The Paradox of Aging in Relativity: Resolving the Twin Paradox with a Twist

[ghwellsjr]

Very nice - thanks George.

Am I right in thinking that you've constructed the same coordinate chart that Dolby & Gull use in the paper linked from #20?

Yes, except that the first time I did this, I didn't realize that it was the same but I was glad to learn that I was not the first to discover this method of making a non-inertial chart.

 

[PeterDonis]

George, this is the best easy visual presentation of the issues in constructing non-inertial charts that I have seen. Great job!

 

[Ibix]

Yes, except that the first time I did this, I didn't realize that it was the same but I was glad to learn that I was not the first to discover this method of making a non-inertial chart.

I actually remember that thread, and not understanding what was going on. Progress!

 

[PhoebeLasa]

[...]
if you want a chart in which the traveler is always at rest, you're going to confront the issue we've been discussing one way or another.

I've just realized that there is a particular similarity between (1) the special relativity scenario where the traveling twin (motionless wrt his home twin at some sufficiently large distance from her) begins and then continues a constant acceleration away from her forever, and (2) the general relativity scenario where someone falls through the event horizon of a spherically-symmetrical non-rotating black hole.

In the case of the black hole (if I understand it correctly), an observer "at infinity" (effectively infinitely far away from the black hole) will say that the age of the person falling toward the black hole will approach a finite limit, and that the person will never quite reach the event horizon, nor will he ever quite reach that limiting finite age. So, for the "infinitely-far-away" observer, the future ages of that falling person never happen. But the falling person himself will say that he passes through the event horizon, and that his age continues to smoothly increase as he continues his falling inside the black hole.

In the case of the special relativity scenario, the traveler [when his "rest" frame (in which he is always at the spatial origin) is determined using the "montage" of the "momentarily co-moving inertial reference frames" (his "MCMIRF" frame)] will say that the home twin's age will continuously increase, but will approach a finite limit. So he will say that she never gets older than that finite, limiting age ... her future ages (which she certainly will say she does reach) never happen at all, according to him.

So, in both the GR black hole scenario, and in the SR infinitely-long-lasting acceleration scenario, the observers (the "infinitely-removed" person in the GR case, and the accelerating traveler in the SR case) are each using a reference frame that doesn't "see" the missing future ages of the other person in each scenario). What I don't understand, though, is why this situation seems to be considered to be acceptable and "valid" in the GR scenario, but is considered (at least commonly on this forum) as unacceptable and "invalid" in the SR scenario.

 

[PeterDonis]

for the "infinitely-far-away" observer, the future ages of that falling person never happen.

No, that's not correct. What is correct is that the far-away observer will never see the future ages of the falling person. But he has no justification for saying that, since he will never see them, they will never happen. Making that unwarranted inference is a common error, but it's still an error.

What I don't understand, though, is why this situation seems to be considered to be acceptable and "valid" in the GR scenario, but is considered (at least commonly on this forum) as unacceptable and "invalid" in the SR scenario.

You are incorrectly describing the situation. The GR and SR scenarios are both the same in this respect: one observer never sees a certain portion of spacetime, which includes the "future ages" of the other observer. That's all there is to it.

 

[Dale]

What I don't understand, though, is why this situation seems to be considered to be acceptable and "valid" in the GR scenario, but is considered (at least commonly on this forum) as unacceptable and "invalid" in the SR scenario.

How can you possibly not understand this point. It has been explained to you over and over and over again. As near as I can tell it has been explained to you every single time that you have posted by multiple people in multiple ways during multiple threads.

A valid coordinate chart does not need to cover all of spacetime. Both charts you describe are alike in this feature

A valid coordinate chart does need to be invertible. The two charts differ in this required feature.

Please read chapter 2 here
http://preposterousuniverse.com/grnotes/

 

[PhoebeLasa]

After reading Carroll's 2nd chapter, I think I've learned two important things:

1) First, I think I've learned that, when spacetime is curved, there is generally no single coordinate system that can cover the entirety of spacetime. When that is the case, it is necessary to smoothly knit together more than one coordinate system, so that all of spacetime can be covered. And in order to guarantee that these multiple coordinate systems can be smoothly knitted together, they must be invertible.

2) Second, I think I've learned that, when spacetime is flat (Minkowskian) everywhere, there is no requirement that all coordinate systems must be invertible, because there is at least one coordinate system which can cover the entirety of spacetime. No knitting-together of multiple coordinate systems is required to get complete coverage, so invertibility of all coordinate systems is not required. In particular, I see nothing in Carroll's 2nd chapter that prohibits the use of the "MMCMIRF" coordinate system (the coordinate system consisting of the "montage" of "momentarily-co-moving-inertial-reference-frames") as the "rest" coordinate system of an accelerating traveler, PROVIDED that the spacetime is flat (Minkowskian) everywhere. The fact that the MMCMIRF isn't invertible, and the fact that it doesn't always cover all of spacetime, does not appear to be disqualifying, according to my reading of Carroll.

I am basing my above thinking on the following quotes from Carroll's 2nd chapter:

Quotes from Carroll in support of my thinking in item #1 above:

"So a chart is what we normally think of as a coordinate system on some open set, and an atlas is a system of charts which are smoothly related on their overlaps."

"We therefore see the necessity of charts and atlases: many manifolds cannot be covered with a single coordinate system."

"The entire manifold is constructed by smoothly sewing together these local regions."Quotes from Carroll in support of my thinking in item #2 above:

"Why was it necessary to be so finicky about charts and their overlaps, rather than just covering every manifold with a single chart? Because most manifolds cannot be covered with just one chart."

"Nevertheless, it is very often most convenient to work with a single chart, and just keep track of the set of points which aren’t included."

 

[Ibix]

Maps must be invertible by definition. It isn't stated in the clearest terms in Carroll's notes, but it is stated. He defines a map as a one-to-one function, then notes that any map is onto its image. A function that is both one-to-one and onto is invertible. Therefore maps must be invertible by their definition. Their is no exception for flat space.

Think about what you are saying for a minute (as I should have done). A non-invertible map means that you can draw the chart but can't use it to navigate, or that you can use it to navigate but cannot draw it. Neither is useful.

 

[pervect]

It might be helpful to look at MTW's "Gravitation section $6.3". You can probably get it through google if you search for "Constraints on size of an accelerated frame" and look at the google book results. "Constraints on size of an accelerated frame" is the section title for section $6.3 of this textbook.

My $.02 is that it appears to be standard practice to insist that coordinate systems assign unique labels to every point - and that this approach seems to me to be less confusing than the alternative of attempting to deal with multiple labels.

If there was some strong benefit to having coordinate systems with non-unique labels, it might be worth investigating the issue in more detail - I'm not aware of any advantages to such a practice.

 

[PeterDonis]

I think I've learned that, when spacetime is curved, there is generally no single coordinate system that can cover the entirety of spacetime.

Yes.

When that is the case, it is necessary to smoothly knit together more than one coordinate system, so that all of spacetime can be covered.

Yes.

And in order to guarantee that these multiple coordinate systems can be smoothly knitted together, they must be invertible.

What you mean here is that the mapping between the two coordinate systems must be invertible. This is quite true: in order to "knit" two charts together, the mapping between them must be invertible in any part of spacetime that both charts cover.

However, that is not the same as having a single coordinate chart, considered as a mapping between points in spacetime and 4-tuples of real numbers, being invertible. The latter sense of "invertible" is the one people have been discussing here.

I think I've learned that, when spacetime is flat (Minkowskian) everywhere, there is no requirement that all coordinate systems must be invertible, because there is at least one coordinate system which can cover the entirety of spacetime.

First of all, even if we apply the word "invertible" to maps between charts, instead of to a single chart (see above), this does not follow. Yes, in Minkowski spacetime any inertial coordinate chart covers all of spacetime (and there are an infinite number of such charts). However, that in no way removes the requirement that if you choose to use some non-inertial chart on a portion of Minkowski spacetime, the mapping between that chart and any other chart that you use to cover the rest of spacetime (which could be an inertial chart or another non-inertial chart) must be invertible, because if you are using a non-inertial chart at all, as Ibix pointed out, you are imposing on yourself the requirement of "knitting" that chart together with other charts to cover all of spacetime, and doing that works the same whether spacetime is flat or curved.

However, none of this has anything to do with the fact that even a single inertial chart covering all of Minkowski spacetime must still be an invertible map between points in Minkowski spacetime and 4-tuples of real numbers. Any map that does not have that property is simply not a valid coordinate chart, period.

 

[Dale]

1) First, I think I've learned that, when spacetime is curved, there is generally no single coordinate system that can cover the entirety of spacetime. When that is the case, it is necessary to smoothly knit together more than one coordinate system, so that all of spacetime can be covered. And in order to guarantee that these multiple coordinate systems can be smoothly knitted together, they must be invertible.

This is correct.

2) Second, I think I've learned that, when spacetime is flat (Minkowskian) everywhere, there is no requirement that all coordinate systems must be invertible

This is not correct. At the top of page 37 Carroll defines "A chart or coordinate system consists of a subset U of a set M, along with a one-to-one map φ : U → Rn, such that the image φ(U) is open in R. (Any map is onto its image, so the map φ : U → φ(U) is invertible.) ". So anything which is not invertible is, by definition, not a coordinate system. Pure and simple. I don't know how he could be any clearer on this point.

"Why was it necessary to be so finicky about charts and their overlaps, rather than just covering every manifold with a single chart? Because most manifolds cannot be covered with just one chart."

That is talking about the overlap between two charts. The definition above is a requirement that the mapping between any single chart and the manifold must be invertible. It has nothing directly to do with overlapping or multiple charts, it is a requirement on a single chart and how it maps to the manifold.

The invertibility of the mapping between each chart and the manifold also implies that (on the region of the manifold covered by multiple charts) the mapping between any pair of charts is also invertible. But even with a single chart the mapping to the manifold must be invertible.

"Nevertheless, it is very often most convenient to work with a single chart, and just keep track of the set of points which aren’t included."

Convenience doesn't negate the definition.

 

[PhoebeLasa]

It might be helpful to look at MTW's "Gravitation section $6.3". [...]
"Constraints on size of an accelerated frame" is the section title for section $6.3 of this textbook.

Starting with the 2nd paragraph of section 6.3 (p.168) of MTW, they say:

"Difficulties also occur when one considers an observer who begins at rest in one [inertial] frame, is accelerated for a time, and maintains thereafter a constant velocity, at rest in some other inertial coordinate system. Do his motions define in any natural way a coordinate system? Then this coordinate system (1) should be the inertial frame x_mu in which he was at rest for times x_0 less than 0 [before he started accelerating], and (2) should be the other inertial frame x_mu' for times x_0' > T' [after the acceleration ends] in which he was at rest in that other frame."

This "natural frame" they are describing above is the MMCMIRF frame. It isn't the Dolby&Gull frame that seems to be preferred on this forum. They don't appear to think that the "natural frame" is "a choice".

 

[PeterDonis]

They don't appear to think that the "natural frame" is "a choice".

They also don't think it's a valid frame. So they are using the term "natural frame" only to show that the "obvious" meaning of that term leads to a concept that does not work for observers who are not inertial for all time. Which is why other concepts, such as the Dolby & Gull frame, are needed if it is desired to have a valid coordinate chart in which an observer who is not inertial for all time is always at rest.

 

[pervect]

Starting with the 2nd paragraph of section 6.3 (p.168) of MTW, they say:

"Difficulties also occur when one considers an observer who begins at rest in one [inertial] frame, is accelerated for a time, and maintains thereafter a constant velocity, at rest in some other inertial coordinate system. Do his motions define in any natural way a coordinate system? Then this coordinate system (1) should be the inertial frame x_mu in which he was at rest for times x_0 less than 0 [before he started accelerating], and (2) should be the other inertial frame x_mu' for times x_0' > T' [after the acceleration ends] in which he was at rest in that other frame."

This "natural frame" they are describing above is the MMCMIRF frame. It isn't the Dolby&Gull frame that seems to be preferred on this forum. They don't appear to think that the "natural frame" is "a choice".

I'm not quite following the last sentence? I have a feeling that this whole idea of a "natural frame" may be the sticking point of this long discussion. What is a "natural frame" and what specific properties does it have? You probably have some idea what you mean by that term, I'm afraid that I don't have a precise idea of your meaning.

Let's go back to the very beginning of the section of MTW:

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 that these words might refer to. The most useful first remark one can make about these words is that, if taken seriously, they are contradictory. The definite article "the" in this phrase suggests that one is thinking of some unique coordinate system naturally associated with some specific accelerated observer ...

((I would quote more, but I have to type it all in, not cut and paste.))

My interpretation of this is that MTW is warning us there aren't any "natural coordinates" to use for an accelerated observer, in the sense that some deisired properties are lacking. Note the use of the word coordinates here. MTW seems mostly consistent about referring to the coordinate systems of accelerated observer (the section title is an interesting exception!), and applying the concept "frames" only to inertial observers. I believe this is conceptually less muddled than talking about "frames" of accelerated observers. I understand what a coordinate system of an accelerated observer might be. If what you might mean by "frame" is synonymous to coordinate system, great. If what you might mean by "frame" is not synonymous to coordinate system, then I'm afraid we have to talk more about in regarding what you mean by a frame , and how it is different from a coordinate system. (A reference might do the trick, here.)

Now, MTW doesn't even mention Dolby & Gull's coordinate system, while they do mention momentarily comoving inertial (MCMI) coordinate system. I would tend to agree that in terms of popularity, MCMI is more popular than Dolby & Gull. I would even say that I personally like it better than Doby & Gull. MTW also mentions in later sections a specific extension of the MCMI idea, called "Fermi Normal Coordinates", that I feel are very important. I tend to think of Fermi Normal coordinates as being "natural", but that's just my personal bias. People seem to have different ideas of what is "natural", and I don't believe it's too productive to argue about this.

Fermi normal coordinates are particularly useful when one wants to use an intuition based on Newtonian physics in some small region of space-time where said intuition gives reasonably accurate results. If that is what one is seeking, I would highly recommend using Fermi Normal coordinates, they are well suited for that purpose. Dolby and Gull's coordinates are not particularly useful (and don't claim to be useful) at giving in a good local approximation to Newtonian physics. Fermi Normal coordinates do have this feature. Does this make them "natural"? It really depends on what you're trying to do, exactly.

The next point that MTW makes is that the MCMI coordinate system doesn't cover all of space-time. They don't mention Dolby & Gull, but if you read the fine print, Dolby & Gull don't claim their coordinate system covers all of space-time either. D&G coordinates cover the region of space-time that can send and receive signals from the accelerated observer - both MTW and D&G acknowledge that this is not all of space-time, though MTW may emphasize the point more.

So the way I interpret MTW's point is this:

Most accelerated coordinate systems do NOT cover all of space-time, as a consequence of the fact that an accelerated observer cannot send and receive light signals to all of space time. Because these coordinate systems don't cover all of space-time, it is misleading to talk about "the coordinate system of an accelerated observer". People "naturally" read the words "the coordinate system of an accelerated observer", and make the incorrect assumption that the resulting coordinates cover all of space-time. But in fact, most coordinates (including MCMI coordiantes) don't have this property.

 

[PeterDonis]

Fermi normal coordinates are particularly useful when one wants to use an intuition based on Newtonian physics in some small region of space-time where said intuition gives reasonably accurate results.

This is true, and Fermi normal coordinates are a very useful tool. But it's important to emphasize the "small region of spacetime" part. Fermi normal coordinates are only intended to cover a small "world tube" surrounding the worldline of the observer. They are not intended to be a coordinate chart covering any significant portion of spacetime as a whole. So, for example, if you were trying to analyze a "twin paradox" scenario, you could not use Fermi normal coordinates centered on the traveling twin's worldline, because those coordinates would not cover enough of spacetime to include the worldline of the stay-at-home twin (at least, not if the traveling twin goes far enough to make the difference in aging significant).

 

[PAllen]

Now, MTW doesn't even mention Dolby & Gull's coordinate system, while they do mention momentarily comoving inertial (MCMI) coordinate system. I would tend to agree that in terms of popularity, MCMI is more popular than Dolby & Gull. I would even say that I personally like it better than Doby & Gull. MTW also mentions in later sections a specific extension of the MCMI idea, called "Fermi Normal Coordinates", that I feel are very important. I tend to think of Fermi Normal coordinates as being "natural", but that's just my personal bias. People seem to have different ideas of what is "natural", and I don't believe it's too productive to argue about this.

It is interesting to note that in the GR context (rather than SR), radar coordinates have a long history. Synge, in the 1950s and 60s proved many interesting facts about them, in particular that they are experimentally indistinguishable from FN coordinates at the scale of wildly moving rocket (within the rocket), and that at the distance they become distinguishable from FN, each has nasty tradeoffs, so a preference is hard to justify.