Minkowski in 5-D: what is the fifth?

[nomadreid]

In the equation for the Minkowski 5-D space in which deSitter space is embedded, I read the equation t^2 -x^2-y^2-z^2-v^2=-H^-2. What is v?

 

[Amanheis]

It's the fifth coordinate. De-Sitter-space is a 4 dimensional submanifold of R^5, or I should say R^(1,4) I guess.

 

[nomadreid]

Yes, I know it's the fifth coordinate. The question was whether one can represent it as any measurable quantity, as the other four are (3 spatial and one time). In string theory, of course, it is a rolled-up tiny other spatial dimension, but is this what is meant when discussing the 5-D Minkowski space of which the deSitter space is a submanifold?
While I am at it, I am still puzzling how one gets a positive curvature out of deSitter space if the metric on the deSitter space is the same as on the ambient Minkowski space, which gives a negative curvature.

 

[Amanheis]

It is just the space that the manifold is embedded in. It is still a 4 dimensional manifold on which you have to introduce coordinates before you do physics on it. It's just easier to describe as a submanifold, but it still represents an ordinary 4 dimensional world.

Just like the 2-sphere is described by x^2+y^2+z^2=1. There are three coordinates in the embedding space, but in the end you just introduce two coordinates to describe the manifold you are interested in, for example phi and theta. It doesn't make much sense to me to ask what "z" is "representing" in this case.

 

[nomadreid]

Thanks. Your example was illuminating.