Gravity in four *spatial* dimensions

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The article “Defining potential and kinetic energy?” Nov 26, 2007 showed it is possible to derive causality of all the forces of nature including gravity 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 only four *spatial* dimensions.

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

For example we observe that we can move or change the orientation of a two-dimensional plane such as the surface of a piece of paper in three-dimensional space independently with respect to each of its axes.  This indicates that the axes of a two-dimensional surface are not rigidly fixed but embedded into three-dimensional space.

However, we also observe that we can move or change the orientation of a three-dimensional volume independently with respect to each of its spatial axes.

This suggests each axis of three-dimensional space may be embedded in a universe consisting of four *spatial* dimensions in a manner similar to how a two-dimensional piece of paper is embedded in three-dimensional space.  In other words the origins of the axes of a four dimensional universe is not rigidly fix to each other but are embedded in it allowing for the independent movement of each of its axis with respect to each other.  Therefore, it would be possible to orient each axes of a “surface” of a three-dimensional space manifold independently of its orientation to the other axes of four *spatial* dimensions. 

For example if we move a two-dimensional piece of paper through three-dimensional space by pushing on its center, its surface will develop a curvature with respect to it because of the drag generated by the space it is moving through.  A two dimensional creature living on its “surface” would not realize the surface of the paper is curved with respect to three-dimensional space because, as mentioned earlier he or she could not “look” in that direction.  However gravity will cause a tangential force to be developed along its surface.

Similarly if a three-dimensional object is moved through a fourth *spatial* dimension, its three-dimensional “surface” will develop a curvature due to that movement.  This is similar to how the surface of the paper developed a curvature due to it movement through three-dimensional space.  It was shown in the article “Defining potential and kinetic energy?” Nov. 22, 2007 this curvature in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension is the causality of kinetic forces.

We also observe that it is possible to curl a two-dimensional surface into a sphere forming a balloon in three-dimensional space because as mentioned earlier the axes of a two-dimensional surface are not fixed to the axis of three-dimensional space.  Additionally we observe that we can increase or decrease the magnitude of the curvature of the “surface” of a balloon by increasing or decreasing its internal pressure.

Similarly, a “surface” of a three-dimensional space manifold can be curled to form a three-dimensional “sphere” in four *spatial* dimensions because axes of its “surface” are not fixed to the axes four-dimensional space.  This is analogous to how a two-dimensional surface can be curled to forum a three-dimensional sphere in three dimensions. 

As was shown in the article “Gravity in four spatial dimensions” Dec. 15, 2007 the force developed by this spherical curvature in a three-dimensional space manifold with respect to a fourth *spatial* dimension is responsible for gravity.

Similar to the spherical surface of the balloon a curvature in a “surface” of three-dimensional sphere will contract or expand if energy/mass is added to or removed from its center.  This will result in increasing or decreasing the magnitude of the curvature in the “surface” of the three-dimensional sphere and the gravitational forces associated with energy/mass in its volume.

This shows that one can derive the spherical geometry of a gravitational field by extrapolating the properties of a three-dimensional environment to a fourth *spatial* dimension.

Later Jeff

Copyright Jeffrey O’Callaghan 2009

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