We have and will shown throughout this blog and its companion book “The Reality of the Fourth *Spatial* Dimension” there are many advantages to assuming space is composed of four *spatial* dimensions instead of four dimensional space time.
One is that is that it gives one the ability to derive a theoretical link between the quantum mechanical and relativistic properties of space, time gravity and its equivalence to accelerated reference frames.
Earlier in the article â€Why is energy/mass quantized?†Oct. 4, 2007 it was shown that one can derive the quantum mechanical properties of energy/mass by extrapolating the laws of a classical resonance in a three-dimensional environment to a one consisting of four *spatial* dimensions.
Briefly it showed the four conditions required for resonance to occur in a classical Newtonian environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in one consisting of four spatial dimensions.
The existence of four *spatial* dimensions would give a space (the substance) the ability to oscillate spatially on a “surface” between a third and fourth *spatial* dimensions thereby fulfilling one of the requirements for classical resonance to occur.
These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital. This would force the “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.
Therefore, these oscillations space, would meet the requirements mentioned above for the formation of a resonant system or “structure” in four-dimensional space.Â
Classical mechanics tells us the energy of a resonant system can only take on the discrete or quantized values associated with it fundamental or a harmonic of its fundamental frequency.
These resonant systems are responsible for the discrete quantized energy associated with the quantum mechanical properties of energy/mass.
This cannot be done in terms of four-dimensional space time because time is only observed to move in one direction forward therefore could not support the bi-directional oscillations required to define a resonant structure.
This article will derive the relativistic properties of space, time and mass by extrapolating those same concepts to four *spatial* dimensions.
In the article “Defining what is time?†Sept. 20, 2007 it was shown that one can unambiguous define time only in terms of the sequential ordering of the casualty of events in a manner that is consistent with both its physical and mathematical properties.
The article “Gravity in four spatial dimensions” Dec. 15, 2007 will show that one can derive it and the rest energy/mass of an object or particle in terms of a curvature in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension just as consistently as one can in four dimensional space-time.
This curvature is analogous to the space-time curvature that Einstein postulated was responsible for gravity.
Finally the article “Defining energy” Nov. 26, 2007 derived all forms of energy in terms of a displacement or curvature in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension. It was shown that kinetic energy can derived in terms of an oppositely directed displacement in its “surface” than the one associated with gravity.
The assumption that kinetic energy is a result of an oppositely directed displacement in a “surface” of three-dimensional space with respect to the one associated with gravity is based on the observation that they are oppositely directed. For example, the kinetic energy of an orbiting satellite is oppositely directed with respect to the gravitational energy associated with the planet it is orbiting. Therefore, if one defines gravity in terms of a “depression” in a “surface” of a three-dimensional space manifold with respect to fourth *spatial* dimension one should define kinetic energy in terms of a oppositely directed “elevation” in that “surface”.
However this means according the definitions given in the article “Defining energy” the total energy/mass of an object would be equal to the sum of the displacements of a “surface” of a three-dimensional space manifold caused by the rest mass of an object and that caused by its relative velocity.
Therefore, the energy/mass of an object would be dependent on its relative motion because one must add the energy of its motion to its rest energy/mass.
This defines the mechanism responsible for why the energy/mass of an object increases when viewed by an observer who is in relative motion to it in terms of the geometry of four *spatial* dimensions.
The following analogy can be used to understand and define the relativistic properties length and time
Assume that two “2 dimensional creatures†are living on the surface of two pieces of paper resting on a desktop.
Also, assume the two creatures can view the surfaces of the other piece of paper, which are separated a pencil.
If the diameter of the pencil is increased, the curvature between the surfaces of the two pieces of paper will increase.
Each of these creatures, when viewing the other piece of paper will only perceive the two-dimensional translation of the three-dimensional curvature generated by the pencil.
Therefore, each will view the distance between two points on the surface of the other as shorter since they will view that distance as a two-dimensional translation of a three-dimensional curvature in the surface of the paper and each will measure the distance between them on their piece of paper as being longer then they would if they viewed it on the other piece.
Similarly, because three-dimensional beings could only “view” a three-dimensional translation of a “curvature” or displacement in four *spatial* dimension caused by the motion of a reference frame they will measure distance or length in them as being longer than they would be if viewed as an observer who is not in relative motion to it.
The “movement” of “time†on both surfaces will also be affected.
Each of the two dimensional creatures mentioned earlier will view the others “time†as moving slower because the three-dimensional curvature in the paper makes the distance between events longer than the two dimensional translation of those events. Therefore, it will take longer for events “move” through a curvature in three-dimensional space relative to the time it would take for them to move through two-dimensional space.
Earlier it was mentioned that the magnitude of the displacement or “curvature” an object generates in a fourth *spatial* dimension is dependent on its velocity.
However as mentioned earlier, we have defined time as only being the measure or the “distance between” the sequential ordering of the causality of an event.
Therefore time will become dilated in reference frames that are in motion because the curvature generated in three-dimensional space by its relative motion three-dimensional beings in that reference frame will view the distance between events to be longer than it with respect to the distances measured in their reference frames would if they were not in motion relative to it. Therefore, they will view time in a reference frame that is in motion relative to them as moving slower than if they were in that reference frame.
The velocity of light is constant despite the relative motion of an observer because the foreshortening or shortening of the length or distance the light travels is proportional to the motion of the observer. Therefore, the velocity of light will be constant in all reference frames despite the relative velocities of the observers to those reference frames because velocity is defined in terms of distance divided by time.
It should be remember this scenario applies to all forms of energy including gravitational because, as the article “Embedded dimensions†Oct. 22, 2007 showed, three-dimensional beings perceive energy in terms of the magnitude of a “curvature” in “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
The Lorentz transformations derived from this theoretical model will take on the same form as the Lorentz transformations derived from Relativity.
This is because this theoretical model postulates that a displacement or curvature in “surface” of a three-dimensional space manifold, with respect to a fourth *spatial* dimension caused by the gravitational or kinetic energy of an object is proportional to the velocity of light.
Therefore, because both Relativity and the above mechanism predict a physical shortening of length and a slowing of time are related to the geometry of space, the form of the Lorentz transformations associated with the foreshortening length and slowing of time will be identical for both of these models.
However, this theoretical model differs from that of Relativity’s in that it defines the magnitude of a foreshortening of length and a slowing or dilation of time in terms of a “curvature” in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension instead of curvature in four dimensional space-time manifold.
As mentioned earlier “Defining energy” derived the mechanism responsible for both both gravitational and kinetic energy in terms of a “curvature” in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
However, it was shown earlier that a curvature in a “surface” of a three-dimension space manifold with respect to a four *spatial* dimension was responsible for length foreshortening and time dilation.
Therefore, because both gravitational and the kinetic energy of relative motion are derived from a common “curvature” in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension they will have a similar effect on physical properties of length and time.
This means both Relativity and this paper predict an observer in a gravitational field will measure the length of an object to be shorter and passage of time to be slower with respect to an observer who is located outside of a gravitational field.
However, as mentioned earlier this paper defines this shortening of length and slowing of time in a gravitational field in terms of four *spatial* dimension instead of four-dimensional space-time manifold.
The “relative” characteristics of a “curvature” in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension associated with kinetic and gravitational energy would also make it impossible for an observer to determine if an acceleration is caused by gravitational or kinetic energy such as that from an exhaust of a rockets engine.
This is because the mechanism defined above indicates the magnitude of a force associated with both gravitational and kinetic energy is related to the absolute magnitude of a “curvature” a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.
Therefore, because a three-dimensional observer can only observe the three-dimensional effects of a curvature in four *spatial* dimensions he or she could not determine whether he or she is in a gravitational field or an accelerated reference frame.
Therefore, both this paper and Relativity make identical predictions with respect to the relativistic properties of space and time and the inability to determine the casualty of acceleration.
However, as mentioned earlier the model presented in this blog is a broader based because it has and will define a common mechanism responsible for both the relativistic properties of space and time and the quantum properties of energy/mass and energy in terms four *spatial* dimension where as one cannot in four dimensional space-time because time as mentioned earlier is only observed to move in one direction forward and therefore a space-time dimension could not support the bi-directional movement required to define a resonant structure.
This shows how one can derive a theoretical link between the quantum mechanical and relativistic properties of space, time, gravity and its equivalence to accelerated reference frames in terms of the continuous properties of four *spatial dimensions because as mentioned earlier one can derive the quantum mechanical characteristics of energy/mass in terms of resonant structure formed by a matter wave on a continuous “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension while defining its relativistic properties in terms of those same continuous properties.
Later Jeff
Copyright 2007 Jeffrey O’Callaghan