The Twins Paradox: Exploring Time Dilations' Effects

[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.

 

[Mister T]

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?

 

[jbriggs444]

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?

LMGTFY

Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour (the orbital period is roughly 12 hours - contrary to popular belief, GPS satellites are not in geosynchronous or geostationary orbits).

 

[binis]

LMGTFY

Each satellite in the GPS constellation orbits at an altitude of about 20,000 km from the ground, and has an orbital speed of about 14,000 km/hour (the orbital period is roughly 12 hours - contrary to popular belief, GPS satellites are not in geosynchronous or geostationary orbits).

That is,by #56,(ds/dt)=(ds'/dt')=14000 km/h?

 

[jbriggs444]

That is,by #56,(ds/dt)=(ds'/dt')=14000 km/h?

Let us assume that you are using special relativity, an unprimed inertial frame where the center of the Earth is at rest and a primed inertial frame where the satellite is instantaneously at rest.

If you were trying to use general relativity then there would be no inertial frames available [though we may have some coordinate systems that are close enough to inertial for our purposes]. If you were trying to use a frame where the satellite is continuously at rest then we are not talking about inertial frames any longer and you'll need to identify a coordinate system before we can agree on what $\frac{ds'}{dt'}$ means.

Note that, in particular, using general relativity to argue that the satellite is "inertial" and then trying to use special relativity to argue that only relative velocities are significant will yield a nonsense result.

In the earth-centered inertial (ECI) frame, the satellite's velocity, $\frac{ds}{dt}$ is 14,000 km/h.
In the satellite's tangent inertial frame, the Earth's velocity, $\frac{ds'}{dt'}$ is 14,000 km/h.

Surely this was already obvious?

 

[binis]

Let us assume that you are using special relativity, an unprimed inertial frame where the center of the Earth is at rest and a primed inertial frame where the satellite is instantaneously at rest.If you were trying to use general relativity then there would be no inertial frames available

Τhe principle of relativity, is that there is no such thing as "earth motion" or "satellite motion", only relative motion.The satellite movement is equivalent of the earth's.Therefore ds=ds'

In the earth-centered inertial (ECI) frame, the satellite's velocity, $\frac{ds}{dt}$ is 14,000 km/h.

Satellite is almost immobile relative to the center of the Earth,differently to a point on the surface.

 

[A.T.]

Τhe principle of relativity,...

Τhe principle of relativity applies to inertial frames. There are no global inertial frames in curved space time.

Satellite is almost immobile relative to the center of the Earth

Not in an inertial frame.

 

[jbriggs444]

The satellite movement is equivalent of the earth's.Therefore ds=ds'

This is completely incorrect. Length contraction is a thing.

 

[Ibix]

Τhe principle of relativity, is that there is no such thing as "earth motion" or "satellite motion", only relative motion.

If both Earth and satellite were moving inertially in flat spacetime, you would have a point. But you must either model the satellite as moving non-inertially in flat spacetime (and don't ask too many questions about gravity) or use a full general relativistic solution in which there are no global inertial frames. Either approach invalidates your application of the principle of relativity.

 

[Janus]

Τhe principle of relativity, is that there is no such thing as "earth motion" or "satellite motion", only relative motion.The satellite movement is equivalent of the earth's.Therefore ds=ds' Satellite is almost immobile relative to the center of the Earth,differently to a point on the surface.

As many posters have already stated, this only applies when dealing with inertial frames. Once you start to make measurements from a non-inertial frame, this changes. If you are in the tail of an accelerating rocket, for example, you will measure a clock in the nose of the rocket ticking faster than your own, even though you have no relative motion between you and it. (conversely, someone in the nose will measure your clock as running slow). And this is not confined to clocks in the rocket or sharing your acceleration. You will measure any clock "ahead" of the accelerating* rocket as ticking fast, the further away, the faster it ticks.
.
Now take this rocket and arrange things so that its circling a clock at a constant speed, ( it is pointing its rockets outward in order to maintain a "forced orbit" around the clock). It is still accelerating, even though its speed remains constant. The acceleration is centripetal and always pointed at the central clock. As such, someone in the rocket would measure the central clock as ticking fast.
Someone at the central clock would be in an inertial frame and it would only be the rocket's circular velocity that would effect how they would measure the rocket's clock tick, and this would cause them to measure it as ticking slow. The point is that, while both the Central observer and rocket observer can lay claim to being "at rest", only the central observer can do so from an inertial frame, the rocket observer's "rest frame" would be a non-inertial rotating one, for which there are additional factors beyond just relative motion for determining relative clock rates.**
With GPS satellites, gravity provides the centripetal acceleration, and the difference in gravitational potential adds an additional complication, but the idea that the only important factor is "relative velocity alone" still does not hold.

* in this case "ahead" means in the direction of the acceleration

** You could also have the central clock rotating so that it maintains the same orientation with respect to the rocket at all times, and consider itself as being at rest within the same rotating frame as the rocket, But now it would be like the Nose clock in the first accelerating rocket example, and would still measure the rocket clock as ticking slow.

 

[binis]

Not in an inertial frame.

But "There are no global inertial frames in curved space time."

 

[A.T.]

Satellite is almost immobile relative to the center of the Earth

Not in an inertial frame.

But "There are no global inertial frames in curved space time."

Why "but"? Did you mean "because"?