that can be defined in a purely local manner (like a question about the age of two twins at the moment they both pass right next to each other), and that questions about simultaneity therefore are not really "physical" or "objective" in this sense, since they are not local.

 

[cfrogue]

I keep telling you that "reciprocal time dilation" only makes sense when you are comparing two different frames, and that it is totally a matter of choice what frames you choose to use in your analysis. If you choose to analyze a pair clocks in relative motion using the two different frames where each clock is at rest, you will see reciprocal time dilation (i.e. in clock #1's rest frame clock #2 will be running slow, and in clock #2's rest frame clock #1 will be running slow). If you choose to analyze these two clocks from the perspective of a single frame, then there is only one answer to the question of which clock is running slower, so there is nothing "reciprocal" here. Do you think SR "demands" that there should be reciprocal time dilation in a single frame? Or do you think SR "demands" that we are forbidden to analyze a situation from the perspective of a single frame, that we are obligated to use multiple frames? If your answer to either of these questions is "yes", then you are just totally confused. And if your answer to both questions is "no", then I have no idea what you mean when you say reciprocal time dilation is "demanded by SR".

I think I have been very clear to choose the Earth frame and the satellite frame.

And, no I do not think SR forbids looking at a problem from one frame.

But, we would not be scientific if we did not conisder the problem from all perspectives.


Thus, SR invokes reciprocal time dilation without experiment evidence to support it by GPS. In fact, GPS would show it is false.

Do you know why this is or where I am wrong?

 

[cfrogue]

I didn't say anything about it being "the only way to describe events". In fact if you were paying attention, you'd have noticed I was saying that the only questions that have objective physical answers are questions that can be defined in a purely local manner (like a question about the age of two twins at the moment they both pass right next to each other), and that questions about simultaneity therefore are not really "physical" or "objective" in this sense, since they are not local.

Should we take a timeout?

You seem emotional.

 

[JesseM]

I think I have been very clear to choose the Earth frame and the satellite frame.

But the GPS system does not do any calculations from the perspective of the "satellite frame", and you imply there is something problematic about this. Why?

And, no I do not think SR forbids looking at a problem from one frame.

So you'd agree the same is true in GR? There is nothing incorrect about doing all the calculations in an Earth-centered frame?

But, we would not be scientific if we did not conisder the problem from all perspectives.

Why wouldn't it be scientific? Did the science gods command us to repeat every calculation from the perspective of multiple frames, even if we have already answered all physical questions about local events using only one frame? And there are an infinite number of distinct frames you could use to analyze every problem--are we not being scientific if we don't use every possible one?

Thus, SR invokes reciprocal time dilation without experiment evidence to support it by GPS. In fact, GPS would show it is false.

Do you know why this is or where I am wrong?

cfrogue, if it seems like I'm getting frustrated with you it's because I am--you keep repeating the same vague and ambiguous phrases without ever seeming to pay any attention to the criticisms I offer, or the questions I ask that are meant to clarify your meaning. Did you not read the last hundred posts of mine where I said reciprocal time dilation only makes sense when comparing multiple frames, and where I pointed out over and over again that it's your choice what frames to use in any given calculation, and that the GPS system calculations are based on the choice to use only a single frame? How could the GPS system show "reciprocal time dilation" to be false when it doesn't even attempt to compare multiple frames, and when it is a priori impossible that calculations done in a single frame (like GPS) could show any reciprocal time dilation? Please actually think about what I am saying here, instead of just ignoring virtually all my comments and repeating the same cliched phrases and questions over and over again.

 

[Phrak]

"SR time dilation" seems to be misidentified, where ever it came from. We are comparing 2 clocks, not 3.

 

[JesseM]

"SR time dilation" seems to be misidentified, where ever it came from. We are comparing 2 clocks, not 3.

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).

 

[Jorrie]

Should we take a timeout?

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?

 

[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?