What really is a continuous symmetry?

Discussion in 'Physics & Math' started by arfa brane, Oct 27, 2020.

  1. QuarkHead Remedial Math Student Valued Senior Member

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    See what happens when you treat time as a parameter? Nonsense and pseudo-philosophical garbage

    In the Special Theory of Relativity a chap called Einstein assumed that that time was a coordinate, albeit distinct from space. Hence the well-known inequalities for any pair if coordinates \(X\,\,X'\)

    \(x^2+y^2+z^2 \ne x'^2+y'^2+z'^2\) (length contraction) and

    \(ct^2 \ne ct'^2\) (time dilation)

    Enter the incomparable Herman Minkowski. He realized that, by contrast, \(x^2+y^2+z^2-ct^2=x'^2+y'^2+z'^2-ct'^2\). He called this spacetime and it nicely leads to this.

    Since there must exist a transformation \(X \to X'\), and if the above holds, one can define a metric, say \(ds^2=dx^2+dy^2+dz^2-dct^2\) that is invariant under this transformation.

    This defines it as an orthogonal transformation - it is a continuous symmetry group called \(SO(3,1)\) - it is a part of the so-called Lorentz Group
     
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  3. Write4U Valued Senior Member

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    And your post made it simpler to understand? All you did is confirm that time is part of spacetime, but it says nothing about any independent existence of time as a separate continuum apart from 3D space.
     
    Last edited: Nov 3, 2020
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  5. arfa brane call me arf Valued Senior Member

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    The same way a point in \( \mathbb R^3 \) can have coordinates, one for each dimension; that is, on a Cartesian coordinate system, with a choice for the origin. Each coordinate is in a space which is linear, and in which a direction can be chosen.

    The manifold we do physics in, has what's known as a causal structure; this is just a relation (isn't everything?) between pairs of events, but covering all events in some history. A pair of events can be causally related, or not.

    Time is not usually considered to be causal. That would imply that a distance can be the cause of an event; a distance would then have to be an event. It just doesn't fly, even with a distance in time.
     
    Last edited: Nov 3, 2020
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  7. Write4U Valued Senior Member

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    But you cannot do that with unidirectional time regardless of multidirectional spatial change
    I agree.
    I agree.
    The measurement of duration always comes after the event or as a current age (temporal point) in an open-ended duration, such as the unfolding (expanding) universe.

    Within the universal space all coordinates have the same associated temporal age. Part of the spacetime continuum.
     
  8. arfa brane call me arf Valued Senior Member

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    You can't choose the direction of time, no.

    That's why if you use a function to map an interval of time to a distance, the distance must also have the same direction; once you have an interval mapped to a distance, it no longer matters which way it "points". An interval of time can't have a negative distance but you're allowed to subtract time (which is not the same thing, although it is equivalent).
     
  9. Write4U Valued Senior Member

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    That's wrong.
    If I drive from A to B and it takes me 1/2 hr, then return from B to A and it takes 1/2 hr, I have returned to my starting point after driving in 2 opposite directions, but it is now 1 hr later. I cannot subtract the return time!!!
     
  10. arfa brane call me arf Valued Senior Member

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    Perhaps you missed the part where I say an interval of time is mapped to a distance, actually to a displacement. The direction for the distance/displacement might be the direction a pendulum is swinging, or the hands on a clock are moving.

    Once, as I say, you have a distance, the direction doesn't matter, and physics says nothing about time "having" to be unidirectional, that's an observation.
    It's one of those observations we're still trying to explain, but clearly if time's direction didn't really matter the universe would be quite different.

    At least, that's What I Tell Myself.
     
  11. Write4U Valued Senior Member

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    Forget the subjective human element, concentrate on the objective physics which do not map in advance.
    Yes it does. Any other solution than a forward unfolding of time along with duration of a causal physical function immediately creates contradictions which cannot be solved by actual physical dynamics.
    However,
    https://en.wikipedia.org/wiki/Philosophical_presentism

    To me that sounds eminently logical and fundamentally uncomplicated.

    And no, in this thought experiment, I did not map the distance (that is a human endeavor). I drove the distance and measured the duration. On the return trip I did not map the distance either, I drove the distance and measured the time of duration. Objectively, each way was an unknown quantity, until the distance was driven and the duration each way was added to arrive at a total measured time of duration (1/2hr + 1/2hr = 1 hr) for the completed round-trip.
    IMO, if time were reversable, the universe would not exist at all. Any chronology is by definition a unidirectional (forward) measurement of duration.
    No physical chronology, no duration, no spacetime, no universe.
    I appreciate your patience and making me examine what I tell myself......

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    Last edited: Nov 4, 2020
  12. arfa brane call me arf Valued Senior Member

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    So the duration is mapped to a distance, in time. The distance you drove and the time 'duration' are what get mapped to each other.

    Likewise 1 second is mapped to 1/60th the distance around a clock face; or it's mapped to a changing number in a numerical display. When you want to do some arithmetic with the distances, the direction of motion in space or time isn't required because distances are always positive. That's all you need.

    \'yeesh'
     
    Last edited: Nov 4, 2020
  13. Write4U Valued Senior Member

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    Yes, but that is not in advance of the trip. The ability to map is a result of the durations of the trips from A to B and from B to to A, consecutively
    Right, when you want to do some arithmetic with the durations, the direction of motion in space isn't required because the durations are always positive (additive). Hence always forward in time, regardless of direction of distance.
    "That's all you need" is sufficient to get agreement from me in this respect.... "Yeesh".../

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  14. Michael 345 New year. PRESENT is 71 years old Valued Senior Member

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    I don't know who is winning (if that is a correct gauge)

    Just glad I'm out and you are keeping the other occupied

    Keep going

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  15. QuarkHead Remedial Math Student Valued Senior Member

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    And with good reason - time (in modern physics - none of which can ignore relativistic effects ) has no existence independent of space.
     
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  16. exchemist Valued Senior Member

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    Quite. Statler and Waldorf, eat your heart out.

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  17. Write4U Valued Senior Member

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    Wonderful! That was the gist of my entire argument.
    We have agreement! That gives me great comfort......

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    Last edited: Nov 4, 2020
  18. arfa brane call me arf Valued Senior Member

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    But you also posted this:
    The map isn't something that exists because you drove somewhere, there isn't any ability involved. It existed before you did, before any human or a solar system existed. Actually, since the beginning, this map has existed, and doesn't need conscious observers.

    It seems that we agree on distances always being positive. You can say you're moving backwards if you drive your car in reverse. I say, no, you're moving forwards with your car pointing in the other direction that it can point, when driving a car.
     
  19. Write4U Valued Senior Member

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    I explained the anthropomorphism involved in the "map/terrain" (subjective/objective), and the example of not planning (mapping) the route but practically driving it, i.e. "performing the function" of chronologies. I think you are agreeing with me but perhaps I have not made my perspective clear and concise.
    Bingo! Of course we agree, it's only logical.......

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  20. arfa brane call me arf Valued Senior Member

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    A mirror reflection is a symmetry; in physics, I think the rule is to compare a physical symmetry with a mirror reflection. In some cases there is a resemblance, in others not so much.

    So one way to classify physical symmetries is to see if they commute with mirror reflections. It seems to be the case that time-symmetry does this, in that any physical experiment can in principle run backwards in time without altering any laws of physics.
     

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