Mathematically defined units - benefits

[FrankM]

What benefits would accrue to the various scientific disciplines if a set of units, specifically, length, time duration, unit of energy and the speed of light were mathematically defined?

 

I realize this is a hypothetical, but I need some conceptual views on the benefits of such a system of units.

[Pyrotex]

What benefits would accrue to the various scientific disciplines if a set of units, specifically, length, time duration, unit of energy and the speed of light were mathematically defined?

 

I realize this is a hypothetical, but I need some conceptual views on the benefits of such a system of units.

No, actually it is not hypothetical at all.

 

There are a number of benefits to doing this. As a start, check out this Wiki on the international system of units.

 

In physics, I have seen cases where the units were mathematically reconfigured so that Newton's constant, G, was defined as 1 and had no units. Not too surprisingly, this makes some calculations and math proofs much easier. I've seen the same thing done with the speed of light, c. And something similar done in electrical engineering so that one or more troublesome conversion factors become 1.

 

Pyro

[FrankM]

No, actually it is not hypothetical at all.

 

I need to be less hypothetical and put in some specifics. Given a mathematically defined value for the speed of light, which implies that length and the duration of time are also mathematically defined, what benefits would accrue to scientific inquiry?

[modest]

Frank, Did you by chance write this:

Abstract: The methodologies used to determine the numerical value of the speed-of light are limited to the precision of the instruments available and to the defined limits of the units of measurement, which are the meter and the second. A mathematical method can be used to define the numeric value for the speed-of-light using a physical science and a mathematical constant which will be independent of the meter and the second, but readily related to those units. The mathematical method will be defined relative to a physical science and a mathematical constant utilizing a trigonometric function that will exploit electromagnetic relationships. The result will be a numeric value for the speed-of-light that has nearly unlimited precision.

http://vip.ocsnet.net/~ancient/MathSOL.pdf

 

The precision of the speed of light when expressed in meters per second is exact. It is exactly 299,792,458 m/s because the meter is defined as the distance traveled by light in free space in 1⁄299,792,458th of a second. It's numerical value is therefore not affected by the precision of any experimental measurement. It is rather the length of a meter that is subject to an experimental error margin.

 

If tomorrow scientists increased the accuracy of laser interferometers and found out light is a little slower than they previous thought then a meter would get a little bit shorter and c would still equal exactly 299,792,458 m/s.

 

~modest

[FrankM]

Yes, I wrote that paper. As you can see by its date it is over 5 years old. There have been some major revisions in subsequent papers, a revision this year benefiting from a Professor of Mathematics review which revealed an issue which I was not aware of it. The issue did not effect the core concepts of the paper, but a significant philosophical position regarding mathematical and physical constants; I incorporated the philosophical position into my latest revision.

 

You have to realize that the French measurements made to establish the meter were fraught with errors, and the straight line resultant was created by Legendre using the least squares process. Just think what would have happened it the French meter was just a little shorter, with the measured value for the speed of light (SOL) ending up close to 314159000 meters per second. The meter length would still have been arbitrary, but it would have provided a value that would be easier to use in equations dealing with physical law. Why set the value of the SOL equal to 1 when you already have it defined by a numeric value that equals a mathematical constant.

 

Still, I need to have some input on what benefits would accrue if the SOL were mathematically defined with near unlimited precision.

[modest]

Yes, I wrote that paper.

 

Good to know. Thank you :)

 

Still, I need to have some input on what benefits would accrue if the SOL were mathematically defined with near unlimited precision.

 

If c were a mathematical constant then I think that would be fantastic. It would quite potentially reveal why many aspects of the universe are what they are rather than seeming arbitrary. My understanding, however, is that the speed of light is a physical constant.

 

I'm assuming you're using the terms like they are established in these links:

• Mathematical constant - Wikipedia, the free encyclopedia
  
• Physical constant - Wikipedia, the free encyclopedia
  

Let's see... first thing that comes to mind is that if c were a mathematical constant it would probably explain why nature is invariant under Lorentz transforms rather than being invariant under Galilean transforms as that depends on c being finite rather than infinite. Like wikipedia says:

If [math]kappa , = , 0 ,,[/math] then we get the Galilean-Newtonian kinematics with the Galilean transformation...

 

If [math]kappa,[/math] is negative, then we set [math]c , = , frac{1}{sqrt{- kappa}} ,[/math] which becomes the invariant speed, the speed of light in vacuum...

 

Only experiment can answer the question which of the two possibilities, κ = 0 or κ < 0, is realized in our world.

Lorentz transformation--Derivation--From group postulates

 

Think of it like this, we usually know why [math]pi[/math] (a mathematical constant) is in the definition of a physical constant or physical law like the Coulomb constant or the Lorentz force law—it's because the surface area of a sphere of radius r is [math]4 pi r^2[/math] so that the intensity of radiation or power of a force is inversely proportional to that. That aspect of the Coulomb constant is a result of the mathematical properties of three dimensional space.

 

If the speed of light could be derived without the help of any physical measurement then I believe there would be many, many physical formula, laws, and constants that would suddenly make sense. Why, for example, is the proportionality constant between units of space and time or between units of mass and energy the particular value that it is? It would be awesome to know that.

 

Are you thinking that you can derive c without using physical constants that depend on measurement? With something like,

[math] \displaystyle { c } = \frac{1}{\sqrt{\mu_0 \epsilon_0}} [/math]

you cannot know the value of both [math]mu_0[/math] and [math]epsilon_0[/math] (vacuum permeability and permittivity) without physically measuring at least one of them. The same is reportedly true of all known derivations of c.

 

~modest

Edited August 28, 2010 by modest
Removed [INDENT] tag because it parsed incorrectly

[FrankM]

If c were a mathematical constant then I think that would be fantastic. It would quite potentially reveal why many aspects of the universe are what they are rather than seeming arbitrary. My understanding, however, is that the speed of light is a physical constant.

 

Slight change, c is not a mathematical constant, it can be derived using mathematical constants. Yes, the speed of light is a physical constant, but you have to understand its measured value is dependent upon the properties of the medium in which they (electromagnetic waves) are permitted to propagate, which includes a vacuum on the earth's surface.

 

One of the revisions in my current paper, where I am trying to describe a few pertinent benefits of mathematically defined physical constants, is citing how this would support the Mathematical Universe Hypothesis.

 

... you cannot know the value of both mu_0 and epsilon_0 (vacuum permeability and permittivity) without physically measuring at least one of them.

 

Once and awhile, one has to challenge long standing "assumptions," else our knowledge of the universe would still be stuck in the Medieval era, and the scientific community is still dragging along some of the baggage from that era.

[Pyrotex]

Lessee, we could define our standard unit of length as the Flumph: the center-to-center distance between the Earth and Moon, averaged over a 17 year Lunar Cycle, is exactly 1.204 Flumph.

This would give the SOL as exactly 1 (as measured in Flumph/Second).

 

This would make some calculations and proofs easier, but would make others more difficult. For example, my height would be 23 microFlumph.

 

Notice that this does not give you "near unlimited accuracy" - it only gives you "near unlimited precision" which is a totally different thing. Precision is the illusion of accuracy.

 

My belief is that all you will get in the long run is convenience. And that convenience will be local to a certain class of problems or proofs. A gain in convenience in one area will probably mean a loss of convenience in another area.

[FrankM]

Since my original posting of this topic, a paper I submitted has been through peer review, a revision and accepted for publication in an IEEE publication; I do not have a publication date as of this posting.

 

You can read about the concepts in the paper on my revised web page.

 

Physical Constants Defined by Mathematical Constants

[FrankM]

The purpose for the dissemination of the postprint is to identify issues that should be corrected before the final version is published. I have received some constructive comments from individuals that I contacted directly. So far, two changes will be made in the wording in the final version, but there have been no challenges to the mathematical process.

 

I was informed that the biggest challenge in gaining recognition of the methodology is getting one of the well regarded individuals in physics or mathematics to refer to my paper in one of their papers or a presentation. I suspect some of the hundred viewers of the 17Aug posting have already recognized the profound changes the concepts used in the methodology can make in scientific discovery. However, there is a tremendous amount of work that has to be done to fully exploit the concepts, primarily in developing a complete system of scientific units that are truly related to physical law.

 

There is a vast bureaucracy circled around SI units, and having been involved at the management level in a large bureaucracy at one time, I realize they will resist and obfuscate as long as they can. James Clerk Maxwell tried hard to dissuade scientists of his era not to adopt the meter as a scientific unit of measure, and he was very influential at that time. The particle theory of the structure of matter had not been developed when Maxwell was making his argument, nor had any of the emissions from atomic level spectra been identified. The quantum theory of matter is well developed at present and spectral emissions from particle level processes are firmly recognized; perhaps the current knowledge will influence the contemporary scientific establishment to take a look at the concepts used in the methodology and the resulting intrinsic units.

 

It would have been presumptuous for me to claim that intrinsic units represent foundational dimensions at the quantum level, but the presence of a particular frequency in the methodology suggests the related dimension, and time unit, has relevance at the quantum level. Perhaps some members of Hypography can identify how and where intrinsic units fit within the quantum structure. Would this qualify Tormod for a Nobel Prize for hosting a forum that established the base dimensions for the quantum structure?

[Tormod]

Would this qualify Tormod for a Nobel Prize for hosting a forum that established the base dimensions for the quantum structure?

 

Of course! :)

[FrankM]

There have been 200 hits on this posting since August 15th when I provided the link to the postprint titled Methodology to Define Physical Constants Using Mathematical Constants.

 

http://www.vip.ocsnet.net/~ancient/

 

That is not very may hits in comparison to some topics on this forum, but I find it remarkable there have been no comments about the mathematical process involved.

[FrankM]

There have been 200 hits on this posting since August 15th when I provided the link to the postprint titled Methodology to Define Physical Constants Using Mathematical Constants.

 

http://www.vip.ocsnet.net/~ancient/

 

That is not very may hits in comparison to some topics on this forum, but I find it remarkable there have been no comments about the mathematical process involved.

 

 

Odd! When I posted the above today, 10 September 2010, it gave a posting date of 28 August 2010, the date of Tormod's post.

[Tormod]

Odd! When I posted the above today, 10 September 2010, it gave a posting date of 28 August 2010, the date of Tormod's post.

 

Very strange. Thanks for reporting that.

 

(Sorry for not commenting on your math - it is far beond beyond me.) :)

[Farsight]

Frank, I saw you mentioned Kuhn and paradigm shift. Take a look at this article on the fine structure constant: http://physicsworld.com/cws/article/news/43657. It isn't constant, it's a "running" constant. It varies. And I'm afraid c varies too. Take a look at the NIST caesium fountain clock and the definition of the second:

 

"Since 1967, the second has been defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of 0 K (absolute zero), and with appropriate corrections for gravitational time dilation."

 

In essence lasers and a microwave cavity are employed to cause hyperfine transitions, which are electron “spin flips” within caesium atoms. The hyperfine transition emits microwaves, which is light in the wider sense. There’s a peak "frequency" in the emitted light, which is found and measured by the detector. There’s something to note about this. I put the word frequency in quotes because frequency is measured in Hertz, which is defined as cycles per second, and the second isn't defined yet. What the detectors essentially do, is count incoming microwave peaks. When they get to 9,192,631,770 that's a second. Hence the frequency is 9,192,631,770 Hertz by definition. We then use the second, and the metre, which is also defined using the motion of light, to measure the motion of light: 299,792,458 m/s.

 

Note the mention of gravitational time dilation in the definition of the second. If you were to take this clock and place it in a region of low gravitational potential, it would be like pressing a slow-motion button. All electromagnetic and other processes would then occur at a reduced rate, including the hyperfine transition and the motion of the resultant light towards the detector. However regardless of this, when the detectors get to 9,192,631,770, that's a second. It's important to realise here that in this situation, the light is moving slower and this is why the second is bigger. We then use this second to measure the motion of light, and we still measure the speed of light to be 299,792,458 m/s. But the seconds are not the same, so the speeds aren't either. The Shapiro delay and the GPS clock adjustment are direct evidence of this, and Einstein spoke extensively about c varying with gravitational potential, but people still interpret c=1 as a physical constant.

[FrankM]

Frank, I saw you mentioned Kuhn and paradigm shift. Take a look at this article on the fine structure constant: http://physicsworld.com/cws/article/news/43657. It isn't constant, it's a "running" constant. It varies. And I'm afraid c varies too. ... but people still interpret c=1 as a physical constant.

 

Of course the velocity of light varies in the real physical world, it will vary depending upon the medium in which it is permitted to propagate. You now have a mathematically defined value by which all measured values can be compared. Isn't that convenient?

 

As for SI units and the way the second is defined, been there, understand the arguments. For a couple years the Introduction to my paper cited the things that SI states define the second, which you cite in your post. I know how the second is defined, and why they chose 9,192,631,770 cycles of the caesium 133 atom. Everything about defining the second is a circular argument to make it fit the astronomical Ephemeris second measurement. The actual measured value was 9,192,631,770 +/- 20 cycles of the caesium 133 atom. They took the average. Note the entry Frequency of Cesium in terms of Ephemeris Time on the leap second page. The Markowitz & Hall and Essen & Parry Physical Review article is no longer accessible from the page.

 

http://www.leapsecond.com/history/

 

This is one of the walls that the scientific establishment has made that is hindering the advance of the scientific enterprise, a man-made definition for a fundamental unit of measure, a duration of time, which is based upon a division of the rotation of a small planet. The methodology described in my paper provides a mathematically defined unit of time. I did not state in the paper that the mathematically defined unit of time is a cosmological time unit.

 

I had forwarded a draft of my paper to a retired Professor of Electrical Engineering, who then forwarded it to a mathematician friend. The mathematician suggested some improvements and thereafter submit it to a peer reviewed scientific publication. I completely rewrote the Introduction, eliminating everything that might be considered a criticism of the Consultative Committee on Units and the vast scientific authority structure associated with SI units, which includes NIST.

 

What is wrong with the mathematical process described in the paper? Are the dimensions that define the triangle pair not conforming to the rules of geometry? Is Euclidean geometry wrong? Is there something wrong with the relationship between wavelength and frequency? The only thing new are the cross products that define the constant of proportionality of a right triangle pair. Is the multiplication wrong? Does it come up with the wrong answer when the mathematically defined duration of time is lengthened to equal that of the duration of the second?

 

Besides identifying two fundamental units of measure, and providing a mathematically derived value for the velocity of light, it provides a fundamentally different way to apply mathematics to physical law. Doesn't that intrigue anyone? The EE professor and his mathematician friend thought so, as did those that were involved in the peer review for IEEE.

 

The physicworld URL does not return a valid page. Kuhn did not describe any paradigm shift scenario wherein a well established mathematical process, such as geometry, was combined with a long accepted basic physical law, such as the relationship between wavelength and frequency.

[Tormod]

The physicsworld link was wrong. Here is the correct one (without the stop):

 

Changes spotted in fundamental constant