Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: LeeE on 21/04/2008 18:16:27
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Hi all,
here's a little conundrum that has been bugging me for a while now.
If you give a little thought to it, it is apparent that an object that has zero length in the direction that it travels can only move in that direction in discrete jumps. That is to say, it cannot smoothly pass through every possible point between two positions but must do so in discrete steps.
Invoking the idea of making infinitely small movements in infinitely small periods of time doesn't solve this because no matter how small the step is it will still be greater than 0. I guess this can be shown mathematically by the statement: 1/infinity > 0
This is all well and good, and provides a nice model for lightspeed objects such as photons etc. where the object does not accelerate up to lightspeed but immediately starts moving at 'c' and, once moving, moves at the same constant velocity (for any given medium through which it travels). It would also fit with the phenomenon where the measured speed of such objects, regardless of the relative motion of the detectors, is always the same.
Time, of course, is also relevant to this. Not only must there be a difference in position for an object to have moved but there must also be a difference in time – if there's no difference in time it means that the object is just simultaneously in two places at the same time as opposed to having moved between the two positions.
If an object has moved between two positions and has done so in a discrete step it's then difficult not to wonder if time also progresses in discrete jumps as well and in fact, I think it could be argued that it does.
For example, if one considers an event occurring in time, we would generally think of it occurring over the period between the time that it started occurring and the time that it had finished occurring. We can look at this as saying that the proportion of the event that has occurred = 0 when the event starts, is 0.5 half way through the duration of the event and is 1 by the time the event has finished.
However, that starting value of 0 brings us back to a similar situation to the zero length object, where any change in value from 0 must be greater than 0 and therefore results in a discrete step. To be sure, in this case we're dealing with a zero thickness boundary rather than a zero length object but in either case there needs to be a discrete gap between them to prevent them from being in exactly the same place.
Actually, thinking about it even further, numbers themselves are also subject to this zero thickness boundary issue. While we can say that the number of values between say 0 and 1 is infinite, each value has zero 'thickness' and therefore every possible value requires a discrete separation from any other number.
Getting back to the conundrum part, once you've started looking into this zero thickness boundary stuff and it's implications, it's difficult to come up with a model for the normal sort of motion we see every day, where objects accelerate and appear to smoothly pass through every point between two positions. For example, for any object that exists, but does not exist in multiple places at the same time, we can say that at any one instant the proportion of it that exists at one location is 1 while at another location it is 0. However, if the object moves from the first position to the second that proportion must change and we're back to needing a step or gap to do so. A further interesting aspect of this is that it doesn't seem to just apply to the objects themselves, where the physical surface could be considered to be imprecise, but also if it is spinning, to it's axis and plane of rotation.
Just as a further bit of background interest, it's interesting to look at the different classes of n-dimensional objects and how they can move in zero length directions.
A zero dimensional object i.e. a point, can move in any direction and it's radiation pattern from a point of origin would be spherical, defining a volume. A one dimensional object i.e. a line, can move in any radial direction out from it's axis, in the shape of a circle and would define an area. A two dimensional object can only move in the direction of it's 'faces' and it's radiation pattern would be a line. A three dimensional object, in three dimensional space, doesn't appear to be able to move at all, nicely describing a point.
A three dimensional object in four dimensional space would have no problem because it would have zero thickness in the fourth dimension and if we apply it to four dimensional space/time it implies that three dimensional objects are moving in discrete steps, at 'c' but in the time dimension.
Heh:) – there's lots more to this – wave-function and indeterminate numbers type stuff seems to be a requirement , but I've gone on enough already and I'd appreciate hearing some other people's thoughts:)
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If you give a little thought to it, it is apparent that an object that has zero length in the direction that it travels can only move in that direction in discrete jumps. That is to say, it cannot smoothly pass through every possible point between two positions but must do so in discrete steps.
Why can't it pass continuously through every position on its way somewhere?
Invoking the idea of making infinitely small movements in infinitely small periods of time doesn't solve this because no matter how small the step is it will still be greater than 0. I guess this can be shown mathematically by the statement: 1/infinity > 0
1/infinity doesn't mean much mathematically. You have to be more precise and say something like "Take the limit of 1/x as x->infinity". That limit is 0.
Actually, thinking about it even further, numbers themselves are also subject to this zero thickness boundary issue. While we can say that the number of values between say 0 and 1 is infinite, each value has zero 'thickness' and therefore every possible value requires a discrete separation from any other number.
If there are N numbers between 0 and 1, then the total "thickness" of each number would be 1/N. Since there's N of these numbers, the total thickness, T, is given by T="thickness" x number of #s = 1/N x N = 1.
If you take all (real) numbers between 0 and 1, it's like taking limN→∞1/N=0. So In that sense, the "thickness" does go to 0. However, you're also taking a total of N numbers, so the equation T=1/N x N = 1 holds, even if you take the limit as N→∞.
A lot of this is interesting. In some sense, if you look at quantum mechanics, things seem to happen in discrete jumps, so there might be something to proposing that space and time are made up of "quanta." Someone else can probably elaborate on work going on in that area. However, from what I understand of your conundrum, it can be easily solved by introducing calculus. Sure, you can make a basic model by saying that things move in discrete jumps, but if you keep squeezing those jumps smaller and smaller and take more and more of them (to make up for them being smaller) you can cover any distance by an "infinite" number of "infinitesimally small" jumps, and in this limit, you end up with continuous motion.
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LeeE; I suggest you read a book on Mathematical Analysis. A lot of what you say involves basic theories of maths. It is very hard stuff, in the end, but it is fascinating and will tidy up your views of numbers etc..
Of course, the fact that you can apply calculus to natural phenomena does not necessarily imply that reality is precisely modeled by what is, essentially, an axiomatic structure. It works only as far as it works.
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LeeE What you are talking about is mathematics and your problem starts the moment you say "an object that has zero length in the direction that it travels" because all the objects that we know, love and can observe in this universe even subatomic particles fail to meet this criterion.
You might decide to say what about an electron or a neutrino these do not have any measurable physical size but they both have fields the electron has an electric field and a weak interaction field and the neutrino a weak interactoion field. both of these exist over a measurable distance.
So as your first premise fails to meet a sensible physical criterion the rest is mathematics as sophiecentaur says and not physics.
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Hmm... where to start responding:)
First of all, this is just a thought experiment type thing. I'm not proposing any sort of theory but as everything we do takes place in the context of scientific knowledge, it has to recognise that knowledge. While it is true that all 'objects' that we consider to actually exist require four dimensions to do so, three for space and one for time, there are certainly 'somethings', for want of a better word, that exist in less dimensions, for example (but avoiding BH singularities for the time being) axis and planes of rotation, or centers of mass. While an axis or plane of rotation, or a CoM may not exist in the physical sense that you could apply a force directly to it, they are still something that can be defined and, by definition, occupy less than four dimensions with the result that at least in one dimension they have zero length.
So perhaps the problem could be put as: if a spinning object is to move in a direction perpendicular to it's axis of rotation, how does that axis of rotation move?
Now obviously, any movement of the axis in that direction requires a displacement greater than zero, otherwise it'll still be in the same place and won't have moved at all, but once it has moved then the distance that it has travelled, however small that might be, will still be greater than the 'thickness' of the axis with the result that there will be a 'gap' between the two positions - you can't have two individual parallel axis actually touching each other without being them being in exactly the same place - they need to be separated and even if the separation is infinitely small, it must still be greater than zero.
I must say that I don't see this as a maths problem but as a logic problem. Maths is great but it only works within rules that make it self consistant. Dealing with infinity is an area where, imho, it doesn't work very well, producing answers that solve but which may mean nothing, for example ∞ + 1 = ∞, 1 * ∞ = ∞, ∞ * ∞ = ∞ etc. At the same time, maths might allow you to work with sizes smaller than the planck length but would the answers actually mean anything?
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Now obviously, any movement of the axis in that direction requires a displacement greater than zero, otherwise it'll still be in the same place and won't have moved at all, but once it has moved then the distance that it has travelled, however small that might be, will still be greater than the 'thickness' of the axis with the result that there will be a 'gap' between the two positions - you can't have two individual parallel axis actually touching each other without being them being in exactly the same place - they need to be separated and even if the separation is infinitely small, it must still be greater than zero.
The problem is that you're assuming that your object is infinitely thin, but that your steps in space can't be. You can't have it both ways.
I must say that I don't see this as a maths problem but as a logic problem. Maths is great but it only works within rules that make it self consistant. Dealing with infinity is an area where, imho, it doesn't work very well, producing answers that solve but which may mean nothing, for example ∞ + 1 = ∞, 1 * ∞ = ∞, ∞ * ∞ = ∞ etc. At the same time, maths might allow you to work with sizes smaller than the planck length but would the answers actually mean anything?
You may not like it, but mathematics can certainly describe infinities in a self-consistent way: see Mathematical Analysis as sophiecentaur suggested.
If you're going to claim that on some tiny scale, treating space and time as continuous breaks down, you need to be much more rigorous in explaining why. Current models of quantum mechanics generally treat space and time as continuous and are extremely accurate. It might be worth your while to read up on String Theory a bit. I know String Theorists are trying to connect the structure of space-time (predicted by general relativity) with quantum mechanics. So there might be some theories about just what space-time is made up of on those tiny scales. I don't know the details, however.
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Planck Length ≈ 1.6 x 10-35 meters
Planck Time ≈ 5.4 x 10-44 seconds
There seem to be some ideas in the physics community that these two units represent the smallest possible measures of space and time (or perhaps the smallest measurable units of space and time). I don't know how well "proven" these ideas are, though.
http://en.wikipedia.org/wiki/Planck_length (http://en.wikipedia.org/wiki/Planck_length)
http://en.wikipedia.org/wiki/Planck_time (http://en.wikipedia.org/wiki/Planck_time)
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The problem is that you're assuming that your object is infinitely thin, but that your steps in space can't be. You can't have it both ways.
Ah - but I'm not saying that. When I say zero length I _don't_ mean infinitely thin - I mean exactly zero. There's an infinite difference between those two terms:)
Why do you say that I don't like the idea that maths can describe infinities in a self consistent way? I said exactly that and gave some examples. Not very useful examples, I'll grant you, but they are all true:)
Once again, and as I said at the start of my first response, this is a thought experiment and not a proposal for a theory. I'm not trying to claim anything at all, contrary to what you seem to think. Yes, String and Superstring Theory does deal with some similar aspects but I'm not trying to discuss String Theory and similarly, the planck units, especially with reference to the planck distance and time units, are not proven to be limitations, although our current understanding seems to be that they are the smallest units at which quantum physics works.
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I just wanted to add this sort of quick semi-planned part two to my original posting on this topic – I'll happily let it die after this:)
I do want to stress that this isn't a theory but is just a thought experiment in logic. However, it is a sort of directed thought experiment and perhaps it could be phrased as 'How do we design a universe?'
At the end of my first posting, it appeared that a three-dimensional object cannot move through three-dimensional space in the same way that an object with fewer dimensions must, but this is only true to a degree.
While lower-order dimensional objects must move in jumps because, having zero length, any movement must require a non-zero separation between two positions, there's no logical reason why three-dimensional objects cannot also move in jumps. In fact, if you think about it, the leading face of a three-dimensional object represents a zero-thickness surface boundary, so while we can't say that there needs to be a gap between a three-dimensional object's two positions, because the positions could overlap, there still needs to be a gap between the two surface boundary positions.
The need therefore, for movement in discrete steps, doesn't come from zero-length but from zero-thickness, or precise boundaries.
Logically then, if a zero-thickness or precise boundary results in stepped movement, with no acceleration and at constant speed, to achieve continuous motion, variable speeds and acceleration it would appear that we require non-zero thickness, or imprecise boundaries.
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This is the whole basis for Calculus. Calculus doesn't deal with zero size steps, it considers what happens, in the limit, when the size of step approaches zero.
All the time, Calculus assumes 'well behaved functions' which are continuous and differentiable etc etc. It is JUST a mathematical model and you can't just hop in and out of it willy nilly. There are plenty of functions which cannot be treated simply using ordinary Maths. You have to apply the reasonability check each time. It is easy to end up with nonsense arguments if you don't. That's why I recommended the Analysis study in my earlier post.
Take String Theory, for instance. It involves many extra dimensions, some on a very small scale through which you keep jumping around in as you blithely think you are moving 'smoothly' from A to B in 'the x direction'.
When you consider something as 'blindingly obvious' as continuous number space you soon get bogged down as to where to fit the irrational and transcendental numbers and the significance of prime numbers and all the other special ones. Maths can turn round and bite you if you're not very careful. [???]
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I think this is an interesting topic. I agree with sophiecentaur that (classically) the problem you describe is "fixed" by introducing calculus and infinitesimal lengths. At some scale, I agree there will be a problem. This is probably the case when you're talking about tiny objects and tiny distances moved. In that case, instead of calculus saving the day, maybe it's uncertainty relations--the inability to precisely define an object's position and velocity would be a big factor in trying to define its motion.
Another interesting connection (if you have some mathematical training) would be to look at what role using the Dirac delta function would play here, since it's the mathematical structure that physicists generally use to say something has zero length.
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Hi, I don't think I disagree with anything you say, regarding describing what's going but that's not the point (I think) I'm trying to make:)
Instead of analyising something to understand it, it's more a case of synthesising something to see what needs to be understood. What (I think) I'm trying to do is start from nothing and see what, after adding something, are the consequences and in this case it seems to me that even while dealing with a very simple rule-driven and non-complicated state of things, where you would expect to be able to describe everything equally simply, you actually need two entirely different ways to describe the posibilities.
In this specific case, we're just thinking about n-dimensional frameworks and the possible type of entities that might exist within them and what actions those entites might perform. So, because I'm thinking in terms of trying to synthesise a 'universe' with similar characteristics to our own, the framework is is an n-dimensional space-time. The fundamental properties of an entity existing within this framework are therefore going to be location and size, and so therefore the possible actions are going to be changes in those attributes.
Now the aspect of this that I find interesting, and a bit surprising, is that even without adding anything else, such as specific properties or values to any of these hypothetical fundamental entities, that is, from just considering possibilities implicit in, and derived from the framework i.e. location and size, we need two different, and perhaps mutually exclusive types of location action turning up - these being precise movement in steps, which could be mapped to lightspeed behaviour, and imprecise continuous movement, which could be mapped to sub-lightspeed behaviour. What's more, this seems to be an immediate requirement, even before any other properties or values are added to the 'mix'.
The question, to me, isn't 'How do I describe what is going on here?', the answer to which is 'Calculus', but 'What is going on here, such that it can be described by Calculus?'
Of course, I might just be spiralling up my own fundament:)
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Thanks for the pointer to the Dirac function - I'll have a look at that and see if I can make any sense of it:)