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How many frames of reference are there in the twins paradox of relativity? Does each velocity get its own frame, or is it each acceleration?

It seems so natural to talk about the frame of reference attached to the rocket for the whole journey of the twin, but also to talk of the change of frames of reference being the cause of a persistent difference in age between the twins.

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2 Answers 2

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It depends on the detailed set-up of the thought experiment. However, if you assume the travelling twin's periods of acceleration are negligibly short, then there are three inertial frames to consider, namely the frame of the Earth, the frame of the outbound twin and the frame of the returning twin. In that set-up, the time dilation effects are symmetric between the two twins throughout both legs of the journey, and the difference in their ageing arises entirely from the travelling twin's switch of reference frame at the turnaround point.

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  • $\begingroup$ Is it possible for the twin's period of acceleration to not be negligibly short within the confines of special relativity? $\endgroup$
    – Cort Ammon
    Commented Jul 26, 2022 at 15:35
  • $\begingroup$ Yes, SR is perfectly capable of dealing with accelerations, but the mathematics gets much more complicated to the point at which I no longer find it fun to discuss! $\endgroup$ Commented Jul 26, 2022 at 15:50
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Different people use the word "frame" in a great variety of ways. The answer to this (and to your other questions about frames) depends on what you mean by the word "frame". Since you haven't told us that, there's no way to answer your question.

At every event in spacetime, there is a tangent space. A basis for that tangent space is called a frame. (You can think of the elements of that frame as pointing in the directions that you plan to call "left", "forward", "up" and "future".)

You can exponentiate a frame to get a coordinate system on all of spacetime (or in the case of general relativity, on a part of spacetime). (Exponentiating essentially means replacing your tangent vectors with curves that travel as closely as possible in the directions that the tangent vectors point. Those curves then serve as ``axes'' for your coordinate system.) People often call that coordinate system a "frame", which is arguably an abuse of language but a common and harmless one (harmless because everyone understands what's really meant). If you like, you can envision this coordinate system as being made out of "clocks and rods", though personally I think this adds nothing and sometimes causes confusion.)

Often it is implicitly assumed that frames are orthogonal (also called "intertial"), and then the associated coordinate system is often called an "inertial frame" (though it's still not literally a frame at all).

Or you can start with a non-orthogonal frame, and you can exponentiate that to get a non-orthogonal coordinate system.

Or you can associate a different frame to every point in spacetime, in a continuous way. The right word for this is "section of the frame bundle", but again, people frequently just say "frame". Or you can do this just for the events that lie along some specific curve, etc.

Or you can construct a coordinate system that is not the exponential of any frame at all, though your coordinate system now determines a section of the frame bundle.

Sometimes people speak of "accelerated frames". It seems to me that different people use this word in many ways --- to mean a literal frame that is not orthogonal, or to mean a section of the frame bundle that is not constructed by parallel transport, or to mean a coordinate system that is not the exponential of any frame.

To make any sense of your questions, you need to decide what you mean. After you have some experience, you can decide for yourself which abuses of language you are comfortable with. Until then, I recommend sticking with a careful, precise definition: A frame is a set of four vectors. Sometimes one identifies that frame with its own exponential (which is now a coordinate system). It's probably best to require your frames to be orthogonal. Any talk of "accelerated frames" or "the frame of an accelerating rocket" is fraught with ambiguity and really ill-advised right now.

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  • $\begingroup$ I absolutely am not able to say what I mean by "a frame" in terms of what you say, because it's way above my head, except to say that it's equivalent to an inertial infinite lattice of clocks and rods with clocks everywhere. What is confusing about a lattice of clocks and rods? $\endgroup$ Commented Jul 26, 2022 at 14:27
  • $\begingroup$ @MatthewChristopherBartsh : The problem with clocks and rods is that students tend to think of them as something that must be added to spacetime and then get wrapped up in thinking about exactly how (at least in principle) one might go about placing the clocks, getting them synchronized, etc. All of this is not just unnecessary, but it distracts from the main point: All of the frames exist automatically as a part of spacetime itself; once you have a manifold, you automatically have tangent spaces, frames, a tangent bundle, a frame bundle, etc. (CONTINUED). $\endgroup$
    – WillO
    Commented Jul 26, 2022 at 18:55
  • $\begingroup$ (CONTINUED) And once you have a metric, you automatically have a notion of orthonormal (or, if you prefer "inertial") frames, etc. If you think of the frame as something that you have to (mentally) add to your spacetime, then you've missed a very important concept, which opens the door to going astray in all sorts of directions. This might be worth the cost if there were anything to be gained by thinking about clocks and rods, but really, there isn't. $\endgroup$
    – WillO
    Commented Jul 26, 2022 at 18:58
  • $\begingroup$ PS : The other big problem with rods and clocks is that they make no sense when you transform from one frame to another, so that each rod or clock turns into a sort of rod/clock hybrid. This is just about guaranteed to obscure the whole idea of relativity for anyone who's just starting out. $\endgroup$
    – WillO
    Commented Jul 27, 2022 at 0:53
  • $\begingroup$ Avoiding talk of accelerated frames, and even avoiding thinking about them, e.g. by finding a way to leave them out of thought experiments seems like a good idea. I don't understand what you mean by a rod/clock hybrid. $\endgroup$ Commented Jul 27, 2022 at 13:28

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