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Well, I suppose we could add time as a 4th. However, apart from that there is no experimental evidence whatsoever for more than 3 spacial dimensions. String Theory and its friends remain theories with negligible experimental evidence in their favor.

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There are many conditions for identifying that a vector field $\vec v$ is conservative or not: $\nabla \times \vec v =0$ A conservative field vector is essentially irrotational. $\oint_c \vec v \cdot d\vec r =0$ Work done by a conservative vector field about any closed path $C$ is $0$. $\vec v=\nabla \phi$ A conservative vector field can always ...

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Yes. 1. option: Another (equivalent) criterion: If the work done on a arbitrarily chosen closed path is zero, then the field is conservative. I.e: $$\oint \vec{F} \cdot \mathrm{d}\vec{r} = 0$$ means $\vec{F}$ is conservative. 2. option: More general definition of curl  \left(\mathrm{curl} \ \vec{F} \right)\cdot \vec{n} = \lim_{A \rightarrow 0} ...

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Extra dimensions in general have to be compact, since a four-dimensional description fits the world we perceive and measure (so far) very well. It is this compactness that sets a length scale. Usually, we assume that the four "regular" space-time dimensions are not compact, i.e. extended infinitely. In higher dimensions concepts like angular momentum, ...

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What is the process, the rule or the method which leads me to understand/calculate the size of these extra dimensions? How can I understand or prove that, taking the String example, the additional dimensions are so small? Where does the number of their magnitude come from? At the present moment we have no experimental evidence that there exist the extra ...

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Historically this has always been the problem with the Kaluza-Klein approach. In the original Kaluza-Klein theory there was no mechanism to determine the scale of the compactified dimension, and indeed one of the criticisms of it was that the compact dimension was unstable and would naturally expand to infinity. In the context of string theory the problem ...

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Speed itself... is a type of "negative and/or anti length" and "negative and/or anti time" dimension and/or probable "quantum" projection of "either/or". By formula, the amount of time used to transverse a distance is reduced by speed. As speed increases, the amount of time used decreases... and/or the length is, by first observer, shortened. In order to ...

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When Brian Greene talks about the shape of the extra dimensions he is using a simple word for some exceedingly complicated mathematics. For example suppose you are trying to compactify just two dimensions - call them $x$ and $y$ for convenience. Compactifying the dimensions means forming them into a loop, but starting from a flat sheet you could loop the two ...

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According to Brian Greene it is possible to calculate the physical constants from the shape of the extra dimensions. This isn't scientific fact I'm afraid. It's perhaps presented as such, but there's absolutely no evidence for string theory, and hasn't been for fifty years. Is it possible to do the inverse, so predict the shape of these dimensions ...

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