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Everything posted by md65536

  1. 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.
  2. 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 th
  3. 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
  4. 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
  5. 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 am
  6. 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
  7. 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
  8. 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 w
  9. 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 t
  10. "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
  11. 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
  12. 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."
  13. 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 dea
  14. 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 prop
  15. 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
  16. 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 diffe
  17. 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,
  18. 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
  19. 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
  20. 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?
  21. 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?
  22. 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?
  23. 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 de
  24. 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 ap
  25. 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 ch
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