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In models such as M-theory with 7 'higher dimensions' plus the 4 macroscopic spacetime dimensions, where do our 4 macroscopic spacetime dimensions reside ordinally? My reason for asking is TV shows such as the 'Fabric of the Cosmos' that alludes to 'lower' dimensions in string theory. I can understand if our 4 are, say the 7th, 8th, 9th and 10th so there are dimensions lower than us but that we cannot readily detect.

Qmechanic
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PlaysDice
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  • I am way out of may region of expertise here, but my first reaction is that this is equivalent to asking which of the three spacial dimension we live in is "first", which is ill-defined. They all come into the interval in either a space-like or a time-like way $(\Delta s)^2 = \sum_{i \in \text{spacelike}}(\Delta x_i)^2 - \sum_{j \in \text{timelike}}(c \Delta x_j)^2$ (or the other sign convention if you prefer, of course) so they are all equivalent. – dmckee --- ex-moderator kitten Mar 20 '14 at 23:59
  • @dmckee. Aren't there specific mathematical definitions of (orthogonal) dimensions that tie dimensions N and N+1 together by things like projection? So, some sort of ordering should be possible, no? And in matrix math the spaces of "points, lines, plane" are based on tensors of 1st, 2nd and 3rd order etc. – PlaysDice Mar 21 '14 at 00:26
  • I think what dmckee is saying is that we can't really say that $x$ is the 1st ordinal dimension or 3rd ordinal dimensions when considering 3D only, how could we possibly say anything at higher dimensions? – Kyle Kanos Mar 21 '14 at 00:28
  • @KyleKanos Thanks. (I was a bit slow in the uptake, thanks dmkee.) So is the answer that M-theory does not (and does not need to) place our macroscopic universe in a particular ordinal range of possible dimensions? – PlaysDice Mar 21 '14 at 00:39
  • My pre-post search missed this answer re compactification that looks useful – PlaysDice Mar 21 '14 at 00:43

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Think of a very thin wire. It is a 3-dimensional object, but for many purposes you can describe it just as a 1-dimensional line or curve. The two remaining dimensions are curled up in a tiny cross section.

In a similar way, the speculations (not the slightest experimental hint exists that it should be so) about our world possibly being higher dimensional rest on the assumption that the remaining dimensions of our world are curled up so that for most purposes, only four of them are observable to us. The diameter of the curled up part is thought to be of the order of the Planck length, a scale far below the best current spatial resolution: http://en.wikipedia.org/wiki/Planck_length

Note that in string theory, the higher-dimensional universe has no boundary, so the wire is an imperfect lower-dimensional analogue. But the surface of an infinitely long wire has no boundary (though only one extra dimension - it is a 2-dimensional universe embedded in a 3-dimensional space) and is a correct 1+1-dimensional analogue of a 4+6 or 4+7 dimensional universe without boundary and with all but 4 dimensions compactified.

There is no ordering of the dimensions, just as the single dimension of the wire is not related to the x- y- and z-axis labeling three spatial dimensions.

Arnold Neumaier
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  • Thanks. Is there a reason for thinking the non-observable dimensions are below us? I'm particularly thinking of the people of Flatland who have no idea whats "above" them in 3D etc. – PlaysDice Mar 21 '14 at 17:02
  • @PlaysDice: I would visualize it as they would be ''around'' us. But words are not really telling as our imagination of higher dimensions is poor. Thus it is up to the imagination of the author to choose a word such as ''below'', ''above'', or ''around''. Note that flatlanders have no way of distinguishing between ''below'' and ''above'', as these dimensions are not perceptible by them - what they perceive is symmetric in the third dimension. – Arnold Neumaier Mar 21 '14 at 17:08