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A Point in Spacetime has the Cardinality of the Continuum

  1. Apr 26, 2005 #1
    Kind of trivial result, but thought it might be interesting. This is part of a wider development which will be described further, either here or in another thread.

    "A Point in Spacetime has the Cardinality of the Continuum"


    Time can play a really neat trick !! To observe time you must have some type of motion or physical activity of some kind. Without some kind of motion it is impossible to observe time. Motion does not create time or cause it to occur. But without some type of motion it becomes impossible to observe or measure time.

    By hypothesis, there are motions in the universe which are so vast (relative to an observer on our scale) that these motions are impossible to observe. Hence, very large scale motions cannot be used to obsere time relative to an observer on our scale. Time becomes impossible to measure or observe in that context, and therefore ceases to exist relative to an observer. The converse also holds for the quantum scale. The point is that time ceases to exist relativistically on the largest and smallest of scales.

    There is an important distinction. Neglecting to observe time will not cause it to cease to exist. But if it is utterly impossilbe to observe or measure in a given context, then indeed it does not exist relative to you - in that context.
    [ seems to agree with Planck]

    If time ceases to exist relative to an observer, then 4D Minkowski spacetime (3+1) loses one of it's dimensions to become the 3rd dimension (3+0). This defines the edges of our universe. Keeping in mind that this effect is relativistic, you immediately derive that the universe is both open and closed. An observer on our scale will think observe that it is closed, but in an absolute sense it is really open. Hence, it is open AND closed, simultaneously.
    [ Having the 3rd dimension also allows you to explain, easily, all kinds of QM weirdness.]

    The fine structure of Minkowski spacetime is therefore a composite of 4D and 3D "points" of some kind, and the only difference between these two types of points is relativistic. The universe has two boundaries, one on the astronomical scale, and one on the quantum scale. These boundaries exist relativistically.

    Knowing that the universe is both open and closed simultaneously, consider a 4D point, P1, in spacetime. Since 3D points may be considered to be "relativistically nonexistent", you can even say that you have uncountably many such points at P1. Such additional points could become existent depending on how P1 is observed, and so a point in spacetime really can be said to have the cardinality of the continuum.

    Discussion 1:

    You can easily say that I bent the rules here because I invoked points which are nonexistent relative to an observer, but by altering the observation, those points could become existent, so it's only partially cheating.

    Also, you could argue that I have "assumed" continuity by stating that the manifold has cardinality of continuum. This would be a difficult statement to make if we were talking about points which are strictly existent, but since we are talking about points which are relativistically nonexistent there is no reason why you cannot define them to be continuous.

    Continuity in spacetime is a relativistic phenomena, and with luck we'll get a little closer to proving it.

    Discussion 2:
    I know that these ideas are extremely bizarre, perhaps especially the part about the "relativistic nonexistence". But I will prove this with one short addendum. This is very easy to grasp, so please read on.

    Consider the double-slit quantum-interference experiment .
    This experiment is easily explained. Time ceases to exist on the quantum scale (relative to an observer), and so the individual wavecrests in this experiment cannot be separated by time. The wavecrests must, therefore, be separated only by distance. The experiment becomes identical to the classical double slit.

    This experiment appears to be something very new. But because time does not exist on that scale, the arrangement MUST be identical to the classical double slit. The difference between successive wavecrests cannot be time, but distance !


    Je le vois, mais je ne le crois pas !!

    Regards -
    The Average Joe.
    Last edited: Apr 27, 2005
  2. jcsd
  3. Apr 30, 2005 #2
    The idea is that time ceases to exist on large and small scales (relative to an observer on our scale).

    We have seen a very natural explanation that the universe is both open and closed simaultaneously.

    We also have an interesting possible explanation of the "double-slit quantum-interference experiment",

    We have an interesting way of looking at time, and an intuitive, classical explanation of why it can behave in seemingly strange ways.

    We have a new approach to continuity in spacetime.

    We also have a weird sort of anecdotal statement about a possible triviality regarding cardinality of point sets in spacetime. And, even though it may be a genuine triviality I think the example has value in this area. Perhaps such statements will always be reducible to trivialities - but I hope not.

    Lastly, we have illustrated why existence itself is relativistic, and this provides a very natural classical explanation of virtual particles, or vacuum energy as they are called.

    The universe is bounded on the largest and smallest scales. The universe continues on past these boundaries in an absolute sense, but relative to you and I these regions simply do not exist.

    The presence of virtual particles is now completely explained in classical terms. You have structure, waves or particles, which exist on or near this scalar boundary. At times it seems that they do not exist at all, and at times they seem to manifest themselves from an empty vacuum.

    Nothing can exist where time is undefined. Existence requires time. Existence is trivial whenever delta_t=0. We observe a boundary where time ceases to exist, and everything which lies beyond that boundary behaves as if time does not exist. The boundary is relativistic, and so too is existence.

    Does this violate conservation of energy ? I dont think so. I tend to think that entropy will reach an equillibrium across the relativistic boundary described here, and that if you find a way to create particles from a vacuum which are stable, then somewhere else in the visible universe energy must be slipping away - perhaps into relativistic nonexistence.

    One more note-
    On the question of whether space is continuous or discrete. It may be the case that space is continuous AND space is discrete, simultaneously.

    This is certainly difficult to even concieve of, but I think that it may be true. With existence being relativistic, the universe being both open and closed simultaneously, can you really say that continuity or discreteness are well defined in a mathematical sense ? Yet somehow there must be an answer - and it might be that space is BOTH ! - which could concievably result from the "scalar relativism" described in this document.

    An Average Joe
    Last edited: Apr 30, 2005
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