Quote:
Originally Posted by Jeffocal
I guess what I am tiring to say is that Einstein defined (I hope I get this right  ) the magnitude of a gravitational potential in terms of a curvature in a surface of a space-time manifold and what we are saying is that it is a result of a curvature in a "surface" of a three-dimensional manifold with respect to a fourth *spatial* dimension. I guess I am asking is there a way of mathematically converting the magnitude of a space time curvature to an equivalent spatial distance. If so could that be used to define the magnitude of the distance a "surface" of a three-dimensional space manifold would be displaced with respect to a fourth *spatial* dimension to cuase an equivalent space-time curviature.
Then could we substitute that value in for the values that represent the space-time curvature in Einstein's field equations to quantify them in terms of four *spatial* dimensions. |
Sorry for the delayed response.
I still don't know what you're looking for exactly. According to general relativity, 3 dimensions of space and one dimension of time are curved. They are all 4 curved. They theory
does not say what they are curved into. If you want to say that they are curved into a higher dimension then that's fine. It would not change the predictions or the structure of the theory at all. It's the difference between extrinsic and intrinsic curvature. Extrinsic curvature is the curvature of a surface into a higher dimension. Intrinsic curvature is a curved surface (regardless of higher dimensions). It makes no difference if the intrinsically curved spacetime in general relativity has a 5th dimension which it is extrinsically curved into or not. Notice:
Quote:
It is important to realize that the local geometry or curvature characterized by (2.4) [Einstein’s field equation] is an intrinsic property of the manifold itself, i.e. it is independent of whether the manifold is embedded in some higher-dimensional space.
General relativity: an introduction ... - Google Books
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The important thing is that space and time are both curved. Whatever else you want to say about what they are curved into is fine, but should not change the particular answers you get with the theory.
~modest
