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

[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.