Implications of the statement Acceleration is not relative

[stevendaryl]

I know there have been long PF threads on this before, and I don't want to start another one, but I don't think this claim is a slam dunk either way. For a good exposition of the view that GR *does* embody Mach's Principle in at least some form, see Cuifolini & Wheeler's book Gravitation and Inertia.

Well, the sense in which it doesn't satisfy Mach's principle is that in flat spacetime, there are still inertial forces and there's still a difference between rotating frames and nonrotating frames, even though there are no distant stars for the rotation or acceleration to be relative to.

 

[PAllen]

Well, the sense in which it doesn't satisfy Mach's principle is that in flat spacetime, there are still inertial forces and there's still a difference between rotating frames and nonrotating frames, even though there are no distant stars for the rotation or acceleration to be relative to.

Agreed.

The best statement of the particular sense it does that I've seen is:

- In a closed universe, the distribution of matter completely picks out which paths are geodesics (in open universe, boundary conditions are crucial; SR universe is open). Thus matter determines what is inertial versus non-inertial motion (in a closed universe).

 

[PeterDonis]

Well, the sense in which it doesn't satisfy Mach's principle is that in flat spacetime, there are still inertial forces and there's still a difference between rotating frames and nonrotating frames, even though there are no distant stars for the rotation or acceleration to be relative to.

Yes, but that just raises the question of whether flat spacetime is a physically realistic solution, since it requires absolutely no stress-energy anywhere. I agree, though, that this is a "sense" in which GR doesn't satisfy Mach's Principle. But there are other senses in which it does. The Cuifolini and Wheeler book goes into this in detail.

 

[GregAshmore]

Which is why I did qualify it, at length, in the post you referenced earlier.

Yes, you did. I didn't appreciate the qualification, in large part because I did not understand it. There is another reason that I did not appreciate it, which I'll mention later in this post.

I think that your opinion is wrong in this case. The first postulate of special relativity is expressly stated in terms of inertial frames. That postulate was later generalized for general relativity, but for problems in special relativity it is reasonable to treat inertial frames as priveliged according to the first postulate.

Whenever a frame is privileged with respect to other frames, the principle of relativity is violated.

For someone who knows what he is doing, the violation may be a harmless convenience. For someone who is in the process of training his mind to think in accordance with the principle of relativity, the violation makes it extremely difficult to discern between truth and error in one's understanding of the subject.

Recall, if you will, the objection raised by the doubter in Taylor & Wheeler, which I quoted earlier. His objection consists of two claims, though he thinks of them as one claim. The first claim is that the principle of relativity insists that the rocket twin can be treated as permanently at rest, and the Earth moving. The second claim is that in the scenario in which the Earth moves, the Earth twin will be younger upon his return. That is the paradox.

The first claim is correct. The second claim is incorrect. The text never deals with the first claim, and therefore never shows that the second claim is incorrect. Instead, the text asserts that it is the "change of direction" of the rocket twin that results in the younger age of that twin. To which the reader instantly replies, "But the Earth changes direction too, when it is the traveler!" At the end of the section, this particular reader feels as though he has been tricked by sleight of hand--and frustrated because he is not capable of crafting a coherent refutation. And, in the case of T&W, insulted, to boot, as "objectors" are always made out to be buffoons.

I've read at least half a dozen explanations of the twin paradox; I recall only Einstein dealing with the case of the resting rocket twin. (Born mentions the gravitational field in passing in the section on SR, but he threw me off by saying that only the rocket accelerates--not addressing the fact that the Earth accelerates when the rocket is at rest. He must have meant proper acceleration, but did not say so. In fairness, he may deal with the resting rocket in the section on GR; I don't recall.)

With regard to the referenced thread, which dealt with the twin paradox, my recollection is that the OP did bring up the case of the stationary rocket early on, and that the non-relativity of proper acceleration was given as the basis for the assertion that only the rocket twin moves, and thus for saying that the rocket twin must be the younger one of the two. Hence my shock. A review of those posts might show that my interpretation of the flow of logic was wrong; I wouldn't be at all surprised. Even so, I read more than 160 posts and did not see the case of the resting rocket dealt with, or any indication that it needed to be dealt with.

Now, my thickheadedness is my own problem, and I make no excuses for it. And, I truly appreciate the effort put forth by all who patiently answer questions on this forum. In the context of that appreciation, I suggest that an approach that is careful to explicitly treat each frame as permanently at rest (as separate cases, of course) would go a long way toward dispelling confusion and training minds to think correctly about relativity.

All that said, this discussion has gone a long way toward solidifying the basics of relativity in my thinking, and (equally important, as it turns out) helping me to understand why the texts are written the way they are. I believe that I will make faster, steadier progress now.

Thank you, all.

You are right to be concerned about this. I think that the modern resolution has been to just leave it alone. The problem is that there are many quantities which could reasonably be called the "gravitational field" and none of them are so important as to clearly demand that they and not the others be called thus.

My personal preference is to call the Christoffel symbols the gravitational field, others prefer to use the Riemann curvature tensor or the Einstein tensor. Still others like to refer to the metric as the gravitational field. Before you can even discuss the "reality" of the field you need to decide what it is that you are talking about. If you have a preference then I would be glad to use your preference in the discussion.

I don't know enough to have a preference. I may get to that point; we'll see. Thanks for the offer.

 

[stevendaryl]

Whenever a frame is privileged with respect to other frames, the principle of relativity is violated.

The principle of relativity is the claim that all inertial frames are equivalent. It doesn't violate that to say that noninertial frames are not equivalent to inertial frames.

 

[PeterDonis]

Whenever a frame is privileged with respect to other frames, the principle of relativity is violated.

Only the generalized principle of relativity, which requires the equivalence principle, so that you can consider gravitational fields to exist in some frames and not others. But treatments of SR and the twin paradox that I'm familiar with always make it clear that they are only dealing with the principle of relativity in its original version, which only applied to inertial frames.

This is not just an arbitrary distinction: inertial frames are physically different, because objects at rest in them feel no force. Objects at rest in non-inertial frames feel force. That's a real physical difference. IMO, the emphasis on inertial frames is meant to focus your attention on which observers feel a force and which ones don't, rather than on who is "at rest" and who isn't.

Recall, if you will, the objection raised by the doubter in Taylor & Wheeler, which I quoted earlier. His objection consists of two claims, though he thinks of them as one claim. The first claim is that the principle of relativity insists that the rocket twin can be treated as permanently at rest, and the Earth moving. The second claim is that in the scenario in which the Earth moves, the Earth twin will be younger upon his return. That is the paradox.

The first claim is correct.

No, it isn't, because T&W specifically define the principle of relativity to only apply to inertial frames. If the objector was going to contest that, he would have to actually contest it; he would have to make some argument in favor of the generalized principle of relativity instead of the one that only applies to inertial frames. He doesn't; he just makes the flat claim that the rocket twin can be treated as being "at rest", which is simply false given the T&W definition.

The text never deals with the first claim, and therefore never shows that the second claim is incorrect.

This is wrong in two ways. First, as above, the text does define the principle of relativity to only apply to inertial frames, so it does deal with the first claim. Second, even if we extend the principle of relativity to apply to non-inertial frames, and allow a gravitational field to exist in some frames but not others, that still doesn't make the second claim correct, because the Earth doesn't feel a force and the traveling twin does. That means the situation is not symmetric, regardless of which frame you use to describe it.

There is also the issue of how to describe the scenario in a non-inertial frame in which the rocket twin is always at rest. In that frame, as we've seen, the twin firing his rocket causes a gravitational field to exist, which disappears when the rocket stops firing. That's a bit weird for a start. But also, there are issues with setting up coordinates in this non-inertial frame. There is no one unique way to do it (the way there is in an inertial frame), and the obvious ways of doing it run into problems; for example, there will be a portion of spacetime that can't be covered by such coordinates, because they would assign multiple coordinate values to the same points in spacetime.

There are ways of dealing with these issues, so that one can compute the elapsed proper time for both twins in the non-inertial frame, but they require some thought. And, of course, when you do get to the point of being able to do the computation, you find that you get the same answer as in the inertial frame: the Earthbound twin ages more.

Instead, the text asserts that it is the "change of direction" of the rocket twin that results in the younger age of that twin.

Perhaps the text should have said that the rocket twin feels a force, instead of that he changes direction. But again, the text makes clear that it is using inertial frames, and the rocket twin does change direction with respect to an inertial frame.

At the end of the section, this particular reader feels as though he has been tricked by sleight of hand--and frustrated because he is not capable of crafting a coherent refutation. And, in the case of T&W, insulted, to boot, as "objectors" are always made out to be buffoons.

I realize that this is really about pedagogy, not about physics; but one does need to pay careful attention to definitions. As I noted above, T&W specifically define the principle of relativity to apply only to inertial frames. You may not like that pedagogical approach, but it seems to be the one that every text on SR takes. I've never seen any text try to start with the generalized principle of relativity. The reason, I think, is that trying to deal with non-inertial frames at the outset brings in a lot of other issues, some of which I alluded to above.

the non-relativity of proper acceleration was given as the basis for the assertion that only the rocket twin moves, and thus for saying that the rocket twin must be the younger one of the two.

It's important to note, once again, that this is not the correct flow of the logic. The logic is that the non-relativity of proper acceleration means that the rocket twin is younger; there is no intermediate step where it is deduced that only the rocket twin moves. The theorem that the free-fall worldline between two given events has the largest elapsed proper time of all worldlines between those events does not require defining an inertial frame in which the free-fall object is at rest. In fairness, I don't know that this was made clear in the other thread.

I suggest that an approach that is careful to explicitly treat each frame as permanently at rest (as separate cases, of course) would go a long way toward dispelling confusion and training minds to think correctly about relativity.

I could see doing this at some point, but I don't think it's a good idea to do it too soon, for the reasons I gave above. Non-inertial frames are not as straightforward as you appear to think. IMO the more emphasis that is put on things that are independent of coordinates and frames, the better.

 

[PAllen]

The principle of relativity is the claim that all inertial frames are equivalent. It doesn't violate that to say that noninertial frames are not equivalent to inertial frames.

An further [for the OP - you obviously know this very well], while Einstein was fond of a 'general principle of relativity', this does not in any way say that inertial and non-inertial frames are equivalent. Instead it says that a non-inertial frame can be considered to be stationary in a peculiar gravitational field. The more common modern view, which is completely equivalent, is that coordinates for an (proper) accelerated observer (in which the observer has zero coordinate motion) have a metric different from an inertial frame, and this causes trajectories of maximal time to involve coordinate acceleration in these coordinates. In other words, the coordinate accelerated Earth trajectory will be computed to pass greater proper time because of the non-trivial metric in these coordinates.

There is no form of principle of relativity that posits equivalence of inertial and non-inertial frames.

 

[stevendaryl]

Whenever a frame is privileged with respect to other frames, the principle of relativity is violated.

For someone who knows what he is doing, the violation may be a harmless convenience. For someone who is in the process of training his mind to think in accordance with the principle of relativity, the violation makes it extremely difficult to discern between truth and error in one's understanding of the subject.

Well, understanding that the principle of relativity means the equivalence between inertial reference frames is pretty critical. If you don't understand that, then you don't understand the principle of relativity.

Here's an analogy from Euclidean geometry: Take a piece of paper. Pick a line to call the x-axis, and pick a perpendicular line to call the y-axis. Call lines parallel to the x-axis "horizontal" and lines parallel to the y-axis "vertical".

Now, if you have a line that is neither vertical nor horizontal, then you can compute its length using the formula

$L = \delta x \sqrt{1+m^2}$

where $m$ is the slope of the line, defined to be $m = \dfrac{\delta y}{\delta x}$

So now, imagine picking two points on the x-axis; them $A$ and $B$. We draw on the paper two different paths connecting the points. Path 1 is a straight line running horizontally from $A$ to $B$. Path 2 starts at $A$, goes off at slope $+m$ until it is equally distant from $A$ and $B$, and then comes back at slope $-m$ until it reaches $B$.

We can use the length formula above to prove that Path 2 is longer than the first, by a factor of $\sqrt{1+m^2}$. But that's a paradox! Because slope is relative: If the slope of Path 2 relative to Path 1 is $+m$, then the slope of Path 1 relative to Path 2 is $-m$. So from the point of view of a traveler following Path 2, Path 1 is the one that has a nonzero slope, and so Path 1 should be longer by a factor of $\sqrt{1+m^2}$. That's a paradox.

But no, it's not. Although slope is relative, a change in slope is not. Regardless of how you pick your x-axis, everyone agrees that Path 2 changes slope half-way, and that Path 1 has constant slope. The slope formula can be used to prove that a path with a constant slope will be shorter than a path with a changing slope, if they connect the same two points.

There is a principle of "relativity of slopes" in Euclidean geometry, but there is no principle of relativity that allows you to treat a straight line the same as a nonstraight line.

 

[PAllen]

GregAshmore, let me describe a physical example to challenge any possibility of ignoring acceleration that you feel. I won't even use light or relativistic affects - just Newtonian physics. However the equivalence of inertial frames, as well as the equivalence principle, can both be considered to apply here (for low speeds and non-extreme gravity).

Consider that Bob is firing a machine gun at Joe, who is luckily ahead of, and moving at the same speed as the bullets. In Joe's rest frame, the bullets are suspended at a distance; Bob is receding so rapidly he is dropping stationary bullets. Now Joe feels a force from the side away from Bob. Bob is seen to slow down, and the bullets speed to Joe (unfortunately). Joe can say he remained stationary and a a sudden gravitational field appeared with unfortunate consequences. Only an observer feeling force will see such pseudo-gravity effects (to use the more common terminology). We call a frame with such pseudo-gravity effects 'accelerated' even though the origin of such a frame has constant coordinate position. It is completely distinguishable from a frame with no pseudo-gravity. Never, ever, did Einstein or any relativist suggest these two types of frames are equivalent.

 

[ghwellsjr]

Recall, if you will, the objection raised by the doubter in Taylor & Wheeler, which I quoted earlier. His objection consists of two claims, though he thinks of them as one claim. The first claim is that the principle of relativity insists that the rocket twin can be treated as permanently at rest, and the Earth moving. The second claim is that in the scenario in which the Earth moves, the Earth twin will be younger upon his return. That is the paradox.

The first claim is correct.

No, it isn't, because T&W specifically define the principle of relativity to only apply to inertial frames. If the objector was going to contest that, he would have to actually contest it; he would have to make some argument in favor of the generalized principle of relativity instead of the one that only applies to inertial frames. He doesn't; he just makes the flat claim that the rocket twin can be treated as being "at rest", which is simply false given the T&W definition.

You guys have totally missed the point that T&W are making. They are using the doubter to show the inferiority (according to them) of explaining SR by using inertial frames. They prefer an explanation using what they call Proper Clocks as defined at the bottom of page 10 in section 1.3 called Events and Intervals Alone!. They are agreeing with the doubter. They want the reader to identify with the doubter and reject any explanation involving inertial frames and adopt their preferred explanation which is that you carry an inertial wristwatch between each pair of events to measure the Proper Time between those two events. They prefer this explanation because they say all observers will agree on the calculation of the Proper Time displayed on a Proper Clock even though they don't actually send a physical Proper Clock between the two events in question. But any observer can use their own rest frame to calculate the Proper Time from the coordinate times and coordinate positions. They are talking about the time-like spacetime interval.

So their ideal explanation of the Twin Paradox is for the stay-at-home twin to have a Proper Clock and for the traveling twin to carry another Proper Clock, a wristwatch, with him on his trip out, and another, or the same, wristwatch on the trip back, an compare times on them. That, to me, is a ridiculous explanation because the twins already had such clocks.

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.

 

[PeterDonis]

They want the reader to identify with the doubter and reject any explanation involving inertial frames and adopt their preferred explanation which is that you carry an inertial wristwatch between each pair of events to measure the Proper Time between those two events.

Note that it has to be an inertial wristwatch, though. See below.

But any observer can use their own rest frame to calculate the Proper Time from the coordinate times and coordinate positions.

For inertial frames, yes, this is clear from their exposition. But IIRC they don't go into non-inertial frames at all, so they don't give any way of doing what you're describing using a single non-inertial "rest frame" for the traveling twin, which is what the doubter is trying to do by saying we can treat the traveling twin as being at rest. You have to use two inertial frames, one outgoing and one returning. So I'm not sure T&W are trying to get the reader to identify with the doubter.

 

[PeterDonis]

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.

 

[Alain2.7183]

There is also the issue of how to describe the scenario in a non-inertial frame in which the rocket twin is always at rest. In that frame, as we've seen, the twin firing his rocket causes a gravitational field to exist, which disappears when the rocket stops firing. That's a bit weird for a start. But also, there are issues with setting up coordinates in this non-inertial frame. There is no one unique way to do it [...]

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).

 

[PeterDonis]

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).

Do you have a reference?

 

[Dale]

Whenever a frame is privileged with respect to other frames, the principle of relativity is violated.

This is simply not correct.

Suppose that I postulated the principle of beans which stated that "the price of all legumes is equal". Now, clearly the statement "the price of lima beans is higher than the price of pinto beans" violates the principle of beans since lima beans and pinto beans are legumes and the principle of beans states that their price should be equal. However, "the price of steel is higher than the price of lettuce" does not violate the principle of beans since neither are legumes. Similarly, "the price of vanilla is higher than the price of peas" does not violate the principle of beans. Although vanilla looks a lot like a bean and is sometimes even called a bean it is not, in fact, a legume, so the principle of beans does not make any statement about its price compared to the price of legumes like peas.

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.

 

[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$$F^\lambda = m \frac{d^2x^\lambda}{d\tau^2}$$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$$F_\lambda = m \sum_{\mu=0}^3 g_{\lambda \mu} \frac{d^2x^\mu}{d\tau^2} + \frac{m}{2} \sum_{\mu=0}^3 \sum_{\nu=0}^3 \left( \frac{\partial g_{\lambda \mu}}{\partial x^\nu} + \frac{\partial g_{\lambda \nu}}{\partial x^\mu} - \frac{\partial g_{\mu \nu}}{\partial x^\lambda} \right) \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}$$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.