Embedded dimensions.

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We have postulate that all the forces in nature can be explained and predicted by extrapolating the mechanics of a three-dimensional environment to four *spatial* dimension.

For example one can explain gravity as Einstein did in terms of a curvature in a “surface” of a three-dimensional space manifold with respect to either a time or a fourth *spatial* dimension.

However one advantage to doing so is that it would allow one to not only derive gravity but also potential and kinetic energy in the same terms.

It should be remember Einstein gave us the ability to chose how to define all forces in terms of their spatial or time characteristics when he qualitatively and quantitatively defined the geometric properties of a space-time universe in terms of the constant velocity of light.  This is because it allows one to redefine a unit of time he associated with energy in his space-time universe to unit of space in a one consisting of only four *spatial* dimensions. 

In other words by defining the geometric properties of a space-time universe in terms of constant velocity of light he provided a qualitative and quantitative means of redefining his space-time universe in terms of the geometry of four *spatial* dimensions.

We will use an analogy of a two-dimensional creature living on surface of a piece of paper to derive a common physical link between gravity potential and kinetic energy in terms of an interaction between three dimensional space and a fourth *spatial* dimension.

A two dimensional creature “living” on the surface of a piece of paper would not be aware the paper he or she was living on existed in a three-dimensional universe because his field of vision would be limited to the surface or length and width of the paper.

Therefore, he or she would not be aware of the existence of the dimension of height or a third *spatial* dimension because he or she could not look in that direction.
As will be shown latter, the energy three-dimensional beings use to activate their senses does NOT travel through a fourth *spatial* dimension but only on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Therefore, similar to a two-dimensional creature, three-dimensional beings would not be aware of it because it is not included in their field of vision.

However, even though we as three-dimensional being will never be able to see a fourth *spatial* dimension we can as mentioned earlier derive its realty by “seeing” the effects it has on our three-dimensional environment.

But before we begin analyzing these effects we must first have an understanding of how the individual dimensions are oriented with respect to each other.

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 in three-dimensional space.

However, we also observe the 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. 

If we move a two-dimensional surface of a 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 the “surface” of the paper 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 will be shown in the article “The Relativity” of four spatial dimensions” Dec. 01, 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.

Potential energy would be define by the physical difference with respect to a fourth *spatial* dimension of two surfaces of three-dimensional space .

This would be analogous to the energy stored in a piece of paper that was at a different height from another one.

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 form a three-dimensional sphere in three dimensions. 

It will be shown in “Defining gravity“ Dec.15, 2007 that 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 pressure or 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 that mass.

(This “curvature” or distortion in the “surface” of a three-dimensional space manifold with respect to the fourth *spatial* dimension is analogous to the space-time curvature that Einstein postulated was responsible for the force of gravity in his General Theory of Relativity.)

One might ask how the geometry of four-dimensional space can be altered from three-dimensional space to account for the forces of nature.

In the article “Defining gravity“ it was shown when mass is converted to energy, it “expands” towards a fourth *spatial* dimension.  This results in increasing the “pressure” on a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension causing it to become distorted with respect to the “vertical” or “W” axis of the fourth *spatial* dimension.

If the “expansion” of mass to energy is directed only along one of the three axes of three-dimensional space, the “pressure” and the force this it causes will result in acceleration along that axis.

As mentioned earlier the article “The Relativity” of four spatial dimensions” this distortion in a “surface” of a three-dimensional space manifold caused by a conversion of mass to energy is responsible for kinetic forces.

Another way of describing how objects in a third spatial dimension can have an effect on a fourth is by comparing the mechanism responsible for their interactions with it to that of a steam engine.

In a steam engine, water expands in the form of steam from the two-dimensional surface of the water.  This expanding steam generates a force that distorts the two-dimensional geometry of the surface of a piston by causing it to move with respect to vertical axis of three-dimensional space.

As mentioned earlier when mass is converted to energy, it “expands”, in the form of energy, from the three-dimensional “surface” of the mass in the “direction” of a fourth *spatial* dimension.  This expanding mass, in the form of energy generates a force on a “surface” of a three-dimensional space causing it to “move” with respect to the “vertical” or “W” axis of the fourth *spatial* dimensional.

This is analogous to how the steam in a steam engine generates the force on the surface of the piston that results in the two-dimensional surface of a piston to move or become distorted with respect three-dimensional space.

We can also use the analogy of a two-dimensional creature to get a better understanding how and why the energy “contained” a three-dimensional mass can cause the forces of nature.

However instead of “living” on the surface of a piece of paper as in the earlier example, the two-dimensional creature will be “living” on the surface of water which will also be considered the surface of a piston in a steam engine.

If the water were heated to the boiling point the steam would expand towards the volume above its surface putting pressure on it causing it to move with respect to vertical axis of three-dimensional space.

However, the two-dimensional creature living on its surface could not directly tell where the steam had originated that was causing its geometry to move with respect to three-dimensional space because, as mentioned earlier he or she could not “look” down in the direction that it was coming from.

But, if he had placed marks on the wall of the piston he or she could indirectly tell the geometry of his surface had changed by looking along the surface at those marks as the piston moved passed them.

He or she could determine the distance the surface had moved because he or she could “see” and count the marks on the wall of the piston as it passed them.  Since the magnitude of the force on the surface of the piston determines the distance it would move the two-dimensional creature would then have a way of determining the total force on the surface of the piston by counting these marks.

Just as a two dimensional creature cannot look down to see the volume of water below its surface where the steam originates from, we as three-dimensional beings cannot look “down” to see the “volume” of mass below a “surface” of a three-dimensional space manifold where energy originates from.

When mass “expands” to energy it generates a force on a “surface” of a three-dimensional space manifold, which results in its “surface” “moving” with respect to a fourth *spatial* dimension.

The “separation” in the “surfaces” of two three-dimensional space manifolds with respect to a fourth *spatial* dimension caused by this movement defines the relative energy differential between two different volumes of three-dimensional space.

In addition, three-dimensional beings could not directly tell the “surface” of a three-dimensional space manifold had “moved” or become distorted with respect to a fourth *spatial* dimension because three-dimensional beings cannot look in the direction of a fourth *spatial* dimension.

It has been and will be demonstrated throughout this blog that all forces of nature are associated with a “curvature”, distortion or a “spatial separation” in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.  The depth or magnitude of this “curvature” determines the relative magnitude of the forces between two points in space.

It should be remember Einstein’s genius allows us to choose to define all systems in either a space-time environment or one consisting of four *spatial* dimension when he defined the geometry of space-time in terms of the constant velocity of light. This interchangeability broadens the environment encompassed by his theories thereby giving us a new perspective on the casual of forces how they physically connected to our observable universe.

Latter Jeff

Copyright Jeffrey O’Callaghan 2007

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