Recently there have been many observations that are extremely difficult to integrate into the currently accepted theoretical models.

Particularly the force called Dark Energy has eluded any attempt make it a part of the "The Standard Model of Particle Physics" one of the most successful theories ever created. 

However what makes this even more troubling is that the Standard model is based on two other very successful theories, that of Einstein General Theory of Relativity and Quantum Field Theory.

Therefore if, despite continued efforts to developed a theoretical understanding of Dark Energy in terms of these theories we still cannot succeed, should we assume that, due to how interconnected these theories are we must discard them and look in a new direction.

Not necessarily because we may be able to understand its causality in terms of our current theories if instead of trying to make Dark Energy conform to them we allow its observed properties to guide us to a more complete understanding their validly.

Most scientists would agree that the best way of determining how one should interpret a theoretical model would be to list all observations regarding the forces in its domain and try to define them in terms of the rules it lays out.

Dr. Michio Kaku
The Physics of the future

For example observations of the expansive force called Dark Energy tell us that three-dimensional space is expanding towards a higher spatial dimension not a time or space-time dimension.  

Therefore, to explain the observed spatial expansion of the universe one would have to assume the existence of a another *spatial* or fourth *spatial* dimension in addition to the three-spatial dimensions and one time dimension that Einstein’s theories contain to account for that observation.

This would be true if Einstein had not given us a means of qualitatively and quantitatively converting the geometric properties of his space-time universe to one consisting of only four *spatial* dimensions.

He did this when he used the constant velocity of light and the equation E=mc^2 to define the dynamic balance between mass and energy responsible for geometric properties of space-time because it provided a method of converting the space-time displacement he associated with energy in a space-time universe to a spatial one in a universe consisting of only four *spatial* dimensions.  Additionally because the velocity of light is constant he also defined a one to one quantitative correspondence between his space-time universe and one made up of four *spatial* dimensions.

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

The fact that the equation E=mc^2 allows us to both qualitatively and quantitatively derive the spatial properties of energy in a space-time universe in terms of four *spatial* dimensions is the bases for assuming as was done in the article “Defining energy” Nov 27, 2007 that all forms of energy, including that associated with the Higgs field 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.

As mentioned earlier it is difficult to integrate the causality of how three-dimensional space can be expanding towards a higher *spatial" dimension into Einstein space-time universe because it does not define a higher spatial dimension. 

However it is easy if one redefines Einstein’s space-time universe, as was done above in terms of four *spatial* dimensions because a higher or fourth *spatial* dimension would be an integral part of its theoretical structure.

Yet it also allows one use Einstein theories and the laws of thermodynamics to understand how and why the expansive force called Dark Energy is causing the spatial expansion of our universe because it gives one the ability to qualitatively and quantitatively define energy in terms of a spatial displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimensions instead of one in a space-time environment.

We know from the study of thermodynamics that energy flows from areas of high to one of low density very similar to how water flows form an elevated or "high density" point to a lower one.

For example, if the walls of an above ground pool filled with water collapse the water molecules on the elevated two-dimensional surface of the water will flow or expand and accelerate outward towards the three-dimensional environment surrounding it while the force associated with that expansion decreases as it expands.

Yet we know from observations of the cosmic background radiation that presently our three-dimensional universe has an average energy component equal to about 3.7 degrees Kelvin. 

However according to concepts developed in the article “Defining energy" (mentioned earlier) the three-dimensional "surface" of our universe which has an average energy component of 3.7 degree Kelvin would be elevated with respect to a fourth *spatial* dimension.

Yet this means similar to the water molecules occupying the elevated two dimensional surface of the water in the pool, the particles occupying a region of three-dimensional space that is elevated because of its 3.7 degree temperature will flow and accelerate outward in the four dimensional environment surrounding it.

This shows how reformulating Einstein’s theories in terms of four *spatial* dimensions allows one to use the laws of thermodynamics to explain what the force called Dark Energy is and why it is causing the accelerated expansion of the universe in terms of the Einstein’s theories.

Many feel that because space is everywhere, the force called Dark Energy is everywhere so therefore its effects will increase as space expands.  In contrast, gravity’s force is stronger when things are close together and weaker when they are far apart. Therefore they feel the rate at which the universe expands will increase as time go by resulting in galaxies, stars, the solar system, planets, and even molecules and atoms could be shredded by the ever-faster expansion. In other words the universe that was born in a violent expansion could end with an even more violent expansion called the Big Rip.

However if the above theoretical model is correct than the magnitude of Dark Energy relative to gravitational energy will not continue to increase as the universe expands but will decrease because, similar to the water in a collapsed pool the accelerative forces associated with it will decline as it expands and yet because the quantity of energy/mass of the universe remains constant through its history its gravitational potential will also.

Therefore in the future the gravitational contractive forces associated with it will exceed the expansive forces associated with Dark Energy because, as mentioned earlier according to this theoretical model its accelerative forces should decrease as the universe expands. This would be true even though its components may be separated by extremely large distances because, as just mentioned if the above theoretical scenario is correct the force associated with dark energy will decease relative to gravity as time goes by.

Recent observations also suggest that early in the universe evolution the gravitational forces exceeded the expansive forces of Dark Energy.

The reason is according the above theoretical model, just after the big bang when the concentration of energy and mass was high, the gravitational forces of the universe’s energy/mass would predominate over Dark Energy because the distance between both its energy and mass components was relatively small.

However as the universe expands its gravitational attractive forces will decrease more rapidly than the expansive force associated with Dark Energy because they are related to the square of the distance between them while those of the expansive forces of Dark Energy are more closely related to a linear function of the total energy of content of the universe.

Therefore after a given period of time the expansive forces associated with Dark Energy will become predominate and the expansion of the universe will accelerate.

However as the universe expands and cools that force will decrease because as mentioned earlier similar to the two-dimensional surface of the water in a collapsed pool, the forces associated with that expansion will decrease as it expands.

This means that eventually gravitational forces will win because, as mentioned earlier the laws of thermodynamics tells us the total accelerative forces associated with Dark Energy will decease and therefore will eventually approach zero, while the total mass content and the gravitational attractive forces associated with it will remain constant as the universe expands even though they may be separated by a greater distant.

Therefore. gravity will eventually win the battle with dark Energy because as was just mentioned the forces associated with it approach zero as the expansion progress while those of gravity remain constant.

There can be no other conclusion if one accepts the validity of Einstein’s theories and the laws of thermodynamics because the theoretical arguments presented are a base solely on their validity.

This shows how one can fully integrate the observed properties of Dark Energy into Einstein General Theory of Relativity while at the same time demonstrating the advantages of allowing observations guide our understanding of our theoretical model instead of forcing them to be subservient to our preconceive theoretical ideas.

However this may also allow gravity to be integrated into the Standard Model if one can reformulate its space-time equations to their equivalent in four *spatial* dimensions as was shown above to be possible.

It should be remember that Einstein’s genius allows us to choose whether to view Dark Energy and the mathematical equations in the Standard Model in either a space-time environment or one consisting of four *spatial* dimension when he defined the geometry of space-time in terms of energy/mass and the constant velocity of light.

Later Jeff

Copyright Jeffrey O’Callaghan 2014


 

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How can we be sure that the mathematical universes we create actually exist in nature?

Paul Adrien Maurice Dirac addressed this issue in a lecture he delivered on February 6, 1939 regarding "The Relation between Mathematics and Physics".

"The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed (or cannot be performed). There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme.

One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up."

But exactly how deep is the connection between the mathematical reasoning we use to predict nature to its reality.  In other words how can be sure the equations we use to "infer the results of experiments that have not been performed" (or cannot be performed) actually defines the reality of the environment that encompasses them

BBC Horizon –Black Holes (2013)

Unfortunately we cannot because, as was just mentioned we have not or never will be able to observe them.

Therefore we must be very sure that the equations we use to predict a "quality of Nature" that is unobservable have a "factual" foundation in the theoretical models they are derived from because it is only way in which we can be connect to true "Nature" of reality defined by that theoretical model.

This is especially true when we use the mathematics of an established paradigm such as the General Theory of Relativity to predict the existence of objects or things such as a singularity which, by definition can never be observed.

For example ESA, at its HubbleSite tells us using Newton’s Laws in the late 1790s, John Michell of England and Pierre-Simon Laplace of France independently suggested the existence of an "invisible star." Michell and Laplace calculated the mass and size – which is now called the "event horizon" – that an object needs in order to have an escape velocity greater than the speed of light. While n 1915, Einstein’s gave us a conceptual basis for their existence when he publish his General Theory Relativity was able to gives for their predicted the existence of black holes.

Later Karl Schwarzschild, when quantified their existence using mathematics based on Einstein General Theory of Relativity discovered that the gravitational field of a star greater than approximately 2.0 times a solar mass would collapse form a "invisible star" of black hole, as it is now called. Additionally he showed those same equation indicated that the mass would continue to collapse even after its formation to a singularity or one dimensional point. 

He was also able to mathematically quantify the critical circumference or boundary in space around it where the strength of a gravitational field will become strong enough to prevent light from escaping and time being infinitely dilated or slowing to a stop.

In other words, as a star contacts and its circumference decreases, the time dilation on its surface will increase.  At a certain point the contraction of that star will produce a gravitational field strong enough to stop the movement of time.  Therefore, the critical circumference defined by Karl Schwarzschild is a boundary in space where time stops relative to the space outside of that boundary.

However unlike a black hole which have been observationally confirmed through the gravitational effects they have on companion stars the singularity which Schwarzschild’s mathematics predicted is at its center has not been observed and never will be because, as mentioned earlier light cannot escape from a black hole.

Yet there are some who say that the mathematics used to predict the existence of a black hole also predicts, with equal certainty the existence of singularities.  In other words by verifying the existence of black holes though observation means that we have also verified the existence of singularities.

However that assumption is correct if and only if the formation of a singularity is consistent with the concepts of Einstein’s General Theory of Relativity because as mentioned earlier that is conceptual basis for the mathematics predicating their existence.

However, it can be shown there is an inconsistency between the mathematics Schwarzschild used to predict the existence of a singularity and the concepts developed by Einstein in his Theory of General Relativity. 

To understand why we must look at how it describes both the collapse of a star to a black hole and then what happens to its mass after its formation.

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 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 observer who is external to its 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 be identical to those predicted by Robert Oppenheimer and Hartland Snyder using conceptual concepts of Einstein’s theory regarding time dilation caused by the gravitational field of a collapsing star

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.)

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 time becomes frozen at the critical circumference.

Therefore, because time stops or becomes frozen at the critical circumference for both an observer who is at the center of the clasping mass and one who is at a fixed distance from its surface the contraction cannot continue from either of their perspectives.

However, 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 reach the critical circumference.  Therefore, the surface of a star and an observer on that surface 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 stop in all reference 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 stars 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 on its surface would become infinitely dilated or stop in with respect to reference frames that were not on it when it reaches its critical circumference. 

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.  Therefore it cannot move beyond the critical circumference because motion cannot occur in an environment where time has stopped.    `

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

Therefore, 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 as many physicists believe and must maintain a quantifiable minimum volume which is equal to or greater than the critical circumference defined by Karl Schwarzschild.

This means either the conceptual ideas developed by Einstein are incorrect or there must be an alternative solution to the field equations based on the General Theory of Relativity that many physicists used to predict the existence of a singularity because as has just been shown the theoretical predications made by them regarding its existence are contradictory to the concepts contained in the theoretical model they are base on.

We agree with Dirac that the connection between mathematics and nature goes far deeper than just being a useful tool in its description.

However as was shown above one must make sure that facts upon which the mathematics is based reliably follow the theoretical model they were development from if we want to use them to understand the "quality of Nature" defined by that model.

Later Jeff

Copyright Jeffrey O’Callaghan 2014


 

The Imagineer’s
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The Imagineer’s
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The Reality
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