Can space-time singularities be treated as mathematical knots occurring in dimensions greater than four? I just drew an analogy with knots in one-dimensional strings. When a rubber-band is looped over again and again, it ultimately forms a clumped-up ball like structure, having 3 dimensions. Similarly, the fabric of space-time may also get looped over and over again to form singularities having dimensions greater than four.
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$\begingroup$ Singularities are a bit more like this: Locally, time seems like a very long piece of string, with events such as the turning of a clock hand evenly spaced on it. But if you were to travel backwards towards the big bang, you would find the hands of the clock turning increasingly fast. It's a bit like taking ever smaller steps as you approach something, only ever halving the distance between you and it in any given step. You can never actually get there. That's a singularity. $\endgroup$– it's a hire car babyOct 15, 2019 at 20:08
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2 Answers
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It depends on your interpretation of space and time. Because you are talking of knots, I shall only consider space, not time, as dimensions. For example, on the quantum scale,one and two dimensions are not PHYSICALLY possible, as you cannot have zero thickness given the quantum fluctuations. But assume that it is possible, at least on a macroscopic scale.If the fabric of spacetime , and if I am right, the tangent space at each point of the energy or mass distribution, gets curved, and then looped over and over, there cannot be any change in dimensions, because when you go on the quantum scale, every dimension that you add (in analogy to the knots you referred to) to that loop, should be actually 3-dimensional. If it weren't so, then the curvature of space time would only be limited to the tangent spaces around, as fa as "one dimension" permits, which is quite senseless, as there is nothing that be defined physically as one dimensional. However, we know that the curvature is spatially extended, and that is what defines the grasp of the gravitational field. In other words, the tangent space to a point on the body of matter, when curved, leads to the bending of another space tangent to it, and it continues, depending upon the mass of the body. Or, when they extend their curvature in a continuous sequence they create a "height" or "width" eventually for the space curved, so that it is no longer 2D, but 3D. So, for your question, if you add loops to the curvature, each dimension added up would eventually become three dimensional when the loop is sequentially repeated. Roughly, it is the same as a string (assume it to be one dimensional)overlapped with itself again and again, until it acquires a noticeable 3D structure, like a reel or a ball. In brief, such clumping would be 3 dimensional on the whole, at least physically, if not mathematically.
Taking the analogy to singularities, they are also similarly 3 dimensional.
My answer, I guess, was philosophical. Pardon me for any mistakes as I probably couldn't explain what I really meant to: Its my first answer. But that's what I think about dimensional structure of curvatures. I welcome any improvement or suggestions, as I have no "proof" from any source about this. Hope this helps!!
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$\begingroup$ Then I must regard the seemingly one-dimensional band as a flexible 3-D cylinder, on the quantum scale. But still, I am not much confident with what you say, because some words are bit too professional for me to understand. $\endgroup$– abstractMay 22, 2014 at 10:55
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$\begingroup$ Yes, but I think that that the 'flexibility" will not be an inbuilt property, it should be a property of the geometry of spacetime depending on the gravitational field... $\endgroup$– GRrocksMay 23, 2014 at 8:44
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You seem to imply that the knotting of the band increases its number of dimensions. But in actual fact the band already has three dimensions because its thickness is non-zero, and the ultimate size of the ball is governed by the sum of all those the thicknesses next to each other.
If your analogy were to be made correct, we would use an elastic band of zero thickness and then when you balled it all up, the volume of the ball would be zero.
answered Mar 30, 2017 at 11:00