Gravitation in N+1 Dimensional Flat Space

[ophase]

We very well know how to calculate curvature in gravitation. But this time i just need a physical explanation to this question on my mind:

"In order to describe n dimensional space with constant curvature why do we need to go to n+1 dimensional flat space? Why don't we use just n-dimensional spherical coordinates instead?"

I know this is about the curvatures of hypersurfaces and every hypersurface is an n-dimensional manifold embedded in an n+1 dimensional space. So there may be some mathematical difficulties in calculation. But isn't there any other methods to describe n dimensional space with constant curvature?

 

[WannabeNewton]

What kind of curvature are you talking about? You have to be specific in this case. For example, the notion of constant sectional curvature makes perfect sense for riemannian manifolds, with no need for ambiance. Constant sectional curvature is what leads to the three possible cases: hyperbolic geometry, elliptic geometry, and euclidean geometry. Do Carmo's text on Riemannian Geometry has an entire chapter on this.

 

[Bill_K]

"In order to describe n dimensional space with constant curvature why do we need to go to n+1 dimensional flat space? Why don't we use just n-dimensional spherical coordinates instead?"

See the Wikipedia page "n-sphere", which gives spherical polar coordinates and also stereographic coordinates for the n-sphere.

 

[bcrowell]

"In order to describe n dimensional space with constant curvature why do we need to go to n+1 dimensional flat space?

We don't. Flat space is different from a space with constant curvature (unless the curvature happens to be zero).

Why don't we use just n-dimensional spherical coordinates instead?"

Curvature is coordinate-independent. Changing from one set of coordinates to some other set of coordinates doesn't make zero curvature become nonzero, or vice versa.

I know this is about the curvatures of hypersurfaces[...]

No, it isn't.

 

[Chestermiller]

We very well know how to calculate curvature in gravitation. But this time i just need a physical explanation to this question on my mind:

"In order to describe n dimensional space with constant curvature why do we need to go to n+1 dimensional flat space? Why don't we use just n-dimensional spherical coordinates instead?"

I know this is about the curvatures of hypersurfaces and every hypersurface is an n-dimensional manifold embedded in an n+1 dimensional space. So there may be some mathematical difficulties in calculation. But isn't there any other methods to describe n dimensional space with constant curvature?

It's easy to envision a 2D curved space immersed in a 3D flat space. Just think of spherical surface or a saddle surface in 3D flat space. In describing gravity, 4D spacetime is curved, and, conceptually, it is simple to imagine (in an analogous way) 4D spacetime immersed in a 5D flat manifold. This simplifies ones visualization of what is happening geometrically, but it is not really necessary for arriving at the correct mathematical formalism for describing gravity and the curvature of spacetime.

 

[ophase]

In other words, I am wondering about why we need to embed an n dimensional sphere to n+1 dimensional flat space, in order to derive the metric? If the reason is really a simplified mathematical formalism, i can say that it is not getting easier at all by this way.

What kind of curvature are you talking about? You have to be specific in this case. For example, the notion of constant sectional curvature makes perfect sense for riemannian manifolds, with no need for ambiance. Constant sectional curvature is what leads to the three possible cases: hyperbolic geometry, elliptic geometry, and euclidean geometry. Do Carmo's text on Riemannian Geometry has an entire chapter on this.

 

[Bill_K]

In other words, I am wondering about why we need to embed an n dimensional sphere to n+1 dimensional flat space, in order to derive the metric? If the reason is really a simplified mathematical formalism, i can say that it is not getting easier at all by this way.

As we've said repeatedly, it is NOT necessary. Did you check out the stereographic coordinates on the "n-sphere" Wikipedia page, like I suggested?