A fundamental issue in Einstein Theory of Relativity is if all motion is relative how can we measure the inertia of a body? Einstein and many others assumed we must measure it with respect to something else. But what if a particle is the only thing in the universe, how can we measure it.

Mach, an Austrian physicist and philosopher developed a principle which some have interpreted as the motion of such a particle’s has no meaning if it was alone in the universe.

In Mach’s words, "the principle is expressed as the investigator must have knowledge of the immediate connections, say, of the masses of the universe. There will hover before him as an ideal insight into the principles of the whole matter, from which accelerated and inertial motions will result in the same way."

Einstein considered Mach prospective so important to the development of General Relativity that he christened it Mach’s principle and used it to explain why inertia originates in a kind of interaction between bodies.

For example, according to General Relativity, the benchmarks for all motion, and accelerated motion in particular, are freely falling observers who have fully given in to gravity and are being acted on by no other forces. Now, a key point is that the gravitational force to which a freely falling observer acquiesces arises from all the matter (and energy) spread throughout the cosmos. In other words, in general relativity, when an object is said to be accelerating, it means the object is accelerating with respect to a benchmark determined by matter spread throughout the universe. That’s a conclusion which has the feel of what Mach advocated. So, in this sense, general relativity does incorporate some of Mach’s thinking.

However, he provided another way of defining inertia that does not require the existence of any other objects but relies only on the geometric properties of space defined in his General Theory of Relativity. In other words, geometry of space itself provides an absolute baseline for inertia.

In physics inertia is the resistance a physical object to a change in its velocity. Therefore, one can define a baseline for its measurement if one can find a universal starting point for it based on objects velocity.

One of the most logical ways to do that would be to use the observable differences between the two types of motion; velocities and accelerations.

For example, velocities transverse the same space or distance in a given time frame while accelerations transverse an exponentially increasing distance over that same time period.

This tells us the primary difference between them is a component of space not time because if one uses the same time frame for both the only thing that distinguishes them is the distance they transverse.

However, Einstein defined the geometry of space and our universe in terms of time therefore, because space not time is, the variable that distinguishes velocities from accelerations, we should look for a way to define motion and it energy purely in terms of its spatial properties.

Einstein gave us the ability to do this when he defined the mathematical relationship between space, time and energy in terms of the constant velocity of light because in doing so, he provided a method of converting a unit of time in a space-time environment to its equivalent unit of space in four *spatial*  dimensions. Additionally, because the velocity of light is constant, he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words, Einstein’s mathematics actually defines two mathematically equivalent physical models of the universe, one consisting of four-dimensional space-time and one of only four *spatial* dimensions.

This allows one to define the energy associated with both accelerations and velocities, in terms of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension as well as one in four-dimensional space-time.

In other words, using the spatially equivalent model of Einstein space-time theories one could define the energy associated with velocities in terms of a linear displacement in the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension because it remains constant as an object moves though space.

While one would define accelerations both gravitational and non gravitational in terms of a non-linear displacement or curvature in that "surface" because, as was mentioned earlier it increases as a object move though space.

In other words, if one defines gravitational accelerations in terms of a positive non-linear displacement or curvature in that "surface" one would define all other forms of accelerations in terms of an oppositely directed displacement or curvature in that "surface".

Additionally, the magnitude of the linear displacements associated with relative velocities is dependent on the energies associated with their movement or momentum while the degree of the non-linear displacement associated with accelerations would also be dependent on the magnitude of the energy required to cause them.

In other words, the greater the relative velocities or accelerations the greater the displacement or curvature in the "surface" of the three dimensional space manifold with respect to fourth *spatial* dimension associated with their motion.

What makes accelerated motion different from velocities is that they do not create an energy gradient in space necessary to activate the human senses or measuring instruments because, as was just mentioned the displacement they create is linear with respect to the "surface" of the three dimensional space manifold with respect to a fourth *spatial* dimension

Therefore, the reason it only makes sense to say that this is moving with respect something is because referencing it to that something provides an energy gradient or differential which can activate measuring equipment or human senses.

However, because Einstein tells us the displacement in the "surface" of a three-dimensional space manifold with respect to fourth *spatial *dimension associated with accelerated motion is non-linear it will intrinsically create an energy gradient between two points space.

This also allows one to define a universal baseline for the measurement of inertia in terms of the linear displacement in that "surface" because as mentioned earlier it defines the energy level of all constant motion.Â

As was mentioned earlier, in physics inertia is a measure of the resistance or force (over a given time period) required to the change the velocity of a physical object. Therefore, to define an absolute benchmark for measuring it one must first define a starting point for the energy gradient that, as mentioned earlier is responsible for acceleration. Additionally to make it universal benchmark that point must be the same of all objects and particles.

Therefore, a universal baseline for the measurement of the inertia in all objects is the linear displacement in that "surface" with respect to a fourth spatial dimension associated with their velocity before a measurement was taken . In other words, one can measure the inertia of all objects by measuring the energy difference (in a given time frame) between its starting displacement in space and its displacement at the end points. In other words, it defines a universal starting point or baseline the measurement of inertia for all objects.

Some have said that one cannot measure the inertia of a particle or object that exists alone in the universe because one cannot reference its movements to anything.  However, referencing its velocity with respect to the universe is not relevant to its measurement because Einstein tells us that the energy of velocity is made up of two parts.  One is the energy of associated with its velocity and the other is that of the energy of it’s rest mass defined by the equation E=mc^2.  Therefore, because the displacement that defines a object is made up of two parts the energy of its rest mass and that of its velocity does not need to be reference to any other object or particle. In other words the mass of the object provides the displacement or baseline for measuring the inertia of a particle or object at rest. Therefore, its movement or velocity or lack of it with respect to the entire universe will not effect that measurement because it is determined only by the energy required to produce a change in its velocity or the displacement the "surface" of a three dimensional space manifold with respect to a fourth *spatial* dimension that is responsible for that change.

This shows how one can derive a universal baseline for measuring the inertia of all particles and objects in terms of the physical geometry of space as defined by Einstein.

It should be remembered that Einstein, by defining the universe’s geometry in terms of the constant velocity of light allows us to choose whether to define inertia either a space-time environment or one consisting of four *spatial* dimension.

Latter Jeff

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

Is it an intrinsic property of space that cause velocities to make sense only by saying that this is moving with respect to something while accelerations or changes in velocity don’t require comparisons to give them meaning?

Newton came to conclusion it was based an experiment involving a bucket of water.

Greene, Brian describes his book "The Fabric of the Cosmos" (Knopf Doubleday Publishing Group) why he though space
this by observing a spinning of a bucket hanging from rope filled with water.  At first, after it is allowed to unwind the bucket starts to spin but the water inside remains fairly stationary; the surface of the stationary water stays nice and flat. As the bucket picks up speed, little by little its motion is communicated to the water by friction, and the water starts to spin too. As it does, the water’s surface takes on a concave shape, higher at the rim and lower in the center,

"Why does the water’s surface take this shape? Well, because it’s spinning, you say, and just as we feel pressed against the side of a car when it takes a sharp turn, the water gets pressed against the side of the bucket as it spins. And the only place for the pressed water to go is upward. This reasoning is sound, as far as it goes, but it misses the real intent of Newton’s question. He wanted to know what it means to say that the water is spinning: spinning with respect to what? Newton was grappling with the very foundation of motion and was far from ready to accept that accelerated motion such as spinning is somehow beyond the need for external comparisons.

A natural suggestion is to use the bucket itself as the object of reference.  As Newton argued, however, this fails. You see, at first when we let the bucket start to spin, there is definitely relative motion between the bucket and the water, because the water does not immediately move. Even so, the surface of the water stays flat. Then, a little later, when the water is spinning and there isn’t relative motion between the bucket and the water, the surface of the water is concave. So, with the bucket as our object of reference, we get exactly the opposite of what we expect: when there is relative motion, the water’s surface is flat; and when there is no relative motion, the surface is concave.

Newton explained the terrestrial bucket experiment in the following way. At the beginning of the experiment, the bucket is spinning with respect to absolute space, but the water is stationary with respect to absolute space. That is why the water’s surface is flat. As the water catches up with the bucket, it is now spinning with respect to absolute space, and that is why its surface becomes concave. As the bucket slows because of the tightening rope, the water continues to spin spinning with respect to absolute space”and that is why its surface continues to be concave."

However, Einstein demonstrated, in his Theory of Relativity that absolute space does not exist, therefore their must exist another reason for the concavity of the water in Newton’s
Bucket.

One of the most logical ways to find it would be to use the observable differences between the two types of motion; velocities and accelerations. 

For example, velocities transverse the same space or distance in a given time frame while accelerations transverse an exponentially increasing distance over that same time period. 

This tells us the primary difference between them is a component of space not time because if one uses the same time frame for both the only thing that distinguishes them is the distance they transverse.

However, Einstein defined the geometry of space and our universe in terms of time therefore, because space not time is, as was just mentioned the variable that distinguishes velocities from accelerations, we should look for a way to define motion purely in terms of its spatial properties.

Einstein gave us the ability to do this when he defined the mathematical relationship between space, time and energy in terms of the constant velocity of light because in doing so, he provided a method of converting a unit of time in a space-time environment to its equivalent unit of space in four *spatial* dimensions.  Additionally, because the velocity of light is constant, he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words, Einstein’s mathematics actually defined two mathematically equivalent physical models of the universe, one consisting of four-dimensional space-time and one of only four *spatial* dimensions.

This allows one to define the energy associated with both accelerations and velocities, in terms of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension as well as one in four-dimensional space-time.

In other words, using the spatially equivalent model of Einstein space-time theories one could define energy associated with velocities in terms of a linear displacement in the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension because it remains constant for a given time interval. 

While one would define accelerations both gravitational and non gravitational in terms of a non-linear displacement or curvature in that "surface" because, as was mentioned earlier it increases for as it moves through time.

In other words, if one defines gravitational accelerations in terms of a positive non-linear displacement or curvature in that "surface" one would define all other forms of accelerations in terms of an oppositely directed displacement or curvature in that "surface". 

Additionally the magnitude of the these displacements for both accelerations or a those associated with relative velocities is dependent on there energies associated with there movement.

In other words, the greater the relative velocities or accelerations the greater the displacement in the "surface" of the three dimensional space manifold with respect to fourth *spatial *dimension
associated with their motion

What makes accelerated motion different from velocities is that they do not create an energy gradient in space necessary to activate senses because, as was just mentioned the displacement they create on the "surface" of the three dimensional space manifold with respect to a fourth *spatial* dimension is linear.

Therefore, the reason it only makes sense to say that this is moving with respect something is because referencing it to that something provides an energy gradient or differential which can activate measuring equipment or human senses.
 
However, because Einstein tells us the displacement in the "surface" of a three-dimensional space manifold with respect to fourth *spatial *dimension associated with accelerated motion is non-linear its movement will intrinsically create an energy gradient between two points space which can activate measuring equipment or human senses.

In other words, the reasons it only make sense to say that this is moving with respect something while changes in velocity or accelerations don’t is because acceleration intrinsically cause energy gradients between different points in space where as velocities do not.

This would also explain the observations Newton made in his bucket experiment because the energy or velocity of the water is different at each point in the bucket.  In others word, because the water near the bucket’s edge is moving faster or is accelerated with respect its center it has more energy and therefore will create a larger displacement in the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension resulting in its "surface" becoming a concave.  In other words, the concavity of the surface of the water in Newton bucket is not caused by an interaction with space or the bucket but due to direct effects Einstein showed the energy associated with the velocity of the water has of the geometry of space.             

However, this also tells us that what makes space space is energy.

For example, what makes the space in a house and its rooms is not a property of that space but is a property of the geometric structure created by its foundation and walls

Similarly, Einstein tells us that what makes space space in our universe is not a property of space but of the geometric structure created by energy.

Latter Jeff

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

Is there lower limit to the size of our universe.  In other words, how many times can the universe and its mass components be divided up into smaller and smaller chunks until it can divided no farther.

The answer would most likely be found in the two dormant theories, Quantum Mechanics and Einstein’s Theories of Relativity which are used by cosmologists and particle physics to define its evolution.

For example, Einstein’s theories say very little about its origins but it does say a lot about how its components interact to create its observable structures and while doing so tells a lot about how they interact to define the lower limit of its size. 

While on the other hand, a few Quantum Mechanical Theories define its evolution and the lower limit to its size in terms of an infinitesimally small point in space-time.  However, it is unable to providing any details about how its components, after its beginnings interact to create the universe, we can observe around us.

For example, one theory called the Big Bang, which is based on the mathematics of Quantum Theory defines its beginnings and the lower limit to its size in terms of the expansion of a point in space-time called a quantum fluctuation while defining its evolution not in terms of how its component interact but in terms of points in space-time that represent positions of all of the particles it contains at the time they are observed.

This technique of using a one-dimensional point to represent a particle or an objects position is similar to how NASA defines the orbits of planet and its space probes. 

For example, they do not use physical size or the volume of a planet to calculate position and interactions with its orbiting components, instead they use a one-dimensional point at its center called the center of gravity to represent those interactions.

Similarly, quantum mechanics does not need to use physical size of a particle to define its position because similar to how NASA can use a point at the center of an object to represent it, it can use a point in space-time that is in the center of a particle to represent its position.  In other words, the fact that Quantum mechanics describes the microscopic environment of particles in terms of one-dimensional points does not mean that they do not have size.

As was mentioned, earlier Quantum Mechanics assumes the universe began as quantum fluctuation which is a mathematically defined as point in space-time.  In other words, it assumes the size of the universe could be, at its beginning smaller than the period at the end of this sentence.

However, Einstein theories tell us a completely different story of its beginning.

For example, it tells us that matter can only compacted so much before the forces of gravity and time stop it from going any further.

This is true even though in 1915, Karl Schwarzschild proposed based on Einstein theories the gravitational field of a star greater than approximately 2.0 times a solar mass would collapse to form a black hole whose which is a region where time stops and neither light nor particles can escape from it.  However, many assumed that the collapse continues until is compacted into a one-dimensional point or singularity in space-time.

One can understand why those that believed that came to the wrong conclusion by analyzing how those forces interact to create a black hole as was done in the previous article "Time is a force more powerful than those of a black hole" published on Aug 31, 2019

Briefly

"In Kip S. Thorne book " Black Holes and Time Warps ", he describes how in the winter of 1938-39 Robert Oppenheimer and Hartland Snyder computed the details of a stars collapse into a black hole using the concepts of General Relativity.  On page 217 he describes what the collapse of a star would look like, form the viewpoint of an external observer who remains at a fixed circumference instead of riding inward with the collapsing stars matter.  They realized the collapse of a star as seen from that reference frame would begin just the way every one would expect.  "Like a rock dropped from a rooftop the stars surface falls downward slowly at first then more and more rapidly. However, according to the relativistic formulas developed by Oppenheimer and Snyder as the star nears its critical circumference the shrinkage would slow to a crawl to an external observer because of the time dilatation associated with the relative velocity of the star’s surface.  The smaller the circumference of a star gets the more slowly it appears to collapse because the time dilation predicted by Einstein increases as the speed of the contraction increases until it becomes frozen at the critical circumference.

However, the time measured by the observer who is riding on the surface of a collapsing star will not be dilated because he or she is moving at the same velocity as its surface.

Therefore, the proponents of singularities say the contraction of a star can continue until it becomes a singularity because time has not stopped on its surface even though it has stopped with respect to an observer who remains at fixed circumference to that star.

But one would have to draw a different conclusion if one viewed time dilation in terms of the gravitational field of a collapsing star.

Einstein showed that time is dilated by a gravitational field.  Therefore, the time dilation on the surface of a star will increase relative to an external observer as it collapses because, as mentioned earlier gravitational forces at its surface increase as its circumference decrease.

This means, as it nears its critical circumference its shrinkage slows with respect to an external observer who is outside of the gravitation field because its increasing strength causes a slowing of time on its surface.  The smaller the star gets the more slowly it appears to collapse because the gravitational field at its surface increase until time becomes frozen for the external observer at the critical circumference.

Therefore, the observations of an external observer would make using conceptual concepts of Einstein’s theory regarding time dilation caused by the gravitational field of a collapsing star would be identical to those predicted by Robert Oppenheimer and Hartland Snyder in terms of the velocity of its contraction.

However, Einstein developed his Special Theory of Relativity based on the equivalence of all inertial reframes which he defined as frames that move freely under their own inertia neither "pushed not pulled by any force and Therefore, continue to move always onward in the same uniform motion as they began".

This means that one can view the contraction of a star with respect to the inertial reference frame that, according to Einstein exists in the exact center of the gravitational field of a collapsing star.

(Einstein would consider this point an inertial reference frame with respect to the gravitational field of a collapsing star because at that point the gravitational field on one side will be offset by the one on the other side.  Therefore, a reference frame that existed at that point would not be pushed or pulled relative to the gravitational field and would move onward with the same motion as that gravitational field.)

(However, some have suggested that a singularity would form in a black hole if the collapse of a star was not symmetrical with respect to its center.  In other words, if one portion of its surface moved at a higher velocity that another towards its center it could not be consider an inertial reference frame because it would be pushed or pulled due to the differential gravity force cause be its uneven collapse.  But the laws governing time dilation in Einstein’s theory tell us that time would move slower for those sections of the surface that are moving faster allowing the slower ones to catch up.  This tells us that every point on the surface of star will be at the event horizon at the exact same time and therefore its center will not experience any pushing or pulling at the time of its formation and therefore could be considered an inertial reference frame.)

The surface of collapsing star from this viewpoint would look according to the field equations developed by Einstein as if the shrinkage slowed to a crawl as the star neared its critical circumference because of the increasing strength of the gravitation field at the star’s surface relative to its center.  The smaller it gets the more slowly it appears to collapse because the gravitational field at its surface increases until it becomes frozen at the critical circumference.

Therefore, because time stops or becomes frozen at the critical circumference for all observers who are at the center of the clasping mass the contraction cannot continue from their perspectives.

Yet, Einstein in his general theory showed that a reference frame that was free falling in a gravitational field could also be considered an inertial reference frame.

As mentioned earlier many physicists assume that the mass of a star implodes when it reaches the critical circumference.  Therefore, an observer on the surface of that star will be in free fall with respect to the gravitational field of that star when as it passes through its critical circumference.

This indicates that point on the surface of an imploding star, according to Einstein’s theories could also be considered an inertial reference frame because an observer who is on the riding on it will not experience the gravitational forces of the collapsing star.

However, according to the Einstein theory, as a star nears its critical circumference an observer who is on its surface will perceive the differential magnitude of the gravitational field relative to an observer who is in an external reference frame or, as mentioned earlier is at its center to be increasing.  Therefore, he or she will perceive time in those reference frames that are not on its surface slowing to a crawl as it approaches the critical circumference.  The smaller it gets the more slowly time appears to move with respect to an external reference frame until it becomes frozen at the critical circumference.

Therefore, time would be infinitely dilated or stopped with respect to all reference frames that are not on the surface of a collapsing star from the perspective of someone who was on that surface.

However, the contraction of a star’s surface must be measured with respect to the external reference frames in which it is contracting.  But as mentioned earlier Einstein’s theories indicate time in its external environment would become infinitely dilated or stop when the surface of a collapsing star reaches its critical circumference.

Therefore, because time stops or becomes frozen at the critical circumference with respect to the external environment of an observer who riding on its surface the contraction cannot continue because motion cannot occur in an environment where time has stopped.

This means, as was just shown according to Einstein’s concepts time stops on the surface of a collapsing star from the perspective of all observers when viewed in terms of the gravitational forces the collapse of matter must stop at the critical circumference.

This contradicts the assumption made by many that the implosion would continue for an observer who was riding on its surface.

In other words, based on the conceptual principles of Einstein’s theories relating to time dilation caused by a gravitational field of a collapsing star it cannot implode to a singularity or a one-dimensional point as many physicists believe because it causes time to freeze at its critical circumference with respect to all observers.  Therefore, a universe whose evolution is governed by his theories must maintain a quantifiable minimum volume which is greater than the one defined by Schwarzschild radius because if it were smaller matter could not move through that boundary in space time and it could not evolve any further." 

However. the same principle must be applied to the size of the universe at its beginning.  In other words, if time stops at the Schwarzschild radius any object or component of a universe smaller than that could not move through it and evolve to form the structures we observed today.

Additionally, Schwarzschild radius also defines the lower limit to size of all subatomic particles because it defines where time would stop at their surface.  Therefore, if they were smaller or even equal to that radius they could not interact with the other particles because time would stop as they approached each other and interaction with other particles would never happen.

In other words, in a universe governed by Einstein’s theories the lower limit to the size of both the universe and the particles it contains is defined by Schwarzschild radius.

Yet this would seem to contract the quantum mechanical description of a particle as being represented as point in space-time without an extended volume.

HOWEVER, THIS IS NOT THE CASE because, as was mentioned earlier the point in space-time that quantum mechanics defines as the position of a particle could be interpreted as the center of wave component of its duality similar to how NASA uses the point at the center of mass of an extended object to determine its position in space-time as was shown in the article published on Jan. 1, 2020 "Particles as standing waves in space-time"

Yet, it is possible that someone with better mathematical skills than me may be able to unify the Quantum universe with Einstein’s by mathematically describing a environment in which the point description of a particle defines the energy center of its wave component of its wave particle duality while showing how that point interacts with their environment based on those properties similar to how NASA uses the center of the energy or gravitational components of planets to how they interact with other each other.

Latter Jeff

Copyright 2020 Jeffrey O’Callaghan

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

A few years after Albert Einstein unified space and time in his (and by now very well tested!) Theory of General Relativity he applied it to the entire universe and found something remarkable. The theory predicts that the whole universe is either expanding or contracting.

Later in 1929 the astronomer Edwin Hubble measured the velocities of a large selection of galaxies and found that the majority of them were moving away from us.  In other words, the universe was expanding.

However, is the universe expanding in space or is it expanding through time? 

To answer this one must first define what time and space are.

Some define time only in the abstract saying that is an invention of the human consciousness that gives us a sense of order, a before and after so to speak.  To physicist’s it is a measure of the relative interval between events which is measured in units of time such as seconds minutes or hours.

However, space can be defined as the arena where events occur.  We use the measurements of inch or meter to define the position of those event in that arena.

As was mentioned earlier, Einstein’s General Theory of Relativity mathematically define the universe in terms of a melding of time with space.  However, as was mention above they are they have vastly different properties.  For example, one is measure in terms of second while the other is in inches or meters.

Therefore, it is very difficult to understand how time which is measured in seconds can have a dynamic effect on space measured in meters.

To this end Marina Corts, a cosmologist from the Royal Observatory, Edinburgh came up to what has come to be called the block universe.

Basically, it asks us to imagine a regular chunk of cement. It has three dimensions but we live in four dimensions: the three spatial dimensions plus one time dimension. A block universe is a four-dimensional block, but instead of being made of cement, it is made of space-time. And all of the space and time of the Universe are there in that block."

We can’t see this block, we’re not aware of it, as we live inside the cement of space-time. And we don’t know how big the block universe we live in is: "We don’t know if space is infinite or not. Or time – we don’t know whether it has a beginning or if it will have an end in the future. So, we don’t know if it’s a finite chunk of space-time or an infinite chunk."

However, picture this presents a problem for cosmologists because if the merging of space and time causes it to become as ridge as a block of cement how can its spatial component be expanding.

It should be remembered only the spatial component of the universe is expanding not time. 

Additionally, because Einstein defined the universe in terms of only four dimensions, one time and three spatial how can we understand its spatial expansion without adding an additional one because a spatial one cannot expand to one made up of time because, as mentioned earlier they have vastly different properties.

Yet, Einstein gave us an alternative when he defined the mathematical relationship between space and time in terms of the constant velocity of light because in doing so, he provided a method of converting a unit of time in a space-time environment to its equivalent unit of space in four *spatial* dimensions.  Additionally, because the velocity of light is constant, he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words, Einstein’s mathematics actually defined two mathematically equivalent physical models of the universe one consisting of four-dimensional space-time and one of only four spatial dimensions. 

Yet, because both of these models are mathematically equivalent and since we cannot physically observe  either a time or a fourth *spatial* dimension, we must look to the effects they would have on the ones we can observe to determine which one of these physical models is correct.

For example, if we were a two-dimensional creature living on the surface of a balloon that was inflating, we could explain its spatial expansion by assuming we were living in an environment consisting three spatial dimensions because they have the same properties as the two dimension surface of the balloon therefore, it could expand through it.  However, we could not explain it by assuming that we were living in an environment consisting of only time and the two-dimensional surface of the balloon because time as mentioned earlier it does not have the properties of space and therefore could not expand in it. 

Similarly, we can explain why our three-dimensional world was undergoing a spatial expansion by assuming we were living in an environment or universe consisting four *spatial* dimensions because it would have the same spatial properties as the three dimension one we live in.  However, we could not if we assume our universe consisted of four-dimensional space-time because time does not have the properties of space and therefore similar to the surface of the balloon it could not expand in it.

As was mentioned earlier  "A few years after Albert Einstein unified space and time (and by now very well tested! ) in his theory of General Relativity" and showed it can be "applied to the entire universe ." Therefore, he also showed that because of their mathematical equivalence, a physical model based on one unifying three-dimensional space with a fourth *spatial* dimension has also been very well tested and could also be applied it to the entire universe.

However, as was shown above his physical model based on four *spatial* dimensions pass an additional test which his space-time model cannot, that of explaining the spatial expansion of our three-dimension environment. 

Later Jeff

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

The arrow of time, is the name reason given to the "one-way direction" or "asymmetry" of time by British astrophysicist Arthur Eddington in the macroscopic universe.  Its direction, according to Eddington, is determined by studying the spatial organization of atoms, molecules, and bodies, and might be drawn upon a four-dimensional relativistic map of the world.

However physical processes at the microscopic level are believed to be either entirely or mostly time-symmetric: if the direction of time were to reverse, the theoretical statements that describe them would remain true. Yet as was just mentioned at the macroscopic level it appears that this is not the case.

The question as to why things appear to different on the microscopic level is an unanswered question.

Many explain the observed temporal asymmetry at the macroscopic level, the reason we see time as having a forward direction, ultimately comes down to thermodynamics, the science of heat and its relation with mechanical energy or work, and more specifically to the Second Law of Thermodynamics. That laws uses the states that the entropy of a system either remains the same or increases in every process. This phenomenon is due to the extraordinarily small probability of a decrease or that a system will return to its original configuration, based on the extraordinarily larger number of microstates in systems with greater entropy. In other Entropy  can decrease or a system can return to its original configuration, but for any macroscopic system, this outcome is so unlikely that it will never be observed in the future.

However, entropy can decrease somewhere, provided it increases somewhere else by at least as much. The entropy of a system decreases only when it interacts with some other system whose entropy increases in the process.

Yet, it is difficult to apply that definition to a quantum environment because Schrödinger wave equation that quantum mechanics uses to determine the position component of a particle when observed does so in terms of a probability distribution over the entire universe.  Therefore, to define an arrow of time for a quantum system in terms of entropy one must show there is a physical connection between the macroscopic space-time environments we live in and a particles position in that probability field when it is observed. 

Unfortunately, we define the spatial components of entropy in our macroscopic universe in terms of the space-time concepts defined by Einstein.  Therefore, to define the arrow of time in the probabilistic world associated quantum mechanics in terms of entropy we must show how it is physically connected to the spatial properties of the macroscopic universe defined by him.

Einstein gave us the ability to do this when he used the equation E=mc^2 and the constant velocity of light to define the geometric properties of space-time because it provided a method of converting a unit of time he associated with energy to unit of space.   Additionally, because the velocity of light is constant, he also defined a one to one quantitative correspondence between the both the relativistic and physical properties of a space-time universe and one made up of only four *spatial* dimensions.

Dong so allow will one to physically connect the probabilities associated with Schrödinger’s wave equation to the Thermodynamic laws that governor the entropy in our macroscopic universe.

For example, the article “ Why is energy/mass quantized? ” Oct 4, 2007 showed one can derive the quantum mechanical wave/particle properties of matter in terms of an energy wave on a "surface" of a three-dimensional space manifold with respect to fourth spatial dimension by extrapolating our understanding of a resonant structure created by a wave in a three-dimensional environment.

Briefly it showed the four conditions required for resonance to occur in a three-dimensional environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in one consisting of four spatial dimensions.

The existence of four *spatial* dimensions would give an electromagnetic wave 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 to oscillate with the frequency associated with the energy of that event.

The oscillations caused by such an event would serve as forcing function allowing a resonant system or "structure" to be established space.

Therefore, these oscillations in a "surface" of a three-dimensional space manifold would meet the requirements mentioned above for the formation of a resonant system or "structure" in four-dimensional space if one extrapolated them to that environment.

In our three-dimensional environment the energy of a resonant system can only take on the discrete or quantized values associated with its fundamental or a harmonic of its fundamental frequency.

Hence, these resonant systems in four *spatial* dimensions would be responsible for the discrete quantized energy quantum mechanical associates with the particle properties of matter.

Yet one can also define its boundary conditions of its resonate structure in the terms of our perceptions of a three-dimensional environment.

For example, in our three-dimensional world, a point on the 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, it could, similar to the surface of the paper oscillate up  or down with respect to a fourth *spatial* dimension.

It is the confinement of the upward and downward oscillations of an energy with respect to a fourth *spatial* dimension by an observation is what defines the spatial boundaries associated with a particle in the article Why is energy/mass quantized? " Oct 4, 2007.

This shows the reason Quantum Mechanics can define matter in terms of a particle/wave duality and why it only presents its particle or position properties when it is observed is because its wave component is only confined to three-dimensional space when an observation is made.

However, as mentioned earlier it also provides a way to physical connect the probabilistic environment defined by SchrÃdinger wave equation to the physicality of Einstein’s relativistic universe.

The physics of wave mechanics tell us that due to the continuous properties of the wave component associated with a quantum system it would be distributed throughout the entire "surface" a three-dimensional space manifold with respect to a fourth *spatial* dimension.

For example, the energy of a vibrating or oscillating ball on a rubber diaphragm would be disturbed over its entire surface while the magnitude of those vibrations would decrease as one move away from the focal point of the oscillations.

Similarly, if the assumption outlined above, that quantum properties of matter are a result of vibrations or oscillations in a "surface" of three-dimensional space is correct those oscillations would be distributed over the entire "surface" three-dimensional space while the magnitude of those vibrations would be greatest at the focal point of the oscillations and decreases as one moves away from it.

(Some may question the fact that the energy wave associated with particle would be simultaneously distributed over the entire universe.  However, the relativistic properties of space-time tell us the distance perceived by objects or particles in relative motion is dependent on their velocity which become zero at the speed of light.  Therefore, from the perspective of an energy wave moving at the speed of light, the distance between all points in the universe along its velocity vector is zero.  In other words, because its electromagnetic wave component of a particle is moving at the speed of light as all electromagnetic0 energy must is it would be distributed or simultaneous exists at every point in the universe along its velocity vector.  There can be no other conclusion if one accepts the validity of Einstein’s theories.)   

As mentioned earlier the article “ Why is energy/mass quantized? ” shown a wave/particle duality of matter can be understood in terms of a resonant structure formed wave energy on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Yet the science of Wave Mechanics tells us resonance would most probably occur on the surface of the rubber sheet were the magnitude of the vibrations is greatest and would diminish as one move away from that point,

Similarly, a particle would most probably be observed were the magnitude of the vibrations in a "surface" of a three-dimensional space manifold is greatest and would diminish as one move away from that point.

This demonstrates that one can interconnect probabilities associated with Schrödinger’s wave equation to the physicality of the Einstein’s Relativistic universe.

As was mentioned earlier the arrow of time is defined in classical system in terms of entropy or the level of randomness (or disorder) of a system and the Second law of thermodynamics which states that there is an the extraordinarily small probability that a system will return to its original configuration, based on the extraordinarily larger number of microstates in systems with greater entropy even though its. 

Additionally, the above discussion also shows one can use the same definition for the arrow of time in a quantum universe as the one used in a macroscopic one because the position of a particle in a quantum can only be determine with respect to other particles in probability field Schrodinger’s equation.  Therefore, due to the fact that there are infinite number of possibilities in the probabilistic universe of quantum mechanics there an extraordinarily small chance of that universe retuning to is original configuration when an observation is made in the future.  

Later Jeff

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

Presently, there is disconnect between our understanding of one of the most mysterious facets of quantum mechanics quantum, that of quantum entanglement and the classical one of separation.

Entanglement occurs when two particles are linked together no matter their separation from one another. Quantum mechanics assumes even though these entangled particles are not physically connected, they still are able to share information with each other instantaneously seemingly breaking one of the most hard-and-fast rules of classical physics and Einstein theories: that no information can be transmitted faster than the speed of light.

Even though it may be hard for some to accept the instantaneous sharing of information over what appears to be long distances has been proven time and time again over the years.

For example, when researchers create two entangled particles, separate them and independently measure their properties, they find that the outcome of one measurement influences the observed properties of the other particle.

This was made possible in 1964, when John Bell showed there is a theoretical limit beyond which correlations can only be explained by quantum entanglement, not classical physics.

However, we must be careful not to jump to conclusions because Einstein gave us the definitive answer as to how and why particles are entangled in terms of the physical properties of space-time even though he was so upset to what he called this  "spooky action at a distance." that in 1935 he along with Podolsky Rosen proposed the following thought experiment which came to be called the EPR Paradox.

In 1935, Einstein co-authored a paper with Podolsky and Rosen highlighted a problem that they felt showed that Quantum Mechanics could not be a complete theory of nature.  This thought experiment came to be called the EPR Paradox. The first thing to notice is that Einstein was not trying to disprove Quantum Mechanics in any way.  In fact, he was well aware of its power to predict the outcomes of various experiments.  What he was trying to show was that there must be a "hidden variable" that would allow Quantum Mechanics to become a complete theory of nature.

The argument begins by assuming that there are two systems, A and B (which might be two free particles), whose wave functions are known.  Then, if A and B interact for a short period of time, one can determine the wave function which results after this interaction via the SchrÃdinger equation or some other Quantum Mechanical equation of state.  Now, let us assume that A and B move far apart, so far apart that they can no longer interact in any fashion.  In other words, A and B have moved outside of each other’s light cones and Therefore, are spacelike separated.

With this situation in mind, Einstein asked the question: what happens if one makes a measurement on system A?  Say, for example, one measures the momentum value for it.  Then, using the conservation of momentum and our knowledge of the system before the interaction, one can infer the momentum of system B.  Thus, by making a momentum measurement of A, one can also measure the momentum of B.  Recall now that A and B are spacelike separated, and thus they cannot communicate in any way.  This separation means that B must have had the inferred value of momentum not only in the instant after one makes a measurement at A, but also in the few moments before the measurement was made.  If, on the other hand, it were the case that the measurement at A had somehow caused B to enter into a particular momentum state, then there would need to be a way for A to signal B and tell it that a measurement took place.  However, the two systems cannot communicate in any way!

If one examines the wave function at the moment just before the measurement at A is made, one finds that there is no certainty as to the momentum of B because the combined system is in a superposition of multiple momentum eigenstates of A and B.  So, even though system B must be in a definite state before the measurement at A takes place, the wave function description of this system cannot tell us what that momentum is!  Therefore, since system B has a definite momentum and since Quantum Mechanics cannot predict this momentum, Quantum Mechanics must be incomplete.

As was mentioned earlier, in response to Einstein’s argument about incompleteness of Quantum Mechanics, John Bell derived a mathematical formula that quantified what you would get if you made measurements of the superposition of the multiple momentum eigenstates of two particles.  If local realism was correct, the correlation between measurements made on one of the pair and those made on its partner could not exceed a certain amount, because of each particle’s limited influence.

In other words, he showed there must exist inequities in the measurements made on pairs of particles that cannot be violated in any world that included both their physical reality and their separability because of the limited influence they can have on each other when they are "spacelike" separated.

When Bell published his theorem in1964 the technology to verify or reject it did not exist. However, in the early 1980s, Allen Aspect performed an experiment with polarized photons that showed that the inequities it contained were violated.

Since then there have been many experiments using the properties of paired of photons and other particles that verify without any doubt that two photons and others particles that are spatially separated can be entangled.

In quantum mechanics it is assumed that the act of measuring the state of one of a pair of entangled particles instantly affects the other no matter how far they are apart.

However, Einstein in his Special Theory of Relativity gives us a classical explanation in terms his theory for the entanglement of two particles.

For example, with regards to the polarized photons mentioned earlier that Allen Aspect used to verify the quantum mechanical interpretation of entanglement his theory tells us that because photons must always be moving at the speed of light they can never be separated with respect to an external observer no matter how far apart he or she perceives them to be.

This is because he tells that that there are no preferred reference frames by which one can measure distance. Therefore, one must not only view the separation of a photon with respect to an observer who was external to them but must also look at that separation from a photon’s perspective.

However, his theory tells the distance between the two photons A and B would be defined by their relative speed with respect to an observer.

Specifically, he told us that it would be defined by

Yet, this tell us that the separation between two photons moving at the speed of light from their perspective would be zero no matter how far apart they might be from the perspective of an observer in a laboratory because according to the concepts of relativity one can view the photons as being stationary and the observers as moving at the velocity of light.

Therefore, according to Einstein’s theory all photons which are traveling at the speed of light are entangled with all other paired photons no matter how far apart or "spacelike" separated they may appear to be to ALL observers.

In other words, the inequities in the measurements made on ALL REPEAT ALL pairs of photons should be violated in a world containing the physical reality of Einstein’s theories because they will influence each other no matter how far they may be separated when viewed from a reference frame other than a photon’s, such as a laboratory.

Up until now we only have addressed the entanglement of photons that are moving at the speed of light.  However, the same the relativistic properties of motion can be applied to explain the entanglement of other particles that are not moving at that speed.

This is because quantum mechanics defines the composition of matter in terms of its wave particle duality.  More specifically, as was shown in the previously article  "Quantum mechanics in a nutshellt look: waves. Look: particles" Dec. 1, 2015 it assumes that before an observation is made matter is propagated though space in terms of its wave properties and only after being observed does it present its particle properties.

In other words, in Quantum Mechanics matter has an extended volume while moving through space which is directly related to the wavelength associated with its particle properties.

This means the wavelengths of two particles in motion will overlap and be entangled if the separation between the end points of an observation as measured from their perspective is less that the wavelength of those particles.

However, as mentioned earlier Einstein tells us that we must use this theory to derive the separation of two moving particles from their perspective and not from the prospective of observers in a laboratory.

Therefore, even though particles may appear to be separated from the view point of a laboratory observer they may not be separated from the view point of the particles that are moving with respect to those observers because of an overlap of their wave properties..

In other words, one does not have to break one of the most hard-and-fast rules of classical physics and Einstein theories: that no information can be transmitted faster than the speed of light because one can use his classical theories to explain how and why particles that appear to be separated can communicate instantaneously.

The illusion is not that entanglement of two spatial separated particles from the perspective of the observers in Allen Aspect experiment mentioned earlier does not exist.  The illusion is that entanglement is not the result of the quantum mechanical properties of matter but instead is the result of the physical reality of Einstein’s Theory of Relativity because it tells us that the separation of particles must be measured from their perspective and not from the perspective of an observer in a laboratory.

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

Richard Feynman the farther of Quantum Electrodynamics believed Thomson’s double slit experiment provided a mechanism for understanding the wave particle duality of energy/mass because it clearly demonstrates their inseparability and provides a mechanisms for understanding how it is propagated through space.

The wave-particle duality postulates that all particles exhibit both wave and particle properties. A central concept of quantum mechanics, this duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the behavior of quantum-scale objects.  Standard interpretations of quantum mechanics explain this paradox as a fundamental property of the Universe, while alternative interpretations explain the duality as an emergent, second-order consequence of various limitations of the observer.

The reason the above-mentioned experiment is so important is because it provides a mechanism for understanding how electromagnetic energy is propagated and why the particle wave dually exists purely in terms of Einstein’s Theory of Relativity.

But before we begin, we must first understand how the electromagnetic wave component of a particle’s duality is propagated through space and time.

One of the difficulties involved in doing so is that we define its movement though space in terms Maxwell’s equations which are based on the interaction between its electric and magnetic components with respect to time not space.  This presents a problem because the particle component of its duality must always be defined by its spatial position when observed. Therefore, to understand how they are related we should attempt to define its movement through space and time in term of its spatial properties.

Einstein gave us the ability to do this purely in terms spatial properties of its electromagnetic wave components when he used the constant velocity of light to defined the geometric properties of space-time because it allows one to convert a unit of time in his space-time universe to an equivalent unit of space in an environment consisting of only four *spatial* dimensions.  Additionally, because the velocity of light is constant it is possible to defined a one to one correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words, by mathematically defining the geometric properties of a space-time universe in terms of the constant velocity of light he provided a qualitative and quantitative means of redefining his space-time universe in terms of the geometry of four *spatial* dimensions.

This gives one the ability to derive the properties of an electromagnetic wave and understand its movement in terms of the spatial displacement that would be created by its observed transverse wave characteristics. 

For example, a transverse wave on the two-dimensional surface of water moves through water because it causes a point on that surface to be become displaced or rise above or below the equilibrium point that existed before the wave was present.  A force is developed by the differential displacement of the surfaces, which will result in the elevated and depressed portions of the water moving towards or become "attracted" to each other and the surface of the water. This results in a wave to move on the surface of the water.

Similarly, an energy wave on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would cause a point on that "surface" to become displaced or rise above and below the equilibrium point that existed before the wave was present.  This would result a wave moving on the "surface" of three-dimensional space.

Therefore, classical wave mechanics, if extrapolated  to four *spatial* dimensions tells us a force will be developed by the differential displacements caused by an energy wave moving on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension that will result in its elevated and depressed portions moving towards or become "attracted" to each other causing it to move through space.

This defines the causality of the attractive forces of unlike charges associated with the electromagnetic wave component of a photon in terms of a force developed by a differential displacement of a point on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, it also provides a classical mechanism for understanding why similar charges repel each other because observations of water show that there is a direct relationship between the magnitudes of a displacement in its surface to the magnitude of the force resisting that displacement.

Similarly, the magnitude of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two similar charges will be greater than that caused by a single one.  Therefore, similar charges will repel each other because the magnitude of the force resisting the displacement will be greater for two charges than it would be for a single charge.

One can also define the directionality of electrical component of electromagnetic energy in terms of the energy associated with its "peaks" and "troughs" that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement because classical Mechanics tells us a horizontal force will be developed by that displacement which will always be 90 degrees out of phase with it.  This force is called magnetism.

This is analogous to how the vertical force pushing up of on mountain also generates a horizontal force, which pulls matter horizontally towards the apex of that displacement.

However, this means that one can define a physical model for the propagation of an electromagnetic field in terms of Einstein’s space-time theory because, as was shown above when he mathematically defined its geometric properties in terms of the constant velocity of light he provided a qualitative and quantitative means of redefining his theory in terms of the geometry of four *spatial* dimensions.

Yet, viewing it in terms of its spatial components also allows one to understand the mechanism responsible for the wave particle duality of a photon as observed in the Thomson’s double slit experiment and why electromagnetic energy always presents itself as a particle when it strikes the detector in the that experiment.

For example, the article, "Why is energy/mass quantized?" Oct. 4, 2007 showed that one can use the Einstein’s theories to explain the quantum mechanical properties of an electromagnetic wave by extrapolating the rules of classical resonance in a three-dimensional environment to an energy wave moving on “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly it showed the four conditions required for resonance to occur in a classical environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in an energy wave moving in four *spatial* dimensions.

The existence of four *spatial* dimensions would give the energy wave associated with a photon 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 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, the oscillations caused by such an event would serve as forcing function allowing a resonant system or "structure" to be established in four *spatial* dimensions.

As was shown in that article these resonant systems in four *spatial* dimensions are responsible for the particle called a photon.

However, one can also use Einstein space-time theories when viewed in their spatial equivalent to explain how the boundaries of the standing wave responsible for creating the resonant system that article indicated was responsible of a particles formation are created.

In classical physics a standing wave is created when the vibrational frequency of a source causes reflected waves from one end of a confined medium to interfere with incident waves from the source.  This interference of the wave energy causes their peaks troughs to be reinforce in the volume they are occupying thereby creating a standing wave.

The confinement required to create a standing wave in space-time or its equivalent in four *spatial* dimensions can be understood by comparing it to the confinement a point on the two-dimensional surface of paper experiences when oscillating with respect to three-dimensional space.  The energy associated with the wave motion of that point would be confined to its two-dimensional surface and would be reflected and interfere with the incident wave when reaches three-dimensional space at its edge. Therefore, a standing would be created by its interaction with three-dimensional space.

In other words, when a wave on the surface of a piece of paper encounters the third *spatial* dimension at its edge it is reflected back allowing a standing wave to be formed on its surface.

Similarly, an electromagnetic wave moving on the surface of three-dimensional space would be confined to it and reflected back to that volume, similar to the surface of the paper if it was prevented from oscillating with respect to a four *spatial* dimensions or four-dimensional space-time.

In other words, the interference caused by the confinement of an electromagnetic wave to three-dimensional space, which is caused by it striking the detection screen in the Thomson’s double slit experiment results in the resonant standing wave to be formed in space called a photon.

That experiment is made up of "A coherent source of photons illuminating a screen after passing through a thin plate with two parallel slits cut in it.  The wave nature of light causes it wave component to interfere after passing through both slits, 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".

When only one slit is open, the pattern on the screen is a diffraction pattern however, when both slits are open, the pattern is similar but with much more detail.  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 Huygen ‘s Fresnel principle that is based 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.

However, the most baffling part of this experiment comes when only one photon at a time impacts a barrier with two opened slits because an interference pattern forms which is similar to what it was when multiple photons were impacting the barrier.   This is a clear implication the particle called a photon has a wave component, which simultaneously passes through both slits and interferes with itself.  (The experiment works with electrons, atoms, and even some molecules too.)"

Even more puzzling is why any attempts to measure which slit that electron passed through cause the interference pattern to disappear.

Yet, as mentioned earlier one can derive the outcome of this experiment by assuming that electromagnetic energy is propagated by a wave on the "surface" of a three-dimensional space manifold with respect to a fourth spatial dimension instead of four-dimensional space-time

For example, the reason why the interference patterns remain when only one photon at a time is fired at the barrier with both slits open or "the most baffling part of this experiment" is because, as was just shown it has an extended spatial volume which is directly related to the wavelength.

This means a portion of its energy can 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.

Additionally, one can also explain why the interference pattern disappears when a detector is added to determine which slit a photon has passed through.  The energy required to measure which slit it passes through interacts with it causing the wavelength of that portion to change so that it will not have the same resonant characteristics as one that passed through the other slit   Therefore, the energy passing thought that slit will not be able to interact, with the energy passing through the other one to form an interference pattern on the screen.

However, as was shown earlier one can also show the reason the interference pattern appears as a particle when electromagnetic wave contacts a detection screen is because striking it results in it being confined to three-dimensional space instead of four-dimensional space-time or four spatial dimensions, thereby creating a standing wave in either four spatial dimensions or four dimensional space-time to be created.

In other words, it clearly shows the reason all forms of energy exhibit both wave and particle properties are because they are physically made up of waves in terms of Einstein’s Theory of Relativity.

The above discussion shows that Richard Feynman was right in assuming that Thomson’s double slit experiment provided a mechanism for understanding the  wave particle duality of energy/mass because it clearly demonstrates their inseparability.

Additionally, it also provides an explanation how and why energy  is propagated through space because it shows the quantum mechanical and wave properties of energy displayed in the double slit experiment can be understood if one assumes they are made up of a resonant system in a moving in a four dimensional space-time manifold or on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension in terms Einstein theories. 

It should be remembered that Einstein’s genius allows us to choose whether to define an electromagnetic wave either a space-time environment or one consisting of four *spatial* dimension when he defined its geometry in terms of the constant velocity of light.

Later Jeff

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

Maxwell defined the propagation of an electromagnetic wave in terms of a field consisting of both electric and magnetic components which continuously interact with each other, forming an electromagnetic wave.

While Quantum Field Theory defines an electromagnetic field in terms of discrete parcels of energy while avoiding the question as to how it moves through space.

Additionally, it cannot explain in terms of a physical model and why an electromagnetic wave always without exception becomes a particle when observed.

Einstein also had a problem of deriving its electromagnetic properties and how they moved through space in terms of a physical model based on his gravitational theories as was documented by the American Institute of Physics .

“From before 1920 until his death in 1955, Einstein struggled to find laws of physics far more general than any known before. In his theory of relativity, the force of gravity had become an expression of the geometry of space and time. The other forces in nature, above all the force of electromagnetism, had not been described in such terms. But it seemed likely to Einstein that electromagnetism and gravity could both be explained as aspects of some broader mathematical structure. The quest for such an explanation ” for a unified field theory that would unite electromagnetism and gravity, space and time, all together — occupied more of Einstein’s years than any other activity.

However, one of the difficulties in understanding the similarities between electromagnetic forces and gravity is that we define its movement though space in terms of an interaction between its electric and magnetic components with respect to time while we define the magnitude of gravitational forces in terms of the physical distance between two bodies.

Therefore, to understand a physical connection between them we should define the interaction of the forces associated with an electromagnetic wave in in terms of distance as we do with gravity.

Einstein gave us the ability to do this when he used the constant velocity of light and the equation E=mc^2 to define geometric properties of forces in a space-time environment because it allows one to convert a unit of time in his four-dimensional space-time universe to a unit of space in a universe consisting of only four *spatial* dimensions.   Additionally, because the velocity of light is constant it is possible to define a one to one correspondence between his space-time universe and one made up of four *spatial* dimensions.

In other words, by mathematically defining the geometric properties of time in his space-time universe in terms of the constant velocity of light he provided a qualitative and quantitative means of define the time-based components of Maxwell’s equations in terms of their spatial counterparts. 

The fact that one can use Einstein’s equations to qualitatively and quantitatively redefine the curvature in space-time he associated with gravitational forces in terms of four *spatial* dimensions is one bases for assuming, as was done in the article “ Defining energy? ” Nov 27, 2007 that all forms of energy including gravitational and electromagnetism can be derived in terms of a spatial displacement in a surface of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This allows one to form a physical image of electromagnetic force and why it moves through space as was done in the article " What is electromagnetism? " Sept, 27 2007 in terms of the differential force caused by the "peaks" and "toughs" of an energy wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly it showed it is possible to derive the electrical and magnetic properties of an electromagnetic field by extrapolating the laws of Classical Wave Mechanics in a three-dimensional environment to a wave moving on a "surface" of three-dimensional space manifold with respect to a fourth *spatial* dimension.

For example, a wave on the two-dimensional surface of water causes a point on that surface to become displaced or rise above or below the equilibrium point that existed before the wave was present.  A force is developed by that differential displacement of the surfaces, which will result in the elevated and depressed portions of the water moving towards or become "attracted" to each other and the surface of the water.

Similarly, an energy wave on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would cause a point on that "surface" to become displaced or rise above and below the equilibrium point that existed before the wave was present.

Therefore, classical wave mechanics, if extrapolated to four *spatial* dimensions tells us a force will be developed by the differential displacements caused by an energy wave moving on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension that will result in its elevated and depressed portions moving towards or become "attracted" to each other resulting as the wave moves through space.

This defines the causality of the attractive forces of unlike charges associated with the electromagnetic field component of a photon in terms of a force developed as the wave moves through four *spatial* dimensions by a differential displacement of a point on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, it also provides a classical mechanism for understanding why similar charges repel each other because observations of water show that there is a direct relationship between the magnitude of a displacement in its surface to the magnitude of the force resisting that displacement.

Similarly, the magnitude of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two similar charges will be greater than that caused by a single one.  Therefore, similar charges will repel each other because the magnitude of the force resisting the displacement will be greater for two charges than it would be for a single charge.

One can also derive the magnetic component of an electromagnetic wave in terms of the horizontal force developed by the displacement caused by its peaks and troughs.  This would be analogous to how the perpendicular displacement of a mountain generates a horizontal force on the surface of the earth, which pulls matter horizontally towards the apex of that displacement.

Additionally, one can derive the causality of electrical component of electromagnetic energy in terms of the energy associated with its "peaks" and "troughs" that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement because classical Mechanics tells us a horizontal force will be developed by that displacement which will always be 90 degrees out of phase with it.  This force is called magnetism.

In other words, it allows one to define a physical model for the propagation of an electromagnetic field in terms of Einstein’s space-time theory.

Additionally, the above conceptual model can be quantified, as was mentioned earlier by using the valid laws of mathematics to transform his space-time equations to their equivalent in four *spatial* dimensions. This equivalence also allows one to explain both electromagnetism and gravity "as aspects of some broader mathematical structure" in terms of the geometry of four *spatial* dimensions or four-dimensional space-time. 

Yet, it also explains why electromagnetic energy when observed always presents itself as the particle called a photon in terms of Einstein’s space-time model.

For example, the article, " Why is energy/mass quantized? " Oct. 4, 2007 showed that one can use the Einstein’s theories to explain the quantum mechanical properties of an electromagnetic wave by extrapolating the rules of classical resonance in a three-dimensional environment to an energy wave moving on surface of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly it showed the four conditions required for resonance to occur in a classical environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in an energy wave moving in four *spatial* dimensions.

The existence of four *spatial* dimensions would give the energy wave associated with a photon 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 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, the oscillations caused by such an event would serve as forcing function allowing a resonant system or "structure" to be established in four spatial dimensions.

As was shown in that article these resonant systems in four *spatial* dimensions are responsible for the particle called a photon.

However, one can also use Einstein space-time theories to explain how the boundaries of the standing wave responsible for creating the resonant system that article indicated was responsible of a particles formation.

In classical physics a standing wave is created when the vibrational frequency of a source causes reflected waves from one end of a confined medium to interfere with incident waves from the source.  This interference of the wave energy causes their peaks troughs to be reinforce in the volume they are occupying thereby creating a standing wave.

The confinement required to create a standing wave in space-time or its equivalent in four *spatial* dimensions can be understood by comparing it to the confinement a point on the two-dimensional surface of paper experiences when oscillating with respect to three-dimensional space.  The energy associated with the wave motion of that point would be confined to its two-dimensional surface and would be reflected and interfere with the incident wave when reaches three-dimensional space at its edge. Therefore, a standing would be created by its interaction with three-dimensional space.

In other words when a wave on the surface of a piece of paper encounters the third spatial dimension at its edge it is reflected back allowing a standing wave to be formed on its surface.

Similarly, an electromagnetic wave moving on the surface of three-dimensional space would be confined to it and reflected back to that volume, similar to the surface of the paper if it was prevented from oscillating with respect to a four spatial dimensions or four-dimensional space-time by an observation.

In other words, when an electromagnetic wave is confined to three-dimensional space by an observation or an interaction with particle like a proton or electron the interference caused by that confinement sets up a resonant standing wave in space which is called a photon.

Additionally, it tells us that the reason the energy of electromagnetic wave always without exception becomes a particle when observed is because of the fact that all observations or interactions with other particles will confine its motion to three-dimensional space thereby creating the resonate system that defined a particle that was shown to be responsible for a particle in the article " Why is energy/mass quantized? " Oct. 4, 2007

As mentioned early, the above conceptual model can be quantified by using the valid laws of mathematics to transform his space-time equations to their equivalent in four *spatial* dimensions.  This equivalence as was shown above allows one to explain both particle and wave properties of electromagnetisms and gravity "as aspects of some broader mathematical structure" in terms of the geometry of four *spatial* dimensions or four-dimensional space-time. 

It should be remembered that Einstein’s genius allows us to choose whether to define an electromagnetic wave either a space-time environment or one consisting of four *spatial* dimension when he defined its geometry in terms of the constant velocity of light.

Later Jeff 

Copyright Jeffrey O’Callaghan 2020

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

It is possible, as this article will show that a standing wave in space-time is responsible for a photon.

A standing waves are created within a medium when the vibration frequency of the source causes reflected waves from one end of the medium to interfere with incident waves from the source. This interference occurs in such a manner that specific points along the medium appear to be standing still. Because the observed wave pattern is characterized by points that appear to be standing still, the pattern is often called a standing wave pattern. Such patterns are only created within the medium at specific frequencies of vibration. These frequencies are known as harmonic frequencies, or merely harmonics. At any frequency other than a harmonic frequency, the interference of reflected and incident waves leads to a resulting disturbance of the medium that is irregular and non-repeating.

In March 1905 Einstein published a paper on the photoelectric effect entitled "On a Heuristic Viewpoint Concerning the Production and Transformation of Light" in which he proposed the idea of energy quanta and postulated light exists as tiny packets, or particles, called photons.

In that paper he stated Energy, in the propagation of a ray of light, is not continuously distributed over steadily increasing spaces, but it consists of a finite number of energy quanta but he did not say why.

Even so many fell the idea of light quanta contradicts the wave theory of light that follows naturally from James Clerk Maxwell’s equations for electromagnetic behavior and, more generally, the assumption of infinite divisibility of energy in physical systems.

Additionally he did not address the issue of how these "tiny packets" of energy called photons can move through space at the speed of light. This present a problem because he showed that energy and mass are equivalent and that the mass of any object or particle moving at the speed of is infinite. 
Therefore, if energy is equivalent to mass one would assume that energy required to move a photon at the speed of light would be infinite.

However, Einstein gave us a way to define electromagnetic energy in a manner which is not only consistent with his theories but also with our classical understanding of nature when, in his General Theory of Relativity he showed that matter can be converted into energy or energy into matter according to the equation E=mc2.

For example Einstein defined the origin of the mass component of particles and all other objects, such as the sun in terms of curvature or distortion in the continuous field properties of space-time not in terms of their particle properties.

QED defines the fundamental unit (quanta) of light as "bundles of pure energy traveling at the speed of light with the unique property of being both particle and wave. However this means that as light moves through space-time the peaks and troughs of its wave properties would cause positive and negative spatial displacements in the "surface’ of space time.

Yet, it is difficult to understand how a spatial displacement can be responsible for of electromagnetism and how and why its wave properties morph to the particle QED defines as the photon when it is observed or interacts with matter because he CHOSE to use time or a displacement in space-time dimension to define mass and energy and not its spatial properties.

Yet he gave us the ability to form a physical image of this how the spatial properties of a photon’s wave packet are responsible for its movement through space when he defined its geometric properties in terms of the constant velocity of light and a dynamic balance between mass and energy because that provided a method of converting a unit of time in a space-time environment to a unit of space in four *spatial* dimensions. Additionally because the velocity of light is constant he also defined a one to one quantitative and qualitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

The fact that one can use Einstein’s equations to qualitatively and quantitatively redefine the curvature in space-time he associated with energy in terms of four *spatial* dimensions is one bases for assuming as was done in the article “Defining energy?” Nov 27, 2007 that all forms of matter and energy can be derived in terms of a spatial displacement in a “surface” of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, this change in perspective gives one the ability to understand how the energy of a photon can move through space at the speed of light why it becomes a particle when interacting with matter in terms of the concepts of his theories.

For example as waves travel through water; they do not take the water with them because as wave arrives it lifts the water particles, they then travel forward, down and back so that each particle completes a circle. Circling movements of particles near the surface set off smaller circling movements below them therefore the waves don’t actually move the water forward. In other words the particles in a wave do not move with respect to space but exchange their potential energy of the water for kinetic energy associated with the wave’s movement.

Similar to wave on water the trough of a light wave would create a point with a positive curvature on a "surface" of the three-dimensional space manifold with respect to a fourth *spatial dimension which would present itself as the potential energy Einstein associated with mass. That point in space would then travel forward and up and back so that each one completed a circle without moving with respect to background of space. As the wave passed this point the potential energy of positive curvature in four "spatial" dimensions Einstein associated with mass would be converted to kinetic energy associated with a moving mass. In other words the wave packet of a photon can move though space at the speed of light because similar to a wave on water light waves do not cause a point in space to move with respect to the background of space-time.

This suggest that light is not electromagnetic wave but an energy wave in space-time which is the result of the potential energy created by the trough of a wave on its "surface" being converted to the kinetic energy associated with its peak thereby causing what is called light to move through space.

However, this also tell us when viewed in terms of their spatial properties that the electromagnetic properties of a light wave are the result of its propagation and not the casualty as is suggested by Maxwell’s equations.

(Later it will be shown in terms of those spatial properties the reason why this wave becomes a particle when interacting with matter but for now we would like to focus our attention on electromagnetic properties of light or a photon’s wave packet)

As was mention earlier a wave on the two-dimensional surface of water causes a point on that surface to be become displaced or rise above or below the equilibrium point that existed before the wave was present. A force will be developed by the differential displacement of the surfaces, which will result in the elevated and depressed portions of the water moving towards or become "attracted" to each other and the surface of the water.

Similarly a matter wave on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would cause a point on that "surface" to become displaced or rise above and below the equilibrium point that existed before the wave was present.

Therefore, classical wave mechanics, if extrapolated to four *spatial* dimensions tells us a force will be developed by the differential displacements caused by a matter wave moving on a "surface" of three-dimensional space with respect to a fourth *spatial* dimension that will result in its elevated and depressed portions moving towards or become "attracted" to each other.

This defines the causality of the attractive forces of unlike charges associated with the electromagnetic wave component of a photon in terms of a force developed by a differential displacement of a point on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

However, it also provides a classical mechanism for understanding why similar charges repel each other because observations of water show that there is a direct relationship between the magnitudes of a displacement in its surface to the magnitude of the force resisting that displacement.

Similarly the magnitude of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two similar charges will be greater than that caused by a single one. Therefore, similar charges will repel each other because the magnitude of the force resisting the displacement will be greater for two charges than it would be for a single charge.

One can define the causality of electrical component of electromagnetic radiation in terms of the energy associated with its "peaks" and "troughs" that is directed perpendicular to its velocity vector while its magnetic component would be associated with the horizontal force developed by that perpendicular displacement.

However, Classical Mechanics tells us a horizontal force will be developed by that perpendicular or vertical displacement which will always be 90 degrees out of phase with it. This force is called magnetism.

This is analogous to how the vertical force pushing up of on mountain also generates a horizontal force, which pulls matter horizontally towards the apex of that displacement.

This shows that one can use the spatial properties of Einstein’s theories to derive causality of the electromagnetic forces of light and how the wave packet Quantum Electrodynamics associates with a photon is propagated through space by extrapolating the laws of classical mechanics in a three-dimensional environment to one consisting of four dimensional space-time or four *spatial* dimensions.

However viewing a light in terms of the spatial instead of the time properties of his theories allows one to understand how why it always appears as a particle when measured or observed.

For example in the article “Why is energy/mass quantized?” Oct. 4, 2007 it was shown one can physical derive photonic properties of light by extrapolating the laws of classical wave mechanics in a three-dimensional environment to a matter energy wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Briefly it showed the four conditions required for resonance to occur in a classical environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial would occur in one consisting of four spatial dimensions.

The existence of four *spatial* dimensions would give a matter wave 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 to oscillate with the frequency associated with the energy of that event.

The oscillations caused by such an event would serve as forcing function allowing a resonant system or "structure" to be established space.

Therefore, these oscillations in a "surface" of a three-dimensional space manifold would meet the requirements mentioned above for the formation of a resonant system or "structure" in four-dimensional space if one extrapolated them to that environment.

Classical mechanics tells us the energy of a resonant system can only take on the discrete or quantized values associated with it fundamental or a harmonic of its fundamental frequency.

Hence, these resonant systems in four *spatial* dimensions would be responsible for the discrete quantized energy associated with the quantum mechanical properties of a photon.

Yet one can also define the boundary conditions required to establish the standing wave component of a resonant system mentioned earlier that is responsible for it its particle properties as defined by QED.

For example in classical physics, a point on the 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, it could, similar to the surface of the paper oscillate “up” or “down” with respect to a fourth *spatial* dimension.

The confinement of the “upward” and “downward” oscillations of the field properties of mass with respect to a fourth *spatial* dimension is what defines the spatial boundaries that enables the formation of a resonant system which the article  “Why is energy/mass quantized?” defined as being responsible for a particle.

When a wave on water comes ashore the energy associated with its wave properties is confined to a specific region of the shoreline.

Similarly when a photon’s wave packet is measured or observed a portion of its wave energy is transmitted to the measuring instrument while some of it may be redirected or reflected similar to a wave striking the shoreline. In other word the energy wave which earlier was define as being responsible for the transmission of light interacts with measuring equipment for the same reason a water wave interacts with the shoreline.

However it also explains why light is always observed as a particle when it encounters encounters a measuring instrument or is observed.

In the quantum mechanical system described above such as light interacting with a particle, resonance only occurs when the frequency at which the force applied is equal or nearly equal to one or a multiple of the natural frequencies of the system on which it acts. In other words light when confined to three-dimensional space by interaction with a particle it will always present itself as the resonant structure that has the energy equal to one or a multiple of the natural frequency of space time. The remaining energy will be radiated through space as light with a lower frequency.

In other words the particle component of light or an electromagnetic wave is not the cause of its interaction with particles but a result of it.

However one can use the above model to explain why photons do not interact with each other because similar to waves on water if their is no obstruction to hinder their movement a wave will not interact with each other.  In other words, photon do not interfere with each other for the same reason that all energy waves do not.

Summing up, Einstein genius allows us to view his theory in either four dimensional space-time or four spatial dimensions. As was shown above changing ones perspective on his theory from time to its spatial equivalent allows one to define light as an energy wave in space and shows the electromagnetic properties are the result NOT the casualty of its propagation but a result of it. Similarly it shows the particle component of light is not the cause of its interaction with a particle but a result of it. 

It is important to note the validity these conclusions cannot be falsified if one accepts the validity of his theories.

Later Jeff

Copyright Jeffrey O’Callaghan 2019

Please visit our Facebook group The Road to unification of Quantum and Relativistic theories if you would like to comment or contribute to our project

One cannot deny that Quantum mechanics, the theory that defines the tiny world of particles and Einstein’s theories, the one that defines what we see through a telescope have been the most successful scientific theories of modern times However, attempts to bring these two theories together and define "A Theory of Everything" have been unsuccessful.

However, the fact that we have been unable to do so suggests that one or both of these theoretical models does not describe the true nature of reality because the world we see through a telescope must have its foundations in the world of the very small therefore they must be connected.

There can be many reasons for this.  One is that foundational assumptions of either or both of them is incorrect.  In other words, the world of the tiny may not be governed by probabilities as quantum theory suggests or the world, we see though a telescope may not be ruled by relativistic properties of four dimensional space-time.

However, there is another possibility that many have over looked is that even though their mathematics makes very accurate perditions of experimental observations they do not accurately define reality of their operating environments.

For example, Einstein mathematically defined the physical structure of the universe in terms of the geometry four dimensional space-time.

However, when using the constant velocity of light and the velocity of objects that do not move at that speed to define its geometric properties he provided a way of mathematical converting a unit of time in a space-time universe to unit of space in one physically consisting of only four *spatial* dimensions.

In other words, their is an equally valid interpretation of his mathematics in terms of only four spatial dimensions.

Since both of these solutions that of four dimensional space-time and four spatial dimensions would yield the same numerical results it gives one a different way of connecting his theories to those of Quantum mechanics based on the physical properties of four spatial dimensions instead of four dimensional space-time.

Quantum Theory on the other hand defines tiny world of particles in terms of the non physical probabilities associated with Schrödinger wave equation which as mentioned earlier no one has been able to physical connect to the space-time universe define by Einstein.

However, the fact that Einstein provided an alternative solution to his mathematics in terms of four spatial dimensions suggests it may be possible to make that connection and therefore define a Theory of Everything by using the alternative solution of four spatial dimension that his theory provides instead of one based on four dimensional space-time.

For example the article "Why is mass quantized?" Oct. 4, 2007 showed one could derive the quantum mechanical properties of energy in terms of a resonant "system" or structure formed by a energy wave on the surface of a three-dimensional spatial manifold with respect to a fourth *spatial* dimension.

Briefly that article showed the four conditions required for resonance to occur in a three-dimensional 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 can be meet in one consisting of terms of four spatial dimension.

Its continuous properties would allow an energy wave on a "surface’ of a three-dimensional space manifold to oscillate with respect to a fourth *spatial* dimension thereby fulfilling one of the requirements for resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital. This would force the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

Therefore, these oscillations in a continuous non-quantized field of energy would meet one of the requirements mentioned above for the formation of a resonant system or "structure" in space.

Observations of a three-dimensional environment show the energy associated with resonant system can only take on the incremental or discreet values associated with a fundamental or a harmonic of the fundamental frequency of its environment.

Similarly the energy associated with resonant systems in four *spatial* dimensions could only take on the incremental or discreet values associated a fundamental or a harmonic of the fundamental frequency of its environment.

These resonant systems in a space-time environment would be responsible for the incremental or discreet energy associated with quantum mechanical systems.

Another requirement for a resonate system to be formed is that the wave must be confined to specific volume of space.

However, one can also define the confinement of the resonant component of a particle and therefore establish a physical connection to the wave particle duality quantum mechanics associates with energy in terms of the relativistic properties of four *spatial* dimensions.

In physics, a point on the 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, it could, similar to the surface of the paper oscillate "up" or "down" with respect to a fourth *spatial* dimension.

The confinement of the "upward" and "downward" oscillations of a three-dimension volume with respect to a fourth *spatial* dimension is what defines the spatial boundaries associated with a particle in the article "Why is energy/mass quantized?".

In other words, an energy wave in four *spatial* dimensions will maintain its wave properties unless it is confined to three by an observation, therefore it it always be view as a particle when an observation is made and any energy left over from the formation of its resonate structure will be radiating from the point of observation in the form of light or an energy wave.

The physics of wave mechanics also tells us that due to their continuous properties the energy waves the article "Why is energy/mass quantized?" Oct. 4, 2007 associated with a quantum system would be distributed throughout the entire "surface" a three-dimensional space manifold with respect to a fourth *spatial* dimension.

For example the energy of a vibrating or oscillating ball on a rubber diaphragm would be disturbed over its entire surface while the magnitude of those vibrations would decrease as one move away from the focal point of the oscillations.

Similarly if the assumption that quantum properties of energy are a result of vibrations or oscillations in a "surface" of three-dimensional space is correct those oscillations would be distributed over the entire "surface" three-dimensional space while the magnitude of those vibrations would be greatest at the focal point of the oscillations and decreases as one moves away from it.

(Some may question the fact that the energy wave associated with particle would be distributed over the entire universe.  However, the relativistic properties of space-time and four spatial dimensions tell that distance perceived by objects or particles in relative motion is dependent on their velocity which become zero at the speed of light.  Therefore, from the perspective of an energy wave moving at the speed of light, the distance between all points in the universe along it velocity vector is zero.  In other words, it’s energy is distributed or simultaneous exists at every point in the universe along its velocity vector.  There can be not other conclusion if one accept the validity of Einstein’s theories.)

As mentioned earlier the article “Why is energy/mass quantized?” shown a quantum particle is a result of a resonant structure formed by an energy wave on the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Yet the science of Wave Mechanics tells us resonance would most probably occur on the surface of the rubber sheet were the magnitude of the vibrations is greatest and would diminish as one move away from that point,

Similarly a particle would most probably be found were the magnitude of the vibrations in a "surface" of a three-dimensional space manifold is greatest and would diminish as one move away from that point.

This shows how, by interpreting Einstein space-time theories in their equivalent four spatial dimension one can connect the non physical probabilities associated with Schrödinger wave equation to the reality of the world defined by him. 

Additional it shows, by changing our interpretation of Einstein’s theories from four dimensional space-time to it equivalent in four spatial dimensions allows one to clearly understand the physical connection between the probabilistic world of quantum theory and the relativistic world of his theories, thereby allowing one to form a Theory of Everything.

Later Jeff

Copyright Jeffrey O’Callaghan 2019