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Supposing I have a piece of paper that can be folded infinitely.

  • In the first $5 \, \mathrm{s}$, I fold it to twice its thickness.

  • In the next $5 \, \mathrm{s}$, I fold it to 4 times.

  • If I fold it to twice its thickness in the $n^{\text{th}}~5 \, \mathrm{s}$, since time increases linearly and thickness doubles in each $5 \, \mathrm{s}$, will I not be able to increase the speed of increase in thickness of the piece of paper to beyond the speed of light?

Qmechanic
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Shashank Shekhar
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    upvoted because this is hilarious – zzz May 24 '18 at 03:06
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    The piece of paper that folds infinitely many times is infinitely thin (e.g. 0 mm). Two times zero is still zero, so the thickening doesn't exceed any speed ;-) – M.Herzkamp May 24 '18 at 10:02
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    Suppose I have infinite strength. I start with 1 kg, 2 kg, doubling every time the amount of weight I am lifting. So finally I am lifting the universe and throw it away with infinite speed. How does relativity prevent it? This question is exactly as meaningful as yours. – Thorsten S. May 24 '18 at 11:20
  • The thickness after n folds will still be at most the original diameter of the paper if it is non-stretchable. – Tom May 24 '18 at 15:13
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    This reminds me of a fellow highschool student many years ago who asked the physics teacher whether a giant with shoulders broad enough to span the radius of Earth's orbit really couldn't clap his hands in less than eight minutes. Naturally our teacher laughed, mumbled something about scaling laws, and then declared that the giant would spontaneously fall apart and be crushed by its own gravity, taking much of the solar system with him. – J... May 24 '18 at 15:20
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    tl,dr; You cannot keep folding it every second, because the ends of the paper would have to move far faster than the speed of light. – RBarryYoung May 24 '18 at 16:47
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    Exercise: Find a flat, circular field 100km in radius. Build a tall, white, circular wall enclosing the field. Mexico will pay for it. Get a dreml rotary tool and a battery powered laser. Attach the laser to the business end of the dreml, get it spinning at 2000 revolutions per second. The spot that the laser makes on the wall is moving at 628000 km per second, twice the speed of light. How is this possible? – Eric Lippert May 24 '18 at 21:08
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    @EricLippert Impossible, ridiculous and easily refuted: Mexico won't pay the wall. – Thorsten S. May 24 '18 at 21:30
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    Suppose Mexico pays for a 1cm high wall. Every year, they double the height of the wall they are paying for. Won't the top of wall they are paying for eventually surpass the speed of light? – user253751 May 24 '18 at 23:24
  • Or: A snowball rolling down a hill doubles in size every minute. Won't it eventually have more than the mass of the solar system? (Did we just find a way to create mass from nothing?) – user253751 May 24 '18 at 23:25
  • I did some math semi-related to this over on worldbuilding once. It may be of interest. – Draco18s no longer trusts SE May 25 '18 at 00:55
  • Voting to reopen because this isn't "non-mainstream physics", it's just a misconception, which I think I addressed quite well in the answers. Asking "why wouldn't experiment X violate physics law Y" isn't non-mainstream physics, it's exactly the kind of question a student should ask in order to learn – N. Virgo Nov 17 '19 at 03:48
  • @Nathaniel I agree it's more a ill-formed question than a nonmainstream one, but it's then duplicate to any of the other many speed-of-light-violation questions we already have (this for instance), so I don't think it's worth it reopening it only for closing it then for a different reason. Besides, maybe one could argue that assumptions of the question (thick infinitely foldable at constant rate sheet) is non-physical/nonmainstream. – stafusa Nov 17 '19 at 11:39
  • @stafusa An unphysical misconception is a very different thing from a non-mainstream theory. Of course the assumptions of this question are unphysical, but evidently that wasn't obvious to the OP at the time of posting. I take "non-mainstream" to refer to incorrect or unlikely theories that are actively promoted, rather than to misconceptions about mainstream theories. – N. Virgo Nov 17 '19 at 14:18
  • @Nathaniel Yes, I also think that's what's mainly meant with the tag. At any rate, I consider it rather a poorly thought-out question (and it doesn't show any previous effort, in my opinion) that doesn't really bring much new to the site and could end up closed for other reasons. Your answer is great, but at any rate my vote had been cast. But I don't feel too strongly about this case: if the question shows up again in the reopen queue I'll abstain from voting. – stafusa Nov 17 '19 at 17:25

2 Answers2

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No, you can't, for a couple of different reasons.

The first is the difficulty of folding paper more than a few times. Mythbusters managed to fold one sheet 11 times I think, using a very large sheet of paper and the help of a steamroller. It took a lot longer than 5 seconds per fold.

The second issue is more fundamental. You could resolve the first issue by just cutting the paper in half and stacking one half on top of the other instead of folding it. But then you have another problem: suppose your stack of paper has reached one light year in height. Next you have to cut it in half and put one half on top of the other to make a two light-year stack.

With some cleverness you can do the cutting as quickly as you want. (For example, you could cut it using a carefully timed laser pulse from far away.) But once it's cut you have to move one half of the stack upward by one light year, so that the bottom of that half lines up with the top of the other half. You can't move the stack faster than light, so no matter how you do this it has to take at least one year. The next iteration will take two years, the next four, and so on, and the top of the combined stack will never move faster than light.

So really the logic of your question has to be reversed: it's not that you can move faster than light if you fold a piece of paper every five seconds, it's that you can't fold a piece of paper indefinitely every five seconds, because doing so would mean moving something faster than light.

(There is a third issue too, which is that every time you cut the paper in half you reduce its size, and eventually you'll just have a stack of atoms that you can't cut. But of course you can always just start with a bigger sheet of paper.)

As David Starkey points out in a comment, you can actually do a factor of two better than this, if you don't mind the bottom of the stack moving as well as the top. Then you can move one half of the stack down at the same time as moving the other up, so each one only has to move half a light year instead of one. But of course this doesn't change the overall argument. Each end of the stack is still limited by the speed of light, so you can't double the height of a one light-year stack in less than 0.5 years.

N. Virgo
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    Technically, after the stack is cut, could you move both stacks in opposite directions to get it done in half the time? – David Starkey May 24 '18 at 14:11
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    @DavidStarkey Yes, good observation. You would move each one half a lightyear in > half a year. Congratulations. ;-) – Peter - Reinstate Monica May 24 '18 at 14:28
  • @DavidStarkey, the time it takes to move the stacks of paper depends on where you base your origin. Is it a 'fixed' point in space or are you standing at the edge of one of the stacks of papers? –  May 24 '18 at 15:14
  • @Geliormth Assuming you want to do it quickly, you are cutting in half ASAP after putting the previous 2 stacks together. In such a scenario, you likely are around the middle of the stack. However, there's no stipulation that you have to do it alone. – David Starkey May 24 '18 at 15:35
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    @DavidStarkey: Yes, so the clock on your wrist won't match the clocks on the wrists of the robot arms you used to manipulate the paper. There's also the question of whether they think the stack of paper is the same height you think it is, but that only varies while they're actually in motion relative to you. – Steve Jessop May 24 '18 at 16:35
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    Also you can't cut the paper as fast as you want. Cutting a one light-year thick stack with a laser takes at least a year (since the light needs to reach the other end). – Graipher May 24 '18 at 20:58
  • @graipher the laser isn't pointing up or down along the direction of the stack, it's off to one side. That way you can easily arrange for the light pulse to hit every piece of paper simultaneously. – N. Virgo May 25 '18 at 00:35
  • @Nathaniel You would still need to turn this laser, or better many small lasers arranged on the side, on. This then still takes at least half a year (assuming your control center is in the middle) because that's the time the information needs to reach the lasers at the end to tell them to turn on. – Graipher May 25 '18 at 05:45
  • There is also the fact that information is also the information of the fact that you are moving the paper. So no matter how long it took you to “fold” your end of the stack, that information will only propagate at a speed < c to the rest of the stack that would slowly catch up. – Fogmeister May 25 '18 at 05:55
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    @Graipher the laser is very far away (many light years), so from its point of view the stack subtends a very small angle and it takes only a short time to turn. The laser does not need or get any information about the stack - it's fired many years before the stack exists and the pulse is timed to arrive at just the right time. If it still seems impossible, see https://en.wikipedia.org/wiki/Lighthouse_paradox – N. Virgo May 25 '18 at 07:53
  • @Fogmeister similarly to my above comment, you don't actually need any information to be propagated if everything is precisely timed. The moving of each part of the stack can just happen at a pre-scheduled time. – N. Virgo May 25 '18 at 07:54
  • @Nathaniel True that way it would work (at least that part). – Graipher May 25 '18 at 07:58
  • @Nathaniel how do you send the timing to each part of the stack? Given that we're talking in astronomical distances does "precise timing" over those scales even have a meaning? It can be proven that in certain circumstances the order of events and so "precise timing" doesn't mean things happen in the same order for everyone. – Fogmeister May 25 '18 at 08:11
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    @Fogmeister yes, precise timing does have a meaning in special relativity. We have to fix a reference frame, and we have to be careful about how everyone working on the project synchronises their clocks, but this is possible to do --- in fact, Einstein wrote quite a bit about how to do this while developing the special theory of relativity. (In general relativity it becomes trickier, but since this is just a thought experiment I'm assuming no large massive bodies are nearby.) – N. Virgo May 25 '18 at 08:19
  • Tried to make this comment on this answer, not the question. – Draco18s no longer trusts SE May 25 '18 at 16:48
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It is a fact of nature that nothing moves faster than c, the velocity of light in vacuum.

One can imagine infinite scenaria for things moving faster than light, simpler than folding a paper. A continuous acceleration in vacuum is the simplest.

It is an observational fact that nothing moves faster than light. Innumerable observations of particle physics and astrophysics have not falsified this statement..

anna v
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    Plural of scenario is scenarios – JAD May 24 '18 at 08:34
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    @JAD it is a borrowed word in english, from italian. https://en.wiktionary.org/wiki/scenaria , it is called "non standard" but I only stick on the "standard model" ;) – anna v May 24 '18 at 08:54
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    @annav There is no such word as scenaria in Italian, where the plural of scenario is scenari (or scenarii/scenarî as an outdated form). – Giulio Muscarello May 24 '18 at 12:00
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    @annav This is not true. Nothing can move faster than light in void, but there are some cases when a particle moves faster than light in a medium (Cherenkov effect). – Martigan May 24 '18 at 12:49
  • @GiulioMuscarello thank you , so it must be a paraphrase in the wiktionary. It is the correct form in greek, so it comes naturally to my hand. – anna v May 24 '18 at 12:50
  • @GiulioMuscarello thanks for the info. It is the correct form in greek, though a word coming back from Italy, the "stage" ( σκηνη is scene, latin scena https://en.wiktionary.org/wiki/scene#Etymology) – anna v May 24 '18 at 12:57
  • +1 for pointing out that the proposed mechanism is needlessly complicated. – Peter - Reinstate Monica May 24 '18 at 14:30
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    @Martigan Sure, things can sometimes move through a medium faster than light can move through that medium. Anna obviously means that nothing can move faster than the speed that light moves through a vacuum, which is such a standard concept that it hardly needs to be spelled out in pedantic detail. – David Richerby May 24 '18 at 15:40
  • Isn't this the whole "a train is traveling at the speed of light and then the engine car is detached from the rest of the train" kind of trick? Where the one object is technically moving faster than the other but they are both moving at the speed of light? – Anthony May 25 '18 at 00:07
  • "Aephraim Steinberg, a quantum optics expert at the University of Toronto, Canada, uses the analogy of a train traveling from Chicago to New York, but dropping off train cars at each station along the way, so that the center of the ever-shrinking main train moves forward at each stop; in this way, the speed of the center of the train exceeds the speed of any of the individual cars." That's what I'm thinking of. – Anthony May 25 '18 at 00:11
  • @Anthony Sorry, I do not understand what "this" you are referring to. Cerenkof radiation? – anna v May 25 '18 at 03:51