dimensions

[1123581321]

Hi,

 

I was wondering if it is known what determines spacial dimensionality. As in, for example our 3D reality (of everyday) and up to the 11th dimension (in hyperspace) ?

 

thanks.

[ajb]

The only theory I know of that predicts the dimensionality of space-time is string theory. This is a little technical, but in order for superstring theory to be well defined as a quantum theory the dimension must be 10. (26 for the bosonic string). If this is what you are asking? Why out of all these dimension we live in only 3 is really a mystery.

 

Or are you asking about the separation of space-time into space and time?

 

A space-time comes with a metric, usually with a Lorentzian signature. That is at a point the metric can be written as [math]diag(-1,1,1 \cdots 1)[/math]. Note you will not in general be able to cast the metric in this form other than at a point. Such a space-time is said to have one time dimension and n space dimensions. In all there is n+1 dimensions.

 

It may also be useful to consider metrics with a Euclidean signature, [math]diag(1,1 \cdots 1)[/math]. In this case there is no time, but the advantage is that the metric is positive definite. This may be essential in the mathematics. Often you can relate results in the Euclidean signature to the Lorentzian signature.

 

Even more generally, one can consider metrics of signature [math]diag(-1, \cdots, -1, 1, \cdots, 1)[/math]. Such metrics come up in twistor sting theory and related things. The space-time has say, m time dimensions and n space dimensions.

 

Now, one cannot really separate space-time into space and time. If you do this in one coordinate system then in general this separation will not be respected when you change coordinates.