md65536

Not theories. I've only talked about the theory of SR in this thread. I think they'd be called different coordinate systems. So much wrong with this. B passing C is an event. OP has set it up so that B's clock shows 4 years at that event, and C's clock shows 4 years at that event. Those are invariants. No change in coordinates can change what B's clock shows at an event that it passes through. It's the time at A, far away, not local!, when B and C pass, that depends on a coordinate system. You already know, that even using the standard coordinate systems of SR (I think we call them Minkowski coordinates?), A's time, when B and C pass, differs depending on which inertial frame you're describing it in. But it can also change if you use an alternative coordinate system. Not theories. B and C disagree on the coordinate time at A, when they pass, yet all observers agree on the time at A when A and C pass. Do you understand how that's possible in SR? I don't know where you're getting this from. It's the coordinate time at A that is relative, when B and C pass. A is not intersecting any of the other given world lines when B and C pass. It is only the time according to a clock that is some distance (not local) from the event, that is relative.

No, I don't think so. How is the measure of speed defined? If other observers measure time AND distance the same as it is measured in the preferred frame, all observers measure the speed of light the same. That's the point I keep trying and failing to make. These values have definitions, you have to go by how they're defined. You can't just use a common-sense definition of speed, or mix-and-match definitions, altering one quantity but not another that is based on it. No. Multiple alternatives that give different values for RELATIVE quantities can be "right". There's no evidence of an ether frame, but lack of evidence isn't proof of non-existence. They haven't been proven false, just proven so-far useless. Everything still works if you decide a frame is preferred. This is a waste of time... ... but for another example, suppose you have an event, and observer A uses coordinates with the origin in one place, and B has the origin in another place, and they get different results for the coordinates of the event. Which observer is right? They both are. Suppose A uses Cartesian coords and B uses spherical, and they get different results. Which coordinate system is wrong? Neither. The LT gives you COORDINATES within a defined system, it doesn't claim that those coordinates are "real". Comparing different ageing at a meeting point is comparing values that are real. You can't come up with working alternative coordinates that make someone older or younger than they actually are at a given local event. You *can* come up with working alternative coordinates that make someone older or younger than the LT says they are relative to some distant event. Different simultaneity conventions that still corresponded with reality as SR and Einstein's simultaneity definition do, would simply give you different time coordinates of distant events. Which is right? Maybe all. Which is useful? I've never seen any better than Einstein's. Which is "real"? There's no theoretical answer and the question is likely meaningless. Analogous to "what are the real coordinates of the event, A's or B's?"

Instead of describing a physical rocket and figuring out how the different points of it accelerate, you can specify how you want the different points to accelerate. If all you're comparing is two points, you can have the two points move independently and then not even care about the physical aspects of a rocket. For example, if you want to see what happens when they perform the same maneuvers, specify that the two points have identical acceleration measured from some inertial frame (eg. Earth frame). Or if you want the rocket to be rigid, use Born rigidity equations. It's especially pointless to describe the physical aspects of the rocket, then ignore the physics in some tiny details (like assuming it's completely rigid with one source of acceleration, which is impossible), and then try to figure out other tiny details of the physics. Sure. If the top and bottom accelerate at the same time and rate according to an observer on Earth, those clocks always read the same from Earth. While accelerating, the bottom clock ticks slower than the top (in their reference frames), this can be verified from the Earth frame just by considering the always increasing time it takes light signals to go from the bottom to the top (takes longer because the top is moving away during the time the light travels) vs top to bottom (takes less time). If the rocket then coasts, Earth says their clocks still read the same. On the rocket, the clocks now tick at the same rate but the rear clock is behind, in agreement with relativity of simultaneity. If the rocket reverses and returns to relative rest with the Earth, still with the same timing and rate of acceleration as measured by Earth, the clocks as always remain the same according to Earth, and now the rocket agrees with that. This would describe the situation in Bell's paradox, where eventually the rocket (fixed length in the Earth frame) rips apart. If you change that, so the clocks don't always have the same velocity as each other as measured by Earth, they can end up still out of sync after returning to Earth's inertial frame.

The time it takes for light to go from points P1->P2 is the same as P2->P1 in a stationary system, by definition. The LT has a mathematical definition, it's not based on measured constants. It has c in it as a constant, which also is a defined (not measured) value. The metre is defined based on c. Which of those are you changing to get a 1-way speed not equal to 2-way speed? Depending on what you change, you'll change the LT. I don't know if it's possible to change multiple things to make a modified LT give the same results as the LT, while having different 1-way speeds of light in the x direction. The relative rates of a moving clock would typically change, the lengths of world lines between events would not. Here's an example. Suppose someone Q decided their own reference frame was privileged, and invented a system based on SR but where all relative measurements used their reference frame. Suppose they're moving relative to twins A and B (and clock C too), such that in their frame, the standard LT says that the time at A when B turns around (or passes C) is 6 years after A and B depart, as measured by A's clock. Then, on the outbound trip, B's clock measures 4 years while A's measures 6, so B's clock ticks at a rate of 2/3. On the inbound, B (and C) measure 4 years while A measures 4. As always, in this experiment A ages 10 years while B ages 8 in the end. These measurements are different than what the LT says for A and B (but it agrees with what the LT says for Q, so you know it isn't predicting something inconsistent with reality). Normally A and B measure things using their own reference frames, but if they adopted this alternative, they'd use a definition where the time between P1->P2 is the same as P2->P1 (because it is in Q's), which is weird for them, because P1 and P2 are not stationary in A's or B's frames, so using normal SR measurements they'd measure those two times to be different. So there's an alternative. It's bad, but it works. It doesn't match reality for A and B in the fact that their systems are not measurably distinguishable as less privileged than Q's, but they're defined to be, and things are measured differently in different frames to match what Q' measures. But they have still have consistent definitions for all the measurements they need wrt. things like the twin paradox, and can confirm them experimentally. I hope I'm not just making things more confusing. The only reason I'd mention alternatives is to figure out what exactly SR says and doesn't say. The twin's ages when they meet, are invariants. The values that are relative can be consistent with various different systems of measurement that give different values. I feel like I'm straying farther and farther off topic trying to clarify what I said earlier, but I'm failing in that because you keep thinking I'm saying something else. PS. If you really get what I'm saying, I think I accidentally described a screwed-up system where the 2-way speed of light is still the same as the 1-way, according to the definitions, even though if you have two points p1 and p2 that are stationary in A's reference frame, A measures the time from p1->p2 as different than p2->p1, because it's using Q's measurements and they're not stationary to Q. Sorry, it's excessively complicated now, but if you get that then you probably understand the way I see this. All these measurements aren't "common sense" descriptions of what we understand as time and distance and speed, they have precise definitions that don't care what common sense says.

Right. Where the clocks intersect, its only events at the other clock (which are now local events) whose coordinate time is independent of how you define remote simultaneity. The solutions to the twin paradox remain the same. You can set all those other clocks everywhere else however you want to (eg. invent some alternative clock sync definition), but that won't affect the time measured by the two twins clocks.

Yes, we all agree on that! Celeritas, you're right on that. However Markus, if you take clocks out of it you're no longer talking about the twin paradox, which concerns ageing. The geometric length of the world lines doesn't depend on the validity of the clock hypothesis (right?) but the twin paradox does. Back to OP's topic, the 3-clock variation also does not depend on the validity of the clock hypothesis. But in either case its fine because we assume the clock hypothesis is true in the twin paradox in any case, which means both OP's experiment and the geometric length measure the same ageing as does a clock that turns around (instantly, in this case corresponding to OP's setup). The situation in terms of geometry is the easiest and most straightforward, leaving no room for debate or confusion about physical aspects, and the lack of effects due to acceleration (that aren't fully accounted for in terms of velocity) is by definition of geometric length (I think). In the twin paradox the lack of effects due to acceleration (that aren't fully accounted for in terms of velocity) is by assumption and experimental confirmation "to very high accelerations".

The point of the twin paradox setup is that it produces a result that's certain and independent of reference frame. You never *have* to compare distant clocks to resolve it. You don't need synchronized clocks, or a synchronization convention, or a definition of simultaneity. In that sense they don't matter. You don't need coordinate times or the LT to resolve the paradox. But if you try, using the LT, you always get consistent results. If you used any alternative that was consistent with reality, it would give you the same results at the events where the twins meet. Any other result isn't consistent with reality. If you use some other transformation or system of coordinates or definition of time where the one-way speed of light is different in different directions, if it was consistent with reality you'd get the same age difference that SR predicts when the twins reunite, but generally a different coordinate time of events at the other clock while they're separated. The coordinate times given by the LT would be different if they used a different definition of time other than Einstein's, where the time of a light signal between two locations is the same in either direction. You say "readouts" given by the LT, I guess you mean the calculated coordinate time in the distant clock's reference frame of the local event "now" ... the calculated coordinate time in the distant clock's reference frame, of an event at the distant clock's location, whose time in the local clock's reference frame is "now"???. At the intersections of two clocks' world lines, the "readouts" wouldn't depend on the one-way speed of light because the distance between the clocks would be 0, and the time of a light signal would be the same regardless. So I guess the answer, as best as I can understand your question, is "no", in general an alternative to the LT would give you different "readouts" everywhere except where the two clocks meet.

Whether it's "true to nature" is neither testable nor even relevant! I'll repeat Einstein's argument: (emphasis mine) That one-way is the same as two-way is by definition. See Einstein's 1905 paper, eg. translated at http://www.fourmilab.ch/etexts/einstein/specrel/www/ where he writes: There's no test that can invalidate a self-consistent definition! IF on the other hand, some test found that SR does not adequately describe reality, then some other definition of time, or a modified definition, might be used instead. For example in curved spacetime, in cosmology etc, that definition isn't used and there is no such definition of a common time (or simultaneity) for different locations throughout the universe. If you think you're debating whether the LT or Einstein synchronization is "correct or incorrect", you're missing the point and wasting your time, because it's correct. But when you speak of what happens at a distant A when B does something locally, you are basing that off of definitions. What's happening at A is outside B's light cones, there is no causal effect, and arguments about what is "true to nature" are not supported by SR as written by Einstein. The difference in ageing when they meet is indisputable, testable, true to nature, geometrically measurable, etc. The time at A when B is far from it is NOT based on "nature", Occam's razor, experiment, etc., it is established by definition. Any true conclusions you make based off of that are true by definition, regardless of reality. As for Occam's razor, if something like the simultaneity of events at A and B can neither be physically proven real nor proven inconsistent, wouldn't it be simpler to neither assume that they're real nor wrong, and accept that it might be just a definition that can "supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled" and not necessarily physical? Also, I don't argue that acceleration plays no role in the twin paradox; B's path involves a turnaround that involves acceleration. I argue it plays no role in OP's experiment. But if you can agree that in the water boiling analogy (two twin cups of water A and B, B is poured into a kettle that's on the stove and it boils)... if you agree that "pouring the water causes it to boil" then I'll agree that "B accelerating causes the difference in ageing", because then I'll understand the intended interpretation of the statement. Please, can we just agree??? Otherwise, I think that the role of acceleration in the twin paradox is the same as the role of pouring the water; something that establishes the necessary conditions for the outcome of the particular experiment.

Agreed. If twins are symmetric, they can't have aged differently at an event through which they both pass (or ever, in a reference frame in which they're always symmetric). Another example of asymmetry is OP's experiment, where the asymmetry is not due to acceleration. I agree, in the twin paradox class of experiments, twin B's acceleration is the cause of the asymmetry. In other experiments, acceleration doesn't cause asymmetry (eg. if two twins both accelerate), in others asymmetry doesn't cause a difference in path length (you can contrive two twins to have aged the same amount when they meet), in others asymmetry has a cause other than acceleration.

Celeritas, I don't want to argue this. I can't prove simultaneity is merely a convention and you can't prove that it's physically real. Einstein's convention does work in the case of the accelerating twin, because there is always a momentary comoving inertial reference frame that it can use. Even in the case of instant acceleration, you can let the twin instantly sweep through all velocities from its outbound to its inbound velocities, and you can instantly sweep through all planes of simultaneity between outbound and inbound. Whether or not there's any physical meaning to that doesn't even matter, because whether you do it that way or not, you're not going to predict any different results. The LT uses Einstein's definition of simultaneity to establish the time at different locations within an inertial frame. I don't understand what you mean when you say the LT is real but Einstein's definition of simultaneity is a convention. When B and C meet (or when B turns around), the time at A is relative. There is no one "then" that you're asking about when you say "Does a clock truly read what it's hands then display?" The clock at A is correct at all times through its journey. Also you talk about what the clock at A "then displays". But you know that B and C when they meet see the same time appearing on A's clock, in accordance with the relativistic Doppler effect. That they disagree on when "then at A" is, can come down to the fact that they disagree on how long it has taken light from A to reach them at that moment (because in this situation, B and C each have A moving at the same relative speed in different directions, and they agree on the rate at which A's clock ticks in accordance with SR). They measure at that moment that the clock at A appears to display the same time, but they can say that the time at A is "really" a different time at that moment because of the "time" it has taken light to reach them. BUT the "time" it has taken light to reach them is based on the definitions that Einstein uses, which are the same that establish whether two distant events are simultaneous or not. So the relative time at A, according to some distant observer, is "really" what the LT says it is, exactly to the extent that Einstein's simultaneity definition is "real". Whether one believes it is or not, it is established by definition, not by physical measurement. That's why the results of the twin paradox and OP's experiment should be incontrovertible; they only require the comparison of proper times. I've been debating the meaning of coordinate time and that's not helpful. The outcome of the twin paradox is as certain as SR is, and doesn't depend on how distant clocks relate, about which I'm not going to agree with others.

Oh right, gravitational potential energy is gravitational potential * mass, so a more massive clock would have higher potential energy, but wouldn't tick faster than a nearby lighter clock. If I worked out the maths I'd make fewer mistakes like this. Do you mean it depends on the position of the reference clock? Is it just the relative gravitational potential (determined eg. by height h and g(h)) of the two clocks that matters? If you have two clocks and all you know is their gravitational potentials relative to some arbitrary common reference point, can you determine their gravitational time dilation?

It's not increased g but lower gravitational potential energy (depth in a gravitational well) that makes clocks relatively slower. It's easy to confuse because with common masses g is typically higher where the potential energy is lower. If you took a pendulum clock tuned for Earth's gravity and suspended it somewhere above the sun where g=9.81 m/s^2, you could confirm that the clock keeps time with a nearby light clock the same way one would on Earth. But it should be deeper in a gravitational well compared to the Earth's clock, so that if you compared the two pendulum clocks operating with an equal g, the one above the sun should be slower. That's the thing about "all other things being equal"; you can make all other things equal and you still get the relativistic effects, so you can rule out mechanical reasons for time dilation.

No, that's false. The twin that turns at B-turnaround (aka BC event) measures the same contracted length between AB and BC that OP's inertial clock B measures between the same events. The distance between AB and BC is 3 light years according to A, and 2.4 light years according to twin B or clock B. If you have a marker at BC that is stationary in A's frame (so that the proper length between A and the marker is the length measured in A's frame), then the marker approaches B (twin or clock) at a speed of 0.6c for 4 of B's years, traveling 2.4 light years in outbound B's frame. Bringing this back to OP's post, I'll agree that your twin B's turn around causes it to follow the shorter world line made up of OP's B and C between the 3 events where clocks pass. Well let's consider that analogy. You have two twins A and B, and the only differences between them is that B turns around and ages less. In the absence of some other form of time dilation (GR), having B turn is necessary to have it age less. Therefore B's turnaround is the cause of it ageing less. Now say you have 2 glasses of water, A and B, and a kettle on the stove, and you pour B into the kettle and it boils. The only difference is that B was the one that was poured and it was the one that boiled. In absence of some other way to heat it, pouring B into the kettle was necessary to have it boil. Therefore, pouring B is the cause of it boiling. I suppose it's possible to interpret that as a true statement. But in isolation it is just misleading. Pouring water causes it to boil. Proper acceleration causes differential ageing. Then you do the equations and you find that the world line AB to BC to AC is shorter than AB to AC, and that the energy used by the kettle heated the water, and the pouring doesn't factor into the maths. Someone points out that if the water started out in the kettle, it would boil in the same time as if it was poured in. Then someone else says that involves hidden pouring. Edit: Maybe I'm dumbing down the argument too much. You're arguing that the change in coordinate time at A's clock when B turns around, is something real (because the LT says it is?) and I'd argue that's just a coordinate effect that disappears with a different system of coordinates or without a definition of simultaneity. Equivalently, you're arguing that Einstein's definition of simultaneity must be physically real? Or, back to OP's experiment, could we say the debate is about whether clock A is physically different to clock B vs. clock C besides how it appears differently to different observers? Would you argue that when twin B turns around (or if clock B were to turn into clock C), that causes a physical change in A? I say it doesn't, that the observed differences are only relative and depend only on the observer. If they're both inertial and their velocities are different, does that just mean that you're choosing a frame of reference where they have those velocites? Otherwise, regardless of the direction of their different velocities, if B's speed is v in A's frame, then A's speed is v in B's frame. The only way to have the twins age the same over a time t is to choose an inertial frame of reference where A's speed is the same as B's speed, right? (Of course there's no inertial frame where A is inertial and B turns to reunite with A, and they have the same speed relative to the observer the whole time.)

"Makes it all happen" is too vague to be meaningful. What's "all"? B's acceleration doesn't determine A's ageing. One might as well say "B's proper acceleration is the magic that makes it work" and "magic" means the part that doesn't show up in the maths, or it's the answer to the "Why?" questions that aren't satisfied with knowing the "what". I don't know why people look for such explanations anyway. We agree everything that's "what" is there in the geometry, and it's there in the 3-clock variation when nothing physical accelerates. Does there need to be more than that? I think it's fair to say that twin B's velocity relative to A is affected by its proper acceleration. That's about all that's needed to explain acceleration's role in the twin paradox. If B undergoes the same proper acceleration but at A's location, there's no change in the coordinate time at A. So someone else might say "It's distance that makes it all happen!" Then someone else might suggest that if B instead orbits A at a very small distance but at 0.6c, you get the same ageing as in OP's experiment, but with B undergoing constant proper acceleration. So neither distance nor proper acceleration alone is making it all happen here. You can scratch a hole in your head trying to figure it out how one or the other is making it all happen. But if all you have is relative velocity and time, you can calculate it.

The top and bottom of the rocket sitting on Earth remain the same distance away from each other, and as specified feel the same gravitational forces. If you want the rocket in space to have the top and bottom remain the same distance away from each other (in their frames), that's called Born rigidity. If you make the rocket Born rigid, the top and bottom will need to have different rates of acceleration, and different proper acceleration. Then the equivalence principle doesn't apply, at least not to say that the space rocket top and bottom are equivalent to the Earth rocket top and bottom. If you want the equivalence principle to apply, just specify that the top and bottom have the same proper acceleration. Don't worry that the spaceship eventually pulls itself apart, you can't constrain everything how you want. There's no contradictions... could the problem be that the two rockets are necessarily different in some way or another? I think the resolution is that the equivalence principle applies locally. it doesn't say that distant clocks and rulers will be equivalent. (Or I suppose it can be applied to the whole rocket if it were in freefall?)

MigL explains why it's not a contradiction and is expected. If you're asking why there is a difference in times at the top and bottom of the rocket, when they should have identical experiences, it's because the difference in time is only relative, not something they experience locally. They actually both experience the exact same thing as each other regarding time: "My clock runs at 1 s/s, clocks above me run faster, clocks below me run slower."

That all sounds great. Your carefulness is appreciated. If we were all careful, we could remove anything that isn't agreeable, like opinions or interpretations. If instead of "In terms of the twin's instant turnabout scenario, it comes from the proper acceleration at B's instant turnabout.", we said "...it corresponds with B's proper acceleration", then I can agree. Sure, it's helpful to consider the experiment from different points of view and different measurements, but when you speak of the time at clock A, when B passes C, then you're comparing distant clocks, and you have to deal with the caveats of that. Then if you want an even closer to complete understanding of it, it is useful to know the reason that we can say the BC event is simultaneous in B's frame with an event at A at time 3.2 years on A's clock, but is simultaneous in C's frame with an event at A at time 6.8 years on A's clock. It comes from the definition of time that Einstein gives in his 1905 paper that establishes special relativity, in the section translated as "Definition of Simultaneity". As Einstein wrote (in Relativity: The Special and General Theory): Anyway, whether the definition of simultaneity has a physical basis or is merely a chosen convention, is something that can be argued. It doesn't matter though. Especially if A's distant clock is never compared to a local clock, the definition of simultaneity can be completely avoided. I also believe that a closer-to-complete understanding of relativity involves being aware of the definition of simultaneity and knowing when that definition doesn't matter (eg. for invariants). A's clock "jumping ahead" is not something that B or C can instantly detect. It's something that's seen gradually over time as C approaches. Actually, a Doppler analysis is easy here because with a speed of .6 c, the relativistic Doppler factor is 2. On its 4-year journey to meet C, B sees A's clock appear to tick at half its rate. It watches A age 2 years. On its 4-year journey to meet A, C sees A's clock appear to tick at twice its rate. It watches A age 8 years. Together they see the full 10 years of A's ageing. When B and C pass, or when a twin instantly turns around, no ageing of A appears to happen then. The instantaneous jump in A's coordinate time, is not something that is measured at that time.

I agree. Still, the 3 clocks or 3 siblings experiment OP describes uses only inertial clocks, only geodesics, following the same inertial sections that an instantly accelerating twin passing through the same events would (measuring the same total geometric length). I suppose I'm claiming that you can add the geometric length of two world lines, and get the total length of a world line made by connecting the end of one to the start of another. I think it is considered. If a twin makes the turnaround, it feels proper acceleration which (in accepted theory) contributes nothing to its proper time, so that feeling won't show up in the ageing equations. In changing inertial frames, its relative simultaneity with events at clock A changes. In the 3-clocks scenario, clocks B and C already have different relative simultaneity with events at A. So we're already accounting for those differences. There's no physical frame-independent difference between B and C, but measurements made in their respective inertial frames, are different. Simply by considering the two inertial frames properly in any relevent equations, everything's accounted. I disagree that the twin paradox implies all those things. If you look at this geometrically, and compare the geometric lengths of two twin's paths, you should be able to see what's important and what's not. If some aspect is not needed in the geometric analysis, then it's not a necessary contribution to the overall and different ageing of the twins. That includes synchronizing clocks, and being at relative rest. Also being twins. All these other aspects illustrate the seemingly paradoxical nature of the different ageing. You could say "the twin paradox" requires some of those things, but "the differential ageing of the twin paradox" does not. Being in close proximity is important because if you compare the time on two separated clocks, you can get different answers for different observers (inertial frames). The experiment described by OP is okay, because you only need to compare clocks at single locations (events), to predict the result. Those events are the moments and locations of the given clocks passing by each other. Yes, Einstein was right. I think that a lot of people who say Einstein was wrong, have no idea what he actually said or wrote.

Actually, if I understand the description of a geodesic with minimal ageing as described in Gravitation, I don't think any exist. It makes sense that a free-fall path over a saddle point is minimized in the sense that any spatial deviation will result in a longer path. However if a non-freefall particle follows the same spatial path, but speeds up and slows down in order to stay "nearby" the free-fall particle, they should have even lower ageing than the free-fall particle. I suspect that there are free-fall paths that do not maximize ageing, but that those paths have neither maximum nor minimum proper time among nearby (in 4D) world lines.

It's described in the first post in this thread. There is no paradox in any twin experiment, only a surprising or confusing result of SR. As described by OP, paraphrased, the length of the inertial path from AB (where A and B pass) to BC plus the length of the inertial path from BC to AC, totals 8 years of proper time (whether or not a single clock follows the entire world line), while the length of the inertial path from AB to AC is 10 years long, for the given speeds and distance. Do you disagree with that? If so, what is your calculation? (Remember that nothing here accelerates.) That difference in path lengths may be similarly surprising? One might argue that this thread proves that it's confusing.

I ask again, what is accelerating? Nobody has described a physical thing accelerating, not OP nor anyone else. OP's experiment describes 3 inertial frames. I disagree with the description of "changing inertial frames" (unless someone can explain what changes inertial frames). For what OP is describing, it suffices to say "we are considering two different inertial frames" to describe clocks B and then C. If OP's explanation relies on a "change" then I disagree with that, because as Markus Hanke has suggested, one can compare the geometric lengths of the observers’ world lines in spacetime, and I say you get the same answer that OP gave. I agree with the critique of OP's wording ("turnaround", "synchronized", "change" etc) because this is a topic that even some people who have expert understanding of relativity will refuse to accept, and anything ambiguous or interpretable in an unfavorable manner will be nitpicked to death. If OP's "explanation" relies on interpretation then I don't care about it. However the results of the description of OP's experiment are predicted as described, by special relativity. That is, unless you purposefully ignore a reasonable interpretation of what OP describes and invent something else (like clocks slowed for some other reason than SR's time dilation, which by the way I think is a ridiculous justification for doubting the predictions of SR). That is why a topic like this is better presented with precise language and only claims that are incontrovertible. On the other hand, I doubt even then, that incontrovertible predictions of SR would be accepted here if they didn't fit with preconceived ideas.

Did you not read OP's description of the experiment? There is no acceleration. Are you able to understand that case? 3 inertial clocks, passing each other at 3 separate events. SR involves many measures that are "invariant", and can be made sense of by all observers, including accelerated ones. Other measurements can be calculated for different observers. An accelerating observer can usually be treated as having a "momentarily comoving inertial frame" at any instant. I disagree. The proper time on a world line is invariant. You don't need to compare two clocks to measure it. Also I think you misunderstand. Of course relativity resolves the twin paradox, neither I nor OP is arguing against that. The measurements described in OP's experiment are 1) completely consistent with special relativity, and 2) the only possible values that are consistent with each other (if you change one of the values like speed, distance, or time, you'd have to change another to keep it consistent). There's no need for alternative explanations, unless you disagree with SR.

I'm trying to figure that out, too. I'm not even sure of any examples of a free-fall world line with a minimum proper time. I think that something like a massive ring or washer would work. If you free-fall through its center, I think that might be a minimum. If you free-fall around it, that would be a maximum (similar to if it was a point mass). Then to make a world line with both maximum and minimum sections, have a test particle orbit a normal mass in an eccentric orbit, and add a minimum part at its apogee (a massive washer to pass through, if that works). Make it eccentric enough that each of the masses has negligible effect when the particle is near the other.

Does that mean that for a world line whose proper time between two events is 4 years (for example clock B's world line as per OP), you wouldn't be able to tell if a clock that measured 4 years between the two events on that world line was running properly in accordance with relativity, or was running too slow (or fast) for some other reason, unless you can compare it side-by-side with another clock?

The different clock rates are a direct prediction of relativity. The proper times along the given world lines are invariant. How does comparing clocks in a single frame have any bearing on that? Can you give an example of how the different clock rates are due to something other than relativity, yet is still consistent with relativity?