An object at rest relative to the Sun cannot be in free fall so how can it be inertial?

@AlfredCentauri In SR Isn't an inertial frame one that is not accelerating? I don't think the OP is looking at things from a GR perspective. Note the tags.

As you know, in SR, there is coordinate acceleration and there is proper acceleration (that measured by an accelerometer). An accelerometer attached to an object at rest in the coordinate system of the Sun reads non-zero acceleration (even in the Newtonian context). Yes, I know that these SR problems ignore gravity but many tend to forget that, and they go on to think that a statement made in that context is generally true. I would at least add an explicit disclaimer about that.

@AlfredCentauri A disclaimer saying specifically what? That in GR we think about inertial frames differently?

I've made the point I wanted to make and that's all I have to say about that.

@AlfredCentauri Oh sorry, I thought you were proposing a suggestion that I could use to make my answer better. I just don't fully understand what it is. I wasn't really trying to make any counterpoints.

Thank you for this answer. I indeed implicitly assumed that the Sun was a better approximation of an inertial frame than the Earth. In that case, it seems to me that you agree about the fact that different experiments could lead to very different results depending on the motion of the rocket?

@DataXplorer I guess? That is a pretty broad statement. Of course if you do something different something different will happen. I don't think I fully follow.

If this is the case, it seems very silly to claim that the twin paradox can be resolved easily with SR, with the result that the twin on Earth is aging faster. Hermann Bondi in his book "Relativity and Common Sense" is doing so, and I found other physics books and articles doing exactly the same. On Wikipedia, the article about "experimental testing of time dilation" does not mention the fact that in some cases (of peculiar motion) the muons could age faster. They rather claim that finding in any case the muons aging slower is confirming the theory.

@DataXplorer it's not silly. Like I say in my answer, the usual Twin paradox just involves one twin that stays in the same frame and another twin that changes frames. The scenario is stated upfront. The Twin Paradox is not just "two people go away and come back together." There are explicit assumptions as to what is going on. If you want to consider other variations, go for it. That doesn't make the usual treatment silly.

@DataXplorer Also keep in mind that the difference in aging only occurs during the acceleration. That is the key. It isn't just that one twin moves relative to another twin.

Actually, many people do not assume the twin in the rocket is changing frame. They just assume "infinite acceleration" for the turnaround, or in the case of Bondi that when two inertial frames cross each other, they synchronize (so instead of a rocket "coming back", there are two inertial frames crossing each other). In both formulations (and I think it is particularly obvious in the first formulation), there is no break of symmetry between the two frames, which makes it very hard to understand how it truly solves the paradox.

@DataXplorer, once you 'get the hang' of SR, the twin 'paradox' seems trivial. (1) There are two events - the moment the twins separate and the moment the twins reunite. (2) There are many (an infinity) of world lines through these two events but one is special - it is the world line with the maximum elapsed time between the events (according to a clock on that world line). (3) That is, the elapsed time along any other world line (according to a clock on that world line) is less. (cont.)

@DataXplorer There is a break of symmetry... the twin on the rocket doesn't stay in the same inertial frame. That is true in both formulations you mention. (Actual twin or "clone twin" that synchronizes coming back)

(4) That special world line is precisely the one for which an accelerometer on the clock reads zero always, i.e., it is the world line of a clock that is inertial. (5) The reading of an accelerometer on a clock on any other world line through the events must read non-zero acceleration along a portion of the world line. (6) All observers agree on the readings of the accelerometers and the readings on the clocks.

I understand that one line is special. It seems however troubling to me that we do as if the Earth was precisely on this line. It is a bit like a "geocentric" model of the universe. And if we do not assume that the Earth is on this line (even if the Earth is very close to it), I think it would be worth mentioning that maybe in very special cases it could be possible that we do not find the usual classical time dilation, but rather a time contraction. This is what makes me think I misunderstood the whole theory from the very beginning.

@BioPhysicist, if from one frame we can see that the other is infinitely accelerating to come back, we can see exactly the opposite from the other frame. That is why I am saying there is no break of symmetry.

@DataXplorer No, it isn't the same. Only one twin changes inertial reference frames, and both twins will agree on this. Please read my answer. Observed acceleration does not mean change in inertial reference frame if this observation is itself not made from an inertial reference frame. The difference is that one twin actually accelerates and one does not. It has nothing to do with "observed acceleration".

@DataXplorer, ditch the Earth and the twins. Just think about two (ideal) clocks (with accelerometers) that are co-located at one event, separate, and are co-located again later. Stipulate that the accelerometer of one of the clocks always reads zero. That clock will show a later time at the second event than the other clock shows regardless.

@DataXplorer I agree with AC: you're getting hung up on using the Earth. Think of the simplification I propose in my answer.

@BioPhysicist, I do agree with that, about the observation being valid only from inertial reference frames. But in that case you are assuming the break of symmetry not because of the motion of the frame, but because you assume from the start that one is inertial. But we do not know any perfect inertial frame (to my knowledge), so the rocket (or the muons) could also a priori be an inertial frame of reference (even if it is then proven not true by some real-world experiment - and not a purely theoretical assumption).

@DataXplorer Like I said, if you want to consider other scenarios that is fine. That doesn't make the Twin Paradox invalid. The Twin Paradox is a thought experiment where one twin stays in the same inertial frame and the other does now. Just because you want to consider scenarios where these assumptions do not apply doesn't make the thought experiment and its treatment invalid.

@BioPhysicist, if you assume from the start that one frame is inertial and not the other, there was never any paradox to begin with (and I am happy with that). The trouble is to find the inertial frame in the real world, and I do not understand why people assume very naturally the Earth is a very good candidate for that (you mentioned yourself that the rotation of the Earth made the Earth not inertial at all).

@DataXplorer Right. The Twin Paradox is not an actual paradox

@AlfredCentauri Why did you delete your answer?

@AlfredCentauri Your answer was very interesting. And it makes me considering learning about GR, to better understand the idea of free fall, etc.

@Dvij D.C I don't understand why the rocket could never be in an inertial frame. Of course as you stated, "the point of a functioning rocket is to be accelerated with respect to the local inertial frame". But if the Earth is not a true inertial frame, it could be (very theoretically, in practice I understand there is no chance it happens) that the rocket strictly follows the motion of a true inertial frame, couldn't it?