Dimensionless Constants

LaurieAG · February 28, 2012

Dimensionless constants have some very interesting characteristics.

http://math.ucr.edu/home/baez/constants.html

Many constants involves units of length, time, mass, temperature, charge and so on. The numerical value of these constants depend on the units we use. The numbers would change if we used different units. Thus, though they certainly tell us something about nature, to some extent they are human artifacts.
On the other hand, certain constants don't depend on the units we use - these are called "dimensionless" constants. Some of them are numbers like pi, e, and the golden ratio - purely mathematical constants, which anyone with a computer can calculate to as many decimal places as they want. But others - at present - can only be determined by experiment. These tell us facts about nature that are completely independent of our choices of units.
...
Constants that aren't dimensionless can be regarded as relating one sort of unit to another. For example, the speed of light has units of length over time, so it can be used to turn units of time (like years) into units of length (like light-years), or vice versa.

In 1964 G. Schlesinger asked Is it false that Overnight everything has doubled in size. If the universe is detectably expanding then someone must have developed a true answer. Could a trio of dimensionless constants be used as an observational constant or immutable yardstick?

This 'thought experiment' shows one possibility. All of the constants on the 3 ratios below can be regarded as time or distance (based on the distance travelled by light in the time). Mass is not part of any ratio used in this 'thought experiment'.

If I started photographing a light around 6 and a bit feet away, and the light was being spun in a circle 2 feet in diameter and I captured the light from the spinning light source in one complete circle the ratio (A) of the time between the rotating source and the observer over the diameter of rotation would be roughly equal to Pi.

In this case the ratio ( B ) of the actual distance between source and observer over the distance travelled by light in a year would be very small and the ratio ( C ) of the observation period over the time it takes for the light to be rotated once will equal one. All observations should have a width of field that covers the complete diameter of rotation of the source being observed.

If I halve the exposure period I get half a circle and capture half as much light and when I double the exposure period I get 2 circles over each other and twice as much light in my photograph. If the light is rotated twice as fast I would expect something that looked similar to when I doubled the exposure period but I would also expect to capture the same amount of light as the original one rotation despite the doubling of the speed of rotation. If I put two lights together I could halve the exposure time and double the speed of rotation to capture a similar amount of light from the original 1 light doing 1 complete rotation. If the light moved at an angle to me I would observe an oval instead of a circle but the amount of light captured would remain the same as in a complete circle.

In this simplest base context A = Pi, B = tiny, C = 1 and the observer will capture one complete cycle. On any scale where C >= 1 the observer will capture at least one complete cycle despite the size of B.

On any scale where A = Pi * x, B >= 1 and C < 1 the observer will only capture the light from B * C = x of one rotation during any observation regardless of the speed of rotation of the same object.

On a galactic year scale where A = Pi * x, B = 230 million and C = 1/230 million you would expect to capture the light from B * C = x rotations or roughly one rotation regardless of the speed of rotation.

On a galactic year scale where A = Pi * x, B = 4.2 billion and C = 1/4.2 billion you would expect to capture the light from B * C = x rotations or roughly one rotation regardless of the speed of rotation.

Only changes in brightness can really make a difference on any scale as the speed of rotation does not change the total amount of light captured from the same source during any similar observation period. I have used figures for convenience, put your own figures in and keep ratio A as Pi * x and you will have a base point to compare observations.

Edited February 28, 2012 by LaurieAG

LaurieAG · February 29, 2012

In 1964 G. Schlesinger asked Is it false that Overnight everything has doubled in size. If the universe is detectably expanding then someone must have developed a true answer. Could a trio of dimensionless constants be used as an observational constant or immutable yardstick?

I will start a thread in Philosphy of Science later in the week to discuss the philosophical aspects of Is it false that Overnight everything has doubled in size.

This link points to a bit of light reading in google books ;) .

http://books.google.com.au/books?id=CPM8AAAAIAAJ&pg=PA134&lpg=PA134&dq=%22overnight+everything+has+doubled+in+size%22&source=bl&ots=N3l0xTnZ6n&sig=V3hX7NZIypp10eyfMwZ9nrUfLFQ&hl=en&sa=X&ei=1VdDT_zgB8WXiQf_ruXSBA&sqi=2&ved=0CEwQ6AEwBQ#v=onepage&q=%22overnight%20everything%20has%20doubled%20in%20size%22&f=false

Edited February 29, 2012 by LaurieAG

LaurieAG · February 29, 2012

Frank Jacksons book, Perception: a representative theory (link in previous post), brings the discussion into the physical sphere of SR.

p 162, The issue is whether we can specify logically possible observable occurrences relevant to the falsifiability of the ND (Night Doubling) Conjecture not the compatibility of such occurrences within the time invariance of the actual laws of nature.

Don Blazys · February 29, 2012

Quoting Laurie AG:

Dimensionless constants have some very interesting characteristics.

You ain't just "whistling Dixie"!

p 162, The issue is whether we can specify logically possible observable occurrences relevant to
the falsifiability of the ND (Night Doubling) Conjecture not the compatibility of such occurrences within
the time invariance of the actual laws of nature.

Personally, I think the "night doubling" question is fundamentally undecidable because

there can be nothing "outside" the universe which can be used as a "measuring stick".

These (dimensionless constants) tell us facts about nature that are completely independent of our choices of units.

I agree, and the two most important dimensionless constants are the fine structure constant [math]\alpha[/math]

and the proton to electron mass ratio [math]\mu[/math]. Now, by far the most accurate determination of

the fine structure constant ever made was by Gerald Gabrielse, and the following derivations

(from another thread) match his exactly, so I hope you don't mind if I use them here.

[math]\alpha_b^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{\frac{2}{2}})}[/math]

[math]\alpha_s^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(6*\pi^{5}*e^{2}-2*e^{\frac{4}{2}})}[/math]

[math]\alpha_L^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]\mu=1836.15267245(75)[/math] is the "proton to electron mass ratio", and

[math]A=2.566543832171388844467529...[/math], is that very special "Blazys Constant"

which generates all of the prime numbers, in sequential order, by the following simple method:

Note that the whole number part is the first prime [math]2[/math], and that:

[math]((2.566543832171388844467529...)/2-1)^{-1}[/math]

is approximately:[math](3.530176989721365539402422...)[/math],

where the whole number part is the second prime [math]3[/math], and that:

[math]((3.530176989721365539402422...)/3-1)^{-1}[/math]

is approximately [math](5.658487746849688216649061...)[/math],

where the whole number part is the third prime [math]5[/math], and so on.

(In short, we divide the approximate number by it's whole number part, subtract [math]1[/math],

and take the reciprocal of the result to get the next approximate number whose whole number part is the next prime!)

Now here's what makes all this this so darn wierd and why I'm certain that

the fine structure constant may have even more information "encoded" into it.

As we all know, the energy scales for atoms are set by the electron rest energy [math]mc^{2}[/math] times powers of [math]\alpha[/math].

Thus, for hydrogen atoms,

the binding energy scale is: [math]mc^{2}*\alpha^{2}[/math],

the fine structure splitting scale is:[math]mc^{2}*\alpha^{4}[/math],

and the Lamb shift scale is:[math]mc^{2}*\alpha^{5}[/math],

But note that the exponents [math]2[/math], [math]4[/math] and [math]5[/math] are also the very last exponents in the derivations of [math]\alpha_b^{-1}[/math],[math]\alpha_s^{-1}[/math] and [math]\alpha_L^{-1}[/math].

And if that isn't wierd enough, [math]\alpha_h^{-1}[/math],[math]\alpha_m^{-1}[/math] and [math]\alpha_L^{-1}[/math] can also be viewed as:

[math]\alpha_b^{-1}=137.035999135=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{((M_n/M_e)*e^{2}-2*e^{\frac{5}{2}})}[/math]

[math]\alpha_s^{-1}=137.035999084=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\frac{1}{2}(M_n/M_e+\mu)*e^{2}-2*e^{\frac{5}{2}})}[/math]

[math]\alpha_L^{-1}=137.035999033=(A^{-1}*\pi*e+e)*(\pi^{e}+e^{(\frac{-\pi}{2})})-\frac{1}{(\mu*e^{2}-2*e^{\frac{5}{2}})}[/math]

where [math]M_n/M_e=1838.6836605(11)[/math] is the neutron-electron mass ratio... yet another dimensionless constant!

So you see, the fine structure constant contains information about prime numbers,

polygonal numbers, other dimensionless constants, energy scales and who knows what else.

Don.

Edited April 5, 2012 by Don Blazys

LaurieAG · March 1, 2012

hi Don,

Personally, I think the "night doubling" question is fundamentally undecidable because there can be nothing "outside" the universe which can be used as a "measuring stick".

I agree, while it is possible to perceive and hypothesize from external viewpoints the real problem is with regards to interpreting observations made from inside the universe in that external context.

I agree, and the two most important dimensionless constants are the fine structure constant and the proton to electron mass ratio .

The fine structure constant contains an electric and magnetic constant component with an embedded unit of 4π.

http://en.wikipedia.org/wiki/Electric_constant

The wiki above diverts to the wiki below.

http://en.wikipedia.org/wiki/Magnetic_constant

The ampere defines vacuum permeability

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2×10−7 newton per meter of length.

Adopted in 1948, the effect of this definition is to fix the magnetic constant (permeability of vacuum) at exactly 4π×10−7 H·m−1.[5] To further illustrate:

Two thin, straight, stationary, parallel wires, a distance r apart in free space, each carrying a current I, will exert a force on each other. Ampère's force law states that the force per unit length is given by[6]

The ampere is defined so that if the wires are 1 m apart and the current in each wire is 1 A, the force between the two wires is 2×10−7 N·m−1. Hence the value of μ0 is defined to be exactly
μ0 = 4π×10−7 N·A−2 ≈ 1.2566370614...×10−6 N·A−2.[7][8]

Don, it would be interesting to see what the equations look like if all of the primary dimensionless constants (4π, c etc) were separated from all of the secondary dimensionless constants and mass was segregated in the latter. If photons had any mass they could not travel at c so the real constant part of the fine structure constant must be the tiny amount of mass the photon must have so that π times another constant does not equal zero. 4π is roughly 12.56 and 13.7/4.3 is roughly π.

Edited March 1, 2012 by LaurieAG

Don Blazys · March 1, 2012

Hi Laurie,

The fine structure constant contains an electric and magnetic constant component
with an embedded unit of 4π.

Interestingly, there are now several theories floating about that

the fine structure constant at infinite energy is [math]\frac{1}{4*\pi}[/math].

Here's one of them:

http://arxiv.org/pdf/hep-th/9904158.pdf

Don, it would be interesting to see what the equations look like if all of
the primary dimensionless constants (4π, c etc) were separated from all of
the secondary dimensionless constants and mass was segregated in the latter.

You sort of lost me here...

"c" (as in "the speed of light") is not a dimensionless constant.

Don.

Edited March 1, 2012 by Don Blazys

sigurdV · March 1, 2012

Interesting quality thread...

LaurieAG · March 2, 2012

Hi Don,

"c" (as in "the speed of light") is not a dimensionless constant.

You are right as c is only used to turn time into distance and vs a vs so it should be on the other side.

You sort of lost me here...

Here's a simple example. In my original post the first ratio was A = π * x and the amount of light captured in the observation was B * C = x.

A = Distance / Diameter = π * x

B = Distance / c

C = Observation Period / Distance

Distance = (π * x) / Diameter

Distance = B * c

Distance = 1 / (C * Observation Period)

( a ) (π * x) / Diameter = 1 /(C * Observation Period)

π * x = Diameter / (C * Observation Period)

π = Diameter / (C * Observation Period * x)

π = Diameter / ((Observation Period / Distance) * (Observation Period * x))

π = Diameter / (x * (Observation Period ^ 2 / Distance))

Corrected

( b ) (π * x) / Diameter = Distance

π * x = Distance / Diameter

π = Distance / (x * Diameter)

Corrected

( c ) Diameter / (Observation Period ^ 2 / Distance) = 1 / Diameter

Diameter ^ 2 = 1 / (Observation Period ^ 2 / Distance)

Diameter ^ 2 = Distance / Observation Period ^ 2

Distance = (Diameter / Observation Period) ^ 2

Edited March 3, 2012 by LaurieAG

LaurieAG · March 3, 2012

Hi Don, please note the corrections in the previous post,

So you see, the fine structure constant contains information about prime numbers, polygonal numbers, other dimensionless constants, energy scales and who knows what else.

The fine structure constant wiki talks about observations of pulsars that give a varying measured constant and even the hubble constant is not exactly constant. msowww.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf.

section 4.3 of www.mso.anu.edu.au/~charley/papers/DavisLineweaver04.pdf is also interesting.

Sign In

Dimensionless Constants

Recommended Posts

LaurieAG

LaurieAG

LaurieAG

Don Blazys

LaurieAG

Don Blazys

sigurdV

LaurieAG

LaurieAG

Don Blazys

Join the conversation

Browse

Activity