New Ideas

 

You’re So Vain

Well, you're where you should be all the time
And when you're not, you're with some underworld spy
Or the wife of a close friend, wife of a close friend, and
You're so vain
You probably think this song is about you
You're so vain (so vain)
I bet you think this song is about you
Don't you, don't you, don't you?

I am so vain.

An article by Ethan Siegel asked why scientist are so hostile to new ideas. https://medium.com/starts-with-a-bang/the-good-reasons-scientists-are-so-hostile-to-new-ideas-27aa237c0375. I have the bad feeling that I triggered this article, I am so vain, because I sent an email to Dr. Siegel about whether the universe could be hyperbolic instead of flat, admittedly a new idea.

Dr Siegel asked some questions in his article.

“What is the problem you’re considering that motivated this idea?

How does this idea compare to the prevailing theory when applied to this specific phenomenon?

How does this idea compare to the prevailing theory when applied to the other major successes of the prevailing theory?

And what are some critical tests that you can legitimately perform (with current or near-future technology) to further discern your idea versus the prevailing theory?

As Richard Feynman once put it so eloquently, “The first principle is that you must not fool yourself — and you are the easiest person to fool.””

Challenge accepted. I will try to answer those questions.

What is the problem you’re considering that motivated this idea?

To start at the very beginning, I was born in Providence, Rhode Island. Oh, not that beginning. How about this one? In my senior year at Brown University, I took a class that made me embark on a career in Transportation Engineering. But that was over 50 years ago, so what was I looking at lately? I was trying to address a problem with the relationship between speed and volume, commonly known as congestion. There was an equation that was simple, but wrong, proposed in the 1930s. There was another equation that was much more complex, at least the relationship was not understandable to me. It was, to use Dr. Siegel’s words, “like Johannes Kepler, who threw away his “beautiful” theory of nested spheres and perfect solids before settling on his “ugly” theory of elliptical orbits that fit the data better,” and I was looking for an ugly theory that might fit the data better. The problem was that the data fit in the normal domain very well where volume on a road is less than the capacity on that road but did not fit the observations in over capacity conditions. While most of traffic engineering is concerned with making things better before capacity is reached, i.e. operations/tactics; I work in the branch of traffic engineering that deals, when, as the great philosopher Jimmy Buffet puts it “Shit hits the fan”, traffic volumes exceed capacity, i.e. planning/ strategy.

In trying to improve, or at least understand, the relationship between speed and volume, I wanted to examine the idea that the relationship, which appeared to be a rotated asymmetrical parabola, might instead be a rotated asymmetrical hyperbola.

This was not the first time that I encountered hyperbolas. The firm at which I have worked for almost 25 years, became known for its ability to explain choices.  In fact one of the early employees of that firm, Daniel McFadden, won the Nobel Prize in Economics for his advances on the theory of choices. A normal distribution based on the percentage making a choice is the logistics distribution. One of alternative names for the logistic distribution is the hyperbolic secant squared distribution. The Cumulative Distribution Function, CDF, of this hyperbolic secant squared distribution is also a scaled version of the hyperbolic tangent.

The hyperbolic tangent looks almost like an exponential association, which is itself the CDF of an exponential distribution, but the exponential association is not normal. I had previously proposed that the relationship between reliability, the ability to achieve an on-time performance, and the mean time on a road was an exponential association.

I was also tasked with looking into improving the way that traffic makes a choice among competing routes. This is when I am forced to employ the old chestnut that coincidences are an example of God’s sense of humor. Remember that course that influenced my career choice. It was taught by the late Dr. Stella Dafermos, who proposed the current method used in those route choices. Here is where I get into a second coincidence. Dr. Dafermos was a member of the Applied Math (popularly known by the students as Apple Math) Department at Brown University. My late mother retired from a position as a clerk in what was effectively the mail room of that department (I was not admitted due to nepotism. She started working there after I graduated). I thus felt a special affinity for that Department. In the mid 1980s I was working at a public agency that had an exceptionally good library. I went looking for articles published by those in that Department at Brown that were also in my field. I came across an article written by a post-doctoral student of Dr. Dafermos, Anna Nagurney. In that article, she explained that the impedance function used in route choice, which worked best with Dr. Dafermos’ method, was a fourth power function of volume, i.e. a parabola, but could not explain why. Shortly before this, the Bureau of Public Roads, the predecessor of the Federal Highway Administration before the formation of the US Department of Transportation, USDOT, had proposed an empirical volume diversion equation, known in the profession as the BPR curve, which was a fourth power function of volume. Despite the warnings of Dr. Alan Horowitz, that this equation had only been observed in conditions less than capacity and should not be used in over capacity conditions, the USDOT’s Travel Model Improvement Program, TMIP, recommended the use of a variation of this curve in the traffic models based on Dr. Dafermos’ method in all conditions.

I participated in a study that related the choices with observations of mean travel time and the planning time (the mean time plus reliability, expressed in units of time). We found that route choice was better correlated with planning time, than with mean time alone. So I was faced with choice (which was based on a hyperbola), mean time (and that is simply the inverse of speed, which appeared to be a hyperbola) and reliability, which also seemed to follow a hyperbola. Given that, I did a Google search on hyperbola to find where else it was used and found a link a paper by Dr. Mabkhout which suggests that the shape of the universe was hyperbolic.

So I did not propose that the universe is hyperbolic. I came across the ideas because I was studying relationships in traffic engineering.

How does this idea compare to the prevailing theory when applied to this specific phenomenon?

The idea that the shape of the universe is hyperbolic means that it is open, i.e. at infinity it will also be infinite, but also that it is not flat, i. e. has a curve. It means that it also should begin at a single point, e.g. a Big Bang. This hyperbolic shape explains cosmic inflation in the early universe, as well as the continued expansion of the universe. It is an “ugly” theory because it explains these without resorting to the “beautiful” additions of Dark Energy and Dark Matter. 

How does this idea compare to the prevailing theory when applied to the other major successes of the prevailing theory?

The hyperbolic universe is consistent with the age and size of the observable universe.  It is also consistent with small scales, i.e. the Planck length is consistent with the Planck energy.

A hyperbolic shape also explains the observations of galaxy rotation while a flat universe does not.

And what are some critical tests that you can legitimately perform (with current or near-future technology) to further discern your idea versus the prevailing theory?

I have been involved in using GPS data as traffic observations. The GPS records the latitude and longitude of a point. The distance between observations is not a straight line, because the earth is not flat, it is the Great Circle Distance, a non-Euclidean solution, because the earth only appears locally to be flat but is in fact a sphere. This same phenomena also have a bearing the shape of the universe. If the universe appears to be flat locally, but is in fact hyperbolic, then Pythagoras’ Theorem, which only applies in flat space, should not be used. Instead, the shortest distance between two points in a hyperbolic universe should use Pythagoras’ hyperbolic theorem. Any rotation by n/2π to n/π where n is odd in a flat universe using conventional trigonometry may change a real number to an imaginary number. However a rotation in a hyperbolic universe will never result in changing a real solution to an imaginary solution because the hyperbolic trigonometry used does not change the set of the solutions. Thus the Lorentz Transform, which in a flat Euclidean universe is √(1-(v/c)²⁾, in a hyperbolic universe would be 1+ln(cosh(v/c)±sinh(v/c)) The results are not appreciably different until the ratio of velocity to the speed of light, v/c, exceeds 85%. ( Acrtually 82.2%, but close enough for government work.)

If the universe is hyperbolic, then it would be random, not deterministic. Determinism requires a flat universe where at a Standard Deviation from the mean of zero, there is only one solution. In a hyperbolic universe, the square root of the variance, which is usually thought to be the Standard Deviation from the mean should always be apprimtely 0.91. You can approach zero but can never reach zero, and thus there is not always a single solution.

If the universe is hyperbolic, curved, then gravity might be an apparent force, where two masses are just following a geodesic to a point of lower entropy. It only appears to be a force because in a flat frame of reference, the masses would not appear to move unless they were acted on by a force.

Speaking of entropy, many of the solutions that appear deterministic in my field are in fact maximum entropy solutions.  That is they are the mesostate that has the greatest number of microstates.   This sounds very complex but is actually very simple.  In a game of craps (i.e. a macrostate), the most probable roll (i.e. a mesostate) is a seven because there are more combinations (i.e. microstates) that total seven than any other roll.  The fact that there is entropy means that there is at least one microstate.  Only when there are no microstates will the be no entropy. The fact that we exist (are a microstate) AND there is entropy, must mean that the universe is random.