We have shown throughout “The Imagineer’s Chronicles” it is possible to define a universe in terms of four *spatial* dimensions in a manner that makes predictions identical with those of Einstein’s General and Special Theories of Relativity.  Additionally we have shown there would be several theoretical advantages to adopting this perspective over that of his theories. 

One it that it allow one to understand why the universe must be flat

A four dimensional universe can be geometrically open, closed, or "flat" and its shape is dependent on the mass and energy within it.

In an opened universe, there is insufficient matter to halt the expansion initiated by the big bang.  This will result in a saddle shape or open universe, which will continue to expand forever.

In a closed universe, the gravitational potential of its mass is large enough to overcome the expansive forces of the big bang.  This will result in the universe having a spherical shape, which would be destined to collapse.

A universe will be flat if the attractive gravitational potential of matter just equals the expansive energy of the big bang.  This will result in the expansion slowing and only stop after an infinite amount of time has passed.

However, recent observations by NASA’s WMAP satellite has shown the universe is flat to within a 2% margin of error.

But why the universe appears to be flat even after 14 billion years of expansion is still a mystery because a flat universe is like the top of a hill.  If you are a little away from it – a bit open or a bit closed – the expansion of the universe soon drives you far away from this value, just as a ball that is a short distance from a hilltop will roll down to the bottom.  Therefore, when the Universe was one second old, it must have deviated from flatness by less than one part in ten-thousand-trillion (10¹⁶).  This is a problem because it is hard to understand how the amount of mass and the energy associated with the expansion could have been adjusted to such precision.

To resolve this issue physicist Alan Guth proposed the universe underwent a very rapid period of expansion increasing it size by more than a trillion in the first few nano-seconds after its birth.  This resolves the flatness problem because its size is magnified by the inflation factor so much that locally it appears flat.

The reason for this can be understood by imagining what a two-dimensional creature who was living on a surface of a balloon would observe regarding the curvature of its surface.  If the size of the balloon were small compared to his field of vision he would notice that it surface was curved.  However, if its size was very large compared to his field of vision it would appear to him to be flat.

Inflation solves the flatness problem because it predicts the size of the universe increased so much in the initial expansion that portion we can observe appears to flat.

However, another reason why the universe appears to be flat is that all of its expansive energy must originate with its mass/energy.  This is because if the universe is a closed system, the first law of thermodynamics tells us the sum of the gravitational potential of its mass/energy and its kinetic or thermal energy is constant.

As mentioned earlier, the curvature of our expanding universe is related to the ratio of total gravitational potential of its mass/energy to the total kinetic energy of its expansion.

However, as was shown in the article "Defining potential and kinetic energy?" Nov 26, 2007 the equivalence between kinetic energy and the gravitational potential of mass/energy can be defined in terms of equation E=mc^2 because it showed that, similar to Relativity mass is equivalent to energy.  Therefore, energy in all its forms must also posses the gravitational potential energy associated with mass.  Additionally the asymmetry of the equation E=mc^2 tells us kinetic energy is oppositely directed from the gravitational potential of mass/energy.

This means a balance must exist between the gravitational potential of the universe’s mass/energy and the oppositely directed kinetic energy associated with its expansion because, as mentioned earlier all of its expansive energy must come from its mass/energy.  However, if one defines "m" as the total gravitational potential of the mass/energy of the universe, the equation E=mc^2 also defines the total energy available for its expansion.  If one then substitutes "c" for "v" in the equation for kinetic energy (KE=1/2mv^2) one arrives at the equation 1/2mc^2.  This equation defines the ratio of the total kinetic energy available to power the universe’s to its mass/energy  This indicates there is a 1 to 1 ratio between the potential energy of the universe’s mass/energy and the total quantity of oppositely directed energy associated with its expansion.

Additionally because the kinetic energy of the universe’s expansion is defined by the non-linear equation  m₀ = sqrt(E²/c⁴ – p²/c²) the ratio of the gravitational potential of its mass/energy will always maintain a 1 to 1 ratio to the its expansive energy if one assumes that its expansion has been as rapid as possible.

This means the universe will be flat now and throughout its history because as mentioned earlier there is a 1 to 1 ratio  between the curvature associated with the gravitational potential of its mass/energy and oppositely directed curvature associated with the kinetic energy of its expansion, which means they will cancel each other.

Therefore, one does not have to assume the universe underwent an inflationary period to explain why it is flat now and has remained that way if one assumes the existence of four *spatial* dimensions, and that it has been expanding as rapidly as possible since the big bang.

This cannot be done in terms of four-dimensional space-time because time or a space-time dimension is observed to move only in one direction forward and therefore could not support the bidirectional movement required to define the symmetry between the potential energy of mass/energy and its kinetic energy.

However, for many this concept of a zero energy universe may sound strange, but it is rather simple to understand.  A ball thrown up in the air has two forms of energy: kinetic and gravitational potential.  If kinetic energy were considered as positive, the potential energy, due to the gravitational pull of the Earth, would be negative.  If the positive portion of the energy beats the negative portion, the ball will escape from Earth.  If the negative energy is greater, it will return.  If the total energy is precisely zero the ball will barely escape – slowing to a stop when it is infinitely far away.

Many would disagree with this concept and point to the observations of Type Ia supernovae (SNe-Ia) which verified the existence of an expansive force or "Dark Energy" that opposes the contractive gravitational energy and is causing the universe to accelerating.  However, physicists have been unable to define a source or mechanism responsible for this energy.

However, as mentioned earlier the equation E=mc^2 defines a dynamic oppositely directed relationship between the gravitational potential of mass/energy and its kinetic energy.  In other words, in a closed system if the mass in a volume decreases the kinetic or thermal energy associated with that volume increase.

Therefore, due to the asymmetry between the quantity of mass and thermal kinetic energy in a closed system we should expect asymmetry between the quantity of forces those systems contain.  In other words, we would expect a force to be generated that is oppositely directed with respect to the gravitational forces associated with mass when we convert mass to energy.  This force would have to added to the linear expansion associated with the force of the big bang therefore, it would be viewed as acceleration.  This force has been give the name Dark Energy and is responsible for the accelerated expansion of the universe.

Therefore, the mechanism responsible for generating the force called Dark Energy would be analogous to one that earlier defined the kinetic energy generated from a mass/energy.  If the universe is a closed system then one must assume the sum of the potential energy of its mass and kinetic or thermal energy must is constant.  This also means that when stars convert mass to energy in nuclear reactions a force must be generated that is oppositely directed with respect to the gravitational forces associated with mass.

However, the "concentration" of this Dark Energy relative to gravitational energy is also defined by the equation E=mc^2c were "c" equals the speed of light.  This means its "concentration" and therefore its strength is 1/c^2 weaker than gravitational forces.

The reason why dark energy is so hard to detect is that, as show above its field strength is so small relative to gravitational.

Later Jeff

The "Shadow’s" of four spatial dimensions

Copyright 2008 Jeffrey O’Callaghan

(In a PDF format)

We have shown throughout “The Imagineer’s Chronicles” there are many theoretical advantages to assuming the existence of four *spatial* dimensions a continuous non-quantized form of mass.

One of them is that it would provide a more logical explanation of the observed properties of Dark Matter than can be found based solely on the quantum mechanical assumption that mass exists only in particle form.

Wikipedia tells us "The first person to provide evidence and infer the presence of dark matter was Swiss astrophysicist Fritz Zwicky, of the California Institute of Technology in 1933.  He applied Newton’s law of gravity to the Coma cluster of galaxies and obtained evidence of unseen mass.   Zwicky estimated the cluster’s total mass based on the motions of galaxies near its edge and compared that estimate to one based on the number of galaxies and total brightness of the cluster.  He found that there was about 400 times more estimated mass than was visually observable.  The gravity of the visible galaxies in the cluster would be far too small for such fast orbits, so something extra was required.  This is known as the "missing mass problem".  Based on these conclusions, Zwicky inferred that there must be some non-visible form of matter which would provide enough of the mass and gravity to hold the cluster together."

Many physicists believe the vast majority of the dark matter is in a non-baryonic form such as neutrinos, and entities such as axions, supersymmetric particles, or WIMPs.

However, none of these scenarios is supported by observations.

Neutrinos because of their mass would be characterized by high random speeds in the early universe.  However, observations of the early universe indicate the matter that condensed to form galaxies was not hot enough to support the energy that would be associated with those high speeds.

The other particles, which could provide the missing mass fall into two classes: those which have been proposed for other reasons but happen to solve the dark matter problem, and those which have been proposed specifically to provide the missing dark matter.

Examples of objects in the first class are axions and the supersymmetric particles.  Their properties are defined by the theory, which predicts them, by virtue of their mass, they can solve the dark matter problem only if they exist in the correct abundance.

The second class of particles contain entities such as the WIMP or "Weakly Interacting Mass Particle" whose properties are not specified.  However, they are assumed to have properties that would allow them to explain the missing mass associated with dark matter along with other "ad hoc" ones that would explain why they have not yet been observed experimentally.

However, the existence of them along with axions and the supersymmetric particles is not based on observations so therefore there is no way to either confirm their existence or that they are responsible for the gravitational force associated with dark matter.

However, there is another theoretical possibility that is based on observations that has been overlooked by the scientific community.

In the article “What is Dark Matter?” Sept 10, 2007 it was shown that one could theoretically explain and predict the gravitational forces associated with Dark Matter in terms of a continuous non-quantized form of mass.  This non-baryonic and non-particle form of mass would have all the observed properties dark matter in that it would not interact with ordinary matter via electromagnetic forces and since it is made up of mass it would add to the gravitational force of the particle matter in the universe.

However, unlike WIMPS its existence is supported by observations.

For example, the observation that energy in itself is not quantized because a photon can have any frequency greater than zero or less than infinity and the equation defining the relationship between mass and energy, E=m*c^2 indicates that mass is completely convertible to one or more photons with energies greater than zero or less than infinity.

This means a continuous non-quantized medium must be available to support the continuous properties associated with mass energy and the electromagnetic spectrum.

Additionally the existence of a continuous non-quantized form of mass is supported by the observation of Davisson and Germer, when, in 1927 they observed that electrons along with other particles are diffracted by crystals.  This is because the only way to explain the observed wave properties of individual particles is to assume they must have the continuous non-quantized geometric substructure associated with a wave.

However, the most significant theoretical advantage to assuming its existence is that it would allow one understand, in terms laws of classical wave mechanics the quantum mechanical aspects of mass and energy.

In the article "Why is mass and energy quantized?" Oct. 4, 2007 it was shown that one could explain and predict the quantum properties of mass and energy in terms of a classically resonating system or "structure" form by a matter wave moving through a continuous non-quantized form of mass. 

Classical wave mechanics tells us a resonant phenomenon will occur with all types of vibrations or waves: there is mechanical resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR), and resonance of quantum wave functions.  Resonant systems can be used to generate vibrations of a specific frequency (e.g. musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies.

However, as mentioned earlier the observation that particles mass has the continuous non-quantized geometric substructure associated of a wave means that according to classical wave mechanics a resonant system will be established is space.  This provides observational evidence supporting the derivation of the quantum mechanical or particle aspects of mass and energy, as was done in the article "Why is mass and energy quantized?" in terms of discrete resonant systems formed by vibrations in a continuous non-quantized form of mass.

The are some who feel that the existence of substance like a continuous non-quantized form of mass permeating space validates the Newtonian concept of absolute space.  However, as was shown in the article “Reference frames” July 1, 2008 this is not true.

Briefly, in  the article "Defining energy" Nov. 26, 2007 it was shown the momentum or energy of an inertial reference frame can be derived in terms the displacement of its three dimensional volume with respect to a fourth *spatial* dimension.

Isaac Newton defined an inertial reference frame as one in which 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.

However, as was shown in the article "Defining energy" the momentum or velocity of an inertial reference frame is defined by a constant displacement with respect to a fourth *spatial* dimension of the continuous non-quantized form of mass defining its volume. 

Therefore, the existence of a continuous non-quantized form of mass does not validate the existence of absolute space because its motion can only be measured relative to other reference frames.

This shows there is viable alternative based solely on experimentally verifiable observations to the assumption that dark matter is made up of exclusively of matter in a particular form.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2010 Jeffrey O’Callaghan

(In a PDF format)

The effort to unify all of the forces and laws of nature or find a "Theory of Everything" has primarily focused on explaining gravity in terms of the concepts contained in quantum mechanical particle theories.

However, it is unfortunate that some have not made an effort to find it by explaining the quantum mechanical properties of particles in terms of gravitational forces because if they had we may have found it by now.

We observe that all particles have mass, which is associated with gravitational force.  However, for the past century the brightest minds of the scientific community have been unable to define how this force can be propagated by a particle using the current quantum mechanical paradigms.  Additionally, even with the recent advancements in observational technologies, no one has observed the graviton or particle that many feel is responsible for the propagation of gravitational forces.

However, the fact that we have not been able to define a unifying mechanism either mathematically or conceptually for the observed quantum mechanical and gravitational properties of nature in terms of the current paradigms may not be related to their content but to how we are attempting to integrate them.

For example, the fundamental assumption of Quantum mechanics is that mass and energy is contained in discrete irreducible units or packets of energy called quarks and leptons.

However, the graviton or particle many physics associated with gravitational force has not, as mentioned earlier been observed.  Some feel that this is due to the fact our instruments are not yet advance enough to detect it but it could also be because gravitational force is not propagated by a particle but by a continuous property of mass/energy.

This conclusion is supported by the observation of Davisson and Germer, when in 1927 they observed that electrons and along with other particles are diffracted by crystals.  This means they must have a wave component because that is the only way to explain how they can generate a diffraction pattern.  However, it also indicates that each individual particle must also have the continuous geometric substructure associated of a wave.

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

In the article "Defining gravity" Dec. 15, 2007 it was shown that one can theoretically derive the relativist properties of motion, gravity and the fact that it is equivalent to an accelerated reference frame in terms of a continuous geometric property of four *spatial* dimensions in a manner that makes predictions identical to these made by both The General and Special Theories of Relativity.

While in the article "Why is mass and energy quantized?" Oct. 4, 2007 it was shown that one can derive the quantum mechanical properties of mass in terms of a classically resonating system generated by a matter wave on a continuous "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.

(In a latter article "The geometry of quarks" it will be shown how and why quarks join together to form these resonant systems in terms of the geometry of four *spatial* dimensions.)

The existence of four *spatial* dimensions would give a "surface" of three dimensional space the ability to oscillate spatially with respect to it 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 three-dimensional space would, according to classical mechanics would generate a resonant system or "structure" in it.  These resonant systems are what defined a particle in the article "Why is mass and energy quantized?"

This cannot be done in terms of four-dimensional space-time because time or a space-time dimension is only observed to move in one direction forwards and therefore could not support the bi-directional spatial movement required to establish classical resonance.

The assumption that the quantum mechanical properties of mass are a result of a continuous properties of space is also supported by Davisson and Germer’s observation that all particles has a wave component because, as mentioned earlier it means they must have the continuous geometric substructure of a wave.

However, as mentioned earlier the properties of gravity can also be also derived in terms of the continuous property of four *spatial* dimensions. 

This suggest that we may have already found a "A Theory Everything" if, as mentioned earlier we had tried to derive the discontinuous quantum mechanical properties of mass in terms of the continuous properties of gravity instead of deriving the discontinuous properties of particles in terms of continuous properties of gravitational forces.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright 2008 Jeffrey O’Callaghan

(In a PDF format)

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

One of them is that it would allow for the theoretical definition of the "boundaries" of a quantum particle in terms of the laws of classical physics.

In the article “Embedded Dimensions” Oct. 22, 2007 it was shown that each point in three-dimensional space is embedded in a fourth *spatial* dimension and energy only propagates along a "surface" of three-dimensional space.

Classical wave mechanics tells us the oscillations generated by a wave on the surface of water is responsible for the transmission of energy.

Therefore, as was done in the article "The Photon: a matter wave?" Oct 1, 2007 one could explain how energy is propagated through three-dimensional space by extrapolating the laws classical of wave mechanics to oscillations in its ‘surface” generated by a matter wave moving on it with respect to a fourth *spatial* dimension.

However, if this were true why is it that we cannot see the fourth *spatial* dimension.

The reason is because, as was mentioned earlier all forms of energy move along a "surface" of three-dimensional space.  Therefore, electromagnetic energy or light could not penetrate and illuminate the fourth dimensional component of space.

This is analogous to how a wave on the two-dimensional surface of water can only "illuminate" or interact with objects that comes in contact with its surface.  Therefore, a fish could not "see" the three-dimensional components of objects above its surface if a wave on that surface were the only way energy or information could be transmitted through its environment.

Similarly, three-dimensional being would be unable to "see" the fourth dimensional component of objects if energy is only transmitted on a "surface" of three-dimensional space.

In Einstein’s General and Special theories of Relativity he showed that one could derive the relativistic properties of motion, gravity, space and time in terms of an interaction of a three-dimensional volume with a fourth time dimension.  But he was unable to derive a mechanism that can explain the quantum mechanical properties of mass and energy.

However, in the article “The “Relativity” of four *spatial* dimensions” Dec. 1, 2007 it was showed, one can derive the relativistic properties of gravity, motion, space and time in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension in a manner that makes predictions identical to those of Einstein’s theories.

(This curvature is analogous to the space-time curvature Relativity assumes is responsible for gravity.)

While in the article “Why is mass and energy quantized?” Oct. 4, 2007 it was shown one can also derive the quantum mechanical properties of mass and energy in terms of classically resonating system generated by a matter wave moving on a “surface” of a three dimensional space manifold  with respect to a fourth *spatial* dimension.

In 1924 Louis de Broglie was the first to realize all particles have an oscillating or wavelike component.  In his paper, “Theory of the double solution“ he attempted to define a causal interpretation of their wave properties in the classical terms of space and time.  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.

In classical physics, a point on a two-dimensional surface of paper is confined to that surface.  However, that surface can oscillate up or down with respect to three-dimensional space.

Similarly, an object occupying a volume of three-dimensional space would be confined to it.  However, as was shown in the article “Embedded Dimensions” it could, similar to the surface of the paper oscillate with respect to a fourth *spatial* dimension.

Classical wave mechanics tells us resonance phenomena will occur with all types of waves or oscillations that occur in confined volumes.

Therefore, the confinement of oscillations in a "surface" of three-dimensional space with respect to a fourth *spatial* dimension would define the geometric boundaries of the resonant system responsible for the quantum characteristics of energy in the article "Why is mass and energy quantized?"

This suggests that one could by theoretically extrapolating the laws of classical physics to a fourth *spatial* dimension define boundaries of a particle in terms of oscillations in its mass and energy components.

This defines the quantum properties of mass and energy in terms of the laws Classical Physics because it shows how extrapolating them to four *spatial* dimension would allow one to theoretically define their "boundaries" of and the reason why energy appears to be quantized.

However, it is not possible to define a classical resonant system in terms of the geometry of space-time because time is only observed to move in one direction forward and therefore could not support the bi-direction movements to define the boundary conditions required for it to occur.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2010

(In a PDF format)

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

One is it would allow one to define a background independent quantum theory.

One of the fundamental difficulties in unifying Quantum Mechanics with Einstein’s General Theory of Relativity is that one is background independent while the other is not.

Einstein tells us the geometry of space evolves with time and the laws of nature have to be expressed in a form that does not assume that space has a fixed geometry.  In other words, they are background independent. 

However, present quantum theories can only be formulated in a fixed geometric background. 

For example, Quantum Field Theory was unified with Maxwell’s theory of electromagnetism by assuming that electromagnetic energy does not interact with the background geometry of space.  In other words, two photons or quanta of electromagnetic energy can move through or past a fixed point in space without interacting with each other.

However, two gravitons or quanta of gravitational energy will interact with each other as they pass a fixed point in space because as mentioned earlier Einstein tells us gravity alters the geometry of space. 

This is why, as mentioned earlier one of the problems physicists encounter in defining quantum gravity is that present quantum mechanical models are background independent while Einstein tells us that gravity is not.

However, as mentioned earlier one can by assuming the existence of four *spatial* dimensions instead of four-dimensional space-time derive a background independent quantum theory with respect to gravity..

In this article "Gravity" Dec 15, 2007 it was shown that it is possible to derive gravity in terms of a curvature in a "surface" of a three-dimensional space manifold with respects to a fourth *spatial* dimensions in a manner that makes predictions identical a curvature in a space time manifold "The General Theory of Relativity assumes was responsible for gravity.

Additionally the article "Why is mass and energy quantized?" Oct 4, 2007 showed it is possible to explain and predict the quantum mechanical characteristics of mass by assuming they are a result of a classically resonating system or "structure" formed 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.

(In article "The geometry of quarks" Mar. 15, 2009 it will be shown how and why quarks join together to form these resonant systems in terms of the geometry of four *spatial* dimensions.)

The existence of four *spatial* dimensions would give a continuous non-quantized form of mass (the substance in classical resonance) 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, according to classical mechanics would generate a resonant system or "structure" on a surface of a three dimensional space.

Einstein derived the relativistic properties of motion by merging the dimensions of space and time.  However in the article “Why Space-time?” Nov. 22, 2007 it was show that one can derive the relativistic motion in terms of four independent *spatial* dimensions.

However, this also means a quantum mechanical model based on the resonant properties of four *spatial* dimensions would be background dependent with respect to its geometry while background independent relative to a curvature in that geometry.  This is because, as the article "Why is energy quantized?" showed the quantum mechanical characteristics of mass are the result of a resonating "structure" on a "surface" of three-dimensional space therefore, their interactions would be independent of the magnitude of a curvature in that "surface".  In other words, two photons or quanta of electromagnetic energy can move through or past a fixed point on curved "surface" of three-dimensional space without interacting with each other because their energy is not dependent on the geometry of space but is only dependent on the energy of associated with their resonant properties.

However, gravity would be background independent of the geometry of four *spatial* dimensions because as was shown in the article "Gravity" it is dependent on the changing geometry or curvature of a point on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension.  Therefore two gravitons or quanta of gravitational energy will interact with each other as they pass a fixed point in space because of they share a property of that curvature.

Therefore, one could define a background independent quantum theory relative to gravity if one assumes as we have done the existence of  four independent *spatial* dimensions.

This is not possible if one defines the universe in terms of four-dimensional space-time because Einstein defined gravity in terms of the interdependence of space and time and therefore on could not separate gravity from its quantum mechanical properties.

Later Jeff

The "Shadows" of four *spatial* dimensions

Copyright 2009 Jeffrey O’Callaghan 

(In a PDF format)

We have shown throughout "The Imagineer’s Chronicles" observations of our environment suggest the universe is made up of four *spatial* dimensions instead of four-dimensional space-time.

One of those observations involves the expansion of the universe. 

The conclusion that the universe is undergoing an expansion is based Edwin Hubble observation, made in 1929 that the light from most galaxies is shifted towards the red end of the spectrum.  The only logical explanation for this is that the universe is expanding.

However, if the universe is expanding what is it expanding towards.  

Einstein’s theories, which are used by most physicists and cosmologists to derive gravity, the relativistic the properties of motion, space, and time are based on the existence of four-dimensional space-time.

However, the expansion of the universe appears to be spatial in nature.  For example, we do not observe three-dimensional space to be expanding towards a time dimension but towards another spatial one because only the distance between galaxies is increasing not the time.

Therefore, its proponents must assume existence of another spatial dimension over and above the three that it already contains because one cannot define spatial expansion of three-dimensional space in terms of a space-time environment consisting of only three spatial dimensions.

In the article "Why Space time?" Sept 27, 2007 it was shown it is possible explain and predict the relativistic properties motion, space, time and  gravity and its equivalence to accelerated reference frames in terms of four *spatial* dimensions in a manner that makes predictions identical to those of Einstein’s Special and General theories of Relativity.

One advantage to this approach is that it is possible to integrate the observation that three-dimensional space is undergoing a spatial expansion into a relativistic theory without have to assume the existence of another dimension as Einstein’s theories require because it could be defined in terms of its "spatial movement" towards a fourth *spatial* dimension.

This means physicists and cosmologists could, by assuming space is composed of four *spatial* dimensions instead of four dimensional space-time explain and predict gravity, the relativistic the properties of motion, space, and time and define what the universe is expanding towards in a single theoretical construct.  Whereas, they must go outside of one based on the existence of four-dimensional space-time and assume the existence of another spatial dimension to account for its spatial expansion.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

(In a PDF format)

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 that 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 predictions 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.

However, if one could, as done in the article "Defining gravity" Dec 15, 2007 define it and the relativistic properties of motion, as was done in the article "The relativity of four *spatial* dimensions" Dec. 01, 2007 in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension instead of one in a four-dimensional space time manifold one could replace the time dimension with a spatial one.

But if true one should also be able to explain how and why the expansion of the universe is so uniform in terms of four *spatial* dimensions. 

As was shown in the article “Embedded Dimensions" Oct. 24, 2007 each axis of three-dimensional space is scalar invariant with respect to a fourth *spatial* dimension.  Therefore, the movement of any point in three-dimensional space with respect to a fourth *spatial* dimension would result in the expansion of three-dimensional space around that point. 

One way of understanding how this 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 that 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 so uniform if one assumes three-dimensional space is expanding towards a fourth *spatial* dimension and that it’s individual axes are scalar invariant with respect to it.

However, this also provides a theoretical connection gravity and the observed properties of an expanding universe because it defines both in terms of a common mechanism related to the existence of a fourth *spatial* dimension.  This is  because if three-dimensional space is scalar invariant with respect to a four *spatial* dimension, the movement of a point in three-dimensional space "inward" with respect to it would result in its "contracting" spherical along all three-dimensional axis.  This contraction would result in a force being directed towards its center, which can be shown, as was done in the article “Gravity in terms of four *spatial* dimensions” Jun 01, 2009 to be responsible for gravity.

This cannot be done as mentioned earlier if one assumes as Einstein did that gravity is caused by a curvature in a space-time manifold because one cannot define the spatial expansion of the universe in terms of only three spatial and one time dimension.

Later Jeff

The *Shadows* of four spatial dimensions

Copyright 2010 Jeffrey O’Callaghan

(In a PDF format)

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 “The Flatness Problem”.   It involves the 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 or 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.

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 displacement caused by that crumpling would be zero because the height of it above its surface would be offset by an oppositely directed one 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".

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 reduction in the displacements above and below its three-dimensional surface as it expanded.

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 as we have done here 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

(In a PDF format)

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 defining a common mechanism responsible for 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.

But since its discovery, no one has been able to define a mechanism that can account for its existence in terms of four-dimensional space-time.

However, one can if one assumes the universe is composed of four *spatial* dimension.

In the article “Defining energy” Nov. 26, 2007 it was shown it is possible to derive all forces including gravitational in terms of a curvature in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimensions in a manner that makes predictions identical to those made by Einstein theories.

(This curvature is analogous to the curvature in a space-time manifold Einstein theorized was responsible for gravity.)

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

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 or converting mass to energy as is does in the nuclear reaction is stars 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 and 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 expansion that occurs in addition to any resulting from the expansive energy associated with the Big Bang therefore it will be viewed as acceleration.

This means that one can explain the causality of forces Dark energy and gravity 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. 

This is because the relative "concentration" of the forces associated with dark energy and gravity would be defined by the equation E=mc^2" because as mentioned earlier a theory based on four *spatial* dimension makes predictions identical to those made by Einstein theories.  Therefore, because, as that equation shows the expansive effects it has on space are considerably less "concentrated" by factor of c^2 than the contractive forces associated with it would be extremely difficult to observe it between gravitationally bound objects.

However, the expansive forces caused by the conversion of mass to energy in gravitational bound objects would be distributed throughout the entire universe causing its overall acceleration.

This shows that one advantages to defining the universe in terms of the geometry four *spatial* dimensions instead of four dimension space-time is that it would 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

(In a PDF format)

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 macroscopic properties of space is based on Einstein’s concept of a space-time manifold.

However, it is difficult to understand how fluctuations in 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 them 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 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 content of quantum fluctuations 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.

Therefore, as mentioned earlier one has to assume that either Einstein’s theories or the concept of quantum fluctuations in invalid.

However, as was done in the article “Defining energy” Nov 26, 2007 defining it 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 quantum fluctuations in terms 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 quantum fluctuations in terms of a space-time manifold because the spatial displacements are oppositely directed, the time differentials associated with their relativistic properties will cancel out.

Later Jeff

The "Shadows" of four spatial dimensions

Copyright Jeffrey O’Callaghan 2009

(In a PDF format)