4 dimensional spacetime manifold question

[closet mathemetician]

I'm having trouble understanding exactly what this manifold is. Let me draw an analogy: Say I have a flat map of the world. The map is a two-dimensional surface with a coordinate chart on it. However, its embedded in a higher three-dimensional space.

So by analogy, is the four dimensional spacetime manifold of Einstein equivalent to our three spatial dimensions ("the map") that is embedded in a higher fourth time dimension, or are all four dimensions and related coordinates (x,y,z,t) part of the surface of the "map"?

If that were the case, and the "map" is curved, then does that mean there must be at least five dimensions?

 

[HallsofIvy]

I'm having trouble understanding exactly what this manifold is. Let me draw an analogy: Say I have a flat map of the world. The map is a two-dimensional surface with a coordinate chart on it. However, its embedded in a higher three-dimensional space.

So by analogy, is the four dimensional spacetime manifold of Einstein equivalent to our three spatial dimensions ("the map") that is embedded in a higher fourth time dimension,[\quote]
Surely this isn't what you meant to say! A flat map of the world isn't embedded in the third dimension, it is embedded in space that has 3 dimensions.

or are all four dimensions and related coordinates (x,y,z,t) part of the surface of the "map"?

Those are just different ways of looking at the same thing.

If that were the case, and the "map" is curved, then does that mean there must be at least five dimensions?

Mathematically, at least, it is not necessary for a space to be embedded in higher dimensions in order to be curved. "Curvature" is an intrinsic property.

 

[closet mathemetician]

Surely this isn't what you meant to say! A flat map of the world isn't embedded in the third dimension, it is embedded in space that has 3 dimensions.

You are correct, I meant embedded in a space that has 3 dimensions.

Let me try again, in terms of ambient and local coordinates. A two-dimensional map embedded in a space that has 3 dimensions has local coordinates n(i), where i=1..2.

The ambient 3-space has n(j) coordinates where j=1..3.

By analogy, can we look at spacetime as a 3-dimensional spatial "surface" or "manifold" with local coordinates of n(i), where i=1..3, embedded in a 4 dimensional space with ambient coordinates n(j) where j=1..4?

I'm thinking of the light cone diagrams where spacetime is shown as three-dimensional. In those diagrams you have a two-dimensional spatial plane moving through a 3-dimensional space, where the third dimension is t, the time dimension. Now supposedly in the light cone examples, the entire 2-d plane is traveling through the time dimension at the speed c. Therefore, any residents of this 2-d plane are all moving through the t dimension at the same rate so they must be on some kind of "surface" or "plane" that travels through the higher dimensional space.

Maybe I'm rambling here, but basically, are all 4 dimensions of spacetime analagous to a "surface" or is there a 3-d "surface" embedded in a higher 4-d space?

And I understand how distance is intrisic to the surface of a manifold, but how is curvature intrisic? I guess I need to go read more about the second fundamental form, etc ..

 

[nrqed]

You are correct, I meant embedded in a space that has 3 dimensions.

Let me try again, in terms of ambient and local coordinates. A two-dimensional map embedded in a space that has 3 dimensions has local coordinates n(i), where i=1..2.

The ambient 3-space has n(j) coordinates where j=1..3.

By analogy, can we look at spacetime as a 3-dimensional spatial "surface" or "manifold" with local coordinates of n(i), where i=1..3, embedded in a 4 dimensional space with ambient coordinates n(j) where j=1..4?

I'm thinking of the light cone diagrams where spacetime is shown as three-dimensional. In those diagrams you have a two-dimensional spatial plane moving through a 3-dimensional space, where the third dimension is t, the time dimension. Now supposedly in the light cone examples, the entire 2-d plane is traveling through the time dimension at the speed c. Therefore, any residents of this 2-d plane are all moving through the t dimension at the same rate so they must be on some kind of "surface" or "plane" that travels through the higher dimensional space.

Maybe I'm rambling here, but basically, are all 4 dimensions of spacetime analagous to a "surface" or is there a 3-d "surface" embedded in a higher 4-d space?

And I understand how distance is intrisic to the surface of a manifold, but how is curvature intrisic? I guess I need to go read more about the second fundamental form, etc ..

The analogy is rather that all 4 dimensions are analogue to a "surface". This is why we talk about *spacetime* being curved. when masses are present, for example, it's not only space but also time that is distorted. But there is no need to imagine an ambient 5-d space in which the 4-dimensional spacetime curves. No extra dimension is required in the context of General Relativity.

Patrick

 

[kryptyk]

Embedding Problem

It can be useful in practice to embed manifolds into (pseudo)Euclidean spaces of higher dimension since the tangent spaces at all points in our manifold will be (pseudo)Euclidean hyperplanes.

Curvature is intrinsic in the sense that these tangent hyperplanes are different at different points of our manifold regardless of how we do the embedding. Flat manifolds have the property that the tangent hyperplane is the same at all points on the manifold.

 

[closet mathemetician]

It can be useful in practice to embed manifolds into (pseudo)Euclidean spaces of higher dimension since the tangent spaces at all points in our manifold will be (pseudo)Euclidean hyperplanes.

Curvature is intrinsic in the sense that these tangent hyperplanes are different at different points of our manifold regardless of how we do the embedding. Flat manifolds have the property that the tangent hyperplane is the same at all points on the manifold.

Different in the sense that they each have a different slope at each point of tangency?

 

[robphy]

It can be useful in practice to embed manifolds into (pseudo)Euclidean spaces of higher dimension since the tangent spaces at all points in our manifold will be (pseudo)Euclidean hyperplanes.

Curvature is intrinsic in the sense that these tangent hyperplanes are different at different points of our manifold regardless of how we do the embedding. Flat manifolds have the property that the tangent hyperplane is the same at all points on the manifold.

It seems to me that one would do this for purposes of visualization only (i.e. to draw the tangent planes of various sample points)..as long as one knows how to read the visualization. Embedding isn't necessary... especially since one is interested in intrinsic quantities.

Comparison of vectors in different tangent planes of a manifold requires a connection on the manifold. Embedding isn't necessary... and it may even be distracting or misleading.

 

[Haelfix]

Embedding is a crutch that you have to learn to get rid off. It doesn't generalize very well, particularly in the cases where we have manifolds that lack much symmetry. Very often we are forced to multiply by two the amount of dimensions (so the amount of degrees of freedom starts growing drastically).

Moreover, I have no good visualization of what those embedded spaces should look like either, so its not clear what it buys you.