# Re: Superstring Theory Explanation

Anders Sandberg (asa@nada.kth.se)
16 Feb 1999 12:22:20 +0100

"Michael S. Lorrey" <retroman@together.net> 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:
> x,t
> x,y
> x,z
> y,t
> y,z
> z,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.

```--
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Anders Sandberg                                      Towards Ascension!
asa@nada.kth.se                            http://www.nada.kth.se/~asa/
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