• Quick note - the problem with Youtube videos not embedding on the forum appears to have been fixed, thanks to ZiprHead. If you do still see problems let me know.

Are physical constants irrational?

Magic Pancake

New Blood
M
Joined
Mar 14, 2009
Messages
6
Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers? I sure can't see a ready fraction in any of them, but then again these numbers are determined experimentally, so how can you tell for sure.

How about the universe itself? Is there a countable number of states for it?As a layman I understand that units of dimensions in the universe are quantified in various Plank units, so that would seem to say that these must be countable, but is this the case?

(Sorry if I mutilated some concepts.)
 
Eddington thought the fine structure constant was 1/136. Later, he changed his mind to 1/137. Both are wrong.

The short answer is that we don't know, and probably never will. Any number with a non-zero error bound can be approximated with a rational number, and most likely we'll never figure out what the physical constants are from first principles--only by imprecise measurements. They certainly "look" irrational, but that's infinitely far away from knowing that they're irrational.
 
Magic Pancake said:
Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers? I sure can't see a ready fraction in any of them, but then again these numbers are determined experimentally, so how can you tell for sure.
Click to expand...

Almost certainly. The thing is, their specific values are expressed in terms of units which are historical accidents, and (at least in terms of details) essentially random. Consider Planck's constant. Here is a quantity which, however fundamental, is expressed in units of J-s, or alternatively, kg-m^2/sec. Look up the historical foundations of each of the 3 units. Meter: based on the then-current understanding of the size of the earth. Kilogram: originally based on the meter (1 gram = 1 cc of water at triple point). Second: 1 sidereal day divided by (24 x 3600).

With units this arbitrary (from a universal perspective), why would you expect the numeric values of any fundamental constant to be other than essentially arbitrary? And since there are infinitely more irrational numbers than there are rational ones, well, you do the math.

The fine constant, of course, is the unitless joker, and it's anybody's guess how that one will turn out.
 
WhatRoughBeast said:
Almost certainly. The thing is, their specific values are expressed in terms of units which are historical accidents, and (at least in terms of details) essentially random. Consider Planck's constant. Here is a quantity which, however fundamental, is expressed in units of J-s, or alternatively, kg-m^2/sec. Look up the historical foundations of each of the 3 units. Meter: based on the then-current understanding of the size of the earth. Kilogram: originally based on the meter (1 gram = 1 cc of water at triple point). Second: 1 sidereal day divided by (24 x 3600).

With units this arbitrary (from a universal perspective), why would you expect the numeric values of any fundamental constant to be other than essentially arbitrary? And since there are infinitely more irrational numbers than there are rational ones, well, you do the math.

The fine constant, of course, is the unitless joker, and it's anybody's guess how that one will turn out.
Click to expand...

If the physical constants are rational, wouldn't it imply that the SI units, no matter how arbitrarily chosen, are multiples of them?
 
Why would one expect the fundamental constants to be rational any more than one would expect the ratio of the circumference to the diameter of a circle be rational?
 
Plank units turn a bunch of physical constants into rational numbers (1s).
 
Perpetual Student said:
Why would one expect the fundamental constants to be rational any more than one would expect the ratio of the circumference to the diameter of a circle be rational?
Click to expand...

On the other hand, why not? Is there some imbalance in the number of rational vs irrational numbers in the universe? What difference does it make to, say, the Plank constant that we happen to know pi is irrational?
 
shadron said:
On the other hand, why not? Is there some imbalance in the number of rational vs irrational numbers in the universe? What difference does it make to, say, the Plank constant that we happen to know pi is irrational?
Click to expand...

well, rational numebrs are countably infinite, irrational numbers are uncountably infinite, so from that alone, we shouldn't expect that there should be a bias towards rational numbers.
 
supaluminal said:
well, rational numebrs are countably infinite, irrational numbers are uncountably infinite, so from that alone, we shouldn't expect that there should be a bias towards rational numbers.
Click to expand...

If you're going to test that theory, don't forget to account for the aleph-null hypothesis! :D
 
Even the irrationals can be separated into those which have a formula and those which don't.

All the integers and rational numbers, and all the irrationals with a formula (which are enumerable) are an infinitely tiny set compared to all the irrationals without any possible formula.
 
Magic Pancake said:
Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers?
Click to expand...
Neither. See for example the NIST page on the fine structure constant. See the bit that says this:

Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant. Indeed, due to e+e- and other vacuum polarization processes, at an energy corresponding to the mass of the W boson (approximately 81 GeV, equivalent to a distance of approximately 2 x 10^-18 m), α(mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental.

It's a "running" constant, which means it varies. So it's neither rational nor irrational. It's often given as α = e²/2ε0hc, where e is the charge of the electron, ε0 is the permittivity of space, h is Planck's constant, and c is the speed of light. The e is described on the NIST website and elsewhere as "effective charge", but IMHO one should bear in mind conservation of charge. The effect of the electron's unit charge e might vary, but charge is conserved. So if α varies and e doesn't, that means ε0 and/or h and/or c vary too. You tend not to hear much about this sort of thing, but see this IOP physicsworld article Can GPS find variations in Planck's constant? Sadly what you do get to hear about is the "fine tuned" constants and the Goldilocks anthropic multiverse, which is woo.
 
Farsight said:
Neither. See for example the NIST page on the fine structure constant. See the bit that says this:

Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant. Indeed, due to e+e- and other vacuum polarization processes, at an energy corresponding to the mass of the W boson (approximately 81 GeV, equivalent to a distance of approximately 2 x 10^-18 m), α(mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental.

It's a "running" constant, which means it varies. So it's neither rational nor irrational. It's often given as α = e²/2ε0hc, where e is the charge of the electron, ε0 is the permittivity of space, h is Planck's constant, and c is the speed of light. The e is described on the NIST website and elsewhere as "effective charge", but IMHO one should bear in mind conservation of charge. The effect of the electron's unit charge e might vary, but charge is conserved. So if α varies and e doesn't, that means ε0 and/or h and/or c vary too. You tend not to hear much about this sort of thing, but see this IOP physicsworld article Can GPS find variations in Planck's constant?
Click to expand...
Doesn't mean it can't be rational or irrational, just because it isn't actually a constant.

Sadly what you do get to hear about is the "fine tuned" constants and the Goldilocks anthropic multiverse, which is woo.
Click to expand...
No it isn't. Its speculative.
 
Tubbythin said:
Doesn't mean it can't be rational or irrational, just because it isn't actually a constant.
Click to expand...
Not sure I follow you there Tubbythin, but it certainly doesn't change that the zero energy value is well defined as a single number and it seems to me that it's a valid question to ask about that number.
 
edd said:
Not sure I follow you there Tubbythin, but it certainly doesn't change that the zero energy value is well defined as a single number and it seems to me that it's a valid question to ask about that number.
Click to expand...

I'm not sure I follow you, in turn.
I was saying that just because the fine structure constant may not be a constant, doesn't meant that we it is necessarily impossible to say whether or not it is (ir)rational (in principle).
 
Does it really matter if they're rational numbers in base 10 SI units?
 
You must log in or register to reply here.

Back
Top Bottom