The reliability of our mathematical universes

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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
Unfortunately we cannot because, as was just mentioned we have not or may not ever be able to conduct 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 them 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 are 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

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