Reality is Reduced to Axioms

Discussion in 'Physics & Math' started by Spellbound, Sep 10, 2014.

  1. przyk squishy Valued Senior Member

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    3,203
    I'm not sure I see the issue. If you have a theory and you can apply Noether's theorem to it, then Noether's theorem will tell you everything that needs to be included in the energy conservation law. For example, if you apply Noether's theorem to relativistic particle physics, it will tell you that you should include mass as part of the definition of total energy.

    I honestly don't know. It's more than just a little bit speculative, you know. At the moment our best guess, from some simple dimensional analysis, is that a TOE would concern physics operating at the Planck scale. That's about fifteen orders of magnitude or more smaller than the characteristic distance scale that the LHC is capable of probing interactions at. That's a lot of room for things we might not yet know about to hide in.
     
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  3. lpetrich Registered Senior Member

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    Let's look at some additional features.

    The interactions in the Standard Model and in general relativity are all local, and most proposed extensions of them also have only local interactions. Local means all the interacting fields are referred to the same points. A nonlocal interaction would involve one field at one point and another field at a different point.

    The issue of local vs. nonlocal appeared when Sir Isaac Newton proposed his law of gravity. He proposed it in nonlocal form, and he tried to argue that he was not dragging in "occult qualities", meaning anything that was too far from what is directly accessible empirically. "Hypotheses non fingo" (I don't make any hypotheses), he wrote.

    However, in 1813, Siméon Denis Poisson devised a formulation of Newtonian gravity that was entirely local. However, if one integrates out the gravitational potential from his formulation, one gets Newton's nonlocal form again.

    Over the early to mid nineteenth century, it became evident that it was best to describe electromagnetism using local equations, and Maxwell's equations are indeed local. Nonlocal ones would be gruesomely complicated in all but the simplest cases, like vacuum.

    Locality has been very successful, but it's hard to construct a good quantum theory of gravity using only local interactions. String theory and loop quantum gravity may be nonlocal, but I'm not not sure.
     
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  5. lpetrich Registered Senior Member

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    Now for space-time. In our current paradigm, GR + SM, space-time is curved with a metric.

    One can show that a completely general metric makes curvature possible. For n dimensions, there will be n(n+1)/2 metric-tensor coefficients. One can try to reduce the metric to a manifestly flat form by a change of coordinates, but there are only n coordinate-change functions available. That leaves n(n-1)/2 functions left over.

    General relativity relates the metric and energy/momentum density/flux by space-time curvature:
    \( G^{\mu\nu} = \kappa T^{\mu\nu} \)
    Alternatives to it have different ways of specifying the metric, though the most successful ones are essentially GR + additional fields that change the gravitational "constant".

    Space-time is locally Minkowskian, with an orthogonal set 1 timelike and 3 spacelike directions. I don't know if it's possible in general relativity to change the numbers of timelike and spacelike dimensions, at least not without creating a singularity. Imagine two regions of space-time with different metric signatures. What is their boundary surface like? I ask that because I once saw a paper on a braneworld cosmology that featured such a signature change, from Minkowskian (our Universe) to Euclidean (all the same: timelike or spacelike). The change was preceded by a "Big Rip" of accelerating expansion.

    That orthogonal set can be used to define 1 timelike and 3 spacelike dimensions.

    So let's see what happens with different numbers of dimensions. With more than 1 timelike dimension, there is no longer a well-defined direction of time, since a particle can travel in circles in the timelike dimensions. With different numbers of spacelike dimensions, we get different sorts of structures in the Universe. With fewer than 3 dimensions, less complexity becomes possible, meaning that we may be unable to emerge in such universes. With more than 3 dimensions, stable orbits become impossible, and thus, structures become impossible. That's because the force law from an interaction field goes as (separation)[sup]1 - (# space dimensions)[/sup].

    So our 3+1-dimensional Universe is convenient for allowing us to come into existence.
     
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  7. lpetrich Registered Senior Member

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    Could there be more space and/or time dimensions. Various physicists have speculated about that, and there are two main classes of speculation.

    Compactification, or the Kaluza-Klein hypothesis. All but our 3+1 dimensions are compactified or curled up into a tiny ball.

    Braneworld universes. Our space-time is a 3+1 surface or membrane or brane, with all the dimensions together forming the bulk. Even though in some scenarios, the extra dimensions are very short.


    There are some theoretical reasons for supposing there to be more than 4 space-time dimensions. Reasons related to string theory. Quantum-mechanical self-consistency constrains how many space-time dimensions they can have. Bosonic strings need 26 dimensions, while supersymmetric strings or superstrings need 10 dimensions. There is something called M theory that lives in 11 dimensions and that relates different superstring solutions, but its nature is very obscure.
     
  8. PhysBang Valued Senior Member

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    2,422
    When he wrote that, he was specifically referring to the idea that he did not know what caused gravity, he was just providing its properties. He was saying to the mechanists of his day that regardless of whether or not everything was caused by physical contact, the phenomena that they had to account for with their underlying causes had to match the properties of gravity as he had established. Occult qualities would be some unknown cause of gravity that Newton did not produce, but which would be different from contact action.
     

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