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Time and Space Continuums

  1. Sep 18, 2009 #1
    I have a degree in economics and have very little background in science or physics but I have been trying to wrap my head around some thoughts tonight. I believe these ideas teeter on the boundary between philosophy and science, but I am hoping there are some widely accepted theories out there to explain.

    The basic idea is that time and space are theoretically continuous but there seems to be a point where this breaks down. With respect to space and travel, how can an object ever arrive at it's destination when always has to go half way before it can get there? I can stand at a fixed point and watch a ball roll by me, so at what point did the continuity argument fail? If space were truly continuous, every object would have a fixed point in the universe because there is always a finer measurement of distance, so you are essentially stuck where you are. Since that is clearly not the case, there has to be a specific point where continuity of space becomes absolutely irrelevant and the object is able move along and through a fixed point. I realize that this argument might not even make a lot of sense to anyone who reads it and has little application to the real world, but I find it intriguing.

    The question is essentially the same with respect to time. There is always a finer measurement of time because it is supposedly continuous, but somehow the seconds come and go. At a certain point, and it seems that it would have to be a very specific point, the ability of what we call time to move forward becomes more powerful then the idea of continuity. What we call a second can come and go, but there are infinite levels of measurement ticking away that became irrelevant at some point, but how can they possibly be disregarded if you want to consider continuity in its purest and most infinite sense. It seems that at some point time vanishes and is lost. I have never read about black holes and the theories behind them but maybe that is what I am getting at. Only a certain amount of space and time are perceivable and or able to exist in our universe, and all that is lost becomes a vacuum of nothingness in a place far away.

    In conclusion, the general idea is that it seems to me like the powers that be in our perception of the world around us are "stronger" than the idea of continuity, even though we are able to conceptualize the infinite nature of a truly continuous existence of time and space.
     
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  3. Sep 18, 2009 #2
    This is best dealt with by mathematical analysis. The problems you dscribe are of the type of Zeno's paradox. This is a copy of a reply I made in a similar thread.


    ------I have copied this extract in full although it may not all be relevant.


    From Hans Reichenbach. The Direction of Time. Pages 5-6. Published in 1956 but this edition in 1971.

    Zeno’s paradoxes of motion have often been discussed. He argues that if motion is travel from one point to another, a flying arrow cannot move as long as it is at exactly one point. But how then does it get to the next point? Does it jump through a timeless interval? Obviously not. Therefore motion is impossible. Or consider a race between Achilles and a tortoise, in which the tortoise is given a head strart. First Achilles has to reach the point where the tortoise started; but by then, the tortoise has moved to a farther point. Then Achilles has to reach that other point, by which time the tortoise again has reached a farther point; and so on, ad infinitum.Achilles would have to traverse an infinite number of non zero distances before he could catch up with the tortoise; this he cannot do, and therefore he cannot overtake the tortoise.

    Concerning the arrow paradox, we answer today that the rest at one point and motion at one point can be distinguished. “Motion” is defined, more precisely speaking, as “travel from one point to another in a finite nonvanishing stretch of time”; likewise, “rest” is defined as “absence of travel from one point to another in a finite nonvanishing stretch of time”. The term “rest at one point at one moment” is not defined by the preceding definitions. In order to define it, we define “velocity” by a limiting process of the kind used for a differential quotient; then “rest at one point” is defined as the value zero of the velocity. This logical procedure leads to the conclusion that the flying arrow, at each point, possesses a velocity greater than zero and therefore is not at rest. Furthermore it is not permissible to ask how the arrow can get to the next point., because in a continuum there is no next point. Whereas for evry integer there exists a next integer, it is different with a continuum of points; between any two points there is another point. Concerning the other paradox, we argue that Achilles can catch up with the tortoise because an infinite number of nonvanishing distances converging to zero can have a finite sum and can be traversed in a finite time.

    These answers, in order to be given in all detail, require a theory of infinity and of limiting processes which was not elaborated until the nineteenth century. In the history of logic and mathematics, therefore, Zeno’s paradoxes occupy an important place; they have drawn attention to the fact that the logical theory of the ordered totality of points on a line—the continuum—cannot be given unless the assumption of certain simple regularities displayed by the series of integers is abandoned. In the course of such investigations, mathematicians have discovered that the concept of infinity is capable of a logically consistent treatment, that the infinity of points on a line differs from that of the integers, and that Zeno’s paradoxes are not restricted to temporal flow, since they can likewise be formulated and solved for a purely spatial continuum.

    As a footnote he adds. For a modern treatment of Zeno’s paradoxes , See Bertrand Russel, Our Knowledge of the External World.

    Matheinste.
     
  4. Sep 18, 2009 #3

    A.T.

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    You can divide a finite distance into an infinte amount of parts. But this division is purelly mathematical and has no physical consequences. Nothing special happens when you pass the half distances.
     
  5. Sep 18, 2009 #4

    dx

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    This 'paradox' has been resolved a long time ago. The confusion arises basically because our intuition is a little lacking when it comes to infinite series.

    First, let's understand what the problem is exactly. To be concrete, we will talk about a rigid rod of finite length. There are always two aspects to problems in physics. One is the physical situation itself, as it is experienced, and the other is the description of that situation in terms of ideas, usually mathematical. The physical rod has no problem being of finite length, so there is no problem there. But in our description of it in a continuous space (E3), there is the possibility of dividing up the length into an infinite number of finite pieces. Here is the problem. Our imprecise intuition leads us to expect that the sum of an infinite number of pieces must be infinite. This is not the case. What has been found is that adding up an infinite number of finite terms can lead to a finite value. For example, 1/2 + 1/4 + 1/8 + .... = 1. You can check this on your calculator. All terms of this series are positive finite numbers, but no matter how many terms of this series you add up, you will not be able to make the sum greater than 1. What actually happens is that the value keeps getting closer and closer to 1.
     
  6. Sep 18, 2009 #5
    The universe does what seems to be impossible on a regular basis, computationally speaking. There are integrals and differential equations used all over in physics which cannot be computed exactly. Physics makes extensive use of trigonometric and exponential functions, too, which unavoidably lead to the use of irrational numbers, for which there are no general algorithms to do arithmetic or even compare (contrast with rationals, which you learned about in elementary school).

    The thing to realize is that we don't really know how the universe actually works "on the metal". Whether or not space is continuous, we don't really know. Our theories tend to say that space is continuous, but because we are only concerned up to a bounded limit of accuracy, it's impossible to know if this is certain.
     
  7. Sep 22, 2009 #6
    I used to think about this when I was a kid. Here's what I came up with. When you cover half the distance, the next half is half as long so it keeps cancelling out until you get to a non-dividable distance. And now I can add a bit. It keeps doing this until you reach the Planck length. Then if you move froward, you're there because you can't divide a Planck length. I don't know much about the concept of a Planck length, but it makes sense.
     
  8. Sep 23, 2009 #7

    Saw

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    This is one of the typical approaches to the paradox. You can see it at the end of the Wikipedia page on “Zeno's paradox solutions”. Its conclusion is: the paradox proves that space and time “are” discrete. I don’t think it is correct.

    Planck length (PL) is 1.616252(81)×10−35 meters. Planck time (PT) is the time required for light to travel, in a vacuum, a distance of 1 Planck length. Suppose Achilles travels at 0.5 c wrt the ground, that is to say, 1 PL every 2 PTs, whereas the tortoise travels at 0.25 c also wrt the ground, that is to say, 1 PL every 4 PTs. While Achilles has traversed 1 PL in 2 PTs, how long has the tortoise travelled in the same time, half-PL? But wasn’t the PL indivisible?
     
  9. Sep 23, 2009 #8
    The modern mathematical resoltion of this type of paradox relies upon the nature of the real number line which is continuous either by construct or by definition, and the correctly defined and applied ideas of rest and motion. To apply the same treatment to space assumes the same continuity of physical space. When using a discrete model of sapce and time then the mathematical resolutions no longer apply.

    However we know from reality that the hare catches the tortoise and the arrow reaches its destination. That seems to me to be a quite convincing proof that this sort of paradox is inccorrecly posed and/or makes invalid assumptions.

    Matheinste.
     
  10. Sep 25, 2009 #9

    Saw

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    Hi Mathe. I was not disagreeing with the standard resolution of the paradox: in modern mathematics, the sum of a geometrical series with infinite terms is, in the limit, a finite amount. I was only objecting the “other” solution that deems the problem solved on the grounds that space and time would not be infinitely divisible but discrete.

    Incidentally, I’ll tell you that the standard resolution, without further development, can leave you as puzzled as the problem itself.

    If, instead of explaining the problem with more literature, you present it in mathematical terms, it may look like this: “Achilles travels at v = 1 m/s, the tortoise at v = 0.5 m/s; the tortoise has a head start = 1 m; while Achilles has displaced to x = 1 m, the tortoise will have reached x = 1+ 1/2 m; when Achilles reaches x = 1 + 1/2 m, the tortoise will be at x = 1 + 1/2 + 1/4 m and so on ad infinitum; so we have two infinite series of which the second seems to have a permanent advantage, albeit an ever-decreasing one; despite that, in the limit, the two series converge, i.e., their sum is a finite number, x = 2; hence Achilles catches the tortoise at position x = 2”. Do you see any difference between this wording of the problem and the usual wording of the solution?

    The structure of a paradox as a reasoning tool is: “You say A, but if you take it to its more extreme consequences, then you get result B, which is absurd and against experience”. Given this, there are three typical reactions:

    1. Normally speaking, with some honourable exceptions, people do not infer this: “oh, then we have to believe that B is possible! It’s against our intuition, but that’s the way the universe is…”

    2. The intended inference is rather: “so there was something wrong in the way we expressed A, some flaw in our concepts; they are phrased in a manner that seems to suggest B; hence we should refine them in order to avoid this undesirable consequence”.

    3. There’s a mid-way reaction, however. A very practical one: “Well, we will not redefine our concepts from scratch, because that would be too cumbersome. What we’ll do instead is to leave them as they are, but introduce some corrections, some gadget, some counter-weight, which is applicable under certain conditions.” That’s what the standard explanation does: the fact that an infinite number of terms add up to a finite number is as puzzling as Zeno’s paradox (in fact, that is Zeno’s paradox!), but if you define adequately the conditions for that, in agreement with experience, and use this new “freak concept” judiciously, the trick works… as long as it does!

    If you ask me about a more ambitious explanation in the line of (2), I don’t have it. I’ve only thought of one that develops (3) a little. It even has some link with SR, so it would not be off-forum. I’ll try to draft it. In the end, it's what AT said,

    but with more words.
     
  11. Sep 25, 2009 #10

    Saw

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    Well, here is the announced text. Please criticise if there’s anything wrong or obscure.

    We have to rely on the distinction between accelerated motion and inertial (unimpeded, free-of-interaction) motion and the distinction between reality and measurement.

    Imagine Achilles has to overcome obstacles as he makes progress. In a thought experiment, certainly, you can imagine that the number of obstacles is infinite. Like Sisyphus, Achilles might be faced with this torture: he has to open one door to traverse the first 1m; then another door pops up in the path for the next 1/2 m; another in the 1/4 m; and so on. Should that display be feasible, Achillles would never reach the tortoise: he would get exhausted and die before he could complete the deed. But it is not: somewhere you must reach a physical limit where you cannot make the doors any thinner. So the answer is, yes, Achilles catches the tortoise, but the solution rests on a finite number of obstacles.

    In purely inertial motion, instead, there are no physical obstacles. What happens then is that we create imaginary obstacles. In order to measure the performance of Achilles and the tortoise, we compare their respective displacements (changes of position) against a unit of change, i.e., the displacement of a ticker inside a clock. Let us imagine that we place one light clock at x= 1 m, another at x =1/2 m, another at 1/4 m and so on. The clocks, to fit in and measure time adequately, would have to be progressively thinner and (since the speed of light is always the same) the room for the ticker would have to be made smaller and smaller. Presumably, we would reach a physical limit as well for the size of the clocks. In any case, we must not forget what we are doing: all this stuff about the clocks is an analogy that tries to mirror what is happening with Achilles. It’s a very good and helpful analogy, but it has limitations. There is a difference between the measurement and the reality being measured. The light is counting, Achilles is not. For the light to count, it must be encapsulated; Achilles is not. Solution? To match the two things, we imagine that the counted numbers are infinite (even if that is physically impossible) BUT their addition is a finite sum, in the limit (even if that looks logically impossible). I called this a “freak” concept because you do not find an equivalent in nature: there are no baskets with an infinite number of pieces of apples that add up to a finite number. But since everything is “happening” here at the abstract, intellectual (mathematical) level, this sort of licences is admissible. If, for the sake of measurement, we confronted Achilles with the imaginary challenge of opening infinite imaginary doors, we can (and we must!) also allow him to succeed in that imaginary task.

    Another example of “freak” concepts: the dual nature (wave and particle) of light. But don’t ask me how or why it works!

    And now the link with SR: if you observe a clock passing by and you measure how long it has taken to do that trip, you get a “coordinate time” that is longer than the “proper time” measured by the clock itself; if the holder of the clock scratches his nose with a frequency = 1 scratch = 1 tick and you want to learn how many times he did it between the two events, your “coordinate time” by itself does not provide the answer; you have to correct it by combining it with your own measurement of distance traversed (“rest length”) through the appropriate formula and thus get the (invariant) spacetime interval between the two events, which is in this case the same proper time measured by the clock. Thus “coordinate time” is also a “freaky” concept: although we still call it time, it doesn’t deliver what you would expect from a concept with that name, that is to say, to tell you what “happens to the holder of the observed clock” (how many “times” he scratches his nose); just like in Zeno paradox we go too far and count to infinity, here we also go beyond the mark and count just too many times; but just like in Zeno paradox we correct by admitting a finite sum, here we square, subtract and square root to get the correct answer.
     
  12. Sep 25, 2009 #11
    This is the heart of the resolution:
    This may be the same "paradox" as the one from anicent Greece where it was asked "How can my arrow, launched across the width of a river, ever get to the other side if there are an infinite number of points along the way??" ( It will take forever to traverse an infinite number of points). This was apparently rigorously resolved after a thousand or two thousand years later ( I forget now) with the development of calculus.
     
  13. Sep 25, 2009 #12
    Einstein's relativity does assume continuous space and time...but quantum theories, like Planck length (10-33 cm) and Planck time (10-43, second posit discreteness.

    Likely we'll have to wait for theoretical unification or experimental confirmation one way or the other; as I understand it, experiemnts so far have shown only three space dimensions and continuous space and time. Stay tuned for more developments. Theories abound including string based theories.
     
  14. Sep 25, 2009 #13

    DrGreg

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    But even quantum theory doesn't say spacetime is discrete in the sense that measurements must be an exact integer multiple of a Planck unit. It is rather that distance is no longer a single numerical value but rather a probability distribution with a mean and standard deviation. The mean could take any value from a continuum, but there are theoretical limits on how small the standard deviation can be.

    That's a gross oversimplification, but, without going into the technicalities of what a Hilbert Space operator is, it gets the idea across.

    I'm no expert in quantum theory and willing to be corrected by someone who is.
     
  15. Sep 25, 2009 #14

    JesseM

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    Quantum theory as it stands does not, but then quantum theory in its current form doesn't deal with spacetime as a dynamical entity as in general relativity; a lot of physicists would probably say it's a plausible guess that in a theory of quantum gravity, spacetime would indeed be quantized in some way (in any case, quantum gravity is where the planck length and planck time would likely have physical significance of some kind, AFAIK they don't have any special significance in quantum field theories aside from the fact that you can arrange certain constants like h and G to get them).
     
  16. Sep 25, 2009 #15

    Saw

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    But, assuming that, what would that mean? Don't you agree with what I said above? If, for example, you measure time with a light clock and there is a limit for narrowing down the time unit (let us say the Planck time)... that does not mean that the motion of the observed object also becomes discrete. It is still continuous, isn't it?

    And if the motion did become discrete, what would that mean? I confess I don't understand the solutions of the Zeno paradoxes that rely on a quantized space or time. If Achilles moves one Planck length and the tortoise moves at 50% his speed but the tortoise cannot move less than Planck length, what does it do? Does it wait until Achilles completes 2 Planck lengths in order to complete 1? How does quantized motion work?
     
  17. Sep 25, 2009 #16

    JesseM

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    Well, we don't have a complete quantum gravity theory and I don't know what the exact answers would be even in speculative approaches like string theory or loop quantum gravity, but from what I've read I'd guess it'd be a mistake to imagine that this is just an issue of clocks, the speculations are that this would be a real revision to how a theory would treat space and time itself (for example, as I understand it, in loop quantum gravity there is no background spacetime at all, just the loops themselves). Also, if it's still a quantum theory, I'd guess it'd be just as much a mistake to imagine paths are "really" continuous even though there are limits to measurement as it would be in standard QM to imagine particles "really" have definite position and momentum at every moment and that the uncertainty principle is just a measurement issue (although this is true in the interpretation of QM called Bohmian mechanics, so I suppose even with a complete theory of quantum gravity there might be multiple experimentally indistinguishable 'interpretations' as well), or that particles "really" have definite spin on all possible axes even though measuring one disturbs others (are you familiar with Bell's theorem and how it rules out the possibility of explaining QM in terms of local hidden variables theories?)
     
  18. Sep 25, 2009 #17

    atyy

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    So he needs dx/dt to exist to resolve the arrow paradox? How about everywhere continuous but nowhere differentiable functions - is there any motion there?
     
  19. Sep 26, 2009 #18

    Saw

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    Yes, that is what I meant, synthetically and hence more clearly.

    “Real” motion, at least inertial motion, is a single indivisible phenomenon. “Measurement” is also a physical real phenomenon but it’s accelerated motion (change of direction), so it is discrete. (This is clear for measurement of time with clocks; it seems that precisely SR makes the idea applicable to length measurement, too.) You can match the two things in calculus by imagining that those discrete measured values are infinitely divisible and, in spite of that, their addition is finite. But that doesn’t make the observed motion less of a single indivisible phenomenon than it was. In fact, what the mathematical solution does, by means of a go-and-return mental trip (first, I allow division; then, infinite division; finally, merger of the infinite values into a finite number) is adapting ultimately, after moving to and fro, to the nature of the observed thing.

    I was roughly aware that what I was saying would touch on the issue of the measurement problem and the different interpretations of QM. I suppose that what I posited above would align me with Bohmian interpretation, as opposed to Copenhagen interpretation? I’ll study those valuable links. I admit this is a thorny question and we are not going to solve it now. In the end, there will be supporters of both interpretations…

    But my problem is that I do not see how the “other” interpretation dissolves this particular paradox at all. It’s not only that I would probably disagree with it if I saw it spoken out, it’s that I’ve never seen a full-fledged statement thereof.

    The one I understand is easy: First, it is hardly understandable that an infinite series of real physical things exists at all. But if it exists, it is logically impossible that it adds up to a finite number. Second, the paradox dissolves because all that (infinite division and finite addition) happens at a mental level, in our imagination, thanks to some cunning mathematical concepts. At reality level, however, there is a single phenomenon (inertial motion) that is indivisible.

    The “other” one… would it then hold that measurement perfectly matches reality? So inertial motion would take place by discrete steps? That could be, who knows, but then it doesn’t seem to solve the paradox. Please note that Zeno’s paradoxes are said to contest the description of motion both in terms of infinite divisibility but also in terms of finite divisibility. His attack is against the divisibility of motion. Once you admit such thing is possible, you are reduced to absurdum. If the divisibility is infinite, you’re faced with the endless loop. If the divisibility is finite, you soon a get a contradiction when you compare motions. Again: If Achilles moves 1 Plank length and the tortoise travels at 50% of his speed, has the tortoise moved 1/2 Planck length?
     
  20. Sep 26, 2009 #19

    Dale

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    That depends entirely on your definition of "real physical things". You will notice that physicists rarely use such words because they don't have clear definitions. You can certainly define "real physical things" such that your statement is correct, but it is just a matter of definition and most such definitions have little impact on whether or not something is measurable. So basically it is a purely semantic statement which is too unclear to mean much from a scientific perspective.
     
  21. Sep 26, 2009 #20
    Good point. I was only quoting Reichenbach and by no means consider myself knowledgeable on the subject. The quoted passage seems, at least, to offer some valid criticisms of Zeno's understanding (or use) of the terms rest and motiont, but I am aware that there are still arguments at various philosophical levels as to the resolution of such paradoxes. My opinion is just a naive, pragmatic one. Motion happens and so there is something wrong with the propostions that lead to conclusion that it does not happen.

    Matheinste.
     
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