The Big Answers

trainLast week, I posed some brain teasers and a riddle about special relativity.

The brain teasers were all solved by multiple commenters; I’ll summarize their answers at the end of this post. The special relativity problem proved trickier; here it is again:

A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?

Now to the answer.

First, “At some point, the train accelerates” is ambiguous. Presumably it means that each part of the train accelerates at the same time, but of course “at the same time” means something different to a train passenger than it does to you and me (the observers stationary relative to the track).

But let’s resolve this ambiguity in the natural way by assuming that the entire train starts moving at the same time as measured by you and me. In that case, we do not see the train shrink. How could we? The front and back ends of each car have, at every moment (as measured by our watches) been moving forward at identical speeds. Given that, the distance between those front and back ends (a measured by our meter sticks) cannot change. Ditto for any couplings between the cars.

What just became of relativity? If the cars are in motion shouldn’t they appear smaller to us than to our friend Jeeter, who’s riding on the train? Sure. But that doesn’t mean we have to see Jeeter’s train car get smaller. In this case, it means that Jeeter sees his car get bigger—because by his watch, the front of his car started moving before the back did, so his train car got stretched out.

But that doesn’t mean Jeeter sees the entire train get bigger. Yes, his car got stretched when the front started moving before the back. But the car opposite him (180 degrees around the track) got shrunk when its back end started moving (according to Jeeter’s watch) before its front.

So nobody has to see the train change size and nobody has to believe the train leaves the track—which is good, because the train doesn’t leave the track.

I have ignored the fact that Jeeter is not in an inertial frame, which complicates the calculation of exactly what he experiences, but I’m nearly sure that the above captures everything important. If you want, replace the circular track with a nearly square track (with slightly rounded corners if you like) so that most of the train passengers are in inertial frames at any given moment.


Now to the brain teasers: For #1, choose a number randomly c from (say) a normal distribution on the real numbers and compare it to the number x that I’ve just revealed to you. If x is greater than c, guess that x is the larger of my numbers; if x is less than c, guess that x is the smaller of my numbers. Your chance of winning is 1 if c is between my two numbers and 1/2 otherwise; this makes your overall chance of winning greater than 1/2.

For #2, Jon Shea’s answer is perfect.

For #3, New Mexico/Colorado is one of many good answers.


93 Responses to “The Big Answers”

  1. 1 1 Al

    I love the answer to #1, it’s so simple, yet elegant.

  2. 2 2 Snorri Godhi

    First of all, I never expected to understand relativity a bit better thanks to a blog post. Thank you.

    Second, a general* question: what is the general fallacy that you [Steve] mentioned in the original post?

    Third and fourth, a couple of specific questions:
    what if observers stationary wrt the track do not see all parts of the train start moving simultaneously?
    and does Jeeter always see his car get bigger? my intuition is that, if he is closer to the back of the car, then he sees the car get smaller, because he sees the back starting to move towards him when the front is still stationary; but I might be completely off the track*.

    *no pun intended

  3. 3 3 Adam

    Which bit of the New Mexico/Colorado border is a circular arc? Or is it just because it is on the curved surface of the earth?

  4. 4 4 Bennett Haselton


    Technically I think your answer to question #1 is incorrect. That was the obvious answer that I went for as well. But the original question says:

    “Devise a strategy S such that P(A,B,S) is greater than 1/2 for all A and B.”

    Your P(A,B,S) is not greater than 1/2 for all A and B, because your probability of winning is only 1/2 if c is not between A and B. (Your *overall* probability of winning is indeed more than 1/2, but that’s not what the question asks for.)

    I believe that the answer Jeffrey posted in the blog comments is correct:

    because it satisfies the requirement that P(A,B,S) is greater than 1/2 for ALL A and B.

    (The reason the question fooled me and I wasn’t able to solve it myself, is because instinctively you think about picking a strategy that will *maximize* the probability of winning. Therefore you would think at first that it makes no sense to adopt a probabilistic strategy, because if you have two options that have different probabilities of winning, you should always pick the one that has the higher probability — or, if they’re the same, then the probability of picking one or the other doesn’t matter. BUT, if you don’t want to maximize the chance of winning, you only want to ensure that it’s greater than 1/2 for all A and B, then a probabilistic strategy can make sense — and Jeffrey’s solution works.)

  5. 5 5 John Faben

    Bennet, you pick c *after* the other person has picked A and B. Then the probability that c is between A and B is non-zero, and so the probability of winning is greater than 1/2.

  6. 6 6 Steve Landsburg

    Bennett: John Faben has this exactly right.

  7. 7 7 Bennett Haselton

    Ah OK, when you said “choose a number randomly c from (say) a normal distribution on the real numbers…”, I didn’t realize that was an actual step *in* the strategy. I thought you meant that you could choose arbitrarily from infinitely many fixed-c strategies, one for every possible value of c on the real line. (Although of course since you said to pick c “from a normal distribution”, that should have made it obvious that it was the former!) Thanks!

  8. 8 8 Philip

    Like Adam, I’d appreciate an explanation of the state border answer.

  9. 9 9 Neil

    I don’t think you can discard the non-inertial frame this easily. Is it not the case that a stationary observer who measures the length of a coach of a moving train on a straight track would find it shorter than an identical stationary coach on an adjacent track? Every physics textbook says so. So how can it be different when the stationary observer measures the coach on a side of the almost square track? Let the problematic corners be far away so that, as far as the stationary observer is concerned, he is looking at a straight track.

  10. 10 10 Steve Landsburg


    Is it not the case that a stationary observer who measures the length of a coach of a moving train on a straight track would find it shorter than an identical stationary coach on an adjacent track?

    This depends entirely on how the train got moving in the first place.

  11. 11 11 Steve Landsburg

    Neil: In case you don’t believe me, I’ve drawn a spacetime diagram that I hope makes everything clear:

  12. 12 12 Dan

    With respect to Neil’s quesiton: What if it is an arbitrarily slow acceleration up to near c? Then wouldn’t the issue of simultaneity of the start be arbitrarily small and thus a non-issue?

  13. 13 13 Steve Landsburg

    Dan: I answered on the assumption that the train “starts moving all at once”. But you ask “What if it is an arbitrarily slow acceleration….”, and now I think the answer depends on specifying exactly what you mean by “it”. If the front and back of the train car follow parallel (curved) world lines, then the distance between them, measured from the ground, still can’t change. (This fixed distance is what I *mean* by “parallel”.)

  14. 14 14 Neil

    Thanks Steve, I will study your diagram. But just so I understand what you are claiming–are you claiming that the Lorentz contraction of a moving object (relative to its length in the observer’s inertial frame) depends on how the object got moving relative to the observer (or the observer got moving relative to the object) in the first place? I have never seen that claim before. For example, the relativity paradox I am most familiar with is the “spear in the barn” paradox and I have never seen a discussion of that paradox where it matters how the spear carrier got moving in the first place.

  15. 15 15 Steve Landsburg

    Neil: I am not claiming that the Lorentz contraction depends on how the train car got moving. The person on the train car says it has length x; the person on the ground says it has length y; the ratio of x to y is determined by the Lorentz contraction. I am saying that the values of x and y, but not their ratio, is determined by how the train got moving.

  16. 16 16 CapitalistImperialistPig

    The “trick” and misleading part of Steve’s relativity puzzle is that the train is not a rigid body in the usual physical sense. In order for every part of the train to start moving simultaneously in the stationary frame it needs to be physically stretched during the acceleration phase.

    For a train accelerated uniformly in its own frame, things are more complicated – observers at different points on the train would be in different inertial frames and see flattened circular tracks, but the flattening would be different for different observers. The stationary observer would see different parts of the train start at different times – and consequent stretching.

    Steve’s diagram would be more illuminating if he showed the acceleration phase for both ends of the train as circular arcs – they would show that the acceleration rates were different for different parts of the train.

  17. 17 17 Neil

    Aha, I get it now. Thanks for the physics lesson. That is a subtlety I had not picked up before. In fact, I think I can now give another explanation.

    Let the stationary observer stand at the center of the track circle. Assume he is spinning. That might make him dizzy, but it doesn’t change his inertial frame, any more than the earth spinning on its axis changes ours. If he spins in the same direction and at the same rate as the train circles the track, he sees the train as stationary, and the cars must measure the same length as when the train is stationary and he is not spinning.

    Of course when he spins he sees the track as apparently moving, but that has no more effect than we see on the apparently moving sun.

  18. 18 18 CapitalistImperialistPig

    The relativity puzzle presented by prof Landsburg is known as Ehrenfest’s Paradox and has been the subject of considerable historical controversy. As the linked Wikipedia article suggests, the “answer” presented here is at best a gross simplification. Consideration of the puzzle shows that a true rigid body can’t exist in special relativity.

    In fact, the geometry of the moving train system is not Euclidean – the Euclidean relation between circumference and radius (C=2 pi r)fails. Observers on the train cannot synchronize their clocks. If the cars and their couplings are kept from stretching, they will fall off the tracks inward.

    Landsburg’s spacetime diagram is misleading because it shows only one dimension of space. The train’s motion in in two space dimensions, and light cones in the train frames are not Euclidean.

    Their is a real word analogue of the trains on a circular track – the bunches of particles circulating in a particle accelerator. As those bunches accelerate to high velocities, they flatten drastically as seen by outside observers. When two balls of opposite moving quarks and gluons collide, they are much flatter than any pancake (or crepe).

  19. 19 19 Steve Landsburg

    CIP: Thanks for the link, but the Wikipedia article makes no sense to me. Among other things, it says that an observer in a boxcar will measure it to be 40 inches long, and then computes the length of the train (according to that observer) by (implicitly) assuming that he will measure *all* the boxcars to be 40 inches long. But I don’t see how this can be right, because, e.g. the boxcar directly opposite him is moving relative to him at speed 2v.

  20. 20 20 Steve Landsburg

    CIP: Per my previous comment, the Wikipedia entry seems to be incorrect. As a separate observation, it also seems to be irrelevant. The story there says that “the train accelerates” and talks about what happens after the acceleration is over. But a large part of my point is that to know what the train looks like to the “stationary” observer, you’ve got to specify *how* it accelerates. The Wikipedia article, if it is to make any sense at all, must make some implicit assumption about this. And whatever that assumption is, it appears to be very odd, because it assumes, e.g. that the front of one car and the back of the next car accelerate along very different paths (they must, in order to appear to separate from each other). It seems to me that no acceleration path in the spirit of the assumption that the train accelerates “all at once” can possibly yield that outcome.

  21. 21 21 Snorri Godhi

    We still have not been told what “the same old fallacy” is: are you saving it to squeeze some other paradox out of it?

  22. 22 22 CapitalistImperialistPig

    Re SL comment 1: In the Wikipedia article, the train is measured by integrating in the comoving frames: Jeeter walks around the train with his ruler measuring until he gets back to his starting point.

  23. 23 23 Steve Landsburg

    Snorri: The same old fallacy is assuming (implicitly, without realizing it) that simultaneity is not relative. In this case, the train starts accelerating “at some point”, suggesting that the entire train starts to move simultaneously. The apparent paradox comes from not thinking carefully about the question: “simultaneously according to whom?”.

    If the “stationary” observer thinks that one entire boxcar started moving “all at once”, then the observer on the boxcar cannot think that, and in fact must think that the front of the car started moving before the back. (See my picture at ). Therefore if the train starts moving “all at once” according to the stationary observer, the cars must be stretched according to observers on the cars. If you forgot that what’s simultaneous to you and me (on the ground) can’t be simultaneous to Jeeter (on the train)—as I did when this thing first started bugging me while I was waking up one morning—-then you get real confused.

  24. 24 24 Steve Landsburg

    CIP: Thanks for the clarification re comoving frames, which now makes sense to me, though I do not think it contradicts my assertion about what Jeeter sees when he stays put on one boxcar.

  25. 25 25 Neil

    CIP: To the best of my knowledge (and googling ability), lorentz contraction has never been observed. The flattening of particles in an accelerator you mention is an inference, not an observation. I have read that there are proposed LHC experiments that could allow observation.

  26. 26 26 jpd

    time contraction is seen all the time in particle
    decay rates vs accelerated particle decay rates.
    spacial contraction has also been verified
    see eg

  27. 27 27 Neil

    jpd: A quote from this article:

    “Next we look at length contraction. While the tests of time dilation may be weak and inconclusive, there has never been any direct test of the Lorentz length contraction. The reason for this is quite simple. In order to test directly for length contraction, we need a combination of very high speeds and very precise measurements. Even as the twentieth century draws to a close, there is no way to test directly for the Lorentz length contraction. But such a test will soon be possible.”

  28. 28 28 CapitalistImperialistPig

    I really like this problem and had several happy ours struggling to understand it, but I don’t think your solution is ideal. The trouble is that it depends on a Procrustean stretching the train in an unphysical fashion to make it fit. If you accelerate the train in a conventional fashion, say by electric motors in each car, you find that the train really does shrink and fall off the track to the inside. To see this quantitatively, imagine not a circular track but a long, thin racetrack shape, with negligible ends and long sides of length L/2. Train and track are each initially of length L. After acceleration to speed v, Jeeter, the observer on the train, measures the track to have Lorentz contracted length L*sqrt(1-(v/c)^2). The part of the train on his side has length L/2, as before, but the part of the train on the opposite side is shortened to length (L/2)*sqrt(1-(2v/c)^2). If we do the arithmetic in the binomial approximation, we see that the total length of the track minus the length of train is (L/2)(v/c)^2. The stationary observer sees each half of the train shortened by (L/4)*(v/c)^2, again in the binomial approximation, so they agree that the train is shorter than the track by (L/2)(v/c)^2.
    The train **does** fall off the track to the inside. Note, by the way, that this form is a close spatial analog of the so-called twin paradox. The key point of asymmetry is the same – one observer gets accelerated and the other doesn’t.
    In addition to the liberties of using the binomial approximation, I also used Galilean addition of velocities, but that’s a higher order correction unless v is close to c.

  29. 29 29 Steve Landsburg

    CIP: Thanks for this. I still think you cannot give “the” answer to this question without specifying exactly how the train gets up to speed.

    Do you object to any part of the following argument?

    1. If I (the observer on the ground) see the train shrink, then I must, at some point, see some part of the train move faster than some other part. (Otherwise, the distances between parts would be preserved, so the length of the train couldn’t change.)

    2. If I am standing at the center of the circle, then the symmetry of the problem suggests that I should, at every moment, see every part of the train moving at the same speed as every other part.

    (Perhaps you object to 2 on the grounds that the motor has to be located at some point on the train and this breaks the symmetry. Is that your point?) (I do not believe it is critical to have the observer at the center of the circle; I am putting him there because it makes the argument simpler.)

    3. Given 1) and 2), the train cannot shrink.

  30. 30 30 Arun

    So if I accelerate a meter stick from rest so that every point on the meter stick starts moving at the same time (according to me), then the meter stick won’t appear to shrink?

  31. 31 31 Steve Landsburg

    Arun: Mostly yes, though you need to be a little more careful about the wording; it’s not just that each point starts moving at the same time but that each point accelerates at the same rate at any given moment. For the extreme case where the acceleration happens all at once, see my diagram at .

  32. 32 32 CapitalistImperialistPig

    Steve, re 1. Not so. Imagine a uniformly contracting circle. Relative motion between parts can be purely radial. In fact, though,that’s not important, because the phenomenon is only sensitive to the fine details of acceleration if you require the train to be strictly rigid. This phenomenon is not purely hypothetical – bunches of protons in a circular accelerator are packed more tightly by virtue of the Lorentz contraction of their spacing relative to the accelerator.

    2. OK, so what? Each length element of the train dL is contracted by the same factor Sqrt(1-(v/c)^2) because each has transverse velocity component v compared to you. For the observer on the ground this is just the same as in my racetrack example. For an observer on the train, the situation is more complex, but if you work through the trig and calculus, I’m confident that you will get a similar answer.

  33. 33 33 CapitalistImperialistPig

    3. Well, there is both theoretical evidence (in simple arguments like mine and more sophisticated GR arguments in radial coordinates) and experimental evidence (from the particle accelerators) that shrink it does, by a hughe factor in particle accelerators.

    I concede that your solution is not a contradiction, but it’s not physical either, because it depends sensitively on the details of the acceleration and because you have to stretch the trains by violence – a la Procrustes. As far as physics, though, you are on the wrong side of both fact and logic.

  34. 34 34 Steve Landsburg

    CIP: Let’s take a simpler problem just so I can be sure of what you are and are not saying.

    Suppose I have a metal rod that is set in motion parallel to itself by subjecting it to a force for some period of time. The force is of constant strength all along the rod. It starts acting all along the rod at a single moment (according to me) and stops acting at a single moment (according to me). As a result of this force, the rod is accelerated to a speed v. Are we agreed that the rod does not shrink (according to me)?

  35. 35 35 Steve Landsburg

    CIP: Let’s use your elongated track of length L. There are L cars on the train, each of proper length 1. The train gets up to speed v. Put alpha=sqrt[1-v^2] and beta=sqrt[1-(2v)^2].

    At a given instant, Jeeter sees A cars on his side of the track, all of length x, and B cars on the other side, all of length y. Let’s see if the train can fit on the track.

    First: The Lorentz contraction tells me that y/x = beta.

    Second: Surely A+B=L.

    Third: For the cars to fit on Jeeter’s side of the track, we must have A x = alpha L/2.

    Fourth: For the cars to fit on the other side, we must have B y = alpha L/2.

    That’s four equations in four unknowns, and there is indeed a solution.

    In that solution, x (Jeeter’s view of his car length) is equal to alpha (1+beta) /(2 beta). (Assuming I’ve done the arithmetic right.)

    Therefore the observer on the ground must see Jeeter’s car length as alpha^2 (1+beta)/(2 beta) , or approximately 1+2 v^4, or approximately 1.

    This is consistent (up to the approximation) with my story in which the observer on the ground sees Jeeter’s car length as 1 due to the uniform acceleration along the length of the train car.

  36. 36 36 CapitalistImperialistPig

    first: Only approximately, since their relative speed is actually not 2*v but 2*v/(1+v*v/c^2) by the Einstein velocity addition law.

    second: false. That is assuming that which you wish to prove. The point is they don’t fit and hence fall off.

    fourth: again, you are assuming that which you wish to prove.

    I repeat – you can only fit them all on the tracks by suitably streching the cars of the train. That’s what the combination of your second and fourth propositions do. Don’t start by assuming anything about the length of the cars in motion, just use their lengths at rest and apply Lorentz contraction to those.

  37. 37 37 Steve Landsburg

    CIP: “Just use their lengths at rest and apply Lorentz contraction to those” doesn’t help.

    The Lorentz contraction gives me the ratio between the length currently measured from the train frame and the length currently measured from the “rest” frame. It tells me nothing at all about the ratio between the length currently measured in the train frame and the length that was measured in the rest frame back before the train left that frame.

    A very large part of my point here is that the Lorentz contraction does not determine the current length of the train in any frame. By asserting that it does, it seems to me that you are—dare I say it?—assuming that which you wish to prove.

  38. 38 38 Steve Landsburg

    first: Only approximately, since their relative speed is actually not 2*v but 2*v/(1+v*v/c^2) by the Einstein velocity addition law.

    AHA! Yes! And when you make this correction, my 1+2v^4 becomes exactly equal to 1, exactly as it should. That v^4 was bothering me; thanks for catching this error.

  39. 39 39 Steve Landsburg

    CIP: It would help a lot if you clearly specified the acceleration path you have in mind for each part of the train. With that we can simply calculate whether your assumptions yield your conclusions. Without it, I don’t know where to begin.

  40. 40 40 Neil

    I must admit, I still suspect that applying special relativity, which presumes inertial frames, to a situation where there is a centripetal force may lead to logical contradictions. Garry Helzer has a paper on “Special Relativity with Acceleration” in the American Mathematical Monthly, March 2000, for those who want to leave intuitive solutions behind. He analyzes the circular train and other acceleration cases there.

    I am still confused by Steve’s belief that the stationary observer needs to know how the train accelerated to measure the moving car’s length. Suppose the observer is in a coma, and his twin marks out the length of a stationary car on the tracks. Later, the observer comes out of his coma to observe the same car moving at a constant speed along the tracks. He has no idea how the car came to be moving. If I understand Steve correctly, he is claiming that the observer will get a different length (relative to the length that his twin marked earlier on the tracks) depending on how the train accelerated. I don’t think this can be true. It implies that if another train with an identical car and the same speed passed by, the car length would be different if the train accelerated differently. Perhaps I misunderstand you, Steve.

  41. 41 41 Neil

    PS–in may example, I am thinking of a straight track where centripetal force is not an issue.

  42. 42 42 Steve Landsburg

    Neil: If I understand Steve correctly, he is claiming that the observer will get a different length (relative to the length that his twin marked earlier on the tracks) depending on how the train accelerated.

    This is exactly what I am saying. If, for example, the front of the train started accelerating five minutes before the back of the train, the train is going to be very long indeed. The forces that cause the acceleration also cause the train to stretch. You can’t know how much the train stretched without knowing what forces acted on it, which is equivalent to knowing how it accelerated.

  43. 43 43 Neil

    Yes, a sudden acceleration would stretch a non-rigid train more than a gradual acceleration, but I assume that you are speaking of something other than deformation in classical mechanics sense (which, like friction, can be assumed away.)

  44. 44 44 CapitalistImperialistPig

    SL – A very large part of my point here is that the Lorentz contraction does not determine the current length of the train in any frame.

    Well you have put your finger on the problem. I am assuming that the parts of the train behave like measuring rods in special relativity – ie, have the same length in any inertial frame they are in. I do that so I can apply Einstein’s theory of relativity, which assumes the same, and that the Lorentz contraction and rest length determine the length as measure in any inertial frame. That sort of assumption, by the way, is fundamental to any theory of measurement that I have heard of.

    You, clearly, are assuming some other theory of relativity. In any theory, however, if you let your train be arbitrarily elastic I should imagine you can make it any length you choose.

    If you want a choice of acceleration details, how about uniformly from rest in the track system for time T? Everyone will agree on when it started, and observers on the train will each get the same stopping time in their proper time (not the same as stationary observers, though)

    Suggest you read a good book on Special Relativity, preferably Taylor and Wheeler’s SpaceTime Physics and come back and try the problem again. Don’t forget to work the problems!

  45. 45 45 Steve Landsburg

    CIP: The theory of special relativity surely cannot tell me that the length of a rod is independent of any forces that have been applied.

    In this problem, as opposed to the simple problems you’re thinking of, the train car changes from one inertial frame to another. Therefore a force must have been applied and therefore the length must depend on the nature of that force.

    Obviously your assumption of “uniformly from rest in the track system at time T” is not the same as my assumption of “uniformly from rest in the station’s system at time T”. It’s not surprising, then, that we get different answers. I haven’t checked yours for correctness, but mine is correct.

    Your assertion the Lorentz contraction and rest length determine the length as measure in any inertial frame seems to be deliberately obtuse. The whole point is that the Lorentz contraction *alone* cannot determine the length as measured in any inertial frame, and the length in the “rest” frame depends on what happened during the acceleration.

  46. 46 46 Steve Landsburg

    Neil: What I am speaking of is exactly what’s illustrated in the spacetime diagram I posted.

  47. 47 47 CapitalistImperialistPig

    Well Steve, I can see that you think you solved some problem, and that you are one of those people who is never wrong. Good luck with your theory of relativity – but try not to confuse it with Einstein’s.

  48. 48 48 Neil

    I read your spacetime diagram explanation. It does not seem to say what you are saying here. There you claim the STATIONARY observer sees the train as the same length at time 1 (when the train is moving) as it was before the train started moving.

    Do we agree on the following straight track example?

    The passenger has a meter stick with him and he measures the length of the car as X before it starts moving. The stationary observer also measures the car and agrees–he marks the length of the car as the distance between two sticks beside the track. After the car starts moving, the passenger uses his (now moving) meter stick and measures the car again. He will get the same length–X. As far as he is concerned, the length of the car is the same. When he passes sticks that measured the length of the car when it was stationary, he will now find (from his moving car perspective) that the distance between the sticks has shrunk to less than X. The stationary observer, meanwhile measures the distance between the sticks as unchanged at X, but when he measures the length of the moving car from his stationary perpsective, he finds it has shrunk to less than X. Both observer measure distances in the OTHER guy’s inertial frame as shrunk, but not in their own.

    This is the explanation I have seen in countless physics texts, so we must agree on this, right?

  49. 49 49 Steve Landsburg

    After the car starts moving, the passenger uses his (now moving) meter stick and measures the car again. He will get the same length–X.

    Neil: As you can see in the diagram (which assumes that the train accelerates uniformly in the frame of the station) this is not true. He will get the length X / sqrt[1-v^2].

    It certainly *is* true that he sees the distance between the sticks as shrunk.

    This is the explanation I have seen in countless physics texts, so we must agree on this, right?

    What do those countless physics books assume about how the train accelerated from one frame to the other?

  50. 50 50 Steve Landsburg

    CIP: I’m frequently wrong, but not on those occasions, such as the present one, when I agree with Einstein. I’m sorry you weren’t able to follow the simple algebra.

  51. 51 51 Neil


    Special relativity is only about inertial frames, so the textbooks are silent on acceleration. The “moving” inertial frame has always been moving. But the point of special relativity, is that the motion is relative, so from the moving frames perspective it is the “stationary” frame that is moving. If one inertial frame is as good as another and motion is relative, how can we say which one has been accelerated?

    I think acceleration poses problems for special relativity, such as in the twin paradox where it is required for the travelling twin to switch inertial frames in order to reunite with the stay at home twin.

  52. 52 52 Neil

    PS. The passenger has to measure the coach length the same when it is moving using his meter stick because his meter stick is moving with the coach. Your analysis implies that motion can be determined absolutely and the point of relativity is that it cannot be. Ignoring accleration, which SR ignores, the passenger in the coach cannot tell whether it he who is moving or the other guy. Suppose the other guy is in another coach on an adjoining track. We’ve all had that experience if the train starts smoothly in a station.

  53. 53 53 Scott


    I’d be curious to know which texts you are reading that imply that the mode of acceleration and/or the forces applied to the body have any relevance to its perceived length. I’m not claiming to be an expert on relativity, nor am I claiming I can make sense of it at all, but I did take a course on it with a world renowned physicist at Stanford. I asked him that exact question, because to my “engineering” mind it must matter, and he seemed to be quite certain that it was not relevant, and it wasn’t part of the calculations that we used.

  54. 54 54 Steve Landsburg

    Neil: If one inertial frame is as good as another and motion is relative, how can we say which one has been accelerated?

    The problem *specifies* that the train accelerates.

    Your analysis implies that motion can be determined absolutely

    No it doesn’t. Why do you think it does?

  55. 55 55 Steve Landsburg

    Scott: Try drawing the spacetime diagram, first on the assumption that the front and back of the train car start moving at the same time as measured from the station and then assuming they started moving at the same time as measured by the train (after it’s gotten up to speed). You’ll see that the perceived lengths have to be different. And indeed this is obvious anyway: If the front of the car starts moving before the back does, then the train has to stretch.

  56. 56 56 Neil

    Steve:”Why do you think it does?”

    Because you claim that the passenger measures the length of his coach as different when it is moving (longer–sqrt(4/3)) than when it was stopped (1). This implies he can determine he is now moving.

    BTW, I long ago decided that if you want to do relativity with accelerated frames, you better figure out how to do it with GR not SR. But GR is not user friendly (I certainly can’t solve it), so people try to apply SR to non-inertial problems. That is the point of my one-G twin paradox problem where I try to, at least, make the non-inertial frames equivalent by using Einstein’s equivalence principle

    You really should check out the Helzer paper–it is all about using SR with acceleration.

  57. 57 57 Steve Landsburg

    Neil: Because you claim that the passenger measures the length of his coach as different when it is moving (longer–sqrt(4/3)) than when it was stopped (1). This implies he can determine he is now moving.

    There is nothing surprising, or anti-relativistic, about being able to tell that you are in motion relative to the frame you were occupying a few minutes ago. When you get in your car, press the accelerator and get up to 60 miles per hour, you are perfectly aware that your state of motion has changed relative to what it was before. Once you’re in your new inertial frame, no experiment can distinguish that frame from any other inertial frame. That’s not the same thing as saying you have to forget how you accelerated from one to the other.

  58. 58 58 Neil

    Wrong. Suppose the passenger measures the length of the car with his meter stick before it starts moving and finds that it is X. Someone knocks him out, and the car starts moving while he is out. He now awakes and wonders if it is he or the other guy that is moving. If he takes his meter stick and measures the car again and finds the length is something other than X, he knows he is moving. That is a violation of SR.

  59. 59 59 Steve Landsburg

    Neil: If he takes his meter stick and measures the car again and finds the length is something other than X, he knows he is moving.

    No. There is no way he can tell he is moving by measuring the length of his car.

  60. 60 60 Steve Landsburg

    Neil: Let me try walking through this:

    You are sitting on a train car, motionless with respect to me. We both measure the train car and agree that it has length 1.

    Now you start to move, getting up to velocity v (relative to me). You measure your train car to have length x and I measure it to have length y. Relativity tells us that y must be less than x by a factor of Sqrt[1-v^2].

    (At the same time, if I am sitting in a train car, I will measure my car to have length z and you will measure yours to have length w. Relativity tells us that w must be less than z by the same ratio—but that won’t be relevant here.)

    Relativity tells us the *ratio* of your measurement to my measurement. It does NOT tell us what your measurement is or what my measurement is. For that we need additional assumptions.

    So let’s make an additional assumption (not the only one possible, but the one I’m making in this example). I assume that *as measured by me*, the front and back of your car both instantly jump to speed v at a single instant. (A more physically reasonable assumption would have them accelerate quickly buy continuously to v.) Then *as measured by me*, the length of your car cannot change, for the simple reason that the front is moving away from me at exactly the same speed that the back is moving away from me, so the distance cannot change.

    Relativity now tells me that if I measure your length to be 1, then you’ll measure it to be something longer, specifically 1/Sqrt[1-v^2].

    If you are sitting at, say, the front of your train, you are permitted to say that you are stationary. But then here’s how the world looks to you: First, by your clocks, the back of the train started moving later than the front did, but of course you’re thinking of the front as stationary, so you’re going to describe that by saying the back of your train moved backward for a while, while the front remained stationary. In other words, something stretched your train!

    I, on the ground, say that your train accelerated. You, on the train, say that a force pulled the back of your train backward while the front remained stationary. You say your train got longer; I say it didn’t. The ratio of the measured lengths is exactly what relativity says it has to be.

    If your train started moving in some different way, we’d get some different story, and then we each have some completely different measurements of your train length. Relativity would still pin down the ratio, but without an explicit story about how you got moving, it cannot pin down either length measurement.

    (Added afterward: Of course there’s a limit to how far a train car can stretch. If it’s stretched far enough, it will snap—according to observers both on the train and on the ground.)

  61. 61 61 Neil

    Steve. Without meaning to resort to authority (but I can think of none better), Einstein expressly assumed rigid bodies in his 1905 paper. To quote: “Let there be given a stationary rigid rod…”.

    All of SR is based on the kinematics of rigid bodies. You could say that Einstein replaced the rigid aether with rigid bodies. And yes, I know that rigidity is problematic under acceleration, but that is another reason why using SR is problematic with acceleration.

    If X rigid meter sticks are laid out along the floor of the coach end to end before it starts moving, when the passenger comes to (he knows nothing about how he came to be on this moving train, he remembers vaguely only that something weird happened in Las Vegas). He now measures the length of the car at X meters looking at the meter sticks on the floor. Perhaps he thinks he is being tricked by the meter sticks already laid out, so he takes out a meter stick he put in his pocket in Las Vegas before he even got on the train, and sure enough, it is exactly the same length as the meter sticks on the floor of the coach. QED

  62. 62 62 Neil

    PS There is a good reason why Einstein considered only rigid bodies in his 1905 paper. Elastic bodies would have undermined the very point he was trying to make. What would have been surprising about elastic bodies changing length?

  63. 63 63 CapitalistImperialistPig

    Try your simultaneous acceleration logic on a train on a straight track. You find once again that …The front and back ends of each car have, at every moment (as measured by our watches) been moving forward at identical speeds. Given that, the distance between those front and back ends (a measured by our meter sticks) cannot change. Ditto for any couplings between the cars. So no Lorentz contraction happens, according to you.

    Now if you are just implying that some special forces stretched the cars, that’s metallurgy, not relativity. If not you seem to be denying the fundamental principle of relativity, the invariance of the space time interval.

    You can insult my grasp of algebra if you like, but this idea you are pushing, that lengths depend on acceleration history, is not the special theory of relativity (barring the kind of extreme accelerations that do metallurgical violence), it’s an invention of your own.

  64. 64 64 Neil


    Here is Einstein’s 1905 paper (in translation).

    Please read–its rigid bodies, rigid bodies,….

  65. 65 65 Steve Landsburg

    CIP: Of *course* a Lorentz contraction happens. The train stretches (in its own frame) so that the passenger on the train car measures the length of that car to be longer than the passenger on the ground measures it to be. That’s a Lorentz contraction.

    The diagram I’ve pointed you to illustrates this very clearly. What do you not understand in the diagram?

  66. 66 66 Steve Landsburg

    Neil: We are allowed to apply Einstein’s theory to examples that are not in his original paper. It wouldn’t be a very interesting theory if we weren’t.

  67. 67 67 Steve Landsburg

    If not you seem to be denying the fundamental principle of relativity, the invariance of the space time interval.

    CIP: My entire example is catpured in a standard spacetime diagram, where of course intervals are invariant. Have you looked at that diagram?

  68. 68 68 Steve Landsburg

    Neil: What would have been surprising about elastic bodies changing length? What was surprising was that the length of a body can be different depending on who’s measuring it. That’s equally surprising whether the body is rigid or elastic.

  69. 69 69 Steve Landsburg

    CIP: You invite me to “try my … logic on a train on a straight track”. This is of course exactly what I did in the diagram I’ve posted, so I infer that you either haven’t bothered to look at it or don’t understand it. The logic you quote: The front and back ends of each car have, at every moment (as measured by our watches) been moving forward at identical speeds. Given that, the distance between those front and back ends (a measured by our meter sticks) cannot change. Ditto for any couplings between the cars. of course applies.

    Do you see an error in that logic? We have two points on the train, a meter apart. The two points begin moving in such a way that at every moment (according to the “stationary”) observer, their speeds are the same. How can that observer possibly think the distance between them has changed?

    If your complaint were that you don’t like to think about that kind of acceleration, that would be your privilege. But you seem to be saying that GIVEN such an acceleration, the two points can somehow magically come closer together (as measured by the observer who sees them always traveling forward at the same velocity). That’s just … nuts.

  70. 70 70 Neil

    I think this is a test to see how long until one of us says, lets agree to disagree.

    My last word–CIP is right, the acceleration stretching of the train is metallurgy–classical physics. It can be made as small as desired by letting the coach be as rigid as technology allows and the acceleration as small as time available for the train to get to speed. It is not part of relativity.

  71. 71 71 CapitalistImperialistPig

    OK, let me eat some (not all) of my words. Call it a limited, modified, partial hang out) Consider a train of cars of length l moving at speed v along a track. At some time t=0 in the track frame each car begins accelerating uniformly.

    Meanwhile, back on the train, observers stationed on the train notice something funny. Do to the relativity of simultaneity, the front of the car in front of him began accelerating earlier than the back by an amount of time dt = l*v*gamma, where gamma = 1/Sqrt(1-(v/c)^2). Also, the back of the car behind him started dt later. This is what Steve Landsberg said, right? Right.

    If the train is circular, and you go around the train, each observer notices that he started later than the guy in front of him and earlier than the guy behind him – very odd, but a manifestation of the fact that these observers can’t synchronize their clocks.

    So how do the cars feel about being stretched like this? Well they resist fiercely and then they either stretch, break, or move inward toward the center of the track. Steve focussed on the details of the acceleration, and decided that the train didn’t shrink. I focussed on the physics of railroad cars being stretched, and concluded that it must.

    It turns out that this is another famous paradox in relativity called Bell’s paradox. The upshot is that uniform acceleration of a rigid body can’t occur in special relativity. An accelerated body either deforms or accelerates nonuniformly.

    Links to more on Bell’s paradox at my site.

  72. 72 72 Jon Shea

    Neil: Steve posted about papers the showed that honest truthseakers can not simply agree to disagree.

    The train on the straight track is a problem identical to Bell’s [Spaceship] Paradox:'s_spaceship_paradox (not to be confused with Bell’s Inequality, which is discussed in The Big Question). In Bell’s Paradox the “front” and “back” of the train are each replaced by a spaceship. The front and back spaceship are tied together with a string, and then they both begin to accelerate at the same rate. Does the string break?

    If you’re not enjoying these questions, then it might make you feel better to know that a lot of people who are smarter than us (including “a ‘clear consensus’ of the CERN theory division”) got this question wrong.

  73. 73 73 Steve Landsburg

    Neil: You are certainly wrong when you say the effect can be minimized by limiting the time of the acceleration, since in my picture the acceleration is instantaneous. If you make the coach sufficiently rigid, the coach will snap.

    Have you taken the trouble to study the diagram that you’re so keen to refute?

    PS: This is certain more relativity than metallurgy, since the stretching is caused entirely by the fact that the front and back of the cars start moving at different times in the train frame, which in turn is entirely a manifestation of the relativity of simultaneity.

  74. 74 74 Steve Landsburg

    Jon Shea: In my picture, the acceleration only lasts an instant, and the cars do or do not snap depending on how elastic they are. If you maintain the acceleration over time, then of course they eventually snap.

    Thanks for the pointer to Bell’s Spaceship Paradox, which I hadn’t known about; I agree that this is exactly what we’ve been arguing about. I’m glad to know that I’m on John Bell’s side, as well as Einstein’s.

  75. 75 75 Neil

    Sure I have, but as i said, you are applying a theory based on rigid objects and uniform motion to elastic objects and accelerated motion. Why do you think Einstein spent 13 years on GR if he thought you could apply SR to accelerated frames? You can’t. You get contradictions, not paradoxes.

  76. 76 76 Neil

    PS If I gave you a proof that 2+2=5, and then when you challenged me, I tell you one I am making an assumption in contradiction of one of Peano’s axioms, you wouldn’t find my proof very convincing. I don’t find yours convincing.

  77. 77 77 Steve Landsburg

    Neil: Now you’re just being silly. The questions *asks* what happens when a train accelerates. How could you possibly give an answer that avoides acceleration?

    Beyond that, special relativity deals with accelerated motions perfectly well. That’s why it enables us to resolve things like the “twin paradox”. It’s simply a matter of drawing curves instead of straight lines.

    There were no assumptions made here beyond standard relativity and what’s given in the problem. I suspect you of not having looked at the picture.

    (And, as I’ve pointed out sixtyfour bajillion times, you don’t have to think about accelerated frames: replace the track with a square, or better yet (per CIP’s idea) an elongated racetrack. )

    PS: You can’t prove 2+2=5 without making some assumption that’s *false*. That would kind of be a key difference here. The conclusion I’m getting is true. The train stretches (or snaps if it can’t).

  78. 78 78 Neil

    IMO, and the opinion of others as well, the twin paradox CANNOT be resolved in SR. Like your “proof”, all of action occurs when the travelling twin turns around (infinite acceleration point) which is outside SR. If you contradict the axioms of a logical system, you cannot then turn around and use that system to deduce a result and say it is contradiction free.

  79. 79 79 Steve Landsburg

    Neil: If you think that the twin paradox cannot be resolved in SR, then you probably don’t understand relativity well enough to be thinking about the train problem.

  80. 80 80 Neil

    Thanks for your opinion, Steve. Not the argument of a truth seeker.

  81. 81 81 Steve Landsburg

    Neil: If you are interested in truthseeking, try asking/answering the following three questions:

    A train car one mile long sits on a track with its front end at milepost 1 and its back end at milepost 0. At 12noon exactly by the stationmaster’s watch, the entire train starts moving forward at a speed of 1 mile per hour (as measured by the stationmaster), and continues forever at that speed.

    1) Where (according to the stationmaster) is the front end of the train at 1PM?
    2) Where (according to the stationmaster) is the back end of the train at 1PM?
    3) What (according to the stationmaster) is the length of the train at 1PM? (Hint: Subtract the answer to 2) from the answer to 1).)
    4) How much has the train shrunk according to the stationmaster?

    These are not hard questions. I’d think that a refusal to confront them was symptomatic of a serious lack of truthseeking.

  82. 82 82 Neil

    I won’t answer that question because it cannot be answered within the logical structure of SR. The properly framed question is the following one:

    At noon a stationmaster is standing by the locomotive of a stationary train. He has previously measured its length at X. An identical train passes the stationary train. At exactly noon, the front of the locomotive of the moving train (it always has been moving–no just starting) is exactly at the front of the stationary locomotive next to the stationmaster. He starts clocking and measuring.

    Now the train passes, blah, blah, and the stationmaster measures the length of the moving train. You can do it with light signals or meter sticks. it is in every intro physics text. It is pointless to do the actual calculation because at 1 mph, the length of the moving train is .999999….times X using the Lorentz tranformation.

    In this properly framed question, I can also calculate how the engineer of the moving train measures the length of the stationary train and he finds it is .99999….times X, which is what he measures to be the length of his moving train. I can also tell you that the stationmaster can measure the passage of time on the moving train by looking at a light clock on board. He measures that time on the moving train is passing more slowly than his very accurate watch says. Not much more slowly at this speed–for every second on the moving train, his watch shows .99999…seconds. The engineer on the moving train checks how fast time is passing according to the stationmaster’s clock. Son of a gun, it is running slow relative to his timepiece—.999….seconds for each of his seconds.

    This is the content of SR. Now if you want, I can calculate the corresponding lengths and times assuming the moving train is 1/2 the speed of light, where it would be more interesting, but you know the formula as well as I do.

  83. 83 83 Steve Landsburg

    I won’t answer that question because it cannot be answered within the logical structure of SR.

    You MUST be joking!!!!!!!!!!!!

  84. 84 84 Steve Landsburg

    Neil—Look. You are *incredibly* confused. You have it in your mind that SR doesn’t tell you how to make measurements from an accelerated frame. Fine. From there you’ve jumped to the nutso belief that SR doesn’t allow you to observe anything that’s accelerated. This is a totally different statement and until now, I’ve never met anyone confused enough to believe it.

    You might meditate on this: The whole *point* of SR was to firm up the foundations of electrodynamics, a subject in which particles are SUBJECT TO FORCES and therefore ACCELERATE. You might meditate on the fact that there are electromagnetic forces, and hence acceleration IN EINSTEIN’S PAPER.

    If SR forbade us to ever think about acceleration, it would be a really really really stupid theory. Fortunately, it doesn’t and it’s not.

    If your position is “acceleration is impossible to think about”, then you’ve graduated from confusion to crankdom. (Another symptom of crankdom is that you *still* haven’t told me what you think is wrong with the diagram I provided—-you’re rejecting the conclusion even though you can’t find a mistake in the argument.)

    I am absolutely through with this discussion.

  85. 85 85 Neil

    If it is, Conan needn’t worry about competition from me.

    SR is an axiomatic logical structure. You cannot deduce true conclusions a logical structure if you contradict one of its a crucial axioms. The axioms of SR include inertial frames and rigid rods. You are askingme deduce a conclusion using SR (Lorentz diagram) while contradicting one of SR’s axioms (letting a train change its inertial frame). I’m not saying you can’t deduce true propositions about a train that changes its inertial frame, just that you can’t do it using only the axioms of SR. You need something more.

    For example, ninety percent of the people who claim to demonstrate the consistency of the twin paradox within SR, actually invoke GR at the turn around point. They say, time slows at the turn around because according to GR time slows for bodies undergoing acceleration. That is not proving the twin paradox within SR. I don’t think it proves it using GR either, but that is another issue.

  86. 86 86 Neil

    I didn’t say acceleration is impossible to think about. That is what GR is about.

  87. 87 87 Neil

    To the best of my knowledge, since its been awhile since I read it in detail, the only place in Einstein’s paper where a force acts to accelerate an object is in part 10 on the slowly accelerated electron. I doubt that accelerating a point particle tells us much about accelerating mile long trains.

    I realize that people do try to analyze acceleration in SR, since GR is intractable. I’m sure crack physicists know a lot of tricks to do this with some confidence (additional things that have to be included above and beyond the Lorentz diagram). I don’t know what happens if you accelerate a train at an instant. Either do you. Its never been done.

  88. 88 88 CapitalistImperialistPig

    Neil – You are wrong in thinking that (a) SR can’t handle accelerations. Einstein was very explicit on that point, and (b)that doing this kind of problem in GR is hard.

    Steve – You are being confusing by posing this problem in terms of instantaneous accelerations, which are needlessly confusing. and especially by posing this problem in terms of a train, which doesn’t normally stretch the way you have it doing. You should replace the front and back of your train by individual spaceships connected by a string the way Bell does. The point, as you say, is that accelerations, gradual or instantaneous, that begin simultaneously in one frame aren’t simultaneous in other frames.

    The reason you get different answers from those physicists get is that your assumption of simultaneous acceleration is unnatural to physicists – the railroad cars we see in ordinary life are rigid objects, not stretchy ones. Your answer to the puzzle, though technically correct, depends on some unphysical slight of hand.

    Incidentally, you have a tendency to do the same sort of thing in economics – start with an unnatural assumption (e.g. future consumption should be valued the same as present consumption*) and reach some conclusion that you like but that most people find absurd.

  89. 89 89 Steve Landsburg

    CIP: The original problem says that “at some point, the train accelerates”. As with all SR “paradoxes”, the trick to seeing through it is to realize that “at some point (in time)” is ambiguous. It still seems to me that the most natural interpretation of that simple everyday language is: “At some point in time (as measured from the station) the entire train accelerates”, but I don’t think it’s terribly unreasonable for you to disagree. In any event, though, it seems to me that the right standard for resolving this paradox is to understand what happens under multiple interpretations of “at some point”.

    Incidentally, I do want to go way back to your suggestion that this has something to do with the Ehrenfest Paradox; despite the superficial similarity, I think it’s quite different. In this problem, we can really do all the interesting analysis from inertial frames, by assuming a square track with rounded corners (as I did) or an elongated racetrack (as you did). The Ehrenfest Paradox concerns the appearance of the *radius* from the point of view of a train passenger, and for this, I think you really do need to consider non-inertial frames. (Though maybe you see a way to do this that I don’t).

    As for this: future consumption should be valued the same as present consumption — I haven’t the foggiest idea what that statement even *means*, and if I ever said any such thing (which I doubt) I bet there was some important context you’ve omitted.

  90. 90 90 Neil

    CIP: I reread (the kinematical part) of “On the Electrodynamics…” last night. I was once again impressed with its admirable clarity.

    You are right, in fact E did consider an accelerated body, but he didn’t describe it as such. He considered an object with “constnat motion” following a circular path in space (not spacetime–Minkowski hadn’t chimed in yet) describing the object as having a constant “velocity”–I assume he meant speed. In fact, he considered a form of the train paradox–by asking what an observer at the pole would conclude about the a clock on the equator. He concluded, based on SR, that the pole observer would observe the equator clock as running slow relative to the clock at the pole. He hadn’t invented GR yet, so he didn’t address the fact that the clock on the equator is undergoing less gravitational acceleration than a clock at the pole because of centrifugal force, which under GR would cause the clock at the equator to run faster. Like everyone since, E wanted to apply a theory based on interial frames to non-inertial frames.

    (Interestingly, in this problem, E did the same thing as the train paradox solvers try to do–consider a polygon, looking only at the sides and ignoring the vertexes, and then lets the polygon deform smoothly into a circle. He adds an assumption to his two principles at this point–”If we assume that the result proved for a polygonal line is also valid for a continuously curved line…” Tricky, tricky. Mathematically, this seems wrong to me. In a circle, every point becomes a vertex, so to speak.)

    Nowhere does he do a Steve thought experiment where he says “what happens if we start a stationary earth spinning”.

    Just the opinion of a “crank”.

  91. 91 91 CapitalistImperialistPig

    SL – So here’s the right argument: First, economics teaches us that everything should be taxed at the same rate to avoid unnecessary distortions. Second, it follows that current and future scones should be taxed at the same rate. Third, therefore there should be no tax on interest.

    OK Steve, so where is my misinterpretation or missing context. It sure seems to me like you are saying that future scones (birds in the bush) should be valued the same as present scones (birds in the hand). On the other hand, your argument also depends on future money be valued much less than present money (100 % daily interest!).

  92. 92 92 Steve Landsburg

    CIP: It sure seems to me like you are saying that future scones (birds in the bush) should be valued the same as present scones (birds in the hand).

    I see that your reading comprehension problems continue.

  93. 93 93 Steve Landsburg

    CIP: More precisely—as is your wont, you’ve responded to what you think someone might have said instead of to what was actually said. What I said was that current and future scones should be taxed at the same rate. (I realize this might have been difficult for you to extract; the secret is to read the words that say “current and future scones should be taxed at the same rate”.) This in turn is not an assumption (as you assert) but the conclusion of a standard textbook argument (as I’ve already explained, but then there’s that reading comprehension problem again).

    So when you ask “where is my misinterpretation”, there are multiple answers. One of those answers is: You just made up that stuff about value. Another is: You wrote “assumption” when the correct word was “conclusion”. You do this a lot.

    (Incidentally, the conclusion is a special case of the more general conclusion that the ratio of the prices of two goods should be equal to the ratio of their values at the margin. No assumption is made about what that ratio is.)

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