Implications of the statement Acceleration is not relative

[PAllen]

In the several descriptions I've seen that use a fictitious gravitational field to resolve the twin paradox from the traveler's viewpoint, I didn't see any ambiguity anywhere ... the procedure gave a specific (unique) answer to the question of how much the home twin ages during the traveler's turnaround (according to the traveler).

An given explanation using this approach will have a specific answer. What may not be stated by the author (but is known by them if they are a knowledgeable author) is that there is no unique and several reasonable choices. All would give the same answer for observations (differential aging over the trip; doppler; exchange of signals; etc.). But different reasonable choices would give different answers as to the distribution of differential aging. Also not stressed at a non-expert level is that the most common treatment of this will not even apply to some twin trajectories; then you have to use one or another less obvious convention.

 

[Mentz114]

In the several descriptions I've seen that use a fictitious gravitational field to resolve the twin paradox from the traveler's viewpoint, I didn't see any ambiguity anywhere ... the procedure gave a specific (unique) answer to the question of how much the home twin ages during the traveler's turnaround (according to the traveler).

(my bold)

Me neither. It looks completely consistent. Using the fictitious force to keep the 'rocket' twin stationary does not change the physics - viz. the traveling twin is non-inertial some of the time but the other one is always in free-fall. Therefore the traveling twin ages less as she should according to the other frames.

 

[PAllen]

(my bold)

Me neither. It looks completely consistent. Using the fictitious force to keep the 'rocket' twin stationary does not change the physics - viz. the traveling twin is non-inertial some of the time but the other one is always in free-fall. Therefore the traveling twin ages less as she should according to the other frames.

Consistency is not the same a uniqueness. There are different, reasonable, choices for simultaneity for the accelerating twin. Each produces a different metric (though they converge near the 'time axis' represented by the accelerating twin), with different statements as to how much of the aging (of the inertial twin) occurs during turnaround (assuming e.g. coast, turn, coast). Further, the most common way this is presented will not work at all for a W shaped traveler trajectory (the simultaneity surfaces will intersect and multiply map the inertial twin world line, preventing you from having any coordinate chart in which to integrate proper time). So then you must use a different set of simultaneity surfaces to handle this case.

 

[Mentz114]

Consistency is not the same a uniqueness.

OK.

There are different, reasonable, choices for simultaneity for the accelerating twin. Each produces a different metric (though they converge near the 'time axis' represented by the accelerating twin),

If both worldlines are specified, is it possible to find a simultaneity choice that enables the WLs to be integrated ?

with different statements as to how much of the aging (of the inertial twin) occurs during turnaround (assuming e.g. coast, turn, coast)

If the worldlines are specified, is the amount of ageing at turnarounds not uniquely defined ? I have to say that I'm not much interested in where the ageing occurs.

Further, the most common way this is presented will not work at all for a W shaped traveler trajectory (the simultaneity surfaces will intersect and multiply map the inertial twin world line, preventing you from having any coordinate chart in which to integrate proper time). So then you must use a different set of simultaneity surfaces to handle this case.

OK, but I was talking about the simplest scenario.

I understand you are advocating caution, but I was addressing the OP's question about a consistent treament of the twins in which the traveling twin remains stationary ( ie has a vertical worldline).

 

[PAllen]

If both worldlines are specified, is it possible to find a simultaneity choice that enables the WLs to be integrated ?

Of course.

If the worldlines are specified, is the amount of ageing at turnarounds not uniquely defined ? I have to say that I'm not much interested in where the ageing occurs.

It is definitely not uniquely defined. Only the observables are uniquely defined. Simultaneity defined by the Einstein convention (two way light signal), and by a simultaneity based on spacelike geodesics 4-orthogonal to the traveling world line tangent, produce quite different answers. The former will work fine for the W trajectory. The latter is the one most commonly used, and will not work at all for the W trajectory.

OK, but I was talking about the simplest scenario.

I understand you are advocating caution, but I was addressing the OP's question about a consistent treament of the twins in which the traveling twin remains stationary ( ie has a vertical worldline).

My point, having seen religious subservience to a convention that is only locally favored, is to stress the non-unuiqueness. Consistency is not a problem. But the non-uniqueness means there isn't one answer to how much ageing of the distant twin occur during turnaround. It really is just as silly as a short line on a piece of paper supposedly having a unique point of view about where the extra length of a longer line is.

 

[Mentz114]

My point, having seen religious subservience to a convention that is only locally favored, is to stress the non-unuiqueness. Consistency is not a problem. But the non-uniqueness means there isn't one answer to how much ageing of the distant twin occur during turnaround. It really is just as silly as a short line on a piece of paper supposedly having a unique point of view about where the extra length of a longer line is.

Thanks for the responses. I guess that finishes off the CADO nonsense.

I have now realized that the OP was bothered because there is no treatment of the twins case with the traveling twin being inertial. As you and others have already pointed out, that is impossible.

 

[GregAshmore]

I have now realized that the OP was bothered because there is no treatment of the twins case with the traveling twin being inertial. As you and others have already pointed out, that is impossible.

No, that is not at all what bothers me. I am fully aware that the traveling twin is not inertial; of course a non-inertial frame cannot be treated as inertial. I am bothered that a theory which is only suited for treating inertial frames is used to deal with a problem involving a non-inertial frame.

 

[Mentz114]

No, that is not at all what bothers me. I am fully aware that the traveling twin is not inertial; of course a non-inertial frame cannot be treated as inertial. I am bothered that a theory which is only suited for treating inertial frames is used to deal with a problem involving a non-inertial frame.

Thanks for the clarification. I've never had a problem with that. We can get some useful results by including acceleration in SR. For instance, the Rindler frame, the Langevin frame, Born coordinates and probably others.

The Lorentz transformation works even if the β parameter depends on time, so we have a transformation from inertial to non-inertial coordinates.

 

[PAllen]

No, that is not at all what bothers me. I am fully aware that the traveling twin is not inertial; of course a non-inertial frame cannot be treated as inertial. I am bothered that a theory which is only suited for treating inertial frames is used to deal with a problem involving a non-inertial frame.

Well, the correct statements are:

- The mathematics of SR is simplest in inertial frames, but all phenomena may be analyzed in such frames, including non-inertial motion.
- There is no such thing as a global non-inertial frame; non-inertial frames are local.
- It is possible, in many ways, to set up coordinates in which a non-inertial world line has constant spatial coordinates of 0. For any such coordinates, you have to transform the Minkowski metric. This transformed metric leads to different formulas for time dilation, light paths, and geodesics. Different choices for such coordinates will produce different answers for coordinate dependent properties, but will produce the same answers as inertial frames for any observations or measurements.

 

[GregAshmore]

The principle of relativity states "The laws of physics are the same in all inertial frames of reference". So statements about non-inertial frames simply cannot violate it, anymore than statements about the price of steel can violate the principle of beans.

No. That is the limited principle of relativity, for the special case of inertial frames. The principle of relativity states that the laws of physics are the same for all frames of reference.

If you deal with non-inertial frames within the confines of special relativity, then you have the same problem that Newton had: There is an absolute quality to acceleration; there is a preferred frame.

I maintain my position that this does damage to the principle of relativity.

 

[PAllen]

The Lorentz transformation works even if the β parameter depends on time, so we have a transformation from inertial to non-inertial coordinates.

Using Lorentz transform with varying β picks out a special class of coordinates with a specific simultaneity convention. If you want to treat more general coordinates, you use a more general transform. In particular, going from inertial coordinates to coordinates based on Einstein (or radar) simultaneity, will not use a Lorentz transform.

 

[PAllen]

No. That is the limited principle of relativity, for the special case of inertial frames. The principle of relativity states that the laws of physics are the same for all frames of reference.

If you deal with non-inertial frames within the confines of special relativity, then you have the same problem that Newton had: There is an absolute quality to acceleration; there is a preferred frame.

I maintain my position that this does damage to the principle of relativity.

There is the special principle of relativity and 'general principle of relativity'. They are different principles, with different physical content. The special principle of relativity, as physical principle, says you cannot detect inertial motion except in reference to other things. The general principle of relativity does not say you cannot detect non-inertial motion. It says you cannot locally distinguish whether your non-inertial motion comes from holding position relative to a gravitational source versus accelerating far from any source.

In general relativity as well, acceleration is distinguishable, and there is a precise mathematical difference between a local inertial frame and a local non-inertial frame in GR: in the former, the connection coefficients vanish, in the latter they do not.

As for laws taking the same form, this is just a matter of the mathematical way you write them (stevendaryl has explained this before on this thread, I believe). If, in SR, you write laws explicitly using the metric and vector/tensor quantities, as you do in GR, then the laws will take the same form in non-intertial coordinates as they do in inertial coordinates. This is still not GR, because there is no gravity involved, nor is the EFE (the equation defining GR) used.

 

[PeterDonis]

The principle of relativity states that the laws of physics are the same for all frames of reference.

You've made this claim several times now. Can you give a reference? You talk as though this is "the" principle of relativity, but that doesn't match what I (and suspect others) know of the history and usage of the term.

Also, arguing about definitions is not the same as arguing about physics. Can you state a *physical* objection that doesn't depend on a particular definition for what "the principle of relativity" says?

 

[Dale]

No. That is the limited principle of relativity, for the special case of inertial frames. The principle of relativity states that the laws of physics are the same for all frames of reference.

No. The principle of relativity as I stated it is the correct one for special relativity (SR). That is the form that it appears as a postulate of SR. The twins paradox is a SR problem, not a GR problem, since it does not use the Einstein Field Equations or curved spacetime.

However, the discussion about inertial vs non-inertial frames is not relevant to the statement "acceleration is not relative". The statement "acceleration is not relative", as we have mentioned, refers to proper acceleration. Proper acceleration is a property of a worldline, not a property of a reference frame.

It doesn't matter what reference frame you use, inertial or not, the proper acceleration is the same in all of them. So, the statement "acceleration is not relative" is about worldlines, not reference frames. I think that you are getting distracted by irrelevancies. The traveling twin has non-zero proper acceleration regardless of what reference frame is used.

 

[Dale]

I am bothered that a theory which is only suited for treating inertial frames is used to deal with a problem involving a non-inertial frame.

This is, IMO, a reasonable objection to make (I have made the same objection previously). The postulates of SR refer only to inertial frames, so how can you use them to make any claim about the physics in non-inertial frames?

Once you know how the physics works in inertial frames, then figuring out the physics in any other frame is simply a matter of performing a change of variables to the coordinates (aka coordinate transform). All of the usual math for doing a chang of variables still applies. Thus, even though the postulates only describe physics in inertial frames, you can use them indirectly to derive the physics in non-inertial frames.

 

[DrGreg]

The principle of relativity states that the laws of physics are the same for all frames of reference.

To repeat what everyone else has said the "principle of relativity" is usually stated in terms of inertial frames only.

So let's consider an example, Newton's second law of motion. The relativistic 4D version of this, for a particle of constant mass, is<br /> F^\lambda = m \frac{d^2x^\lambda}{d\tau^2}<br />when measured in any inertial (Minkowski) coordinate system. This is pretty simple and almost the same as the non-relativistic version.

However in non-inertial coordinates, the equation becomes<br /> F_\lambda = m \sum_{\mu=0}^3 g_{\lambda \mu} \frac{d^2x^\mu}{d\tau^2}<br /> + \frac{m}{2} \sum_{\mu=0}^3 \sum_{\nu=0}^3 <br /> \left( <br /> \frac{\partial g_{\lambda \mu}}{\partial x^\nu} + <br /> \frac{\partial g_{\lambda \nu}}{\partial x^\mu} - <br /> \frac{\partial g_{\mu \nu}}{\partial x^\lambda}<br /> \right) \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}<br />You don't need to understand the meaning of this, just observe that it's very complicated.

So, yes the laws of physics can be expressed in a form that is the same in all frames, inertial or non-inertial, but such expression is much more complicated than the inertial-frame-versions of the laws.

 

[harrylin]

(my bold)

[...] the traveling twin is non-inertial some of the time but the other one is always in free-fall. Therefore the traveling twin ages less as she should according to the other frames.

Note that such reasoning does not generally hold, as I mentioned before: in the original variant by Langevin both are in free fall. Still it is the traveler who ages less (he didn't in 1911 account for gravitational time dilation but that isn't pertinent and gravitation at the turn-around only enhances the effect).

 

[harrylin]

No, that is not at all what bothers me. I am fully aware that the traveling twin is not inertial; of course a non-inertial frame cannot be treated as inertial. I am bothered that a theory which is only suited for treating inertial frames is used to deal with a problem involving a non-inertial frame.

It's just the same with Newton's mechanics. Its laws refer to inertial frames, but nothing prevents from deriving from those laws the corresponding ones for accelerating frames (e.g. coordinate accelerations such as Coriolis).

No. That is the limited principle of relativity, for the special case of inertial frames. The principle of relativity states that the laws of physics are the same for all frames of reference.

If you deal with non-inertial frames within the confines of special relativity, then you have the same problem that Newton had: There is an absolute quality to acceleration; there is a preferred frame.

Likely you mean absolute frame. That was also Langevin's argument although neither Newton nor he saw that as a problem (note: he was one of the most prominent relativists in France). However, for some time Einstein considered that to be a problem. Historically that appears to be the central issue of the twin paradox.

I maintain my position that this does damage to the principle of relativity.

Einstein tried to get rid of that issue with GR, but didn't really succeed. Perhaps you refer here to the introduction in his 1916 paper(in particular §2)?
- http://web.archive.org/web/20060829045130/http://www.Alberteinstein.info/gallery/gtext3.html

It would be good if physics textbooks discussed this topic, but I don't know any that does.

 

[PAllen]

Note that such reasoning does not generally hold, as I mentioned before: in the original variant by Langevin both are in free fall. Still it is the traveler who ages less (he didn't in 1911 account for gravitational time dilation but that isn't pertinent and gravitation at the turn-around only enhances the effect).

An orbit is only possible with gravity. In the SR context, you would have to treat gravity as a force, which means the orbit is non-inertial. In the case of GR, the issue is that there are multiple free fall paths connecting the two end points. One of them is an absolute maximum of proper time (the radial out and back path). The other (orbit) is only a 'local' maximum.

It is a trivial mathematical fact that in flat spacetime, a geodesic=inertial path is an absolute maximum of clock time.

 

[harrylin]

[..] the orbit is non-inertial. [..]

Obviously! Thanks for the elaboration.

 

[stevendaryl]

Note that such reasoning does not generally hold, as I mentioned before: in the original variant by Langevin both are in free fall. Still it is the traveler who ages less (he didn't in 1911 account for gravitational time dilation but that isn't pertinent and gravitation at the turn-around only enhances the effect).

Just to expand on this point, when you include gravity, there can be two freefall paths with different amounts of aging. For example, you can imagine two different orbits around the earth: one is circular, and another is highly elliptical. If you choose them carefully, you can get the time period for one elliptical orbit to be the same as the time period for an integer number of circular orbits. So twins following these orbits would depart and reunite without ever accelerating.

 

[PAllen]

Just to expand on this point, when you include gravity, there can be two freefall paths with different amounts of aging. For example, you can imagine two different orbits around the earth: one is circular, and another is highly elliptical. If you choose them carefully, you can get the time period for one elliptical orbit to be the same as the time period for an integer number of circular orbits. So twins following these orbits would depart and reunite without ever accelerating.

and there is always at least one free fall path that is an absolute maximum of proper time along all possible paths between chosen events.

 

[Alain2.7183]

I guess that finishes off the CADO nonsense.

What is CADO?

 

[Mentz114]

What is CADO?

See this topic

https://www.physicsforums.com/showthread.php?t=490163

 

[GregAshmore]

I will take the time later, probably on the weekend, to go through the details of your responses and see how much of the detail I am able to grasp. My math in special relativity is only just getting into four-vectors--and at the moment I've left off that to start from the beginning in Katz' Intro to SR.

Really, though, the objection I have raised regarding the treatment of the twin paradox in SR is at such a basic level that it can be expressed with no math at all. I object as strenuously as I do because I believe that the net result of such a treatment is to prevent beginners from obtaining a confident grasp of the fundamental concepts of relativity and moving on. Instead of helping them along, it confuses them and makes them question whether they have learned anything at all. (I speak of "them" instead of "me" because I would expect that many who post questions on this forum are in a situation similar to mine: having a college degree with some technical content, and working independently to understand relativity.)

Perhaps the best way to make my point is to go back to the objector in Taylor & Wheeler. Why does the objector say that we ought to be able to consider the rocket to be still and the Earth moving? Nobody comes up with that question on his own. He makes that assertion because that's what every popular book about relativity says. And those books speak accurately, in the general sense. That is the allure of relativity: to learn how they pull it off, how they make sense of having the rocket stay in one place as the Earth moves away and back again. (And, often enough, there is, in addition to the desire to learn how they pull it off, the suspicion that they won't be able to. People can hardly be blamed for that, given the strangeness of the proposition.)

The self-taught person soon learns that the math of relativity in the general sense is far out of reach of the beginner; he will have to start with special relativity and hope to move on to general relativity eventually. But always in his mind is that goal, to understand how it is that the rocket can be legitimately understood to be at rest, and the Earth moving.

To reduce this to a single phrase, I'll repeat Einstein. The passenger in the braking train says, "I am permanently at rest." This is the axiom of relativity: Everyone is permanently at rest, yet everyone uses the same laws of physics. The corollary is, "You can't tell which object is really moving."

What happens when the twin paradox is treated within special relativity? The case of the resting rocket is not considered. Why? Because the rocket accelerates. The logical conclusion, while never explicitly stated, is that the rocket is really moving and the Earth is really at rest. Both the axiom of relativity and its corollary are violated, yet the author acts as though nothing has happened, and all is well in the world of relativity.

As if that were not bad enough, the basic rules of the game are violated--or at least stretched beyond the comprehension of the beginner. Special relativity is for inertial frames. There is no place for reversal of direction in an inertial frame. So how do we get the rocket back to earth? We have it jump from one inertial frame to another. What is the meaning of "jumping frames"? Then, too, proper time is defined as the time between events which occur at the same place. How does one jump frames while remaining in place? Furthermore, if one is in the same place throughout the episode, doesn't that mean that one is at rest? Yet the rocket cannot be resting, because it accelerates--which can only happen if it is moving.

When the teacher violates the rules (or stretches them, if you prefer) without careful explanation of the motives and dangers of so doing, the student probably does not learn the rules, and almost certainly does not learn to test every possible action against the rules. So, for example, I proposed that the case of the resting rocket/moving Earth ought to be drawn on the spacetime diagram. Of course it can't, if the rules are strictly observed. But then, if the rules are strictly observed, the traveling rocket can't be drawn on the spacetime diagram either.

It doesn't help matters at all to say that proper acceleration is invariant. The invariance of proper acceleration is not a reason to exclude consideration of the case of the resting rocket, precisely because proper acceleration is invariant. The resting rocket is at rest in a gravitational field; it does have proper acceleration at the same time it does not have coordinate acceleration. That's part of the seemingly magical feat, isn't it, to show how it is possible to accelerate while remaining permanently at rest?

The crux of the problem is that the resting rocket cannot be considered within the confines of special relativity. The solution to the problem is obvious: ditch special relativity. Better to explain that the math is too difficult for beginners, and the origin of the gravitational field must be left open for now, than to cause all the problems noted above.

 

[stevendaryl]

The crux of the problem is that the resting rocket cannot be considered within the confines of special relativity. The solution to the problem is obvious: ditch special relativity. Better to explain that the math is too difficult for beginners, and the origin of the gravitational field must be left open for now, than to cause all the problems noted above.

Clearly, it's better not to teach anything to someone if he's a beginner, because he's likely to get confused. Just joking!

I have to disagree that the mathematics of relativity is beyond the beginner. Most of the mathematics of relativity is no harder than the mathematics of planar geometry, which people do learn in high school.

I think it's true any time you teach any subject that the interested student can come up with questions that really cannot be answered without going far beyond the beginner level. I don't think that's such a bad thing. It leaves a little mystery that requires more thought and more work to resolve, then that's an incentive to go on and learn the advanced stuff.

But I don't understand what you mean by "ditch special relativity". Do you mean: don't teach relativity, or do you mean only teach General Relativity? It would be a big mistake to do that, because General Relativity relies on an understanding of Special Relativity.

 

[stevendaryl]

The crux of the problem is that the resting rocket cannot be considered within the confines of special relativity.

That is simply not true. The boundary between what's "Special Relativity" and what is "General Relativity" is a matter of naming, but the only thing beyond Special Relativity you need to understand accelerated frames is calculus.

 

[Mentz114]

Yet the rocket cannot be resting, because it accelerates--which can only happen if it is moving.

I see why you're upset. But an object can be at rest and also accelerating. The equivalence principle let's us say that if we can feel our weight, we are accelerating.

When the pseudo-gravitational field is invoked to bring the 'rocket' twin to rest, the occupants of the rocket have dx/dt = 0, but they feel weight because of the gravitational field. Therefore they are still accelerating while 'at rest'.

I think the rest of your exposition is thus based on a misunderstanding. Not to mention somewhat misguided and wrong.

 

[Dale]

I object as strenuously as I do because I believe that the net result of such a treatment is to prevent beginners from obtaining a confident grasp of the fundamental concepts of relativity and moving on.

None of that makes your objection correct. That beginners struggle with a concept does not make the concept wrong nor does it provide any justification for the recalcitrant student.

If you had any substance in your post, I apologize, but I missed it amongst all of the irrelevant angst. Maybe you can try to make a more concise post.

 

[Dale]

The invariance of proper acceleration is not a reason to exclude consideration of the case of the resting rocket, precisely because proper acceleration is invariant.

The problem is one of uniqueness. If I say "Bob's frame" in SR that phrase has a unique meaning if Bob's proper acceleration is 0. If Bob undergoes proper acceleration then the phrase "Bob's frame" no longer has a unique meaning.

We can certainly discuss the rockets frame, but first you have to tell us exactly what you mean by that. There is no standard meaning.

This approach I prefer is this one: http://arxiv.org/abs/gr-qc/0104077 (although they go a little overboard in some of their descriptions)

 

[PeterDonis]

Everyone is permanently at rest, yet everyone uses the same laws of physics.

This is true, but you appear to be mistaking it for another statement, which is not true: "Everyone is permanently at rest, therefore everyone is equivalent." Two observers are only equivalent if they are experiencing exactly the same observables, which includes proper acceleration. Two observers experiencing different proper accelerations (for example, one free-falling off a cliff and one standing at rest at the bottom of the cliff) are not equivalent. They both use the same underlying laws of physics, but they are realizing different and inequivalent particular solutions of those laws.

ditch special relativity. Better to explain that the math is too difficult for beginners, and the origin of the gravitational field must be left open for now

And this leaves us with...what, exactly? If we ditch SR, but we also say the math of GR is too difficult for beginners, how do we do any physics at all?

I understand why you are frustrated; you see the goal (the statement I quoted at the top of this post), but you don't see why we are taking such an apparently roundabout path towards it. The reason is that nobody has found a better path. Perhaps there is one; but you seem to be advocating no path at all, which doesn't strike me as an improvement.

 

[danR]

And this leaves us with...what, exactly? If we ditch SR, but we also say the math of GR is too difficult for beginners, how do we do any physics at all?

I understand why you are frustrated; you see the goal (the statement I quoted at the top of this post), but you don't see why we are taking such an apparently roundabout path towards it. The reason is that nobody has found a better path. Perhaps there is one; but you seem to be advocating no path at all, which doesn't strike me as an improvement.

On the pedagogical side of it:

Would I be throwing a monkey-wrench (that's 'spanner' for all the aluminium-talkers across the pond) in the works to suggest that Einstein et al.'s idea that a 5th dimension might be needed for relativity, and that it might have the added benefit of an absolute frame of reference? The reason I (rather counterintuitively) suggest this is I once asked one of my profs if this 5th dimension was Euclidean—flat. He said it was flat.

Then we strip out 2 superfluous dimensions from the discussion and present things in well-crafted diagrams, videos, and 3-d videos.

But isn't that what people have been doing for years? Good pictures and diagrams?

 

[PeterDonis]

Would I be throwing a monkey-wrench (that's 'spanner' for all the aluminium-talkers across the pond) in the works to suggest that Einstein et al.'s idea that a 5th dimension might be needed for relativity, and that it might have the added benefit of an absolute frame of reference?

I'm not sure what you're referring to here, but I suspect it's Kaluza-Klein theory:

http://en.wikipedia.org/wiki/Kaluza–Klein_theory

Einstein liked this idea because it held out a hope of unifying gravity and electromagnetism. However, it hasn't panned out in the form he liked it, although there are similar elements in current theories.

However, this...

The reason I (rather counterintuitively) suggest this is I once asked one of my profs if this 5th dimension was Euclidean—flat. He said it was flat.

...makes me wonder, because in K-K theory, the 5th dimension is not flat; it's a circle. (In fact K-K theory can be thought of as attaching a tiny circle to each point of 4-dimensional spacetime.) Can you give more details?

 

[harrylin]

I see why you're upset. But an object can be at rest and also accelerating. The equivalence principle let's us say that if we can feel our weight, we are accelerating. [..]

With "accelerating" and "equivalence principle" you likely mean something else than Einstein, while the OP bases his understanding on Einstein's explanations. According to Einstein's GR an object that is accelerating relative to an inertial frame can be held to be constantly at rest and thus not accelerating (of course, a=d²x/dt²).

 

[harrylin]

I will take the time later, probably on the weekend, to go through the details of your responses and see how much of the detail I am able to grasp. My math in special relativity is only just getting into four-vectors--and at the moment I've left off that to start from the beginning in Katz' Intro to SR.

If you go through Einstein's 1905 paper you won't find any four-vectors. Such tools can be handy, but you don't need them for understanding that theory.

Really, though, the objection I have raised regarding the treatment of the twin paradox in SR is at such a basic level that it can be expressed with no math at all. I object as strenuously as I do because I believe that the net result of such a treatment is to prevent beginners from obtaining a confident grasp of the fundamental concepts of relativity and moving on. Instead of helping them along, it confuses them and makes them question whether they have learned anything at all. (I speak of "them" instead of "me" because I would expect that many who post questions on this forum are in a situation similar to mine: having a college degree with some technical content, and working independently to understand relativity.)

There isn't a singular "the treatment" of the twin paradox in SR. While mathematically they all agree here are many different treatments, and you can choose the one that best matches your way of thinking. For example for my own understanding and intellectual satisfaction I had to get to the bottom of it by digging into the old papers so that I understood how the question arose.

Perhaps the best way to make my point is to go back to the objector in Taylor & Wheeler. Why does the objector say that we ought to be able to consider the rocket to be still and the Earth moving? Nobody comes up with that question on his own.

Yes indeed; however that's rather well explained in Einstein's 1918 paper, which you read.

[..] The self-taught person soon learns that the math of relativity in the general sense is far out of reach of the beginner; he will have to start with special relativity and hope to move on to general relativity eventually. But always in his mind is that goal, to understand how it is that the rocket can be legitimately understood to be at rest, and the Earth moving. [..]

Right. And as you now realize, in reality that goal was never reached. I don't know anyone who holds that for example the Earth is constantly "truly in rest".

[...] What happens when the twin paradox is treated within special relativity? The case of the resting rocket is not considered. Why? Because the rocket accelerates. The logical conclusion, while never explicitly stated, is that the rocket is really moving and the Earth is really at rest. Both the axiom of relativity and its corollary are violated, yet the author acts as though nothing has happened, and all is well in the world of relativity.

Note: the Earth is also not "really at rest" in SR. How well do you understand classical mechanics? For smooth learning of SR a good understanding of classical mechanics is important.

As if that were not bad enough, the basic rules of the game are violated--or at least stretched beyond the comprehension of the beginner. Special relativity is for inertial frames. There is no place for reversal of direction in an inertial frame. So how do we get the rocket back to earth? We have it jump from one inertial frame to another. What is the meaning of "jumping frames"? [..]

Once more: please make sure to have a good understanding of classical mechanics. It uses inertial reference frames and one can switch between them, but it's necessary to understand what one does and what that means. For example, it's common to "jump" to a center of mass" frame. Are you familiar with that concept? If not, then you are in fact "jumping" (skipping) lessons. Usually the result is confusion and lack of understanding.

[..] the rocket cannot be resting, because it accelerates--which can only happen if it is moving.

In 1916 GR one may pretend that the rocket is not moving, and instead it is at rest in an induced gravitational field - probably you did not yet read the physics FAQ.

[..] That's part of the seemingly magical feat, isn't it, to show how it is possible to accelerate while remaining permanently at rest?

No, that's two different ways of viewing the same physical situation; and regretfully it's made more difficult to comprehend ("magical"?) due to the introduction of new terms that lead to descriptions that are at odds with earlier ones.

The crux of the problem is that the resting rocket cannot be considered within the confines of special relativity. The solution to the problem is obvious: ditch special relativity. Better to explain that the math is too difficult for beginners, and the origin of the gravitational field must be left open for now, than to cause all the problems noted above.

A much more pragmatic way of dealing with the issue would be (indeed, it's the common approach of textbooks):
- we do not need to consider the rocket to be in rest all the time
- just use SR for the problem

 

[ghwellsjr]

This is not the first time someone has become confused by T&W's exclusive explanation of SR. I do not recommend the book, it does more harm than good.

I've recommended the book here before, but when I learned SR from it, it was in the context of a class, with a teacher teaching from it. I can see how that might make a difference; T&W's language is somewhat idiosyncratic (like that of MTW--I suspect it's Wheeler's influence), and it might come across better when there's a teacher to interpret, so to speak.

The second edition of T&W's Spacetime Physics came out in 1992. I suspect you went to school before that and used the original edition. The two books are as different as night and day. The original edition does not have objectors presented as a buffoons, as Greg puts it in post #39. It has no mention of a Proper Clock (that I could find) which is their preferred method of analyzing scenarios in the second edition.

Perhaps the best way to make my point is to go back to the objector in Taylor & Wheeler. Why does the objector say that we ought to be able to consider the rocket to be still and the Earth moving? Nobody comes up with that question on his own. He makes that assertion because that's what every popular book about relativity says. And those books speak accurately, in the general sense. That is the allure of relativity: to learn how they pull it off, how they make sense of having the rocket stay in one place as the Earth moves away and back again. (And, often enough, there is, in addition to the desire to learn how they pull it off, the suspicion that they won't be able to. People can hardly be blamed for that, given the strangeness of the proposition.)

Greg, I'm telling you, T&W's objectors are buffoons. T&W are glad you see them as buffoons. They don't want you to understand the answer to the objectors' questions. They want you to simply understand the Twin Paradox by their preferred method which is to use Proper Clocks, which is their unique term for the spacetime interval.

Look at their summary on page 131 where they say that "each of the three [inertial reference] frames...has a perfectly consistent and nonparadoxical interpretation of the sequence of events". But then instead of showing how that works, they quote an objector, "all these different [inertial] reference frames sure do complicate the story" and they respond with "Exactly! These complications arise because observations from anyone frame are limited and parochial. All disagreements can be bypassed by talking only in the invariant language of spacetime interval, proper time, wristwatch time."

But at least on the next page they say in very bold letters:

DO WE NEED GENERAL RELATIVITY? NO!

The self-taught person soon learns that the math of relativity in the general sense is far out of reach of the beginner; he will have to start with special relativity and hope to move on to general relativity eventually.

The math of Special Relativity is very simple and so are the explanations that T&W denigrate. In fact I presented it all in the thread that you referenced in your first post:

So this morning I searched on this forum, wishing to avoid being the 9,488th person to ask about the twin paradox. I found this thread.

Go to the second page and post #23. Please study it. It has very simple math. I believe that you can understand it. If you have any questions, please ask. Please don't dismiss it just because T&W dismiss it.

 

[Mentz114]

With "accelerating" and "equivalence principle" you likely mean something else than Einstein, while the OP bases his understanding on Einstein's explanations. According to Einstein's GR an object that is accelerating relative to an inertial frame can be held to be constantly at rest and thus not accelerating (of course, a=d²x/dt²).

I don't agree with this. For instance, the equations show that an object at rest at a constant r in the Schwarzschild vacuum feels a force - and thus is not inertial, nor moving ( relative to the field).

In 1916 GR one may pretend that the rocket is not moving, and instead it is at rest in an induced gravitational field

Yes, but it will still be non-inertial. Are you saying that in this scenario the rocket feels no acceleration ?

 

[PeterDonis]

The second edition of T&W's Spacetime Physics came out in 1992. I suspect you went to school before that and used the original edition.

Yes, that's right, I did.

The two books are as different as night and day.

I had wondered about that, because I don't have the original edition (lost my copy years ago), and when I got the second edition I did think "huh?" quite a bit while reading it, because it didn't seem to match my memory of the original. It's a shame that it seems to have changed so much.

 

[stevendaryl]

But at least on the next page they say in very bold letters:

DO WE NEED GENERAL RELATIVITY? NO!

I would make an even stronger claim: For resolving the twin paradox in flat spacetime, GR is not needed, and DOESN'T HELP AT ALL. The theory of General Relativity, in the case of flat spacetime reduces to Special Relativity. So if spacetime curvature is negligible, then there is no difference between solving the problem using GR and solving the problem using SR. You could say that you're solving the problem using mathematical techniques developed for GR, but you're not using any physical principles that go beyond SR.

(Some people say that the clock hypothesis, that clocks measure proper time, goes beyond SR. I think it's a matter of definition of what an "ideal clock" is. It's surely the case in SR that proper time is a physically meaningful quantity. Whether or not we can build mechanical devices that can measure it is an engineering question.)

 

[danR]

...makes me wonder, because in K-K theory, the 5th dimension is not flat; it's a circle. (In fact K-K theory can be thought of as attaching a tiny circle to each point of 4-dimensional spacetime.) Can you give more details?

I've just bothered now to Google it, and once you subtract Kaluza from the search almost all roads lead to crank-dom. So apparently the prof had one inexplicable gap in his knowledge, or he completely misunderstood the question. After looking at his C.V. now I see his specialty is almost exclusively in QM, so it might have been a bit of both.

 

[GregAshmore]

By "ditch special relativity", I did not mean to say that SR should not be taught at all. I meant that SR should not be used to prove that the Twin Paradox is not a paradox.

Specifically, I mean that SR should not be used to prove that what I earlier called the "second aspect" of the Twin Paradox is not a paradox. That is the claim that when the episode is observed from the "permanently at rest" frame of the rocket, the Earth twin will be younger than the rocket twin. The paradox then is that both twins are "younger than the other", which can't happen in reality.

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

It has been pointed out that only the rocket experiences proper acceleration (which is, I believe, essentially the same thing as saying that the rocket frame is not inertial). If it can be shown that proper acceleration affects the operation of clocks, then there would be no need to consider the case of the permanently resting rocket. Is it claimed that proper acceleration affects clocks?

If proper acceleration does not affect clocks, it is necessary to consider the case of the resting rocket in order to prove or disprove the claim made about the case of the resting rocket. The case cannot be considered in SR; therefore the claim cannot be addressed within the confines of SR.

Unless, of course, it can be shown that the observer in the rocket cannot legitimately consider himself to be permanently at rest.

With respect to that, in my previous post I said, "But always in his mind is that goal, to understand how it is that the rocket can be legitimately understood to be at rest, and the Earth moving."

To which there was this reply:

Right. And as you now realize, in reality that goal was never reached. I don't know anyone who holds that for example the Earth is constantly "truly in rest".

And a few remarks later:

A much more pragmatic way of dealing with the issue would be (indeed, it's the common approach of textbooks):
- we do not need to consider the rocket to be in rest all the time
- just use SR for the problem

Harrylin, did you mean to say that the observer in the rocket cannot legitimately consider himself to be permanently at rest?

How would others in the discussion answer this question: Can the observer in the rocket legitimately consider himself to be permanently at rest? [edited to remove the misplaced word 'cannot']

 

[stevendaryl]

By "ditch special relativity", I did not mean to say that SR should not be taught at all. I meant that SR should not be used to prove that the Twin Paradox is not a paradox.

Well, there is nothing besides SR to prove that. As I said in a different post, General Relativity is the SAME theory as SR, in the case where there are no significant gravitational masses. GR doesn't tell us anything about the twin paradox beyond what SR tells us.

It has been pointed out that only the rocket experiences proper acceleration (which is, I believe, essentially the same thing as saying that the rocket frame is not inertial). If it can be shown that proper acceleration affects the operation of clocks, then there would be no need to consider the case of the permanently resting rocket. Is it claimed that proper acceleration affects clocks?

What you can prove from SR alone is that if two clocks start at the same starting point, travel at different velocities, and end up at the same end point, then the one that accelerates will have the shortest proper time.

 

[stevendaryl]

If proper acceleration does not affect clocks, it is necessary to consider the case of the resting rocket in order to prove or disprove the claim made about the case of the resting rocket. The case cannot be considered in SR; therefore the claim cannot be addressed within the confines of SR.

What you are saying is just not true.

 

[PeterDonis]

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

This is not true; you can use curvilinear coordinates in SR. I believe someone already pointed that out earlier in this thread. What you can't do is use curvilinear coordinates to describe a non-inertial frame and expect everything to work exactly the same as it does in an inertial frame. It won't. But as long as you're in flat spacetime, you're within the domain of SR.

For an example of a non-inertial coordinate chart in flat spacetime, which is perfectly valid within SR, try Rindler coordinates:

http://en.wikipedia.org/wiki/Rindler_coordinates

Note that the Rindler coordinate chart does not cover all of spacetime; that's one key difference between it (and most non-inertial charts) and an inertial chart.

Is it claimed that proper acceleration affects clocks?

No. But proper acceleration does make the "rest frame" of the accelerating object non-inertial.

How would others in the discussion answer this question: Can the observer in the rocket cannot legitimately consider himself to be permanently at rest?

He can, but as I said above, he can't expect his "rest frame" to work like an inertial frame, because it isn't one.

 

[GregAshmore]

Greg, I'm telling you, T&W's objectors are buffoons. T&W are glad you see them as buffoons. They don't want you to understand the answer to the objectors' questions. They want you to simply understand the Twin Paradox by their preferred method which is to use Proper Clocks, which is their unique term for the spacetime interval.

Look at their summary on page 131 where they say that "each of the three [inertial reference] frames...has a perfectly consistent and nonparadoxical interpretation of the sequence of events". But then instead of showing how that works, they quote an objector, "all these different [inertial] reference frames sure do complicate the story" and they respond with "Exactly! These complications arise because observations from anyone frame are limited and parochial. All disagreements can be bypassed by talking only in the invariant language of spacetime interval, proper time, wristwatch time."

I will look again at the text.

But at least on the next page they say in very bold letters:

DO WE NEED GENERAL RELATIVITY? NO!

As you see in the above post, I disagree on this point. If a claim is made about the case of the resting rocket, that case must be considered in order to prove or disprove the claim, except as noted in the above post.

The math of Special Relativity is very simple and so are the explanations that T&W denigrate. In fact I presented it all in the thread that you referenced in your first post:

Go to the second page and post #23. Please study it. It has very simple math. I believe that you can understand it. If you have any questions, please ask. Please don't dismiss it just because T&W dismiss it.

The math of the Lorentz transform is very simple, and it is sufficient for simple problems such as the pole-in-barn paradox. (The interpretation of the results is not so simple, though.) But four-vectors and other, even more advanced math constructions do come up with regularity in these discussions.

I'll reread post #23.

 

[Mentz114]

The problem with treating this aspect of the twin paradox in SR is that the case of the resting rocket cannot be considered. In SR, only observers in inertial frames can consider themselves to be at rest.

As has been pointed out to you in earlier posts, both these sentences are false. It is possible to find coordinates in which the accelerating twin remains stationary but non-inertial. In these coordinates the accelerating twin has less proper time than the 'earth' twin. No problem. Changing coordinates will not change the invariant proper times.

 

[GregAshmore]

What you can prove from SR alone is that if two clocks start at the same starting point, travel at different velocities, and end up at the same end point, then the one that accelerates will have the shortest proper time.

Not quite. What you are proving is that when the inertial frame is considered to be at rest, and two clocks start at the same starting point...

You have not addressed the claim made about what happens when the rocket is considered to be at rest.

 

[stevendaryl]

As you see in the above post, I disagree on this point [about whether GR is needed]

But it isn't a matter of opinion. You're just wrong. GR adds nothing to the calculation that isn't already in SR. GR is the SAME theory as SR in the limit in which there are no significant masses present.

 

[stevendaryl]

Not quite. What you are proving is that when the inertial frame is considered to be at rest, and two clocks start at the same starting point...

That's completely wrong. It doesn't matter what you "consider" to be at rest.

 

[Mentz114]

GregAshmore, you are starting to talk nonsense, and ignoring any post that you can't refute by handwaving. It is possible to find coordinates in which the accelerating twin remains stationary but non-inertial. In these coordinates the accelerating twin has less proper time than the 'earth' twin. No problem. Changing coordinates will not change the invariant proper times.

Are you denying this ?