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  1. I just realized the topic was split, but I wanted to sum it up to bring this back to the original topic anyway. Saying that it is "harder" to accelerate an object that is moving fast (as measured from some reference frame), can instead be said more precisely: A certain change in velocity of an object in a frame in which it is moving fast, requires a greater change in velocity (ie. higher rate of acceleration or acceleration for longer time) in a frame in which it is moving slowly, because the difference in the velocities is not the same in different frames of reference.
  2. Can you clarify "same accelerations"? It's confusing because a given acceleration wouldn't be the same in the two frames. Do you mean the same proper acceleration, which would be measured as relatively minor in the muon frame? Otherwise suppose the "quite considerable" acceleration is of 0.001 c, then accelerating that much in the muon's frame would be measured as an acceleration of 0.001 c (ie. quite considerable, not relatively minor). Ugh, there's always complicated and simple ways to look at anything in relativity. Complicated: Say you are at rest relative to Earth, and are able to easily accelerate to 0.001 c relative to Earth. While at rest on Earth, you are also traveling at say .999 c toward a muon. If you want that "same acceleration" of 0.001 c relative to the muon: You can't accelerate 0.001 c toward it (measured in its frame), but you can accelerate away (or decelerate) so that you're traveling at .998 c toward it. If you did this starting from at rest on Earth, you'd accelerate 333.556 times as much relative to the Earth (using more than that many times as much energy?), so you're now traveling at 0.333556 c relative to Earth (determined using composition of velocities formula). Simple: When you're at rest you have a rest frame, and your mass is the same regardless of the relative velocities of other moving objects. In each of their rest frames, you're the one that's moving. You feel the same regardless of their motion (you can't distinguish between being at rest and being in a moving inertial frame), and proper acceleration is equally easy no matter which inertial frame you're in; the faster you go, the same it always feels. The difference is that a large change in velocity in your frame may be small in another frame, due to the way velocities add in special relativity, so you have to accelerate greater and greater amounts to achieve some specific acceleration in the other frame. (Is this simplification valid? Can the change in total energy of an object as v approaches c, be found using composition of velocities formula, or is there something more to it?)
  3. If you were the body and accelerating yourself, do you know how this would feel or what you would measure? For example, if you're on a rocket accelerating away from Earth at ever-increasing speeds. I think that understanding that would help understand the issue. (Hint: a simple rocket model implies constant proper acceleration, and is probably an easier answer. If instead you suppose constant acceleration relative to Earth you'd measure different things. You could also imagine accelerating in short rocket burns and then describe what you measure in between; what differences do you notice between subsequent burns?) Another question that might help understanding: How would you describe your body's total energy (or "relativistic mass") in the frame of a muon generated by a cosmic ray in the atmosphere, relative to which your speed is over .999 c? In what way is it hard to accelerate yourself relative to it? This isn't a trick question.
  4. Changed my mind! No, I think the interpretation of "James knows only a minus d" is the right one. There are no numbers with a-d=7 that have a unique product. Therefore he knows that James doesn't know the answer. I think 8221 might work in the end? There was the idea that 4421 or 4222 would then also work. However, a-d=3 can be eliminated by 5552 (a unique product), and a-d=2 by 3111. If 0's are not allowed, then 8211 fits all the criteria. However I end up with 3 working solutions: 8211, 9881, and 9981 all seem to work.
  5. The solution also works if Jack/P1 doesn't know a and d but just d-a, so the puzzle could be interpreted either way. The less information Jack is given, the harder it is for us to reason about it, but the easier it is to eliminate possibilities once we figure it out, because for Jack to be certain that James/P2 doesn't know the answer implies that the info Jack actually has must be substantial. If d<=4, then we know that d-a>4, eliminating possibilities like 3222 earlier.
  6. Because not all of the frames will agree on the simultaneity of events. You definitely can define a "now" and you can do it however you want to (a foliation of spacetime would make sensible instants of time throughout space, but the problem is you can foliate it in many different ways, so essentially your "now" would be completely arbitrary). To be useful, you would want to be able to say that all the events in your "now" are simultaneous. Someone in a different inertial frame would not agree that those events are simultaneous, and would have no reason to accept your personal definition of "now".
  7. On second thought I think what I crossed out is true, and possibly equivalent to what I implemented.
  8. I think I've solved it and the answer surprised me. I'll restate the puzzle as I solved it: There's some number with digits abcd, with 0<=d<=c<=b<=a<=9. Person P1 knows a and d. P2 knows a*b*c*d. P3 knows a+b+c+d. P1 says "A) I don't know the digits but B) neither does P2". P2 says "C) I don't know the digits but D) neither does P3." P2 says "D) I know that P3 doesn't know the digits, and C) I don't know the digits either." --- I believe this order is important because P3 can know the answer while the statement is being given. P3 says "E) I didn't know the digits but F) now I do." It does not seem to need to say more but it could say now P1 knows or now everyone knows. Solution: A) a and d must be different or it would know. B) The range must contain some number with ambiguous factorization. For example, 9*1=3*3. If the range contains 1 and 9, or just 3, then it is still good. Do this for all possibilities and you find the range must contain 0, 2, 3, or 4. If it contains 1 we already know it must also contain either 0 or 2. So, eliminate all possibilities where d>=5. C) Eliminate all remaining possibilities that have a unique product. I think that a key element here is the concept of common knowledge. At each step, everyone can eliminate any possibilities that are based on what was said, but not necessarily the possibilities that the others could eliminate. Anyway, that might not be important yet... D) P2 can eliminate all possibilities that add up to a unique sum. E,F) Here is the key to the puzzle and where most of the possibilities are eliminated. P3 can also eliminate all possibilities that add up to a unique sum. However, since it now knows what the answer is, it must be that it was possible for P3 to assume that P2 knew the digits. The answer must be a combination whose sum is not unique sum, but where all other combinations that add up to the same sum have a unique product. Edit: No wait, that's not what I implemented! I looked for combinations that P2 knew had a unique sum *among* only possibilities that have no unique product, but which P3 did not know had a unique sum. Very confusing! I'm not sure I got that right or implemented it right, but I end up with only one possibility left and that is
  9. Intuition would be to minimize non-diagonal segments, so AjkC (However it's easy to add bad diagonal paths that would break the strategy.)
  10. If the solution were 9xy1, the product would be the same as xy33, so Jack could claim that James doesn't know the digits. (Then, 9621 adding up to 18, and 9431 adding up to 17, both with products 108, might be possible reasons why John can eliminate 4333.)
  11. What about 6332? Its product is also 108. It's the only such one that adds up to 14. If John supposes the answer is 4333, then James would have considered 6332 as a possible solution, and wouldn't have been certain that John wouldn't know the answer??? Is that enough to rule out John thinking it's 4333? Or would we have to also show that every other possibility that James might suppose that John might suppose, doesn't have 2 or more combinations that add up to 14? But wait... 8411 adds up to 14... So John considering 4333 as an answer might have considered that James knew both 6332 and 8411 add up to 14, so could say that John didn't know the answer, meaning that 4333 is still a possible answer to John?
  12. Yes. If the range is known to be [3, 4], then Jack cannot know that James doesn't know the answer, as he states. If the range is [3, 4] then Jack knows that James must know the answer. The only way that Jack can assert that James doesn't know the answer is if the range includes factors that can multiply to the same product in different ways, eg. 9*1 or 3*3 (requires a range of at least [1, 9]) or 8*2 or 4*4 (range at least [2, 8]). First ruling out 0, the smallest range for which this works is [1, 4]. *We* know this much after Jack's statement, so everyone else (John, James) are able to know this after his statement too, so John can rule it out. No wait... my line of reasoning is wrong and might not go anywhere useful...
  13. Yes that makes sense. That's what happens at speeds low enough to neglect relativistic effects. Clock synchronization can be "achieved by "slowly" transporting a third clock from clock 1 to clock 2, in the limit of vanishing transport velocity." Then you'd say these clocks measure the same time. Worse than conflated and confused, time dilation was completely ignored, and relativity was presented over simplistically. They cared about getting Einstein's hair right; the science wasn't important. I think it's terrible, because a lot of people base their understanding of things on fiction, then go on to be a politician who decides NASA's budget or what to do about climate change etc., all the while perpetuating "the science wasn't important." (Rant cont.) Fictional science is probably fine, but writers throw in real concepts or jargon to be more interesting or sound legitimate or whatever. How many people could earnestly debate multi-world theories, without ever taking a physics course? Where do they get their information?
  14. What do you mean by time reference? In the situation described, with no one stopping, each observer would measure the other's clock ticking slower, in addition to seeing it tick much slower due to delay of light.
  15. No, Insignificance gets relativity wrong. However the quote above was cut off right before it gets bad. I bolded the main wrong part, and strikethrough'd ... yuck. The character is only talking about delay of light and has neglected time dilation entirely, but explains it as if it's time dilation. In the striked part, the character is describing differential ageing or total time, and describing it as a differential rate of time (which is constant at a fixed relative velocity).