"Michael S. Lorrey" <email@example.com> writes:
> John Clark wrote:
> > I don't know, last I heard 10 dimensions were sufficient but things are changing
> > so fast it's hard to keep up. The gravitational field at a point can not be expressed
> > with a single number, you need 10 numbers (dimensions) for each point.
> > For this reason the gravitational field must be a tensor field. The number 10 comes up
> > because there are 10 and only 10 ways time and the three dimensions of space can
> > be expressed in pairs. I don't know why you'd need 11 dimensions.
> John, please explain this. I count the following:
> for dimensions x,y,z, and t:
> so unless you count x,x y,y z,z and t,t, then you only have six pairs. Am I missing
> something? If you count double pairs like that, why not reverse pairs as well? I must be
> thinking too literally....
The reason is that these pairs are due to the metric tensor g. g represents how distances are measured at different points in spacetime, you can view it as a matrix of numbers (which may vary from point to point). The squared distance is given by
ds^2 = sum_ij g_ij dx_i dx_j
where ds is the distance between two infinitesimally close points, dx_i their coordinate difference along coordinate i and I use a TeX like notation for subscripts. g_ij means the component of g in the ith row and jth column.
For normal euclidean space g is zero except for g_11, g_22, g_33 and so on, the distance between two points is given by Pythagoras' theorem: ds^2 = dx^2+dy^2+dz^2
In Lorentz space (as in special relativity) time is positive and space-coordinates have the opposite sign: ds^2 = c^2 dt^2-dx^2-dy^2-dz^2 where c is the speed of light.
So the double pairs in some sense correspond to the usual distance measures, while the other pairs are more of "mixtures" that occur when the coordinate system is not perfectly aligned with the curvature of space (locally, you can always set up coordinates to make the tensor diagonal, but it doesn't work in general globally).
Now, g is symmetric (that is part of the definition, actually),
g_ij=g_ji, and in 4-dimensional space that means you can only have 10
However, I'm not entirely certain the 10 components of g explain why
there are 10 (11) dimensions in string theory, the last paper I read
about it ("M-theory for laymen" or something similar, available from
xxx.lanl.gov) just made me more confused.
However, I'm not entirely certain the 10 components of g explain why there are 10 (11) dimensions in string theory, the last paper I read about it ("M-theory for laymen" or something similar, available from xxx.lanl.gov) just made me more confused.
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