The article "Defining energy" Nov 26, 2007 showed it is possible to derive all the forces of nature including gravitational in terms of a curvature or displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, some believe that gravitational forces because of their spherical properties cannot be defined in terms of the existence of only four *spatial* dimensions.

This would be true if the movement of three-dimensional space is restricted orthogonally to a fourth *spatial* dimension.

Observations of our three-dimensional environment tell us we can move and rotate a two-dimensional surface independently with respect to each axis of three-dimensional space and that its individual axes are not required to be orthogonal to each other.

We know this because the surface of a balloon can be rotated in any direction with respect to three-dimensional space and the individual axis of its surface are not required to be orthogonal to each other.  Additionally we observe that when a force is applied to it with respect to three-dimensional space every point on it either or contracts or expands while the angles the individual dimensional axis of that surface make with each other shift to conform to that curvature.

We have shown throughout "The Imagineer’s Chronicles" that observations suggest the existence of a fourth *spatial* dimension which has properties similar to those of the three-dimensions that make up our environment.  This means the axes of a three-dimensional sphere should interact with a fourth *spatial* dimension in manner similar to the way a two-dimensional surface interacts with three-dimensional space.

Therefore, based on observations of our three-dimensional environment we can assume the individual axes of four-dimensional space are not required to be orthogonal to each other.  Additionally they also indicate that every point on a "surface" of a three-dimensional manifold that defines a volume could either expand or contract with respect to a fourth *spatial* dimension while the orientation of the axis of that volume could shift to conform to the curvature associated with that expansion or contraction.

This indicates it would be possible to define the spherical nature of a gravitational field in terms of a spherical curvature in a "surface" of three-dimensional space with respect to a fourth *spatial* dimension while defining the causality of gravity in terms of a contraction or expansion of that surface. 

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2009 Jeffrey O’Callaghan
 

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