md65536

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?

But the geometric lengths of the world lines OP describes between AB (where A and B pass) and BC is 4 years, between BC and AC is 4 years, and between AB and AC is 10 years, calculated using special relativity. Are you getting a different answer, or not able to get that answer, or are you proposing a different cause other than relativity that is giving you that answer?

But what we're talking about extremizing here is the proper time over the whole world line, which is the integral you posted. The "given curve" that we're extremizing is all the possible proper times of nearby world lines. If the freefall world line passes through sections where proper time is maximized, and sections where proper time is minimized, is the whole world line a maximum or minimum? If you can nudge the world line in one section and increase the proper time of that section, then the whole is not a maximum (neither local nor global). If you can nudge it in another section and decrease the proper time of that section, then the whole is not a minimum.

OP's description still has 3 clocks passing by each other. What physically changes? It's not cheating to set initial conditions to make measurements easier. Only one scenario is described, ie. 4 years of proper time measured by C at a speed of .6 c relative to A. Whether clock C reads 0 or 4 years or 500 when it meets B, it will read 4 years later when it meets A. You get the same answer either way; the path from AB to BC to CA is 8 years long, regardless of how the clocks are set. Also regardless of how many clocks are used to measure sections of that path. Can you show how to apply turnaround acceleration using the experiment described by OP?

This statement seems to be causing confusion and is unnecessary. Wikipedia instead calls it 'at the point corresponding to "turnaround" of a single traveller,' which emphasizes that nothing turns around. "there is a change in the frame of reference" sounds like a description of something physical happening. Nothing physical changes frame of reference. There's only a change in the frame of reference that we're considering, and it doesn't have to be described as a change. Instead maybe something like, "for the inbound leg, we're considering a different reference frame." That doesn't change the experiment that you described. What changes reference frame? Nothing physical that you've described. Another way to emphasize that nothing changes at the BC meeting point, is to not set clock C there at all. One way to do this is to set C beforehand so that it will already have the time you want (ie. same time as B) when it passes B. Another way is to just add up the proper times of clocks B and C clock between the events, without even mentioning or caring what they were set to. I see it as using 3 clocks to describe the differential aging along two different world lines, which is the heart of the twin paradox. You could have a single physical object follow the same path, and it would be a true twin paradox experiment. If you run OP's experiment alongside that, OP's is a way to measure a twin paradox experiment. It *is* a measure of the ageing in a twin paradox experiment.

I read it again and I think it's exactly what OP posted. What difference do you see in our descriptions?

3 clocks pass by each other at 3 events. What changes frames?

This all looks okay to me. This has been discussed in the past and someone referenced a paper written in the 1900s that describes the same experiment, but I can't find it quickly. Wikipedia has references, see https://en.wikipedia.org/wiki/Twin_paradox#Role_of_acceleration I don't like using the word "synchronized", but it's used in wikipedia. You must be careful because none of the clocks remain synchronized for any length of time, which I think does not satisfy the definition of synchronized. I'd just say the clocks are set the same. The path length that you describe, between meetings of AB, BC, and CA, has an invariant length of 8 years. This is true whether or not a real object follows it. If you compare path lengths, you'll find that what you're describing is true; one path length between AB and CA is 8 years, and another is 10 years. Describing things in terms of path length is incontrovertible. However even if you do this, people will disagree.

What is accelerating? For how long? Is it an infinite acceleration? Where is the acceleration accounted for in formulas? Do you have acceleration every time you consider a new frame of reference? This has been done countless times with the relevant formulas. This description is sufficient because it's a description of the result of those formulas. Here v = .6c and gamma = 1.25, anyone who properly applies the formulas will find that the description checks out. The proper time between when A and B meet, and B and C meet, plus the proper time between when B and C meet, and C and A meet, is 8 years.

I'm not sure if that's helpful, relating to maximal ageing. It's a very simple example but it's obvious the math becomes very complicated quickly with other examples. If you're using x as a representation of time so that you can speak of world lines, and then proper time would be an integral of the square root of dx^2 - dy^2, so that curved lines have a lower value than comparable straight lines, and line slope is less than 1 (analogous to speed of light) so that only positive proper times are considered, then that all makes sense. I just noticed that the Taylor/Wheeler reference above has more than one definition of the principle of maximum ageing for GR, so I'll give a later one: "Gravitation" mentions geodesics analogous to "mountain passes" as an example of a path of minimal aging, where those analogous to mountain peaks are maximal. That it's a minimum is due to the spacetime geometry, not due to how you choose the signs in your metric. As a very rough analogy, if you're making a straight-line (on a map) trip across a mountain, the longest possible trip is over the peak. If you're making a straight-line trip between two mountains, the shortest possible path is over the saddle-point of the mountain pass. I think a real-world example relating to extremal ageing, is a free-fall path that passes straight through the middle of two identical gravitational masses. If that's an example of a saddle point, then it's possible that this is a path of least ageing. (Edit: I don't think that's an example of minimal ageing. Instead just assume there's some path of minimum ageing.) Nudging a stone off this path, and then back onto the original world line, would only increase the ageing of the pushed stone. If the principle of maximal ageing still holds, it might do so only because there is no other path along pairs of adjacent local frames. I'd guess that means, if it's a path of least ageing and also a path of max ageing along pairs of local frames, then any deviation from the path involves effects of spacetime curvature that make it not a local inertial frame. But that seems quite suspicious. I think their definition simply doesn't apply to all free-fall world lines. As well, since the world line can be extended into other regions of spacetime apart from the "saddle-point", it must be possible for the same world line to pass through sections of max ageing and sections of min ageing. This seems like a problem for both Taylor/Wheeler's definition and Misner/Thorne/Wheeler's: "tau = [math] = (an extremum for timelike world line that is straight in each local Lorentz frame along its path, as compared to any 'nearby' variant of this line)," (Gravitation, p. 316), because both define the proper time over the entire world line as a maximum or extremum. Here I'm posting before understanding it enough, but I don't see how that problem can be resolved with the given definitions. Yes, in M/T/W's example, a star falling through the plane of a disc galaxy ("Gravitation", Fig. 13.2), vs being thrown up from the plane, could meet later. Both are geodesics but with different initial conditions. In my example, a stone falling straight between two masses could be met by a stone orbiting around them, both in freefall with different initial conditions.

The thrown stone is in freefall and ages the most among any nearby paths. What I meant by the first statement is that you can find other arbitrary non-freefall paths that age less than the stone sitting on top of the mountain, even one that includes sections of freefall as per OP's example. Eg. if you dropped a stone off the mountain in freefall, but then brought it back up so that you could make a definite comparison of their proper times, it could have aged less than the stone on the mountain, depending on how you do it (but this example involves a non-freefall path and doesn't violate the principle of max ageing). Also... technically nothing's stopping you from comparing two distant (in space and/or time) worldlines any way you want to, and coming up with different answers, but that's not going to violate the principle either. If it's worth it for you; exposure to more maths would help me. One thing I was stuck on is the idea and meaning of globally non-unique geodesics. A wiki page cites Misner/Thorne/Wheeler's Gravitation, p. 316 and I looked it up and it seems to contradict a couple of things from this thread. However I don't want to bring up the details until I understand it better, which I might not do. I doubt it affects the basic understanding of all this.

Alright, I have no further dispute and I think I'm getting hung up on details I don't understand. But back to the original problem, which is that a stone fixed on the top of a mountain will age more than some others, even though it isn't in freefall. However, another stone thrown straight upwards from the top of the mountain, with any velocity that keeps it nearby (no escape velocity etc.), will return to the same spot. Having a freefall path, it will have aged more than the fixed stone did between the throw and the landing. So you can vary the initial conditions and get different geodesics and let that determine the events A and B between which you're comparing different path lengths. Or you can fix A and B and get a locally unique geodesic and unique initial conditions. Another thing the principle doesn't mention is the notion of the two events A and B. All of these details are not aspects of the principle, they're just things we're using to properly specify a particular case that we're applying the principle to.

Feel free to bring us back to the original examples etc., I've kind of gone off into possibly irrelevant details. No. I don't know if you're trying to say, paraphrased, "A given free particle follows a path of maximum ageing, and a given particle has initial conditions." Which is obviously redundant. Or, are you saying "A given free particle follows a path of maximum ageing among all possible nearby paths with the same initial conditions." I can very roughly prove that this is redundant if you doubt it. Either way, the principle doesn't say anything about initial conditions and it doesn't imply that its application depends on initial conditions. I feel like we're having a conversation like this: P1: Given some initial conditions, the velocity of a particle is dx/dt. P2: That doesn't depend on initial conditions. P1: Do you understand that a specific particle that has velocity will have initial conditions? Otherwise, I'm missing the point of why you're talking about initial conditions, especially since limiting the initial conditions will exclude some of the applications of the principle already discussed in this thread.

My experience has been that yes, experts tend to correct their own errors when they're realized, and that it requires an expert to correct an expert (even if it's the same person) because expert replies are far more trusted and accepted than others, and that experts tend not to acknowledge errors in other experts' posts. I agree that a good way to disagree is to ask a question (like, aren't the number of photons sent and received the same?) because even when you're wrong, you're not wrong in asking! But too much stuff like "I need time to understand" says you think the problem is with yourself and that there's no reason for the other person to reconsider what they wrote. When other people read that, and compare a reply labelled "Expert" to one that says "I don't understand", it's easy to brush off the latter.

That means if you're talking about the galaxy as an object moving away from you through flat spacetime, then the light energy it emits and the light energy you receive will be the same in a given frame of reference? But if you're talking about spacetime expansion, you can't treat the source and receiver as having a common frame of reference (or can you speak of a global frame of reference for other laws etc.)?

Digest it? You mean the answer given was wrong and you're pretty sure of that? Is there some unwritten rule on this site that it's poor etiquette to disagree with a resident expert? If people ask questions and see a wrong answer from a resident expert, and no one authoritative corrects it, they tend to trust the wrong answer. 1. They don't, which you know. 2. That's right. 3. I don't know, hopefully someone else does. Photons have kinetic energy, which is different in different frames. There's no conservation of an object's energy between different frames. Where conservation of energy comes in is... in either frame, the galaxy loses the energy of each photon that leaves (ie. different in the 2 frames), and I guess is pushed in the opposite direction of each photon a tiny amount (different in each frame). The red-shift also lowers the intensity of the light (or the number of photons received per your second), so the energy you receive is less than 100 joules (I think you receive 100/n^2 watts for n seconds, 100/n joules total?).