An Alternative to a singularity?

[Jeffocal]

Originally Posted by modest

I don't really follow what you're saying there. The speed of light works as a conversion factor between space and time. To switch from space units to time units (or from time to space) you multiply or divide by the speed of light. There is no conversion factor between spatial dimensions, so this once again shows that time is a little different from the other dimensions.

~modest

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.

Jeff

PS Would it be possible to use the fact that magnitude of time dilation and length foreshortening are connected in terms of the same space-time curvature to quantify an equivalent one in a four dimensional manifold???

[modest]

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:

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

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

[Jeffocal]

Maybe I should have used the term distorted instead of curved because and I think Einstein would agree that would be more accurate way of describing how both the geometry of space-time and four *spatial* dimensions is altered by mass.

Einstein derived equation that quantified the magnitude of length contraction and time dilation based on the total energy content of an object. The greater the velocity of an object the greater its energy and therefore the greater the relative time dilation and length shortening that object will experience due to the energy of its velocity.

However, correct me if I am wrong but this also means that Einstein defined the causality of all mass and energy in terms of a geometric distortion of both a spatial or length and time component of a space-time manifold.

What I am trying to says is does the fact that he analytically defined the effects energy has on space or length and time in terms of a common mechanism related to energy provide us with an connection between his space-time geometry and one of four *spatial* dimensions.

Would this provide us with a way of analytically determining if it would be possible to transpose his space-time geometry to four *spatial* dimensions without altering any of its predictive ability.

Jeff

[modest]

I'm sorry Jeffocal. Despite giving it my best, I just can't make sense of what you are saying.

Perhaps someone else has insight....

~modest