We have shown throughout The Imagineer’s Chronicles there would be many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four-dimensional space-time.

One of them is it would allow for a common theoretical explanation of gravity and how space is expanding.

In 1915, Albert Einstein wrote the General Theory of Relativity, which defined how gravity works.  He showed gravity could be explained in terms of a curvature in a "surface" of a four-dimensional space-time manifold. 

However, when he applied his theory to the whole universe, he found that it predicted space should not be stable.

Later In 1929 Edwin Hubble, determined the redshifts of a number of distant galaxies and their relative distances and found it increased as a linear function of their distance.  The only explanation for this observation is that the universe was expanding.

These observations made by Hubble confirmed Einstein theoretical prediction that his four-dimensional space-time geometry was unstable and that it was expanding.  However, neither he nor Einstein could define how or in what direction this expansion was occurring.

This is because Hubble’s observation showed three-dimensional space was not expanding through a time dimension but in a spatial one.  Therefore, one must add dimensions to Einstein’s gravitational theory to explain how three-dimensional space can be undergoing a uniform spatial expansion.

One possible explanation is that each axis of three-dimensional space is expanding outwards towards another spatial dimension.  This means the universe must be composed of at least six spatial dimensions to allow each of them to individually expand towards another dimension.

This type of expansion is the mathematical basis for the Standard Model of physics which for the past 25 years, has given us a complete mathematical description of the particles and forces that shape our world.  Its predictions have matched experimental data, decimal place for decimal place; with so much precision that many feel it is the ultimate theory of matter and energy.

But if it were true that the uniform expansion of space is a result of each dimension expanding towards another they should be able to explain how they interact.  This is because if each dimension was able to move independently towards another dimension we would expect expansion of space to show some non-uniformity.

However, there is another possibility that has largely been ignored by physicists that would answer this question and provide a connection between an expanding universe and gravity.

If the dimensions that comprise our universe are scalar invariant with respect to a single fourth *spatial* dimension the movement of any point in three-dimensional space with respect to a fourth *spatial* dimension would result in the uniform expansion of three-dimensional space around that point. 

One way of understanding how this would explain the uniform expansion of the universe would be to compare it to the expansion of a balloon’s surface. 

As a balloon is inflated, the length and width of its surface expands around a point on it.   However the magnitude of this expansion is not defined by its movement through three-dimensional space but by the bending or curvature caused by its movement in it.  This is illustrated by the fact that the surface of a balloon only stretches when its curvature is increased by inflation and not when one moves it in its entirety to a different point in space.

This shows the why the expansion of each axis of the surface of a balloon is scalar invariant with respect to its movement in a third dimension.

A similar effect would occur if three-dimensional space were scalar invariant with respect to a fourth *spatial* dimension.  Its movement "outward" with respect to a fourth *spatial* dimension would result in a uniform stretching of the three-dimension around a point in three-dimensional space.

Therefore, one could understand how space is expanding and eliminate the problem of having to explain why, as mentioned earlier it is uniform if one assumes three-dimensional space is expanding towards a single fourth *spatial* dimension and that it’s individual axes are scalar invariant with respect to it.

In the article "Gravity in terms of four *spatial* dimensions" it was shown it is possible to explain gravity in terms of a curvature or contraction in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

If three-dimensional space was scalar invariant with respect to a fourth *spatial* dimension, the movement of a point in three-dimensional space "inward" with respect to it would result in it "contracting" spherically along all three-dimensional axis.  This contraction would result in a force being directed towards its center.  This contraction in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension can be theoretically shown, as was done in the article Gravity in terms  of four *spatial* dimensions to be responsible for gravity.

However, this also provides a theoretical connection between gravity and the observed properties of an expanding universe because it define both the terms of a common mechanism related to the existence of a fourth *spatial* dimension.

Later Jeff

The *Shadows* of four spatial dimensions

Copyright 2010 Jeffrey O’Callaghan

We have shown throughout The Imagineer’s Chronicles there would be many theoretically 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 integration of the quantum and wave properties of matter and energy into a single theoretical model. 

In 1924 Louis de Broglie was the first to theorize that all particles are, in part composed of a transverse wave.  In his paper, “Theory of the double solution“ he attempted to define a causal interpretation of their wave properties in classical terms.  He later abandoned it in the face of the almost universal adherence of physicists to the theories presented by Born, Bohr, and Heisenberg regarding the uncertainties and probabilistic interpretation of quantum particles.

One of the difficulties he may have faced in this endeavor is that he assume along with most other scientists of his day the universe was composed of four-dimensional space-time.

This presented a problem because observations of a space-time environment indicate a time dimension can only move in one direction, forward.  Therefore, it could not support bidirectional movement required for the propagation of a transverse wave. 

However, if he had assumed, as we have done in The Imagineer’s Chronicles the universe was composed of four *spatial* dimensions he may have been able to define the transverse wave properties of particles in terms of sinusoidal oscillations or displacements in a "surface" of three-dimensional space with respect to a fourth *spatial* dimension.

Classical mechanics tells us transverse oscillations in a surface of water will form resonant systems or structures that when view from distance would appear to be made up of discrete units of water with finite boundaries.

Summarily if the microscopic transverse waves theorized by de Broglie were a result of oscillations in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension they would, as was shown in the article "Why is mass and energy quantized?" Oct. 4, 2007 form resonant systems which when viewed from a macroscopic perspective would have the discrete boundaries associated with particles. 

He then could have defined a causal interpretation of the Quantum Mechanical equation for a particles energy; E=hv (where "h" is Planck’s constant "v" is a particles frequency and "E" is the magnitude or its energy) in terms of the classical properties of waves.

Classical mechanics tells us that the energy of a resonant system is quantized in terms of multiples of the harmonics of the fundamental frequency of its environment. 

Therefore, he could have interpreted the equation E=hv as defining the quantization of a particle’s energy in terms of a resonant system with a fundamental harmonic "h" of an environment consisting of four *spatial* dimensions..

However, this would have also allowed him to define a casual mechanism responsible for Planck’s constant, the uncertainty principal and the probability functions of Quantum Mechanics in terms of classical mechanics.

As mentioned earlier, classical interpretation of Planck’s constant would be that it defines magnitude of the incremental energy deference between harmonics of a resonant frequency of four *spatial* dimensions.

Additionally there would be an inherent uncertainty in one’s ability to define the exact position or momentum of a particle because it would be distributed over the finite volume associated with the wavelength of its resonant frequency.  Therefore, one could only define its specific position or momentum in terms of an uncertainty related to where relative to its wavelength an observation is made.

The reason why quantum mechanics must rely on probably functions to define particle interaction is because they are composed of waves that do not have a ridged structure.  Therefore, because their wave components vary with time one can only define their interactions in terms of a probably function related to how their time varying wave components would interact.

This shows that one can explain and predict all of the quantum mechanical properties of particle if one assumes they are made up of transverse waves in the classical terms of the geometry of four *spatial* dimensions.

However, as mentioned earlier this cannot be done if one assumes space it made up of four-dimensional space-time because the geometry of space-time cannot support the transverse wave component Louis de Broglie associated with particles.

Later Jeff

The "Shadows" of four *spatial* dimensions

Copyright 2009 Jeffrey O’Callaghan

We have shown throughout “The Imagineer’s Chronicles” there are many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four-dimensional space-time.

Understanding the origins of the force called of Dark Energy is one of them.

On May 30, 2009″ physicsworld.com” reported in the article Dark Energy “New evidence has confirmed that the expansion of the universe is accelerating under the influence of a gravitationally repulsive form of energy that makes up two-thirds of the cosmos.”

As mentioned in the physicsworld.com article some researchers suggest Dark Energy is caused by a uniform sea of quantum zero-point energy.  However the problem with this proposal is the zero-point energy density would have to be precisely tuned to a value that is an unbelievable factor of 10¹²⁰ below the theoretical prediction and as of yet no one has been able give a reason why that is consistent with theory.

While others have suggested modifications to Einstein’s General Theory of Relativity.

But Einstein, because of his preconceived feeling that the universe was static tried to integrate a mechanism into it to counter the destabilizing forces of gravity.  He realized a static universe is not sustainable in General Relativity because if the universe were static at one instance in time, then gravity, due to its attractive properties would cause it to collapse.  Therefore, he added a term to it that would counteract it. 

This term, which effectively gives the vacuum a repelling force, came to be called a cosmological constant.

However, he could not develop a theoretical argument to justify incorporating it.  Therefore, to satisfy his preconception that the universe was static he simply added it in an “adhoc” manner.

But if its author could not modify the General Theory of Relativity to incorporate an accelerative force into its theoretical structure it would seem highly unlikely that anyone else would be able to.

However, it may be possible to understand the origins of Dark Energy and theoretically integrate it with gravity and the relativistic properties of space, time and motion if one defines their casualty in terms of the existence of four *spatial* dimensions instead of four dimensional space-time.

Observations of Dark Energy indicate it opposes gravity and that it is causing a *spatial* not a time component of the universe to expand.

The article “The “Relativity” of four spatial dimensions” Dec 1, 2007 showed one could theoretically derive the relativistic properties of space, time, motion and the equivalence between gravitational and accelerated reference frames in terms of a curvature or deformation in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, theoretically assuming gravity is a result of a curvature in four *spatial* dimensions would allow one to seamless integrated the repulsive properties of Dark Energy with gravities attractive properties.

(This curvature is analogous to a curvature in a four dimensional space-time manifold Einstein predicted was responsible for gravity.)

If a gravity is a result of a curvature in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension, reducing the mass of an object would reduce the magnitude of that curvature.  However, this would result in the expansion of the “surface” three-dimensional space with respect to a fourth *spatial* dimension analogous to how removing the coils or curvature in a rope causes a physical expansion of its two dimensional length when measured with respect to three-dimensional space.  Since this expansion would be in addition to the expansion associated with the Big Bang therefore, it would be perceived as an acceleration.

Additionally the article Defining energy showed one could also derive the temperature or kinetic energy of particles in terms of a curvature in “surface” of a three dimensional space manifold with respect to a fourth *spatial* dimension.  Therefore, as the universe cools the magnitude that curvature would decrease and cause three-dimensional space where that cooling took place to expand for the same reason as a conversion of mass to energy does.  This expansion would be perceived as an acceleration because it would be viewed in addition to the linear expansion associated with the big bang.

This demonstrates one can use a relativistic theory on based on the existence of four *spatial* dimensions to guide them to an understanding of the properties of Dark Energy rather than basing it on expectations as Einstein had to.

The reason Einstein could not integrate it into his theories was that time or a space-time dimension does not have spatial properties required to support the spatial expansion associated with Dark Energy.

Since the total mass in the universe is decreasing due to the nuclear reactions taking place in stars, the curvature associated with its gravitational field would also be decreasing.  This means, according to this theoretical model the magnitude of Dark Energy would be proportional to the quantity of mass the universe losses due to the nuclear reactions that occur within it.

However, the “concentration” of Dark Energy relative to gravitational energy would be, according to this mechanism defined by the equation E=mc^2c.   This means its strength should be 1/c^2 weaker than gravitational forces.  Therefore, because gravitational forces are much stronger than those of Dark Energy: the space between gravitationally bound objects would not appear to be expanding.

One can also use a theoretical model based on the existence of four *spatial* dimensions to understand why the rate of acceleration appears to be increasing and predicts there should be a crossover where it exceeds the gravitational deceleration caused by the mass density of the universe.  As the universe expands due to the energy of the big bang it mass density decreases, therefore the accelerative properties associated with mass being converted to energy would be larger in proportion to the decelerative gravitational forces. 

This shown that one can seamlessly integrate the observed properties of Dark Energy into a relativistic theory based on the existence of four *spatial* dimensions and why one cannot based four-dimensional space-time.

Later Jeff

The “Shadows” of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

We have shown throughout "The Imagineers Chronicle’s" that observations of our environment suggest that space is composed of four *spatial* dimensions instead of four dimensional space-time.

If true we should be able to define both the wave and particle nature of particles as observed in the "The double-slit experiment in terms of the geometry of four *spatial* dimensions. 

Wikipedia, describes the double slit experiment as being made up of

"A coherent source of photons illuminating a thin plate with two parallel slits cut in it striking a screen after passing through them.  The wave nature of light causes the light waves passing through both slits to interfere, creating an interference pattern of bright and dark bands on the screen.  However, at the screen, the light is always found to be absorbed as discrete particles, called photons.

"Normally, when only one slit is open, the pattern on the screen is a diffraction pattern, a fairly narrow central band with dimmer bands parallel to it on each side.  (See the top photograph to the right.)  When both slits are open, the pattern displayed becomes very much more detailed and at least four times as wide.  (See the bottom photograph to the right.)  These facts were elucidated by Thomas Young in a paper entitled "Experiments and Calculations Relative to Physical Optics," published in 1803.  To a very high degree of success, these results could be explained by the method of Huygens–Fresnel principle that based its calculations on the hypothesis that light consists of waves propagated through some medium.  However, discovery of the photoelectric effect made it necessary to go beyond classical physics and take the quantum nature of light into account.

It is a widespread misunderstanding that, when two slits are open but a detector is added to the experiment to determine which slit a photon has passed through, then the interference pattern no longer forms and the experimental apparatus yields two simple patterns, one from each slit, superposed without interference.  Such a result would be obtained only if the results of two experiments were superposed in which either one or the other slit is closed.  However, there are many other methods to determine whether a photon passed through a slit, for instance by placing an atom at the position of each slit and monitoring whether one of these atoms is influenced by a photon passing it.  In general, in such experiments the interference pattern will be changed but not be completely wiped out.  Interesting experiments of this latter kind have been performed with photons and neutrons.

Restriction to the two experiments in which either both slits are open or one slit is closed has given rise to the idea of wave-particle complementarily (to be distinguished from wave-particle duality) according to which a microscopic object (photon, electron, etc.) would manifest itself as a particle in the which-way experiment but as a wave in the interference experiment.  This idea has been felt to be counterintuitive by those not being content with an instrumentalist interpretation of quantum mechanics in which that theory is accepted as just describing phenomena without providing explanations.

The most baffling part of this experiment comes when only one photon at a time is fired at the barrier with both slits open.  The pattern of interference remains the same, as can be seen if many photons are emitted one at a time and recorded on the same sheet of photographic film.  The clear implication is that something with a wavelike nature passes simultaneously through both slits and interferes with itself — even though there is only one photon present.  (The experiment works with electrons, atoms, and even some molecules too.)

A clue to the true nature of a photon and other particles is the fact its wave energy is contained in discrete packet or particles when striking photoelectric material."

The article, "Why is energy quantized?" showed it is possible to explain and predict the duality or particle / wave properties of matter and energy in terms of a classically resonating system or "structure" generated by a wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth "spatial" dimension.

This means one could explain "quantum nature" of the photoelectric effect by assuming light’s energy is propagated by discrete packets generated by a classically resonating system formed by a wave moving on a "surface" of a three-dimensional space manifold.  Therefore, as mentioned in the Wikipedia article the light that strikes a photoelectric plate "will always … be absorbed as discrete particles, called photons".

Additionally, because the quantum properties of photons and particles can be derived in terms of a resonant "structure" generated by a wave means that one should be able to define the results of result of the double slit experiment in terms of a classical property of their wave components.

The reason why the interference patterns remains when one particle at a time is fired at the barrier with both slits open or "the most baffling part of this experiment" is because, as mentioned earlier it is made up of a resonant "structure" who’s volume would be directly related to the wavelength of that system.

This means a portion of the energy of one photon could simultaneously pass both slits, if the diameter of its volume exceeds the separation of the slits and recombine on the other side to generate an interference pattern. 

It also explains why, as was mentioned in the Wikipedia article the interference pattern remains when two slits are open and a detector is added to the experiment to determine which slit a photon has passed through, a direct contraction of the Copenhagen interpretation of quantum mechanics.

The Copenhagen interpretation of quantum mechanics demands that when a detector is added to the experiment to determine which slit a photon has passed through the interference pattern can no longer form and the experimental apparatus should yield two simple patterns, one from each slit, superposed without interference.

The fact that the interference pattern occurs even if a measurement is made is because if one alters the energy or wavelength of a wave moving through one of the slits by a relatively small amount compared to its original wavelength classical mechanics tells us it will be able to interact with the wave moving through the other slit to form a slightly different resonant system with a slightly different interference pattern on the other side. 

Therefore, there should be a quantifiable minimum value of interaction between a measuring device and a particle that will permit the interference pattern to be reestablished on the other side.   Additionally there should be a direct relationship between the energy of a particle and when and if an interference pattern is generated when one tries to measure which slit it moves through in the double slit experiment.

Richard Feynman the farther of Quantum Electrodynamics or "OED" realized the significance of this experiment because it demonstrates the inseparability of the wave and particle properties of particles and felt a complete understanding of quantum mechanics could be gleaned from carefully thinking through its implications.

The above article demonstrates why.

It shows the quantum and wave properties of mass and energy displayed by the double slit experiment can be understood if one assumes they are made up of a classical resonant system that have a discreet volume which is related to the wavelength of a wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This understanding cannot be derived if one defines the universe in terms of four-dimensional space-time because time is observed to move in only in one direction forward and therefore, it cannot support the bidirectional movement required to support the transverse wave motion associated with particles in terms of classical physics.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2009 Jeffrey O’Callaghan

History has shown the most successful theories are those based on what has come to be called Occam’s razor or that "the explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory".

Newton applied it when he defined the properties of planetary motion in terms of a single concept called the law of gravity which assumes that every point mass attracts every other point mass by a force pointing along the line intersecting both points.

He could have complicated it by assuming the existence of other perimeters such as a repulsive force acting between each point mass and then make a proportional increase the magnitude of the attractive force so that their movements corresponded to observations but he chose not to.  However, a law of gravity based on this assumption would be just as successful in predicting their motion as the one he proposed.

However as technologies advance and more accurate measurements of planetary motions were made scientists realized not all them can explained by Newton’s laws.

This led to the acceptance of more universal explanation of planetary motion based on Einstein’s assumption that space and time can be merged into a single entity called space-time.

However, Einstein also applied Occam’s the principal of the eliminating all assumptions "that make no difference in the observable predictions of the explanatory hypothesis or theory" because he based his redefinition of Newton’s laws only on the single assumption that space and time can be merged to form space-time.

Unfortunately the assumption that space-time is responsible for the macroscopic motion of the planets cannot be used to explain the movement of microscopic or quantum particles.  To predict their movements Quantum theories had to assume they are defined by the probably they will be found at specific point at a specific time and that there is an infinite number of paths a particle can take while moving between two points in space. 

However, this presents a problem for scientists who believe, as Occam’s did that "the explanation of any phenomenon should make as few assumptions as possible because presently it takes two assumptions instead of one to completely explain the phenomenon of motion.

But we may find the key to why we have not been able to explain motion in terms of a single assumption by understanding the reason science develop theories. 

The purpose of a theory is to explain and predict the physical relationships and interactions between different aspects of our observable environment.  Therefore, the validity of the assumptions use to define those relationships should be based or have a foundation in our observable environment.

For example, Einstein conceptually defined the movement of planets by assuming a physical interaction of time with the dimensional properties of space even though no one has ever observed time to have physical properties.

Therefore, the assumption that a time or space-time dimension physically exists has no bases in our observational environment. 

Similarly, Quantum Theories assume the movement of a particle is defined by the mathematical probability it will be in a specific place at a specific time.  However, even though this may give a very accuracy analytical solution to its position, it defines its movement in terms of the properties of an abstract of a mathematical environment.  

Mathematics is an abstract tool that physicists can use to analytically quantify the validity of a physical model of the environment.  However, the abstract environment defined by the equations of Quantum Mechanics is unobservable.  Therefore, as with a space-time dimension the assumption made by Quantum Theory of the physical existence of a probabilistic mathematical environment have no bases in our observational world. 

We have shown throughout "The Imagineer’s Chronicles" that one can explain and predict the movements of both planets and particles by assuming they are result of an interaction between our observable three-dimensional space and a fourth *spatial* dimension.  However we define the existence and properties of a fourth *spatial* dimension by extrapolating the observable properties of three-dimensional space to it.

Therefore, the assumption the movement of both planets and particles is defined by the existence of a four *spatial* dimension has a foundation in the observable environment of three-dimensional space. 

However, we could have chose to complicated it by assuming, as String theorists have, that up to ten dimensions are required to explain the movement of particles and planets.  We did not because even thought the existence of ten dimensions can explain the phenomenon of planetary and particle motion six of them can be eliminated because "they make no difference in the observable predictions of the explanatory hypothesis or theory" 

Later Jeff

The "Shadows" of four spatial dimensions 

Copyright Jeffrey O’Callaghan 2009

We have shown throughout “The Imagineer’s Chronicles” there are many theoretical advantages to defining the universe in terms of the existence of four *spatial* dimensions instead of four-dimensional space-time.

One is that it would allow for the development of a more logical and consistent solution to the flatness problem than the presently accepted inflationary one.

Wikipedia defines the cosmological fine-tuning within the Big Bang model of the universe as being responsible for The Flatness Problem.  Such problems arise from observation that some of the initial conditions of the universe appear to fine-tuned to very "special" values, and that a small deviation from these values would have had massive effects on the nature of the universe at the current time.

In the case of the flatness problem, the parameter, which appears to be fine-tuned, is the density of matter and energy in the universe.  This value affects the curvature of space-time, with a very specific critical value being required for a flat universe.  The current density of the universe is observed to be very close to this critical value.  Since the total density departs rapidly from the critical value over cosmic time, the early universe must have had a density even closer to the critical density, departing from it by one part in 10⁶² or less.  This leads cosmologists to question how the initial density came to be so closely fine-tuned to this ’special’ value.

The problem was first mentioned by Robert Dicke in 1969.  The most commonly accepted solution among cosmologists is cosmic inflation, the idea that the universe went through a brief period of extremely rapid expansion in the first fraction of a second after the Big Bang; the flatness problem is one of the three primary motivations for inflationary theory.

However, if one defines mass and energy in terms of only four *spatial* dimensions as we have done in “The Imagineer’s Chronicles” one can derive a classical mechanism that can predict and explain why our universe appears to be flat without having to resort to exotic or "adhoc" hypotheses such as Cosmic Inflation.

In the article Defining energy Nov 26, 2007 it was shown the "quantity" or density of matter in the universe could be derived in terms of a curvature or displacement in "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension while the density of its energy can be derived in terms of an oppositely directed displacement in that same surface.  

This curvature is analogous to the space-time curvature Relativity defines as being responsible for mass and energy.

However, it differs in that one derives mass and energy in terms of unidirectional displacement in the geometry of space-time while the other in terms of a bi-directional one in a "surface" of a three dimensional space manifold with respect to a fourth *spatial* dimension.

But if it were true the density of mass and energy are the result of oppositely directed curvatures in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension the "flatness" of our universe would be an intrinsic property of its existence and would not require the fine-tuning of any of its components to explain it.

If one crumples a piece of paper that was original flat and views its entire surface the overall magnitude of the curvature caused by that crumpling would be zero because the height of the curvature above its surface would be offset by an oppositely directed curvature below its surface.  Therefore, if one views its overall surface only with respect to its height its curvature would appear to be flat.

However, if density of mass and energy are a result of opposite directed displacements in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension their overall density would appear to be flat because, similar to a crumpled piece of paper the "depth" of the displacement below its "surface" caused by mass would offset by the "height" of the displacement caused by energy.

Many proponents of the Big Bang Model assume it began from the expansion of mass and energy around a one-dimensional point.  However, if we are correct in assuming that mass and energy are a result of oppositely directed curvatures in a "surface" of a three-dimensional space manifold, the universe must have been flat with respect to their density at the time of the Big Bang.  This is because a one-dimensional point would have no "vertical" component with respect to a fourth *spatial* dimension and therefore the "surface" of three-dimensional space originating from it would be "flat" with respect to it.

However, if the universe was flat with respect to the density of mass and energy in the beginning it would remain flat throughout its entire expansive history because its expansion would result in a proportional decreases in the displacement above and below its three-dimensional surface.

This would be analogous to why the overall flatness of a crumpled piece of paper does not change if one smoothes or stretches it because that would result in a proportional decrease in the height of the wrinkles above and below its original surface.

This is not possible if one defines universe in terms of four-dimensional space-time because time moves only in one direction forward and therefore cannot support the bi-directional movement required define the apparent flatness our universe in terms of its geometry.

Therefore, if one theoretically defines mass and energy in terms of four *spatial* dimensions one does not have to assume that their densities were fine turn to ’special’ values when the universe began to expand but only that it was flat when the expansion began as is hypothesized by the Big Bang theory.

Jeff

The Shadows of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

to completely define the reality of our universe?

Einstein was able to define the relativistic properties of motion and the equivalence of gravitational and accelerated reference frames in terms of the geometry of space-time.  However, as Lee Smolin mentions in Chapter Three (page 49) of his book "The Trouble with Physics" he was unable to physically connect them to electromagnetic energy. 

Einstein himself acknowledges this when he said, "I have often tortured my mind in order to bridge the gap between gravitation and electromagnetism.”

One of the reasons may be because he and many other physicists defined electromagnetism in terms of a transverse wave while defining gravity in terms of a time component of a space-time dimension.  However, time is observed to move only in one direction forward therefore it cannot support the bi-directional spatial properties of the transverse wave associated with electromagnetism.  Therefore, it was not possible to define a physical connection between them because they were based on different aspects of the environment.

But as Lee Smolin also mentions (page 39) Gunnar Nordstrom discovered electromagnetism "pops out" of Einstein’s theory if one increases the dimensions of space by one.  In other words, just by adding an extra dimension of space you get a unification of gravity with electromagnetism that was consistent with Einstein’s Special Theory of Relativity."

However, Einstein’s theories predicted that light would be bent by gravity while Nordstrom’s did not. 

In 1919, Arthur Eddinngton observed that light was bent by gravity thereby verifying that Einstein’s theory was more consistent with observations and that gravity was not the result of an extra dimension as Nordstrom had postulated.

Since then there have been many attempts to unite gravity with electromagnetism simply by adding dimensions to Einstein’s space-time manifold.

The most promising of these is called string theory, which attempts to define all of the observed properties of our universe in terms of as many as ten dimensions.

However, as is pointed out on page 51 of "The Trouble with Physics" all attempts at unifying physics through extra dimensions suffer from the same problem.  There are a few solutions that lead to the world we observe but there are many which do not.  One has to set the initial conditions, which are found by observing our world to determine which solutions define what we observe.  The use of the circular methodology means its validity is not based on its theoretical structure but on its flexibility.

But it may be possible to develop a theoretical connection between electromagnetism, gravity and its equivalence to accelerated reference frames if instead of adding a spatial dimension as Nordstrom did to Einstein’s space-time we replace its time component with a spatial one. 

In the article "The Equivalence Principal: an alternative" July 15 2008 it was shown the observed properties of gravity and its equivalence to an accelerated reference frame can be derived in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension as well as a curvature a space-time manifold as is suggested by Einstein’s theories.

Additionally the article "The "relativity" of four spatial dimensions" Dec 2007 showed one can derive a mechanism responsible for the relativistic properties of motion, space, and time in terms of the existence of four *spatial* dimensions that is consistent with Einstein’s Special Theory of Relativity.

While the article "The Photon: a matter wave?" Oct. 1 2007 showed one could derive the particle, wave and electromagnetic properties of light in terms of oscillations or spatial displacements in a "surface" of a three-dimensional space manifold with respect to fourth *spatial* dimension.

This suggests that assuming the universe is composed of four *spatial* dimensions instead of four dimensional space-time may allow one to theoretically "bridge the gap between gravitation and electromagnetism" because it may enable one to define a common mechanism responsible for their properties.

Additionally, if one can derive gravity in terms of a spatial curvature in a "surface" of a three-dimensional manifold with respect to a four *spatial* dimension and electromagnetism in terms of oscillations in that "surface" means it would be bent by gravity for the same reason it would be if it existed in Einstein’s space-time maniflod.  

This means one may not have to increase number of spatial dimensions "in order to bridge the gap between gravity and electromagnetism" as Nordstrom did because it would be an integral part of the theoretical structure presented in The Imagineer’s Chronicles, whereas it would not be part of one based on Einstein’s space-time theories.

Therefore, the answer to the question as to "How many dimensions do we need to completely define the reality of our universe?" may be four.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

We have shown throughout “The Imagineer’s Chronicles” there are many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four-dimensional space-time.

One is that it allows for the defining a common mechanism responsible for both dark and gravitational energy

On May 30, 2004, physicsworld.com reported, in the article Dark Energy "New evidence has confirmed that the expansion of the universe is accelerating under the influence of a gravitationally repulsive form of energy that makes up two-thirds of the cosmos.

However, since its discovery no one has been able to define a mechanism that can account for its existence.

In the article “Defining energy” Nov. 26, 2007 it was shown it is possible to derive all forms of energy including gravitational in terms curvature or distortion in a "surface" of a three-dimensional space manifold with respect to a fourth spatial dimensions.

However, if it were true that all forms of energy are related to a geometric property of four *spatial* dimensions one should be able to define a causal link between dark and gravity energy in terms of four *spatial dimensions.

If gravity, as mentioned earlier is a result of a curvature in a three-dimensional space manifold with respect to a fourth *spatial* dimension, then reducing the mass of an object would reduce the magnitude of that curvature.  This would result in a physical expansion of that three-dimensional space manifold with respect to a fourth *spatial* dimension thereby increasing the volume of the universe.  This is analogous to how removing the coils or curvature in a rope causes a physical expansion of its length with respect to three-dimensional space.

However, this is one of the properties of Dark Energy in that it causes a physical expansion of space and since this expansion occurs in addition to any resulting from the expansive energy associated with the Big Bang it will be viewed as an acceleration.

This means that one can explain the causality of Dark and gravitational energy in terms of a common mechanism related to the geometry of four *spatial* dimensions.

This is extremely difficult to do in terms of four-dimensional space-time because the expansion of a time or a space time-dimension cannot explain the spatially expansive properties of Dark Energy.

The fact that the effects of Dark Energy are only observable between bodies that are not gravitational bound can also be explained by assuming it’s causality is a result of the geometry of four *spatial* dimensions. 

If both dark and gravitational energy are related to the existence of four *spatial* dimensions then, as will be shown in a latter article "Why E=mc^2" the "concentration" of Dark relative to gravitational energy would be defined by the equation E=mc^2".  Therefore, because the expansive effects it has on space are considerably less "concentrated" by factor of c^2 than the contractive forces associated with mass it would be extremely difficult to observe it between gravitationally bound objects.

This shows that one of the advantages to defining the universe in terms of four *spatial* dimensions instead of four dimension space-time is that it allow one to establish a logical and consistent theoretical link between the observed properties of both dark and gravitational energy

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2009 Jeffrey O’Callaghan

We have shown throughout “The Imagineer’s Chronicles” that observations of our environment suggest space is composed of four *spatial* dimensions instead of four dimension space-time.

One of these observations involves the theoretical existence of quantum fluctuations.

Quantum fluctuations are a theoretical construct based on the Uncertainty Principal.  It states one cannot know the precise amount of energy contained in a microscopic volume of space. Therefore, over a short enough time intervals the energy levels in an empty volume of space can fluctuate.  Some physicist believe these fluctuations can provide enough energy to generate virtual particle antiparticle pairs so long as they annihilate each other within the interval of time specified by the uncertainty principal so there is no net creation of matter.

Even though the existence of quantum fluctuations is based solely on theoretical arguments, it may be beneficial to examine the consequences to our understanding of physical structure of the universe if they do exist.

Presently the only viable theory that defines gravitational energy  is based on Einstein’s concept of a space-time manifold. 

However, it is difficult to understand how the "surface" of a space-time manifold can produce the energy theorists associated with quantum fluctuations.

This is because Einstein defined the rate at which time "moves" in terms of the energy content of a volume and according to his space-time concepts time must always move slower in an environment with a different energy content relative to another.

However, this presents a problem for those who define quantum fluctuations in terms of symmetrical energy distortions in the space-time manifold defined by Einstein. 

As mentioned earlier, according to Einstein’s space-time concepts time is dilated or always "moves" slower in volumes that contain a different energy content than the ones from which they are observed.  This means the time dilation associated with quantum fluctuations will be cumulative throughout the evolution of the universe even though the energy associated with the particle antiparticle pairs is not.  Therefore, because of the random nature of these fluctuations we should observe random time dilations in volumes were they occur relative to others throughout the universe.  However, this means we should observe the velocity of light or position of the spectral emission lines to fluctuate randomly to match the random fluctuations associated with a quantum vacuum.

Since the velocity of light is constant in all volumes and we do not observer random fluctuations in its spectral emission lines we must conclude that either quantum fluctuations do not occur or the space-time concepts of Relativity are invalid.  

Additionally according to Einstein’s concepts the high energy density associated with quantum fluctuations means time in volumes were they occur would "move" significantly slower than in the adjacent volumes.  This means the time interval specified by the uncertainty principal to assure that no net creation of matter occurs would be different inside of the volume where they occur relative to the adjacent volumes.  This sets an upper limit on the magnitude of the energy of particle antiparticle creation because higher energy means it would take longer for them to annihilate each other relative to the adjacent volume.  Therefore, to satisfy the time requirement of the uncertainty principal of no net creation of matter relative to the adjunct volumes means, that according to Einstein’s concepts there must be an upper limit to the magnitude of their energy.

However, as was done in the article “Defining energy” Nov 26, 2007 defining energy in terms of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would allow one to define the particle antiparticle pairs associated with quantum fluctuations in terms a symmetrical spatial displacement in that manifold and their annihilation in terms of the canceling out of those spatial displacements.

Additionally defining time dilation in terms of a spatial displacement as was done in the article "The relativity of four spatial dimensions"  Dec 1, 2007 eliminates the problem associated with the cumulative effects causes by defining it in terms of the properties of a space-time manifold because the spatial displacements are oppositely directed, the time differentials associated with them will cancel out.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

We have shown throughout "The Imagineer’s Chronicles" there would be several theoretical advantages to define the universe in terms of the existence of a continuous non-qauntized form of mass and four *spatial* dimensions instead of four-dimensional space-time.

One of them is that it would allow physicists to derive the relationship between gravity an quantum mechanics in terms of the classical laws of physics.

The article "Why is energy quantized?" Oct, 4 2007 showed the properties of a particle could derived in terms of a classically resonating system made up oscillations in a continuous non-quantized form of mass on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

There are 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.

The existence of four *spatial* dimensions would give a continuous non-quantized mass component of space 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.

However, these oscillations in a continuous non-quantized form of mass caused by such an event could generate a resonant system or "structure" to be established in a continuous non-quantized form of mass.  These resonant structures are responsible for the quantum or particle properties of mass and energy.

It is very difficult to define the quantum properties of mass and energy in terms of the continuous properties of a four-dimensional space-time manifold because time is only observed to move in one direction forward.  Therefore, it could not support the bidirectional movement or spatial oscillations required to define the resonating structures modern science associates with particles.

The article "Defining energy?" Nov. 26, 2007 shown one can explain and predict gravity in terms of a displacement or curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This curvature is analogous to a curvature in four-dimensional space-time Einstein assumed was responsible for gravity.  However it differs in that it defines gravity in terms of the classical concepts of spatial dimensions instead of a non-classical one of them being merged with time.

As mentioned earlier, a particle’s properties can be derived terms of a classically resonating or "standing" wave on a "surface" of a three-dimensional space with respect to a fourth *spatial* dimension while gravity can be define in terms depression or curvature in that same surface.

However, both of these mechanisms have a common element related to the existence of a continuous non-quantized form of mass four *spatial* dimensions.

Therefore, it is possible, using the laws of classical physics to derive a theoretical link between the continuous properties of gravity and the discontinuous or quantum properties of mass in terms of the existence of a continuous non-quantized form of mass and four *spatial* dimensions.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2009 Jeffrey O’Callaghan