How to resolve the contradiction in twin clocks?

[PeterDonis]

I think Samshorn has a point that a clock is defined as a device that reads proper time.

In which case the statement "a clock reads proper time" is true. In most discussions of the topic, such as the discussion ghwellsjr was having with nitsuj in this thread, simply stating that is sufficient.

If you really want to get into the nitty-gritty of *how* clocks read proper time, then concerns like those Samshorn raised might be relevant; but Nugatory gave good responses to those concerns, which indicate why, most of the time, just saying "a clock reads proper time along its worldline" is sufficient.

 

[PeterDonis]

a non-ideal clock still provides a perfectly good time coordinate; it labels each point on that worldline with a unique value and with appropriate choice of simultaneity convention will supply a time coordinate for points off that worldline as well. All that's going on is that the imperfections of the clock are encoded in the value of $g_{tt}$ along its inertial worldline - when the metric tensor is expressed in coordinates in which the clock is providing the t coordinate. This can still be a perfectly flat spacetime; the non-unity metric components are compensating for the less than ideally simple choice of coordinates..

Would another way of stating this be to say that the time an ideal clock reads can be used as an affine parameter along its worldline, whereas the time a non-ideal clock reads can't? (The latter can still be used as a time *coordinate*, but since the scaling of the clock varies along its worldline, its reading can't be used as an affine parameter, for which I believe the scaling has to be constant.)

 

[Dale]

The readings of an ideal clock give us both coordinate time in that coordinate system and proper time; as WbN points out above they're equal so discussing which the clock is measuring is sterile.

In my opinion, the distinction in such a case is that the proper time is only defined along the worldline of the clock whereas the coordinate time is defined over the whole coordinate chart. They are equal where both are defined, but since they are defined for different regions of the manifold they are not the same.

 

[Enigman]

Thanks for the corrections. I'll have to admit I haven't quite come to terms with the twin paradox yet - I've read Feynman who simply dismissed me as a cocktail party phillosopher saying symmetry breaks down at acceleration and only inertial frames are relative. How does the acceleration affect the time isn't mentioned in his lecture. (I issued the lectures from library after returning that ancient book of intro to "modern" physics.)
So you are essentially saying that proper time can be measured even in non inertial frames? I am going to look a bit into that for now that and twin paradox.
And I am kicking myself for that line about 2u.
(Also as a clarification I started reading SR from 9th grade
from children's biographies of einstein, documentries and such like. It's only last weekend I read an official text on SR for the first time. Am in btech 1st yr now.)
Regards
P.S. Calculations of Janus are somewhere in the beginning of the post. Sorry about the confusion.

 

[Nugatory]

Would another way of stating this be to say that the time an ideal clock reads can be used as an affine parameter along its worldline, whereas the time a non-ideal clock reads can't? (The latter can still be used as a time *coordinate*, but since the scaling of the clock varies along its worldline, its reading can't be used as an affine parameter, for which I believe the scaling has to be constant.)

I think it's still an affine parameter, just nowhere near as convenient as proper time. The metric coefficient captures the scaling when you remember to express it in the appropriate coordinate system.
(This is a pragmatists's answer, not a mathematician's. If a mathematician says I'm wrong, they're right, but it doesn't stop me from evaluating my line integrals just as I always did).

 

[Nugatory]

So you are essentially saying that proper time can be measured even in non inertial frames? I am going to look a bit into that for now that and twin paradox.

Yes. Indeed all of special relativity works just fine in non-inertial frames as long as the spacetime is flat (If not flat, then there are significant gravitational forces at work and you have to use the methods of general relativity).

There are several reasons why people often haven't realized this:
1) Non-inertial frames require appreciably more complicated math which tends to obscure the basic concepts; so most basic texts use only inertial frames in their examples. It's easy to jump to the conclusion that the inertial frame is a necessary as well as a sufficient condition for applying SR.
2) There aren't that many situations in which considering an SR problem from a non-inertial frame contributes any new insight; so again you don't seem it done very often. (The Rindler solution is one of the more important exceptions, but it's not generally considered an introductory-level problem).
2) Just about every explanation of GR uses the equivalence principle between acceleration and gravitation to introduce GR. It's easy to think that if problems involving gravity require GR, and if there's an equivalence principle between gravity and acceleration, then problems involving acceleration must also require GR. This syllogism is bogus, but awfully tempting.

 

[atyy]

I don't understand your post. First you agree with the idea "that a clock is defined as a device that reads proper time" and then you say that it is "not defined" that clocks "read proper time".

Also, Special Relativity cannot account for gravity and therefore it cannot account for a pendulum clock.

If we define a clock using atomic vibrations, then we because we have equations for how the atom interacts with gravity, we can derive that the clock reads proper time. So given an atomic clock, that a clock reads proper time is derived, rather than put in by hand at the start. This is still not exactly right, since real atomic clocks are not just isolated atoms.

The more traditional way and very proper way of doing things is to define an "ideal clock" as a device that reads proper time. Here atoms are not specified at all in the definition of a clock. Given such an abstract definition, the theory must at least give some assurance that an ideal clock can be built. Typically one says that proper time is coordinate invariant, so it could be the output of a device traveling on the worldline. The argument is also given that since acceleration is absolute in relativity, acceleration can be sensed and corrected for.

Either point of view leads to the same experiemental predictions, so it's just a matter of taste.

Also, Special Relativity cannot account for gravity and therefore it cannot account for a pendulum clock.

Yes, that would be a better example for GR in which clocks still read proper time. An example for SR would be a non-ideal clock like a wristwatch that's been run over by a truck.

@Peter Donis and @Nugatory - yes, I agree with your points.

Thanks for the corrections. I'll have to admit I haven't quite come to terms with the twin paradox yet - I've read Feynman who simply dismissed me as a cocktail party phillosopher saying symmetry breaks down at acceleration and only inertial frames are relative. How does the acceleration affect the time isn't mentioned in his lecture. (I issued the lectures from library after returning that ancient book of intro to "modern" physics.)
So you are essentially saying that proper time can be measured even in non inertial frames? I am going to look a bit into that for now that and twin paradox.
And I am kicking myself for that line about 2u.
(Also as a clarification I started reading SR from 9th grade
from children's biographies of einstein, documentries and such like. It's only last weekend I read an official text on SR for the first time. Am in btech 1st yr now.)
Regards
P.S. Calculations of Janus are somewhere in the beginning of the post. Sorry about the confusion.

In the twin paradox, a clock is defined to be an ideal clock, one which reads proper time. A clock that reads proper time is not "directly" affected by acceleration, in contrast to a pendulum which is "directly" affected by acceleration. Proper time can be read in non-inertial frames, because it is a property of one's trajectory in spacetime. It is the spacetime analogue of distance or the number of rotations a wheel makes when traveling from San Francisco to Los Angeles - that number doesn't depend on whether you use latitute and longitude to describe the path you took in space. The number of rotations the wheel makes depends on the spatial route one took. Similarly, proper time is simply "spacetime distance", and the proper time of the two twins is different because they took different spacetime paths even though they started and ended at the same event.

The nonintuitive thing is that the formula for spatial distance is d²=x²+y² (straight line in space), whereas proper time is T² = -t²+x²+y² (straight line in spacetime), with a minus sign instead of a plus. For curved paths, you use the same formula but cut the line into little pieces which are essentially straight, and add up the results from all the pieces.

 

[PeterDonis]

I think it's still an affine parameter, just nowhere near as convenient as proper time.

It's certainly not as convenient, but the reason I question whether a non-ideal clock's time can be an affine parameter is that affine parameters are supposed to be linearly related to each other. An ideal clock's time, which is just proper time, is an affine parameter; but the point of a non-ideal clock is that its period is not constant, so the time it keeps would not be a linear function of proper time.

If a mathematician says I'm wrong, they're right, but it doesn't stop me from evaluating my line integrals just as I always did).

Yes, I wasn't questioning the fact that, no matter how poorly a non-ideal clock keeps time, you can still set up coordinates and define a metric using its reading as the time coordinate, and use those to do integrals--including the integral that gives proper time. (The practical problem here is that, if you don't know the exact relationship between the non-ideal clock's time and an ideal clock's time, you don't know what the actual metric coefficient $g_{tt}$ should be. But in principle you can always put an ideal clock next to the non-ideal one to find that out.)

 

[nitsuj]

I don't understand your post. First you agree with the idea "that a clock is defined as a device that reads proper time" and then you say that it is "not defined" that clocks "read proper time".

The observation is not part of the physics being observed. So musing over what the clock that has length dependent displays reads while in comparative motion isn't going to yield anything regarding the physical processes of the clock itself. The clock isn't different because it has been observed

A ruler at rest with you is a proper length. Same goes for the clock, I am unsure how else to word it. It is such a blatant point, but was raised to make the distinction between these measuring devices in motion are not the same as when at rest.

for the quoted part remember clocks are not perfect. I'm gunna assume you agree that a clock doesn't "read" proper time at all, it displays it. The variance between the two could be idealized away, even my mechanical watch that loses minutes over days is accurate enough for my scheduling . Sometimes extremely accurate measures of proper time are needed(CERN Neutrino mearement, probably gravity wave detection ect), sometimes it's just a fun distinction to make.

 

[PeterDonis]

A ruler at rest with you is a proper length. Same goes for the clock, I am unsure how else to word it.

Try these wordings:

A clock (an ideal clock, if we need to be precise about that) measures proper time along its worldline.

A ruler (an ideal ruler, if we need to be precise about that) measures proper length in its instantaneous rest frame.

It is such a blatant point, but was raised to make the distinction between these measuring devices in motion are not the same as when at rest.

But the distinction here is in you, not the devices. The clock and ruler don't know that they are "in motion", because motion is relative anyway; they're in motion relative to you, but not relative to themselves.

In other words, the distinction you are making is in the observation, not in the thing observed; but you said that the observation is not part of the thing observed. So it would seem appropriate to choose wording that makes it clear that the distinction is in the observation; but your wording seems to me to obfuscate that issue.

 

[nitsuj]

Ah, okay

 

[ghwellsjr]

I've seen this discussion before, and to me it always comes down to people (myself included) not being completely clear about the distinction between coordinate time along the worldline of an observer at rest at the spatial origin of a coordinate system and proper time along the same worldline.

I'm surprised that you express confusion (or at least not being completely clear) about the distinction between Coordinate Time and Proper Time after you gave such an excellent explanation of them both in post #45:

- Proper time, which is invariant and what the position of the clock's hands (or the progress of any physical process: fraction of a radioactive sample that has decayed between two observations, number of oscillations of a cesium atom between two observations, number of my hairs which have turned gray between two observations) measures.
- Coordinate time, which is different for different observers using different coordinate systems (also known as "frames of reference"). The Lorentz transformations describe how to convert one observer's coordinates, including coordinate time, to another observer's coordinates in a way that preserves the laws of physics and especially ensures that the relationship between the position of the hands of the clock and each observers' coordinate time is consistent with the physical process moving the hands of the clock.

What am I missing?

Also, I should have included you in the list of experts who state that Proper Time is what a clock measures.

 

[ghwellsjr]

Thanks for the corrections. I'll have to admit I haven't quite come to terms with the twin paradox yet

The Twin Paradox is easy--just pick an Inertial Reference Frame (IRF) to specify the scenario, then the Proper Time for each twin is a function of their speed in the IRF, the faster they travel, the slower their clock ticks. Therefore, if only one moves, he will accumulate less time than the other one who remains stationary. Since they must start out together and end up together, that means the one who moves must accelerate (at least change direction).

- I've read Feynman who simply dismissed me as a cocktail party phillosopher saying symmetry breaks down at acceleration and only inertial frames are relative.

I think he must mean that only IRF's can be transformed into other IRF's using the Lorentz Transformation process. IRF's are the standard frames in SR. If you want to veer off into non-inertial frames, then you have to establish your own convention or state which of several conventions other people have promoted you wish to use. My favorite is the convention established by using Radar Methods. In fact, I don't use any others.

How does the acceleration affect the time isn't mentioned in his lecture. (I issued the lectures from library after returning that ancient book of intro to "modern" physics.)

Acceleration doesn't have to affect the time, it only has to affect the direction of the traveling twin. It's the speed according to the IRF that affects the time.

So you are essentially saying that proper time can be measured even in non inertial frames?

I think what you are meaning to ask about is non inertial observers (twins). One of the twins is non inertial because he accelerates. I don't think anyone ever sets up a Twin Paradox scenario by specifying the coordinates of the stay-at-home twin in the non inertial coordinate system of the traveling twin. I have always seen the traveling twin specified in the IRF of the stay-at-home twin. For example, one twin travels away from the stationary twin at some speed for some period of Coordinate Time or Proper Time or for some distance and then turns around and returns at the same speed. That makes it very easy to determine the aging difference. But then the question is asked about other frames (which cannot change the answer). If the other frame is also an IRF, then you can use the Lorentz Transformation. If the other frame is non inertial, then much more work must be done but it is possible.

I am going to look a bit into that for now that and twin paradox.
And I am kicking myself for that line about 2u.
(Also as a clarification I started reading SR from 9th grade
from children's biographies of einstein, documentries and such like. It's only last weekend I read an official text on SR for the first time. Am in btech 1st yr now.)
Regards
P.S. Calculations of Janus are somewhere in the beginning of the post. Sorry about the confusion.

 

[ghwellsjr]

The more traditional way and very proper way of doing things is to define an "ideal clock" as a device that reads proper time.
...
An example for SR would be a non-ideal clock like a wristwatch that's been run over by a truck.

The comment that I asked justin about was:

A ruler in comparative motion is not a "proper" ruler, same goes for the clock.

Now I assumed that his comment would be in contrast to this comment:

"A ruler in comparative rest is a "proper" ruler, same goes for the clock."

EDIT: It turns out I don't have to assume a contrasting statement, justiin provided it in post #69:

A ruler at rest with you is a proper length. Same goes for the clock, I am unsure how else to word it.

I didn't think his comment had anything to do with a broken ruler or a wristwatch run over by a truck. I assumed that he was saying that a ruler at rest measures proper length but a ruler in motion does not and that a clock at rest measures proper time but a clock in motion does not. That's what I asked him for clarification about.

 

[ghwellsjr]

The observation is not part of the physics being observed. So musing over what the clock that has length dependent displays reads while in comparative motion isn't going to yield anything regarding the physical processes of the clock itself. The clock isn't different because it has been observed

The observation is very much a part of the physics being observed. It is fundamental to the Principle of Relativity. If you discount the observations of clocks in motion, then you cannot have a complete Principle of Relativity. It's precisely because two clocks in relative motion both see the other ones clock ticking symmetrically at a different rate compared to their own that we have the Principle of Relativity. And it's the Principle of Relativity applied even to light that leads to the Lorentz Transformation which completely overhauled physics from Newtonian-based to Einsteinian-based.

Of course the clock isn't different because it has been observed, but our physics is different and therefore our understanding of the physical processes of the clock is different because scientists have made careful measurements of clocks in motion (I'm using "clock" in the broadest sense of the term).

A ruler at rest with you is a proper length. Same goes for the clock, I am unsure how else to word it. It is such a blatant point, but was raised to make the distinction between these measuring devices in motion are not the same as when at rest.

True, but a ruler that has been accelerated so that it has a different length (according to its original rest IRF), still has a Proper Length and can be used to correctly measure the length of any object that is comoving with it. A clock that has been accelerated so that it has a different tick rate (according to its original rest IRF), still has a Proper Time and can be used to measure the passage of time of anything comoving with it.

I referred to the original rest IRF in both these situations but keep in mind that if you're talking about a ruler or a clock that start out at rest in an IRF and then are accelerated to a constant speed in the same IRF, it is always possible to pick another IRF in which the objects are at the same speed before and after the acceleration and so there is no change in the Proper Length of the ruler or the Proper Time of the clock.

or the quoted part remember clocks are not perfect. I'm gunna assume you agree that a clock doesn't "read" proper time at all, it displays it. The variance between the two could be idealized away, even my mechanical watch that loses minutes over days is accurate enough for my scheduling . Sometimes extremely accurate measures of proper time are needed(CERN Neutrino mearement, probably gravity wave detection ect), sometimes it's just a fun distinction to make.

No, I don't agree. I don't even know what you're talking about or why you're talking about this. It has nothing to do with the subject you and I have been discussing.

 

[nitsuj]

The observation is very much a part of the physics being observed. It is fundamental to the Principle of Relativity. If you discount the observations of clocks in motion, then you cannot have a complete Principle of Relativity. It's precisely because two clocks in relative motion both see the other ones clock ticking symmetrically at a different rate compared to their own that we have the Principle of Relativity. And it's the Principle of Relativity applied even to light that leads to the Lorentz Transformation which completely overhauled physics from Newtonian-based to Einsteinian-based.

It over hauled Newton physics because Newton physics is wrong. My point is inline with your retort to "The observation is not part of the physics being observed." Who cares about symmetry? That only has purpose for creating the metric. Nothing in your post explained how an observation of a physical process is part of that physical process.

Of course the clock isn't different because it has been observed, but our physics is different and therefore our understanding of the physical processes of the clock is different because scientists have made careful measurements of clocks in motion (I'm using "clock" in the broadest sense of the term).

What do you mean by "But our physics is different." To your point regarding relative motion there is nothing "different" heck it's symmetrical.

True, but a ruler that has been accelerated so that it has a different length (according to its original rest IRF), still has a Proper Length and can be used to correctly measure the length of any object that is comoving with it. A clock that has been accelerated so that it has a different tick rate (according to its original rest IRF), still has a Proper Time and can be used to measure the passage of time of anything comoving with it.

Yuppers, and goes without saying. What typically doesn't go without saying, is this distinction between the very very well known proper length and that the concept is often carried over to length contraction incorrectly. The point is that a moving ruler is not a "traditional" measuring stick. as simple as that. There is no need to bring up the point that motion is relative. That's should be implicit if discussing relativistic effects.

I referred to the original rest IRF in both these situations but keep in mind that if you're talking about a ruler or a clock that start out at rest in an IRF and then are accelerated to a constant speed in the same IRF, it is always possible to pick another IRF in which the objects are at the same speed before and after the acceleration and so there is no change in the Proper Length of the ruler or the Proper Time of the clock.

Yup again motion is relative, and again is implicit here.

No, I don't agree. I don't even know what you're talking about or why you're talking about this. It has nothing to do with the subject you and I have been discussing.

It was with respect to the quoted part in that post. Clocks are imperfect, nuff said. So while a clock or anything for that matter experiences proper time as we define it, getting a clock to accurately display this as an incremental reading is complicated...it's all just so fast and keeping up with the constancy of c with a display is pretty tricky I am sure.

Sorry for quoting & replying to your post in a complicated way/

 

[phyti]

What do you mean by that? The literal readings on an actual clock need not correspond to the proper time along the worldline of the clock. (Actual clocks run fast or slow.) Do you just mean that an ideal clock, i.e., a device constructed in such a way as to read the proper time along its worldline, will read the proper time along its worldline? True enough, but circular. Or do you mean that the temporal state of a clock will progress in proportion to proper time, even though this may not correspond to the literal readings on the clock? If so, then it doesn't need to be a "clock", you could just as well refer to any physical system, and then you would need to say what is meant by "temporal state", which you can't define with reference to elapsed proper time or it is circular.

Telling people that "every clock measures proper time" is not good, because it could only be literally true if we simply defined 'proper time' to be whatever any clock reads, which of course would be utterly incoherent. That's what beginners tend to think you must mean, which totally sends them down the wrong track.

There is a non-circular way of correctly saying what you are probably trying to say, but it's quite a bit more subtle and complicated than just saying "every clock reads proper time".

I agree.
"every clock reads proper time" (for the observer moving with the clock). (requires context)
Einstein said the time of the event is simultaneous with a clock event (position of the hand) located at the event. The poor word choice 'proper' relates more to etiquette or social behavior.
It's about location, so why not define it as local time, in keeping with his additional statements regarding A time (local) and B time (distant).

 

[Dale]

The poor word choice 'proper' relates more to etiquette or social behavior.
It's about location, so why not define it as local time, in keeping with his additional statements regarding A time (local) and B time (distant).

The term "proper time" is standard terminology. The term "local time" was used by LET to denote coordinate time in the non-aether frames, so I think that a different term is preferable. However, it is a purely semantic preference with no physical content whatsoever.

 

[Dale]

The observation is not part of the physics being observed.

I disagree with this statement. Observations are always made by some physical process. You do try to minimize the impace that the observation has on the system being observed, but I don't think that you can say that they are completely separated.

 

[nitsuj]

I disagree with this statement. Observations are always made by some physical process. You do try to minimize the impace that the observation has on the system being observed, but I don't think that you can say that they are completely separated.

You're absolutely right, the observation itself is a physical process connected to the whatever was being observed, all else is "elsewhere".

It's clear enough when in context; your point border lines hyperbole.

My observation of time dilation doesn't mean that clock is "broken" or otherwise any different physically then if I hadn't observed it at all.

 

[ghwellsjr]

It over hauled Newton physics because Newton physics is wrong. My point is inline with your retort to "The observation is not part of the physics being observed." Who cares about symmetry? That only has purpose for creating the metric. Nothing in your post explained how an observation of a physical process is part of that physical process.

What do you mean by "But our physics is different." To your point regarding relative motion there is nothing "different" heck it's symmetrical.

I just meant that our understanding of physics since Einstein is different than before because he applied the Principle of Relativity to all of physics and not just part of it.

 

[Dale]

It's clear enough when in context; your point border lines hyperbole.

Fair enough. I admit, I had not been following the conversation between you and ghwellsjr carefully, so I read it somewhat out of context.

 

[phyti]

The term "proper time" is standard terminology. The term "local time" was used by LET to denote coordinate time in the non-aether frames, so I think that a different term is preferable. However, it is a purely semantic preference with no physical content whatsoever.

"Bald eagle" is standard/common terminology, but the eagle isn't bald!

I'll stick with local time, which implies a clock with the observer, and not remotely located.
The ether is ignored today, and it's use in the 1900's does not prohibit the use of 'local' in a different context now.

 

[ghwellsjr]

"Bald eagle" is standard/common terminology, but the eagle isn't bald!

I'll stick with local time, which implies a clock with the observer, and not remotely located.
The ether is ignored today, and it's use in the 1900's does not prohibit the use of 'local' in a different context now.

And what is your preferred term for the eagle?

 

[WannabeNewton]

I just wanted to add this to the other two references I gave (this one is taken from Malament's text): http://postimg.org/image/hig2mjgyr/

 

[nitsuj]

I just meant that our understanding of physics since Einstein is different than before because he applied the Principle of Relativity to all of physics and not just part of it.

I don't know enough about the entire subject of physics to say with authority, but I see PoR as emergent from the geometry of spacetime, remember regardless of comparative motion physics is the same. I'd agree in that sense PoR, specifically spacetime, is "applied" to all physics, maybe better said as all physics happens within spacetime; but that seems like a silly thing to say.

Isn't PoR merely the geometric symmetry of the physics being observed ?

 

[ghwellsjr]

I don't know enough about the entire subject of physics to say with authority, but I see PoR as emergent from the geometry of spacetime, remember regardless of comparative motion physics is the same. I'd agree in that sense PoR, specifically spacetime, is "applied" to all physics, maybe better said as all physics happens within spacetime; but that seems like a silly thing to say.

Isn't PoR merely the geometric symmetry of the physics being observed ?

No, the geometric symmetry of spacetime emerges from Einstein's first postulate, the PoR, and his second postulate, the propagation of light at c independent of its source.

 

[Dale]

"Bald eagle" is standard/common terminology, but the eagle isn't bald!

I'll stick with local time, which implies a clock with the observer, and not remotely located.

As long as you realize that you are deliberately inviting miscommunication by this approach.

 

[atyy]

I don't know enough about the entire subject of physics to say with authority, but I see PoR as emergent from the geometry of spacetime, remember regardless of comparative motion physics is the same. I'd agree in that sense PoR, specifically spacetime, is "applied" to all physics, maybe better said as all physics happens within spacetime; but that seems like a silly thing to say.

Isn't PoR merely the geometric symmetry of the physics being observed ?

Essentially yes. One can say that the principle of relativity emerges from the Poincare symmetry of the physical laws, where the Poincare symmetry is the isometry group of the Minkowski spacetime geometry.

The historical route went in the other direction - the Poincare symmetry was inferred from the PoR and the invariance of the speed of light. If one uses only PoR without requiring a speed limit, then Newtonian physics is allowed. Exactly how to infer Poincare symmetry from the PoR and a universal speed limit has subtleties which Fredrick once discussed in a very long thread, and whose details I can't remember.

 

[atyy]

I just wanted to add this to the other two references I gave (this one is taken from Malament's text): http://postimg.org/image/hig2mjgyr/

Yes, it's a problem in Newtonian physics too. I like Stephani's definition: time is what makes the laws of physics true.