Curved Space - Embedding

Embedding

One of the defining characteristics of a curved space is its departure with the Pythagorean theorem. In a curved space

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The Pythagorean relationship can often be restored by describing the space with an extra dimension. Suppose we have a non-euclidean three dimensional space with coordinates . Because it is not flat

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But if we now describe the three dimensional space with four dimensions we can choose coordinates such that

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Note that the coordinate is not the same as the coordinate .

For the choice of the 4D coordinates to be valid descriptors of the original 3D space it must have the same number of degrees of freedom. Since four coordinates have four degrees of freedom it must have a constraint placed on it. We can choose a constraint such that Pythagorean theorem holds in the new 4D space. That is

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The constant can be positive or negative. For convenience we can choose the constant to be

where now is positive and .

We can now use this constraint to eliminate the artificial fourth coordinate . The differential of the constraining equation is

leading to .

Plugging into the original equation gives

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This form is usually not particularly appealing and so a coordinate transform is often applied:, . With this coordinate transformation

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