Is the Twins Paradox Equivalent to GPS Time Dilation?

[Nugatory]

note to self: must remember not all IFR's are at rest!)

Back up a bit, to “must learn what a frame is”. A frame, whether inertial or not, is a rule for assigning coordinates to events. Saying that a frame is or is not at rest makes no more sense than saying that the Pythagorean Theorem is at rest - it’s not a thing that moves.

 

[Buckethead]

By jumping into GR before you have got the basics of SR you have got yourself in an almighty tangle here! What a confusion of ideas. I'm not sure where to start to try to disentangle that.

Maybe I'm just thinking of GR or rather relativity in a gravitational field as indicated by my example as being just another straightforward effect with regard to clock rates. For example in SR you can calculate time dilation using the Lorentz formula, very simple. Is it not also the case that given a pure example of two objects not moving relative to each other in a gravitational field an easily calculatable situation if you want to know how the two clocks vary over time? Or is it far far more complex than that?

What does a claim that an inertial frame of reference is "at rest" even mean?

From what I've learned in this thread, to me it means an IFR that is in flat spacetime. i.e. you can use the Lorentz transform to calculate differences in clock rates.

I think your best bet is to ignore General Relativity entirely and work on getting Special Relativity (or even Newtonian mechanics) firmly understood first.

Not sure my brain is that capable :)

 

[PeroK]

Is it not also the case that given a pure example of two objects not moving relative to each other in a gravitational field an easily calculatable situation if you want to know how the two clocks vary over time? Or is it far far more complex than that?

Before analysing one of these problems, I suggest you pick a coordinate system and stick with it. Part of the problem is that you jump between an undefined global coordinate system and local (inertial) coordinate systems.

The lack of global inertial reference frames in GR means you need an even more disciplined approach than you do in SR.

 

[Buckethead]

Back up a bit, to “must learn what a frame is”. A frame, whether inertial or not, is a rule for assigning coordinates to events. Saying that a frame is or is not at rest makes no more sense than saying that the Pythagorean Theorem is at rest - it’s not a thing that moves.

As in "from Bob's frame of reference or Alice's frame of reference"? But what do you call something that is an IFR that is not changing direction? Can't you call that a frame at rest? What would you call it?

 

[Buckethead]

Before analysing one of these problems, I suggest you pick a coordinate system and stick with it. Part of the problem is that you jump between an undefined global coordinate system and local (inertial) coordinate systems.

The lack of global inertial reference frames in GR means you need an even more disciplined approach than you do in SR.

In a simple relative motion case I think of frames like this: From Bob's frame, Alice's clock is ticking more slowly. From Alice's frame, Bob's clock is ticking more slowly. So, yes, I guess I can see, in my example of a satellite orbiting a planet, I would have to do the same thing. From the planet bound clock, the orbiting clock does this, but from the orbiting clock, the planet bound clock does that.

 

[PeroK]

In a simple relative motion case I think of frames like this: From Bob's frame, Alice's clock is ticking more slowly. From Alice's frame, Bob's clock is ticking more slowly. So, yes, I guess I can see, in my example of a satellite orbiting a planet, I would have to do the same thing. From the planet bound clock, the orbiting clock does this, but from the orbiting clock, the planet bound clock does that.

As far as I was aware, Bob and Alice didn't get as far as the course on GR. Unless you're talking about Bob Schwarzschild and Alice Eddington-Finkelstein.

 

[jbriggs444]

In a simple relative motion case I think of frames like this: From Bob's frame, Alice's clock is ticking more slowly. From Alice's frame, Bob's clock is ticking more slowly. So, yes, I guess I can see, in my example of a satellite orbiting a planet, I would have to do the same thing. From the planet bound clock, the orbiting clock does this, but from the orbiting clock, the planet bound clock does that.

There is no inertial frame of reference in which the satellite clock is always at rest. So you can't put Bob on the satellite and pretend that there is an inertial frame where Bob is always at rest.

You can assemble coordinate systems in which Bob is always at rest. But there is no unique unambiguously determined way to go about doing so. And no matter what approach you take, the resulting coordinate system will have things that do not work as you might expect. None of them amount to an inertial frame. Simplistic reasoning needs to be tempered with caution.

 

[Buckethead]

There is no inertial frame of reference in which the satellite clock is always at rest. So you can't put Bob on the satellite and pretend that there is an inertial frame where Bob is always at rest.

You can assemble coordinate systems in which Bob is always at rest. But there is no unique unambiguously determined way to go about doing so. And no matter what approach you take, the resulting coordinate system will have things that do not work as you might expect. None of them amount to an inertial frame. Simplistic reasoning needs to be tempered with caution.

OK, I think I can see why that is the case. I suppose you could do it for one small moment in time and space, but they you would need to recalculate immediately for the next increment in time or change in direction. Similar to when the twin in twin paradox turns around, but instead of just once, constantly, as the direction of motion for the satellite changes.

This also explains why calculating the difference in time between an orbiting clock and a stationary one is doable (and why its asymmetric) in the same way that a twin paradox is calculable once you include the turn around point (where the shift in the Earth clocks time suddenly jump from behind to ahead). The orbiting clock scenario is just a bunch of tiny little twin paradoxes.

 

[PeterDonis]

I have been thinking this whole time that something like an orbiting object following a geodesic through spacetime is at rest.

"At rest" is relative to a choice of coordinates. There is a choice of coordinates in which the orbiting object following a geodesic through spacetime is at rest--in fact there is such a choice of coordinates for any object. (In fact, as @jbriggs444 points out, there are multiple ways to construct such coordinates for any object.)

Those coordinates just won't be a global inertial frame--since there are no global inertial frames at all in curved spacetime.

(note to self: must remember not all IFR's are at rest!)

This is nonsense. The concept of "at rest" doesn't even make sense for a frame. It only makes sense for objects relative to a choice of coordinates.

(note to self: IFR's can have different spacetime geometries!)

This is nonsense as well. The concept of "spacetime geometry" doesn't even make sense for a frame. It only makes sense for a spacetime.

I was thinking about the simplest example of an orbiting satellite and two clocks, one of which is placed at the center of an ideal planet (perfectly round, even density) and the other on the satellite in a perfectly circular orbit.

In such a case can I assume that there is no relative velocity between the satellite and the fixed clock?

The concept of "relative velocity" makes no sense in a curved spacetime for objects that are not co-located.

You can choose coordinates in which both clocks are at rest; you can choose coordinates where one clock is at rest and the other isn't; or you can choose coordinates in which neither clock is at rest. That's all you can do.

is there a gravitational field at the center of this ideal planet.

It depends on what you mean by "gravitational field".

The field must be null, but probably not non-existant and as a result must be considered as having a different spacetime geometry than flat space.

This is mostly word salad. The only part that makes sense is that yes, the spacetime geometry at the center of the planet will not be flat.

 

[PeterDonis]

Maybe I'm just thinking of GR or rather relativity in a gravitational field as indicated by my example as being just another straightforward effect with regard to clock rates.

I guess you didn't catch this statement of mine in an earlier post:

there is no single rule you can apply that will explain the asymmetry in all cases. There just isn't. Looking for one is a waste of time. It isn't there.

 

[pervect]

The classic demonstration of Peter's point, "It just isn't there", is the Einstein's train thought experiment in special relativity, which illustrates that different inertial frames have different ideas of which events occur "at the same time". This is known as the relativity of simultaneity, and has been discussed a lot on the forums.

Unfortunately, many many people struggle to grasp this concept. I am not sure how to help, really - pointing out references on the topics is the best approach I've thought of, though the success rate seems rather low. An honorable mention goes to Scherr (et al) paper, "The challenge of changing deeply-held student beliefs about the relativity of simultaneity", however its not clear how to apply the concepts suggested by the paper in a forum environment. I do like to mention the paper frequently, even at the same time I suspect that not many actually go out and read it.

 

[Frodo]

Neil Ashby's https://www.researchgate.net/publication/26386594_Relativity_in_the_Global_Positioning_System gives an excellent account of the relativistic corrections made in the GPS system.

The correction for the Sagnac Effect is amusing. The design team accepted the Sagnac Effect and designed for it but also allowed for the correction to be swicched out. A flight test validated the decision to accept it.

 

[binis]

Summary:: Twins paradox and GPS equivalent?
In a thread I started awhile back, in the common twins paradox scenario, it was indicated to me that the actual time (paraphrasing) on Earth for any given time in the ship is basically “undefined” (as it can’t be verified) and/or time dilated (ticking slower) for the trip out and then shifted to having advanced quickly for the trip back resulting in an older twin on Earth. In other words, it will be symmetrical (time dilation as seen from either twin) but not “real” or “verifiable” until the trip is complete in which case there in an asymmetry in rate of time resulting in the Earth twin being older (due to the frame change of the space ship).

Only relative velocity is physically meaningful. We can choose to interpret either ship or Earth to be at rest. So,why the trip back is not resulting in the ship twin being older?

However….In a GPS system, the time dilation is real and verifiable and is asymmetric. The orbiting satellite clock ticks more slowly, all the time, and this can be measured at any time.

How can be measured? Every signal gave off the satellite's clock towards Earth is redshifted due to velocity causing an apparent time dilation

Because of this the GPS clock is adjusted for the speed difference until the clocks both tick one second per second Earth time.

Linear velocity v is by definition the runned space s per time t or v=s/t If the time passed in a specific point on the earth`'s surface is t, the dilated time passed on satellite is t'. What is the speed? Is it s/t or s/t'? Only relative velocity is physically meaningful. We can choose to interpret either satellite or Earth to be at rest.

 

[Grasshopper]

Man, there is so much to learn, so many subtleties. So many misrepresentations in the pop-science discussion of the topic. Perhaps necessary to an extent to communicate some of the general ideas in a way that might inspire awe in the lay audience, but it’s beginning to seem to me that those same communications are the author of tremendous confusion among people trying to take their understanding further, and surely these communications are the cause of tremendous frustration to those wanting to share the true details of relativity.

Ugh.

 

[jbriggs444]

Only relative velocity is physically meaningful. We can choose to interpret either ship or Earth to be at rest. So,why the trip back is not resulting in the ship twin being older?

The ship accelerates. There is no interpretation in which the ship is always at rest in a single inertial frame.

How can be measured? Every signal gave off the satellite's clock towards Earth is redshifted due to velocity causing an apparent time dilation

How can you measure the rate of a remote clock? One could, for instance, have the remote clock send a radio signal and encode its current clock reading in the signal. One could have a ground station receive the signal and thereby determine the remote clock reading at the event of transmission. And, of course, one can put a clock on the ground station.

Then one could have two such transmission events and two corresponding reception events. The ground station could compare the elapsed time per the received transmissions against the elapsed time per the ground clock and compute a time dilation rate.

Would there be a red shift due to velocity? Yes. Could you normalize this out of the measured results? Yes.
Would there be a blue shift due to altitude? Yes. Could you normalize this out of the measured results? Yes.

Could there be an additional red or blue shift due to Doppler? Yes. Could you account for this as a change in radio path length? Yes. This is roughly how GPS works. You measure the discrepancy between various measured results of reading remote clocks, use that information to infer some radio path lengths and thereby infer the position of the receiver.

One actually dispenses with the ground clock on the GPS receiver, since the availability of multiple remote clocks makes it unnecessary. Atomic clocks are bulkier and more expensive then GPS receivers.

 

[PeterDonis]

Only relative velocity is physically meaningful.

Wrong. Proper acceleration is also physically meaningful.

We can choose to interpret either ship or Earth to be at rest.

But we cannot choose to interpret the ship as being at rest in the same inertial frame during the entire scenario in the standard twin paradox scenario. We can do that for Earth in that scenario.

 

[PeterDonis]

Every signal gave off the satellite's clock towards Earth is redshifted due to velocity causing an apparent time dilation

Redshift and "apparent time dilation" are not the same thing.

Also, the frequency shift "due to velocity" is not always a redshift; it can be a blueshift, or no shift at all, depending on the direction and magnitude of the velocity.

Also, if you are going to try to analyze the observed frequency shift in signals from GPS satellites, you need to also take into account the gravitational blueshift due to the change in altitude.

But in any case, all of that is irrelevant to the operation of GPS, because GPS does not work by having the receiver measure the observed frequency of light signals from satellites.

Only relative velocity is physically meaningful.

Wrong. Altitude, which leads to gravitational time dilation, is also physically meaningful.

We can choose to interpret either satellite or Earth to be at rest.

But we cannot choose to interpret the satellite as being at rest in the same inertial frame for an entire orbit. We can do that for the Earth, or more precisely the center of the Earth (since the GPS system also has to take into account the Earth's rotation).

 

[binis]

Also, the frequency shift "due to velocity" is not always a redshift; it can be a blueshift, or no shift at all, depending on the direction and magnitude of the velocity.
Also, if you are going to try to analyze the observed frequency shift in signals from GPS satellites, you need to also take into account the gravitational blueshift due to the change in altitude.
But in any case, all of that is irrelevant to the operation of GPS, because GPS does not work by having the receiver measure the observed frequency of light signals from satellites.

I didn't talk about frequency shift . Any kind of signals could be used,they will appear an apparent delay due to the redshift/blueshift (doppler) effect.

 

[jbriggs444]

I didn't talk about frequency shift . Any kind of signals could be used,they will appear an apparent delay due to the redshift/blueshift (doppler) effect.

I am not sure what this "apparent delay" thing is supposed to be. The adjective "apparent" implies that it is a coordinate effect. So it would be natural to assume that we are talking about the relativity of simultaneity. But the doppler effect is real, not apparent. So that can't be it.

 

[Nugatory]

Any kind of signals could be used,they will appear an apparent delay due to the redshift/blueshift (doppler) effect.

Redshift and blueshift don’t cause an apparent delay, they just change the frequency.

 

[binis]

So it would be natural to assume that we are talking about the relativity of simultaneity.

Yes,this is.Doppler effect on the "ticking" rate (frequency) of the clock as it transmitted by signals.

 

[PeterDonis]

Yes,this is.Doppler effect on the "ticking" rate (frequency) of the clock as it transmitted by signals.

The Doppler effect is not the same thing as relativity of simultaneity.

 

[binis]

Would there be a red shift due to velocity? Yes. Could you normalize this out of the measured results?

I think no,because you don't know the velocity (post 43)

Could there be an additional red or blue shift due to Doppler? Yes. Could you account for this as a change in radio path length?

I don't think this is really taken into account.Please provide a link if you know.

 

[PeterDonis]

I don't think this is really taken into account.

All of the issues you are raising are taken into account in the operation of the GPS system. The people who designed and implemented it were well aware of relativity and its implications for what they were doing.

Please provide a link if you know.

See the reference given in post #42.

 

[binis]

Thanks.And what about the last query in #43? What is the satellite's angular velocity? Is it ω/t or ω/t' ?

 

[jbriggs444]

Thanks.And what about the last query in #43? What is the satellite's angular velocity? Is it ω/t or ω/t' ?

The last query in #43 has to do with speed, not angular velocity. I quote it here:

Linear velocity v is by definition the runned space s per time t or v=s/t If the time passed in a specific point on the earth`'s surface is t, the dilated time passed on satellite is t'. What is the speed? Is it s/t or s/t'? Only relative velocity is physically meaningful. We can choose to interpret either satellite or Earth to be at rest.

Speed would be the magnitude of velocity. Velocity would be incremental displacement divided by elapsed time. Important is that the displacement and the time are taken from the same coordinate system. So the answer is that the speed of an object in the non-primed coordinate system is given by:$$|\frac{ds}{dt}|$$while the speed of an object in the primed coordinate system is given by:$$|\frac{ds'}{dt'}|$$

For the separate question of angular velocity, conventionally $\omega$ is already an angular velocity. Dividing it by a time interval would yield something like an angular acceleration. However, consistency demands that one pick a frame of reference and stick with it. It is normally improper to mix measured values from the primed and unprimed frame. So angular acceleration in the unprimed frame would be $$\frac{d \omega}{d t}$$while angular acceleration in the primed frame would be $$\frac{d \omega'}{d t'}$$ The quantity "rapidity" "celerity" is the most commonly used violation to the heuristic about not mixing measurements from different frames. It is computed as distance traveled in the stationary frame divided by elapsed proper time for the moving entity. Importantly, this means that "celerity" is not the same thing as velocity.

[Thank you, @DrGreg]

 

[DrGreg]

It is computed as distance traveled in the stationary frame divided by elapsed proper time for the moving entity.

On a technicality, that's actually the definition of "celerity" (also known as "proper velocity", a name that I consider to be misleading), but the general point you are making applies to both rapidity and celerity.

 

[Herman Trivilino]

In a thread I started awhile back, in the common twins paradox scenario, it was indicated to me that the actual time (paraphrasing) on Earth for any given time in the ship is basically “undefined” (as it can’t be verified) and/or time dilated (ticking slower) for the trip out and then shifted to having advanced quickly for the trip back resulting in an older twin on Earth. In other words, it will be symmetrical (time dilation as seen from either twin) but not “real” or “verifiable” until the trip is complete in which case there in an asymmetry in rate of time resulting in the Earth twin being older (due to the frame change of the space ship).

This is not a correct understanding of the symmetry of time dilation. And that misunderstanding is at the root of your misunderstanding of the twin paradox. First, you need to understand the role played by simultaneity in the symmetry of time dilation. Another thing that might help you is to draw a spacetime diagram of the situation. When you do that you immediately see that the twins take different paths through spacetime, that those paths are not of equal length, and therefore they each experience different amounts of elapsed time.

 

[vanhees71]

Another important point in the twin paradox is that they have to synchronize their clocks at the same place when one of them starts to travel and then have to compare their clocks when meeting again at the same place. Then their clocks show each of the twins' proper time. This "clock postulate" has been confirmed by precision measurements with high accuracy in many ways.

 

[binis]

Important is that the displacement and the time are taken from the same coordinate system.
It is normally improper to mix measured values from the primed and unprimed frame.

Right,but only relative velocity is physically meaningful. We can choose to interpret either satellite or Earth to be at rest.And the relative velocity is one,having one value (magnitube=speed).What is the value?