Understanding Einstein's Thought Experiment & Question 10 of the Relativity Quiz

[εllipse]

I just took the relativity quiz here: https://www.physicsforums.com/quiz.php?quizid=28 and I goofed on question 10. Could someone explain it to me? Below I've posted what the review of the quiz said about question 10 along with some of the previous questions for context.

Question 8

Assume Stella leaves Earth and travels exactly one light-year away (as measured by Terrance) at 0.99 c (as measured by Terrance), sending laser flashes every second back to Terrance.

In that time, Terrance's watch has ticked off an entire year. Since Stella is sending periodic flashes to Terrance, and he has received fewer than a year's worth of flashes, Terrance concludes that Stella's clock has elapsed less time than his own.

Can it now be said that Terrance has aged more than Stella?

...

Question 9

If Stella then stops, turns around, and very slowly returns to Earth, can she be said to have aged less than Terrance?

...

Question 10

The question was:
Can Stella argue that she's been at rest the whole time in her rocket ship, while it was Terrance who flew away for a light-year at 0.99 c?
Your answer was: Yes, because anyone can declare him/herself at rest.
Correct answer: No, because Stella felt accelerations while Terrance did not.
Your answer was: wrong
Explanation of the correct answer:
There are many ways to explain why Stella and Terrance did not take symmetric trips, but the easiest is just to consider the accelerations. If Stella felt accelerations at the appropriate times, then her velocity must have been changing, and she must have been the one who left Earth, turned around, and came home.

In more advanced mathematical discussions of this "twin paradox," it can be easily shown that the so-called "proper time," the time elapsed on clocks that Terrance and Stella carry around with them, is smaller along the path that Stella takes through space-time than along the path taken by Terrance.

This seems to conflict with what I've read about GR. In one of Einstein's thought experiments he imagined a man in a chest with a rope tied to the top being pulled with a constant acceleration in one direction. Einstein said:

Relying on his knowledge of a gravitational field (as it was discussed in the preceding section), the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time...

...Ought we to smile at the man and say that he errs in his conclusion? I do not believe we ought to if we wish to remain consistent; we must rather admit that his mode of grasping the situation violates neither reason nor known mechanical laws. Even though it is being accelerated with respect to the "Galileian space" first considered, we can nevertheless regard the chest as being at rest.

- Relativity: The Special and General Theory, Section 20

This seems to support that Stella can call herself at rest. The only argument I can think of is that Stella's acceleration would generally be thought to not be constant and maybe that's why we can't call her at rest. Although it is possible for Stella to go from moving away from Earth, come to a stop, and return to Earth with a constant acceleration, maybe the question was taking the most likely situation in which Stella changes her acceleration multiple times during the course of the trip, and this is why she can't call herself at rest. The only other reason I can think of why Stella can't call herself at rest is if modern physics has replaced the conclusions of Einstein's acclerated chest thought experiment, and it's more acceptable now to call such "gravitational fields" that aren't the result of matter an indication of change in motion. Can someone clear this up for me?

 

[James R]

Ellipse:

Can Stella argue that she's been at rest the whole time in her rocket ship, while it was Terrance who flew away for a light-year at 0.99 c?
Your answer was: Yes, because anyone can declare him/herself at rest.
Correct answer: No, because Stella felt accelerations while Terrance did not.

I agree with you. You are right. The quiz answer is wrong.

Anybody can declare himself to be at rest. Everybody has a "rest frame".

The differences in final elapsed times in the twin paradox depend on whose frame of reference is inertial, and whose is not. Stella's is a non-inertial frame. Terrance's frame is inertial.

 

[robert Ihnot]

Question 9

If Stella then stops, turns around, and very slowly returns to Earth, can she be said to have aged less than Terrance?

I thought the answer should be not until she is reunited with Terrance, (the clocks can not be compaired) but how the answer is written, it is an unquallified "yes," because when she is reunited she will be seen to be younger.?

 

[robert Ihnot]

James R: I agree with you. You are right. The quiz answer is wrong.

Anybody can declare himself to be at rest. Everybody has a "rest frame".

The differences in final elapsed times in the twin paradox depend on whose frame of reference is inertial, and whose is not. Stella's is a non-inertial frame. Terrance's frame is inertial.

But since Stella felt a change in acceleration, she must recognize she is not at rest.

 

[learningphysics]

Question 9

If Stella then stops, turns around, and very slowly returns to Earth, can she be said to have aged less than Terrance?

I thought the answer should be not until she is reunited with Terrance, but how the answer is phrased it is an unquallified "yes," because when she is reunited she will be seen to be younger.

Yes. I made the exact same mistake. Once she is reunited with Terrance then she can be said to have aged less.

The explanation seemed to presume that they compare the clocks once they are reunited... but it seemed to me that the question suggested that Stella was on the return trip, not that she had arrived at Earth.

 

[robert Ihnot]

learningphysics: Yes. I made the exact same mistake. Once she is reunited with Terrance then she can be said to have aged less.

Good! Then we are outvoting any other explanation. #9 was the only one I missed!

 

[jdavel]

Ellipse:



I agree with you. You are right. The quiz answer is wrong.

Anybody can declare himself to be at rest. Everybody has a "rest frame".

The differences in final elapsed times in the twin paradox depend on whose frame of reference is inertial, and whose is not. Stella's is a non-inertial frame. Terrance's frame is inertial.

James R,

The quiz answer is right. Everybody does not have a "rest frame".

If you're in a rest frame then experiments done in your frame will match the results of those done in any other rest frame. But if Stella starts an experiment on her way out and finishes it after she starts to turn around, she won't get the same result as she would have in a rest frame.

 

[εllipse]

James R,

The quiz answer is right. Everybody does not have a "rest frame".

If you're in a rest frame then experiments done in your frame will match the results of those done in any other rest frame. But if Stella starts an experiment on her way out and finishes it after she starts to turn around, she won't get the same result as she would have in a rest frame.

Can you explain why Einstein calls the chest at rest? Experiments in the chest will not be the same if the chest is inertial.

 

[jdavel]

ellipse,

Yes, because, in the case described by E, the guy in the chest concludes that he is in a g field "which is constant with regard to time..." Stella can't come to that conclusion.

 

[James R]

robert Ihnot:

But since Stella felt a change in acceleration, she must recognize she is not at rest.

No. She recognises that she is at rest, but something weird is going on with her local gravity.

jdavel:

The quiz answer is right. Everybody does not have a "rest frame".

Of course they do. Every person is always at rest relative to him or herself. The definition of "rest frame of an object" is the "reference frame in which the object is at rest".

If you're in a rest frame then experiments done in your frame will match the results of those done in any other rest frame.

Replace "rest frame" with "inertial frame", and you're right.

There's nothing in physics which says the laws of physics are the same in all rest frames. They are the same in all inertial frames, though. Not every rest frame is inertial, but some inertial frames are also rest frames.

 

[pervect]

The quiz answer is right. Everybody does not have a "rest frame".


Of course they do. Every person is always at rest relative to him or herself. The definition of "rest frame of an object" is the "reference frame in which the object is at rest".

The quiz is right.

While everyone is at rest relative to him or herself, the definition of a rest frame given above is incorrect.

A 'frame' is a very special sort of coordinate system, one in which there are no pseudo-forces (to use Newtonian language), or in which the Christoffel symbools are all zero (to use more precise but less generally understood GR language).

A coordinate system in which an accelreated observer is at rest is not a frame, because it does have pseudo-forces (and non-zero Christoffel symbols).

There are other difficulties with the coordinate system of an accelerated observer besides the fact that it is not a frame. Such a coordinate system only covers a small local region of space and not all of space-time as a truly well behaved coordinate system should.

There are two reasons for this - one reason is the existence of an event horizon in the coordinate system of an accelerated observer, and another is the fact that coordiante lines cross each other, giving rise to an unacceptable situation in which one event has multiple coordinates when the coordinate system is made too large.

 

[jdavel]

James R.,

You said: "...The quiz answer is wrong. Anybody can declare himself to be at rest. Everybody has a "rest frame"."

To which I said: 'The quiz answer is right. Everybody does not have a "rest frame"."

To which you said: "Of course they do. Every person is always at rest relative to him or herself. The definition of "rest frame of an object" is the "reference frame in which the object is at rest"."

To which I say: Touche!

However...I still think the quiz answer is wrong! But to make my point, I should have argued with the quiz answer "Anybody can declare himself/herself to be at rest". Instead of your translated version "Everybody has a rest frame" (even though pervect has now said they're the same thing).

Stella can not declare herself to have been at rest the whole time. She knows otherwise! Constant (wrt time) acceleration is indistnguishable from constant gravity. But time varying acceleration (what we have here) is not. Gravity alone can not produce time varying acceleration that is detectable without reference to the mass that's causing the gravity. Even if some huge mass suddenly appeared outside Stella's space ship, she wouldn't know it; she'd just be moving in free fall instead of at constant velocity. Until she crashed into the huge mass, all her experiments would continue to give the same results.

 

[learningphysics]

The quiz is right.

While everyone is at rest relative to him or herself, the definition of a rest frame given above is incorrect.

The question made no mention of "rest frame".

Can Stella say that she's at rest during the entire trip?

 

[pervect]

The question made no mention of "rest frame".

Perhaps not - but the answer offered sure did, and it made a hash of it, I'm afraid.

The original question was, IIRC

Question 8

Assume Stella leaves Earth and travels exactly one light-year away (as measured by Terrance) at 0.99 c (as measured by Terrance), sending laser flashes every second back to Terrance.

In that time, Terrance's watch has ticked off an entire year. Since Stella is sending periodic flashes to Terrance, and he has received fewer than a year's worth of flashes, Terrance concludes that Stella's clock has elapsed less time than his own.

Can it now be said that Terrance has aged more than Stella?

I will assume that this is the question you had in mind (the thread has diverged a lot).

The answer to this question is no. The issue of whether or not Stella can consider herself to be "at rest" in some general philosophcial sense is not really releveant to the answer.

What you need to do to prove that Terrance has aged more than Stella is to prove that the twins cannot be reunited in such a way that Terrance is younger than Stella. (The question was a little vague, that's the way I interpreted it, however).

But there is at least one way to re-unite the twins in which Terrance is the older. This is for Stella to change her mind, and re-accelerate to her original velocity, and for Terrance to accelerate to an even faster velocity to rejoin Stella.

 

[learningphysics]

Perhaps not - but the answer offered sure did, and it made a hash of it, I'm afraid.

The original question was, IIRC

This was the question I was referring to:

"Question 10

Can Stella argue that she's been at rest the whole time in her rocket ship, while it was Terrance who flew away for a light-year at 0.99 c?"

 

[pervect]

Ah, OK. In the context of special relativity, the answer is "no". A person who is "at rest" doesn't feel any accelerations or other pseudo-forces. Stella does feel these things, sho she can't be considered to be "at rest".

You can consider in this context that "at rest" means "in an inertial frame of reference", which means that acceleration is not allowed.

 

[cefarix]

same here...i got question #9 wrong because I thought stella hadn't reached Earth yet

 

[εllipse]

Ah, OK. In the context of special relativity, the answer is "no". A person who is "at rest" doesn't feel any accelerations or other pseudo-forces. Stella does feel these things, sho she can't be considered to be "at rest".

You can consider in this context that "at rest" means "in an inertial frame of reference", which means that acceleration is not allowed.

Isn't the question referring to general relativity though? There are GR related questions before Question 10, so I think it's meant to take into context both SR and GR.

 

[εllipse]

Really, I don't care about the semantics of the question. I just wanted to know what the correct view was. From reading these responses, it seems Stell can call herself at rest, which is all I really wanted to know.

 

[jdavel]

From reading these responses, it seems Stell can call herself at rest, which is all I really wanted to know.

Nope. A lot of the responses say she can't, and they're right.

 

[pervect]

It's true that Stella can assign herself coordinates of (x=0, t=tau) in some local coordinate system - note that this coordinate system does not qualify as a frame.

However, this coordinate system will be limited in it's size - it will not cover all of space and time as one might naievly expect.

Explaining this point in detail is difficult, which is why it is preferable to work special relativity problems using the assumptions of special relativity only. When people attempt to use the axioms of General Relativity without fully understanding some of the subtle points, they can get themselves (and others) very confused.

In the context of special relativity, it's pretty clear that the correct answer is that Stella cannot consider herself to be stationary. If this quiz were going to be on a national test or something, it might be worthwhile to have the point clarified in the wording of the question, to explicitly indicate that only SR is to be used.

 

[James R]

pervect:

In the context of special relativity, the answer is "no". A person who is "at rest" doesn't feel any accelerations or other pseudo-forces. Stella does feel these things, sho she can't be considered to be "at rest".

I still disagree with you.

A person who is "at rest" doesn't change their coordinate position with time. That is the only thing "at rest" means.

A person, by definition, is always at rest in his own rest frame. (See comments on the word "frame" below.)

You can consider in this context that "at rest" means "in an inertial frame of reference", which means that acceleration is not allowed.

That may have been the intent of the question, but "at rest" is not the same as "in an inertial frame of reference", in my opinion. If the question had been phrased as asking whether Stella could regard herself as being in a single inertial frame of reference for the whole trip, then the answer would have been an unequivocal "No". But that's not what was asked.

A 'frame' is a very special sort of coordinate system, one in which there are no pseudo-forces (to use Newtonian language), or in which the Christoffel symbools are all zero (to use more precise but less generally understood GR language).

Do you have a reference for that?

I have never heard the term "frame" (short for "reference frame") used with that particular restriction. I would be interested to hear who uses the term that way (apart from yourself, of course).

There are other difficulties with the coordinate system of an accelerated observer besides the fact that it is not a frame. Such a coordinate system only covers a small local region of space and not all of space-time as a truly well behaved coordinate system should.

There are two reasons for this - one reason is the existence of an event horizon in the coordinate system of an accelerated observer, and another is the fact that coordiante lines cross each other, giving rise to an unacceptable situation in which one event has multiple coordinates when the coordinate system is made too large.

I'll take your word for this for now. You may well be right, but it is irrelevant to the question at hand, anyway.

It's true that Stella can assign herself coordinates of (x=0, t=tau) in some local coordinate system ...

In the context of special relativity, it's pretty clear that the correct answer is that Stella cannot consider herself to be stationary.

I think these two statements are contradictory. But then, we disagree on the meaning of "stationary", it seems.

 

[pervect]

Well, it's clear to me that if the quiz is a quiz on special relativity, Stella can not consider herself to be at rest anymore than she can if she were asked the same question in a quiz on Newtonian mechanics.

So the question comes down to - is this quiz a quiz on SR?

I just checked the title of the quiz, and it doesn't mention specifically that the quiz is about SR, one has to rely on context.

So if I were assigning point grades for some reason, I might give half a point for a yes answer if I were in a good mood (not currently the case - it's allergy season :-) ) because the question could be misconstrued with enough effort.

 

[pmb_phy]

Do you have a reference for that?

I do. See MTW page 467. No \Gammas at a point <i>P</i> in spacetime then there is no symbols then the G-field vanishes t that point. The converse is not required to be true. <br /> <br /> Pete

 

[c_helm@yahoo.com]

I am not well versed, and my answer can be dismissed here (and even gentle corrections would be greatly appreciated).

I would split the difference. Stella can claim SHE is at rest, but the ship she is on is not. Terrence can also claim HE is at rest, but the Earth he is on is not. Therefore, both S and T are at rest locally, but not in relation to each other. So Stella has to consider herself NOT at rest for the experiment, only inside the ship itself.

Okay, now I have confused myself... going back to music composition now...

 

[εllipse]

So if I were assigning point grades for some reason, I might give half a point for a yes answer if I were in a good mood (not currently the case - it's allergy season :-) ) because the question could be misconstrued with enough effort.

Yay!

So, pervect, even if the quiz is about SR, for future reference, if the quiz question (10) had been meant in the context of GR, you're saying that "yes, Stella can say she is at rest," is correct? I mean, I'm not trying to split hairs here (even though it appears it has come to that), I just want to understand.

 

[James R]

pervect:

Well, it's clear to me that if the quiz is a quiz on special relativity, Stella can not consider herself to be at rest anymore than she can if she were asked the same question in a quiz on Newtonian mechanics.

I say ANYBODY can say they are at rest all the time. It's just a choice of a coordinate system.


pmb_phy:

No \Gammas at a point P in spacetime then there is no symbols then the G-field vanishes t that point. The converse is not required to be true.

I have no argument with that.

 

[pervect]

Yay!

So, pervect, even if the quiz is about SR, for future reference, if the quiz question (10) had been meant in the context of GR, you're saying that "yes, Stella can say she is at rest," is correct? I mean, I'm not trying to split hairs here (even though it appears it has come to that), I just want to understand.

The thing to understand is that the coordinate system of an accelerating observer comes complete with some big red warning stickers.

The stickers are hard to read because of the fine print, but every time a manufacturer ships you an "accelerated observer coordinate system" kit, it comes complete with these big red warning labels :-).

Here's a portion of the warning label - you can see why the print is so small, the warning is rather long :-)

It is very easy to put together the words "the coordinate system of an accelerated observer," but it is much harder to find a concept these words might refer to. If takeen seriously, they are self-contradictory. The definte article "the" in the phrase suggests that one is thinking of some unique coordinate system naturally associated with some specified accelrated observer, such as one whose world line is given in <equation refereence>. If the coordinate system is indeed natural, one would expect that the coordiantes of any event could be determined by a sufficiently ingenious observer by sending and receiving light singnals. But from <figure reference> it is clear that the events corresponding to one quaretr of all spacetime can neither send light signals to, nor receive light signals from, the specifried observer. Another half of spacetime suffers lesser disabilities in this respect

<snip> [ed: The lesser disabilities are described, 1/4 can send but not receive, 1/4 can receive but not send.]

It is hard to see how the observer could define in any natural way a coordiante system covering events with which he has no causal reltionship, which he cannot see, and from which he cannot be seen!

 

[James R]

Another quick comment:

It is hard to see how the observer could define in any natural way a coordiante system covering events with which he has no causal reltionship, which he cannot see, and from which he cannot be seen!

You, right now, sitting (approximately) at the origin of an inertial reference frame, can have no causal relationship to events outside your light cone. Yet you can still assign coordinates to those events in a meaningful way. So, this "problem" is not unique to accelerated systems.

 

[pervect]

In an inertial frame, if I wait, the distant event eventually comes within my light-cone. The same cannot be said about the accelerated observer, who never sees events beyond the so-called "Rindler horizon" - at least not as long as he continues to accelerate.

It's fairly common in GR textbooks to compare the Rindler horizon to that of a black hole - they share many similarities.

Another issue that I didn't type far enough to reach is the problem of coordinate lines crossing in the coordinate system of an accelerated observer.

Imagine a sheet of graph paper where the 'x' and 'y' axis lines were curved, and two curves of constant 'x' intersected each other. This situation is pathological because it maps multiple coordinates to one event. This situation occurs when attempting to define "the" coordinate system of an accelerated observer, at least when the usual definition is used. (The usual definition is to use the coordinates of an instantaneously co-moving inertial observer as the coordinates of the accelerated observer). When this definition is used to define a coordinate system for an accelerated observer, it has a maximum size of roughly c/a, c being the speed of light, and a being the acceleration.