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

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