Michael (Tony) A. Sakovsky Sr.       07/01/2001

Preface:

This document attempts to provide a unique explanation of why a forth spatial dimension is required to completely explain our universe and why our universe appears to be 'short' this spatial dimension.  It does this by primarily using logic as a tool and is based upon current findings in physics and other scientific areas as evidence. It draws the reader down a path that is logically irrefutable and is understandable by the common man.  Higher level mathematics are NOT required to understand or explain the concepts herein nor is one required to understand such things as vector spaces and tensor fields to understand how our universe works at it's most basic level.

As we postulate that our universe must have a forth spatial dimension, as did by Kaluza-Klein, and if this forth spatial dimension is NOT a 'hypersurface' (Hypersurface is a trick used by physicists  to add an extra dimension without actually having to explain it's properties or origin!) in which our three perceived spatial dimensions are embedded then we must ask WHY do we ONLY perceive three spatial dimensions and one time dimension?  We also assume that the 4th dimension is NOT compactified for reasons which will become clear later.

Within this document an excepted explanation is never taken at face value and we attempt to dig deeper for answers regarding these and other questions.  Sometimes this requires re-asking the 'simple' questions or a postmodern approach.  We re-ask such questions as; 'what is time?', or 'what is the most basic constituent of matter?'.  These may seem like questions with obvious answers to some people but surprisingly physicists CANNOT really answer these questions yet.

Answering the Unanswerable:

In postulating that a forth spatial dimension DOES in fact exist then, not only will it finally be possible to answer these  unanswered questions (and many others), but it brings us closer to understanding what reality actually is. Coincidently the following unanswered questions are a few of the driving forces behind the reason for my research into a 'real' explanation of our world. 

- Heisenberg Uncertainty Principal: certain measurements disturb the system and push the wavefunction into a superposed state or a state where the attributes of the wavefunction are unknown.  (WHY is this? It seems illogical when talking of discreet particles. And in reality it IS illogical when discreet particles are measured)
- Wavefunction Collapse: Tightly tied to the Uncertainty Priciple, once the observation is made exactly what is and exactly when does the Wavefunction collapse?  Or as stated more accurately when and what exactly causes one eigenfunction to become dominant while the others become more unknown?
- Quantum Behaviour: Why must particles exist at certain energy levels before they will interact with other particles?  For example; a photon can exist at any energy level but must be at specific energy levels before interaction occurs?
- Gravity: what is it? why is it? Although physics/science has advanced incredibly over the last 200 years not one explanation which predicts gravity exists today. Predict being the KEY term.  Many so called theories attempt to explain gravity but none can actually predict it! Why? 
-  Recent anomolies; found in the GPS; accelarated universe expansion; slight miscalculations of sattelite trajectories; new findings that radioactive decay is based on the earths sidereal day and relative position in the galaxy; etc.   ---  No current theory offers a prediction for any of these.
- Matter/Antimatter created at the BIG BANG event.  Why is there more matter than anti-matter left over?  Is there ANY anti-matter left over which occurs naturally?  - Easily explained via the definition of vector masses using the 4th spatial dimension!  - Sakorov: "If the Big Bang is to produce unequal amounts of matter & anti-matter then the symmetry of CP (charge conjugation (C) and parity (P) ) must be violated." - This is assumption leads us down the wrong path of assuming that there is actually a breaking of the CP symmetry or that this 'explanation' is proper for why there is more matter than anti-matter.  Before anyone should accept the fact that symmetry is being broken we should require stronger experimental evidence.  At this point we simply do not know that equal amounts of anti-matter have NOT been produced because our present technology does not allow us to find it.
- Molecules are non-chiral; Physics is not the same when all particles of an atom or molecule are replaced by their corresponding anti-particles and then viewed in a mirror. - New frontier experiments:  SLAC (BaBar) in USA, KEK (Belle) in Japan (1999).  -- Why is this direct violation of symmetry allowed?  This CANNOT be answered by the Standard Model of today's physics.  To answer this requires a 4th spatial dimension.

Symmetry is defined as: sym·me·try (sm-tr)  n., pl. sym·me·tries. 
1. Exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis. See Synonyms at proportion. 
2. A relationship of characteristic correspondence, equivalence, or identity among constituents of an entity or between different entities: the narrative symmetry of the novel. 
3. Beauty as a result of balance or harmonious arrangement. 

Symmetry dicates that the actions, aspects or effects of anything be symmetrical.  For every action there is an equal and opposite reaction. This is the law of symmetry coming into play in Newton's Third law of motion.  Not only must the effects be symmetrical but the aspects of the total system must be equal and opposite.  For example; reflection in a mirror, or gravity (think about planetary gravitation; a planet pulls in equal an opposite directions, the planet itself is round and symmetrical. 

Symmetry exists in EVERYTHING.

This is sometimes hard to understand or except in our world where objects seem to exist asymmetrically.  An apple or a cloud or a rock is NOT symmetrical by our crude methods of measuring.  We must delve deeply into the structure of the universe in order to see the true symmetry in all the objects around us.

But in order to understand the universe more completely we must ask ourselves the following questions; 

At what level does true symmetry actually occur?  - We shall look deep enough into the structure of space and matter to see where the first and truest level of symmetry occurs.

Is Symmetry is THE most powerful force in our universe?  Once the answer to the first question is answered it becomes obvious that it IS the most powerful force in the universe.

Is Symmetry actually defined by the conservation laws? And could they possibly be written symmetrically as follows:
Conservation of Energy/Mass
Conservation of Linear/Angular momemtum
Conservation of Positive/Negative Charge
Conservation of Leptons/Baryons
They are written in this slightly unorthodox way to demonstrate possible hidden symmetries within these laws. Manisfestations of symmetry are; magnetism - north and south poles; spatial - up, down, left, right, back and forth; waves - crest and trough.  Time?  Not ever proven but a nice mathematical exercise for now.  We measure time by the cyclical changes of a system, for example; The decay of a cesium atom in an atomic clock, or the swinging of a pendulum, or the rotation of the earth. Since time can ONLY be measured by comparing the cyclical changes of a system and comparing them with the change to another system how could you reverse change on the proper scale so that it is completely reversed? Is it possible to completely control the energy flows and the matter in a system or universely so that the changes happen in exactly the opposite way? This violates the laws of entropy and because of this it would require significantly more energy to reverse a system then what originally went into it.  Mathematically it is possible to reverse entropy but is it possible in reality? And isn't reality exactly what we are trying to describe and understand?

Some theories associate the amount of symmetry with levels of entropy; for example the theory of Ugly Symmetry does this (Shu-Kun Lin, lin@mdpi.org, http://mdpi.org/lin/, MDPI, Molecular Diversity Preservation International, Sangergasse 25, Basel CH-4054 Switzerland.)  Which simply stated says that higher levels of symmetry are associated with equally higher levels of entropy/distribution/disorder.  When any system is examined on a large enough scale this is simply NOT true or said differently; the total forces acting on the system being examined must be accounted for in the formulas.  The Ugly Symmetry theory and many theories like this only consider a small portion of the effects within and on the system in question.  They also deal with ideal gases and ideal environments.  (Ideal meaning that there is no perceived outside influences).  This is where the theories go off track.  By doing this the theories become complex exercises for the mind and are minimally applicable to reality. 

A simple exercise which demonstrates why higher symmetry denotes lower entropy is as follows:  Imagine a substance, it can be a gas of the same atoms or different types of atoms or molecules and atoms mixed, it does not matter, in space removed from any perceptable gravity field.  This gas would over time fall under the influence of gravity of the individual atoms and coalesce into a larger body, a planet, planetoid, rock, star, or anything in between given enough starting matter.  With enough time and mass this body would possibly form a blackhole.  Each being more symmetrical (spherical) then the latter.  And the sphere is the closest example of perfect symmetry we have.  But the blackhole is, as I will demonstrate later, an example of the ultimate symmetry both internally and externally.

And since symmetry is the greatest universal force we will demonstrate how spatial dimensions MUST adhere to the law of symmetry and are, in fact, the pure essence of symmetry and also how spatial dimensions contribute to the symmetry in EVERY effect that manifests itself to us. 

DIMENSIONS:

We live in a world in which we perceive three spatial dimensions.  Or put another way, three degrees of freedom of movement manisfest themselves to us.  Typically and for brevity we refer to them as 3D.  Typically the degrees of freedom are measured relative to each other.  They are said to be orthogonal to each other, or offset by 90 degrees from each other. We call them typically up/down, left/right, backward/forward, or many other words which are typically used to describe a direction relative to us and always paired with a description of the opposite direction.  The dimensions, each examined separately and taken by themselves are symmetrical, meaning there is always two directions in which a body may move or travel relative to it's current position and motion; ie left/right is one dimension, up/down is another.  This is the Law of Symmetry coming into play regarding the separate spatial dimensions.

But now we must ask ourselves why there are only three perceived dimensions.  Three is not a symmetrical number.  By definition it is an odd number and cannot be made symmetrical by using whole integers.  (It also introduces us to the problem of why molecules can be non-chiral but that is a topic for a future discussion.)  Why not two spatial dimensions? or four? or sixteen for that matter?  Why does it appear that the number of spatial dimensions does NOT adhere to the Law of Symmetry. The answer is actually simple; the number of dimensions MUST adhere to the Law or Symmetry, we are simply missing some key perspective.  We know for a fact that there are more than two dimensions since we can percieve three.  Our senses force us to preclude that there is any less than three.  Since we know that there are at least three and we MUST adhere to the law of symmetry then we must assume or, at least, start with the assumption that there MIGHT possibly be more and that it must be an even number, so we must begin with four dimensions since four is our first symmetrical number above three.

Immediately most people will say that time is naturally the forth spatial dimension and they may ask why have I not mentioned time up to this point?  They may also cite Einstein and use his Theory of Relativity to argue (usually with evangalistic fervor) that Einstein proved that time is a spatial dimension.  What he proved though was that there is a perceived relationship between time and space NOT that time is a spatial dimension.  While time is a dimension in that it is a measurable quality, one must remember that this paper is discussing spatial dimensions. Time is definately not part of the three spatial dimensions since it does not fit into the definition of a spatial dimension in which it has degrees of freedom of movement.   One argument that physicists will bring up at this point is that time manifests itself to us in a different way.  But yet there is evidence that this is not true, that it is simply a preceived effect on spatial distances, and there is no true effect.  They may offer mathematical explanations which attempt to prove that time is spatial because the time variable fits into MANY formula, BUT they NEVER explain how they made the intuitive jump to explain HOW time is spatial. They only succeed in confusing the concept of spatial dimensions further since, as stated before, time IS a dimension, just NOT a spatial one.

I will go into some depth later regarding what time is and how it does interact with our spatial dimensions.  It is an important factor when preceptions between multiple reference systems are to be compared and since we are going to speak initially about a single reference frame we have no need to introduce time yet.  Einstein's theories of relativity are effective in demonstrating what the perceived effects will be  mathematically so there is no reason to re-invent the wheel so to speak.

At the outset it is important to state a few facts regarding spatial dimensions.  Most of the facts are obvious but some are not, so we must clarify as follows:
- A spatial dimension should be defined as a measured change in distance along an arbitrarily choosen axis.  Whether it is measured or not is inconsequential and so a dimension can be stated as the ability of on object to have it's position changed. 
- A spatial dimension is a degree of freedom of motion - Up/Down, Left/Right, Back/Forth.
- Spatial Dimensions MUST adhere to the law of symmetry.

Based upon observation one can make the statement that we live in a universe which consists of spatial dimensions.  We have the ability to move in three ?degrees of freedom?.  This was demonstated in the previous section on Symmetry.  Again we ask 'Why only three degrees of freedom?'.  Why are we restricted to three?  What process or law governs this?  To answer these questions and more we must examine what a spatial dimension really is.  We must do this by taking one dimension and adding from there.

To begin I will list the current known spatial dimensions and discuss each in turn.  Some are incorrectly classified as spatial and I will logically demonstrate why:

* - It is important to state that we are discussing spatial dimensions in the context of singular existence with no higher dimensions in the context of the discussion.  For each spatial dimension we must remember that NOTHING exists outside or inside the actual dimensional universe unless defined by another dimension.  We must not let our minds trick us into adding extra dimensions or phenomena where there is no need to since this would only obscure the logic we are unfolding here.  We are only discussing proved observations within this section.

Dimension Zero / Point of Existance:

Geometry:
Represented by a point - ?.?  This is the ?dimension of existence?.  It is not a spatial dimension simply because it does not agree with the definition of having ?degrees of freedom?.  It is wrong to assume that there is anything ?inside? or ?outside? of the point.  To do so would be adding another dimension to the ?point of existence? and it would immediately become a ?line?.  The human mind is a wonderful thing and it gives us the ability to imagine almost ANYTHING except the difficulty enters when we try to imagine a ?point? that has no spatial size. This means that there is NOTHING else EXCEPT the ?point? itself.  It is difficult because our human minds tend to try to build on concepts by associating the concepts with things our minds know how to process readily.  So it is easy for our minds to ?trick? us by attributing other aspects to the point dimension. 
Symmetry:
The Zero Point Dimension/Dimension of Existence is symmetrical in that it either exists or it does not exist. ie. By definition symmetrical is equal and opposite, just as 0 is symmetrical with 1 (I specifically did NOT use -1 since -1 is defined as the absence of something or the potential for something whereas 0 defines the nothingness and there is logically NO potential within a zero point dimension).  The Zero Point Dimension is the very essence of existence is symmetrical with non-existence.  I will leave it up to the philosophers to discuss how non-existence can be the ?equal and opposite? of existence.

Dimension One / Single Dimension Continuum:

Geometry:
Basically we imagine this as a single line ?___________?  with no perceived beginning or end.  Every point on the line being 180 degrees from every other point on the line.  This statement identifies the fact that there can be no curves or twists in the line since if there was a curve or bend then there would be certain points that are not 180 degrees from other points.  This fact proves that the line can have no other spatial dimensions, it has only one degree of freedom.  To demonstrate; a line represents the ?x? axis on the cartesian coordinate system, if a choosen point was not 180 degrees from another choosen point on the line then that point would lie outside the ?x? axis and would be considered ?extended? into the ?y? axis.  This would mean, by definition, that there is a second dimension and this is not allowed for a one dimensional existence/universe.

Continuity:
An important observation to make at this time is the fact that the only change allowed within this dimensional universe is along the ?x? axis.  We cannot allow ourselves to assume or imagine that a line is made from separate ?points of existence?.  We CAN allow ourselves to ?choose? a point for our own purposes, mathematically or imagninatively, but this does not make the point real in the context of this one dimensional universe.  A spatial dimension is not divisible because it is defined as a single continuum.  If the process of dividing a spatial dimension were allowed in reality then the dimension would not be continuous and continuity is a required property for a spatial dimension.  Imagine moving your hand or body through some random place and it instantly appears in another distant place with no apparent connection between.  The choosen body part would be disconnected from the rest of your body( and immediately fall off! ;-o).  There would be no logical or mathematical way to maintain any physical processes across a discontinuous distance. This would also violate the law of casuality.

Change:
Now if change is only possible in one spatial dimension then we must define what ?type? of change is allowed.  If the is a ?single degree of freedom? we must ask then ?what?? would move through a single dimension.  This thing would have to be, at most, of a single dimension itself and it would have to ?exist? since every ?imagined? point is a ?point of existance?.  We could imagine each point of existance moving back and forth through the ?x? dimension as in a longitudinal wave.  This would nessecitate that the spatial dimension itself be elastic to some degree.  The elasticity would define how large a wave could actually be.  It would define it?s ?wave function? or it?s frequency, speed and amplitude.  One can imagine this as how quickly the spatial dimension would recover from the distortion or stretching and bunching of a wave propogating through it.

Symmetry:
A single spatial dimension CANNOT be symmetrical with itself.  It is not equal and opposite to itself.  It CAN be symmetrical ?within? itself but not by itself.  At any imagined point within this dimension the dimension extends equally to the ?right? x+, and ?left? x-.  Mathematically this calls to mind Dekinds Cut Thereom where when a dividing point is choosen on the line there is an infinity to the ?left? as well as to the ?right?.  Simply stated, for any length traveled to the ?left' you can travel an equal length to the ?right?.  If a single spatial dimension is not symmetrical with itself then one must ask what is needed to exist that IS equal and opposite?  This MUST be another spatial dimension.  It would be opposite by being, what we call, 90 degrees or orthogonal from the original dimension.  This leads us to the next dimension......
 

Dimension Two / Two Dimensional Continuum:

Geometry:
We can imagine this spatial dimension as an infinite sheet of material but with NO thickness.  It has length and width but no height.  It extends into the ?x? and ?y? directions but not the ?z? direction.  Every line within the dimension is 180 degrees from every other line.  The same logic applies to this two dimensional continuum as applies to the single dimension.  Again, there are no curves or twists on the surface as this would mean that there would be ?some? line which is not 180 degrees from another line.  If somewhere along the length of a line it was NOT 180 degrees from any other line then by definition it would have to be extended into the next dimension, in this case the ?z? dimension.  So, again, when we speak about a continuum with only two spatial dimensions then speaking of curves or twists is not allowed.
Change:
Change is allowed ONLY along the ?x? and ?y? dimensions.  Again the only type of change allowed is the stretching or flexing of the spatial dimensions along the ?x? and the ?y? axis.  There is no ?degree of freedom? along the ?z? axis.  The only ?things? that could exist in this continuum would be longitudinal waves.  The spatial dimension itself would be compressed and rarefied to allow the transmission of energy in any direction. Transverse waves do not exist in this dimension except only as manisfestations of longituinal waves.  Since the degrees of freedom are only along the ?x? and ?y? axis the waves