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1kWe have shown throughout this blog that there would be many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four dimensional space- time.
One is that it would allow for the theoretical derivation of the relativistic of properties of three-dimensional space and the inertial properties of energy/mass in terms of a four dimensional Newtonian universe.
In the article “Defining energy†Nov 27, 2007 is was showed all forms of energy including kinetic and gravitational can be derived in terms of a displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension while the article “Defining time?” Sept. 20, 2007 showed defining it only in terms of a measure of distance between the sequential ordering of the causality of events would provide an unambiguous definition that is consistent with both physical and mathematical observations of it.Â
However, as was shown in the article “Embedded dimension” Oct. 22, 2007 we as three-dimensional beings can only perceive the cord of an arc generated by a displacement in a “surface” of three-dimensional space with respect to a four *spatial* dimension.Â
This means the Newtonian concept of absolute three-dimensional space would not exist in a universe consisting of four *spatial* dimensions because, as that article showed three-dimensional beings are only able to observe cord of the arc generated in a “surface” of three-dimensional space by gravity or the kinetic energy of a moving reference frame. Therefore because the cord of an arc is shorter than the arc itself, three-dimensional beings will view the distance between two points or events in those reference frames to be shorter than would be if they were not in relative motion or in a gravitational field. Hence, the Newton concept of absolute length would not exist because length in all reference frames would be influenced by gravity or their relative velocities in a four dimensional environment.
However, the passage of time will also be affected in reference frames that are in relative motion or in a gravitational field if one defines it in terms of measure of the “distance between” the sequential ordering of the causality of an event as was done in the article “Defining time?” because the distance between events in them will be greater due to the curvature in three-dimensional space caused by their gravitational field or relative motion.
Therefore, time will become dilated in reference frames that are in motion or in a gravitation field because the curvature generated in three-dimensional space by the displacements those article associated with gravity and relative motion will cause the distance between events to be longer in it than it would be for an observer who is outside of it. Therefore external observers will view time in each other’s reference frame to be moving slower with respect to their time because of the greater distance those events must travel.
Therefore, the Newtonian concept of absolute time would not exist in a universe consisting in a four *spatial* dimensions.
This shows how one can derive the relativistic properties of three-dimensional space in terms of a universe consisting four dimensional Newtonian space.
Yet one can also derive the causality of the classical concepts of inertia, mass and momentum by extrapolating the Newtonian laws of three-dimensional space to a fourth.
Isaac Newton defined inertia as being responsible for why an object at rest will remain at rest, and one in motion will remain in motion in a straight line at a constant speed.
In the article “Why Space-time?” Sept. 27, 2007 it was shown that one can derive the energy associated with the rest mass of an object or particle in terms of the “depth” of a displacement caused by a curvature in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension. Additionally it was shown one can derive the causality of all forces including gravitational in terms of an interaction of mass with the slope of a curvature in a “surface” of a three-dimensional space caused by the displacement.
(This curvature is analogous to a curvature in a four-dimensional space-time manifold Einstein theorized was the causality of accelerations.)
As mentioned earlier the article “Defining energy†derived the energy associated with the velocity of an object in terms of the displacement with respect to a fourth “spatial” dimension of a volume of three-dimensional space containing it.
Therefore, the inertia or momentum of an object would, according to the concepts presented in those articles be defined by the sum of the displacements in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension associated with its rest mass and its relative velocity.
The first component would be the displacement in a “surface” of a three-dimensional space associated with the rest mass of an object, whose magnitude as the article “Why Space-time?” showed is defined by the magnitude of its mass. While second would be the momentum or the magnitude of a displacement that the article “Defining energy” showed was the casualty of the energy or momentum of its relative motion. (The momentum of an object at rest with respect to other objects is zero so the displacement of three-dimensional space with respect to those objects would also be zero.)
Therefore to define the total energy of an object or particle one would have to add the displacements in a “surface” of a three-dimensional space manifold associated with the energy of its rest mass to that associated with its relative motion.
However, as mentioned earlier the article “Why Space-time?” showed that all accelerations are caused by an object or particle interacting with a curved “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
Therefore, if the causality of the momentum inertia of a reference frame is a displacement of a “surface” of a three-dimension space manifold it would tent to stay rest or ones in motion would tend to stay in motion unless it interacted with a force or as the article “Why Space-time?showed a three-dimensional “surface” that was curved with respect to a fourth *spatial* dimension or a force.
This defines a causal link between the inertia, mass, momentum and force in terms of a displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
Additionally it allows one to understand the casualty of Isaac Newton law of inertia or why an object at rest will remain at rest, and an object in motion will remain in motion in a straight line at a constant speed by extrapolating his laws of three-dimensional space to a fourth.
Einstein could define the casualty of gravity but not inertia in terms of a four-dimensional space-time manifold because the “surface” of three-dimensional space can be curved but not displaced with respect to a space-time or a time dimension.
This shows how one can derive the relativistic and inertial properties of energy/mass in terms of a four dimensional Newtonian universe
Later Jeff
Copyright Jeffrey O’Callaghan 2011
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Many thanks, seen the post on google, nice post fella