Pi (π) in physics

[Rade]

Is there any important predictive information from the observation that Pi (π) is a multiple of 8 in the equations for the:
Cosmological constant
Einstein field theory of general relativity

but Pi (π) is a multiple of 4 in the equations for the:
Heisenberg uncertainty principle
Coulomb's law for the electric force
Magnetic permeability of free space
:confused:

Also, why Pi (π) in all these fundamental equations ?

[chroot]

Uh... what do you mean by Pi(n), and what makes you think there are "equations" for things like the cosmological constant? What are you talking about?

- Warren

[billiards]

I think you're confused with your use of language. Pi is not a multiple of 8, I guess you mean something is multiplied by 8*pi?

Pi is a geometrical number, it is the ratio of a circle's circumference/diameter. In physics pi is normally used when considering shape.

[Rade]

Sorry, massive confusion with the question. See the "physics equations" for Pi at the link below and, as billiards says, note that two equations have 8*Pi and three have 4*Pi--so the OP flows from this observation--please do not read more into the OP question than asked:
http://en.wikipedia.org/wiki/Pi#Physics

[disregardthat]

Well, I guess the reason pi is included in many equations is because the equation have something to do with a circle in some way.

[robphy]

consider the dimensionality of the "space" associated with your quantity, then derive from first principles [and assuming the appropriate "spherical" symmetry] the quantities you seek, following [without arithmetic simplification] the naturally occurring multiple of \pi.

[Rade]

consider the dimensionality of the "space" associated with your quantity, then derive from first principles [and assuming the appropriate "spherical" symmetry] the quantities you seek, following [without arithmetic simplification] the naturally occurring multiple of \pi.Thank you for your time--are you saying all five equations in the OP question "assume" spherical symmetry in the domains in which they apply ? ---that is, the HUP equation only applies to a "space" with spherical symmetry ??

[Chris Hillman]

Hi, Rade,

Your question as stated confused everyone. I think you mean to ask this:


Is there any important predictive information from the observation that π is [multiplied by] 8 in [the Einstein field equaton] of general relativity, but π is [multiplied by] 4 in the equations for Coulomb's law for the electric force


The answer is related to the fact that, roughly speaking, Coulomb's law is part of Maxwell's theory of EM, which is a vector theory ("photons are spin one", in the lingo), while gtr is a (second rank) tensor theory ("gravitons are spin two", in the lingo). If you have heard of GEM formalism (gravitoelectromagnetism), this establishes a detailed mathematical analogy between weak-field gtr and Maxwell's theory, this "dictionary" also involves the odd unexpected factor of two, for essentially the same reason.

(Actually, what I just said is potentially misleading, but it would probably only confuse you if I tried to explain why, so never mind.)

To answer your question, the differing spin is terribly important; it means for example that the effect of gravitational radiation on a test particle (charged or otherwise) is completely different from the effect of EM radiation on a charged test particle.

[Rade]

...The answer is related to the fact that, roughly speaking, Coulomb's law is part of Maxwell's theory of EM, which is a vector theory ("photons are spin one", in the lingo), while gtr is a (second rank) tensor theory ("gravitons are spin two", in the lingo). If you have heard of GEM formalism (gravitoelectromagnetism), this establishes a detailed mathematical analogy between weak-field gtr and Maxwell's theory, this "dictionary" also involves the odd unexpected factor of two, for essentially the same reason.(Actually, what I just said is potentially misleading, but it would probably only confuse you if I tried to explain why, so never mind.) To answer your question, the differing spin is terribly important; it means for example that the effect of gravitational radiation on a test particle (charged or otherwise) is completely different from the effect of EM radiation on a charged test particle.Thanks--yes, your modification of the OP question does address two of the five equations--but I do not follow the physics of your answer, and you do not develop the argument mathematically to show how 8*pi vs 4*pi are derived--so, unless someone disagrees with your explanation, I will accept what you say and it would appear that the two of the five OP equations have been accounted for.

[Rade]

Well, I guess the reason pi is included in many equations is because the equation have something to do with a circle in some way.Clearly, in "many" equations--but I refer to five fundamental equations of physics and ask why would Einstein equation and HUP (and the three other equations in OP#1) limit themselves to prediction within the domain of "a circle"--:confused--this cannot be the entire story--can it ?

[Dbjergaard]

At the risk of sounding foolish, I would like to ask why it matters that you have to multiply pi by 4 or 8 in order to calculate a quantity. Could you not also ask why pi happens to be 3.1415926... why isn't pi 4.1415926? or 3.1514926? I understand that there could be some significance to 4 or 8, but if those numbers didn't come from anywhere (ie those were the numbers that were needed for the equation to agree with observation) then it makes no sense to ask why you need to use those particular numbers.
I'm probably wrong, but please don't post back with "your wrong you fool, I don't have time to explain why because your too stoopid to figure it out, and clearly I would confuse you!"

[gone gator]

Thank you for your time--are you saying all five equations in the OP question "assume" spherical symmetry in the domains in which they apply ? ---that is, the HUP equation only applies to a "space" with spherical symmetry ??

no, i expect robphy was noting that the geometry in 3 dimensions is different that that in two or four or "n". when computing the volume/area/&c of an n-dimensional sphere you'll get some integer multipliers (ratios) along with pi.

At the risk of sounding foolish, I would like to ask why it matters that you have to multiply pi by 4 or 8 in order to calculate a quantity. Could you not also ask why pi happens to be 3.1415926... why isn't pi 4.1415926? or 3.1514926? I understand that there could be some significance to 4 or 8, but if those numbers didn't come from anywhere (ie those were the numbers that were needed for the equation to agree with observation) then it makes no sense to ask why you need to use those particular numbers.
I'm probably wrong, but please don't post back with "your wrong you fool, I don't have time to explain why because your too stoopid to figure it out, and clearly I would confuse you!"

not a foolish question at all. in fact these numbers are fixed, for example, by the dimension of the space and the inverse square nature of the "laws" of physics. William Paley (the guy with the watchmaker arguments against evolution) used the fact that if it wasn't a "2" in the inverse square law we wouldn't be here. it was one of his more robust arguments, a bit harder to discard than the evolution of the eye... but in short, yours was not a silly question at all: it does appear that it is important many of these numbers "are what they are" beyond mere geometry.

[billiards]

At the risk of sounding foolish, I would like to ask why it matters that you have to multiply pi by 4 or 8 in order to calculate a quantity. Could you not also ask why pi happens to be 3.1415926... why isn't pi 4.1415926? or 3.1514926? I understand that there could be some significance to 4 or 8, but if those numbers didn't come from anywhere (ie those were the numbers that were needed for the equation to agree with observation) then it makes no sense to ask why you need to use those particular numbers.
I'm probably wrong, but please don't post back with "your wrong you fool, I don't have time to explain why because your too stoopid to figure it out, and clearly I would confuse you!"

I'm no mathematician but the way I think about it is pi happens to be *fishes out calculator* 3.141592654.... because that's the symbols we have to write that number. It would be very hard to write pi out with roman numerals but if you could it would still be the same thing essentially i.e. the ratio of a circles circumference/diameter.

No matter how big the circle is this number is always the same, it is a constant of proportionality if you like and it has been discovered around the world by many of the great mathematical civilisations independantly of each other.

[gone gator]

No matter how big the circle is this number is always the same, it is a constant of proportionality if you like and it has been discovered around the world by many of the great mathematical civilisations independantly of each other.

and not only just civilisations on earth... arguably pi would be the same in all universes which had maths.

but do you think that make pi more or less fundamental than, say, planck's constant? or the cosmological constant?

[billiards]

I dunno? I just discovered that the permability of free space=4*pi*10-7.

I guess this is what the OP was getting at, why does this number have pi in it? I have to admit that I have no idea...... any physicists out there?

[Claude Bile]

I dunno? I just discovered that the permability of free space=4*pi*10-7.

I guess this is what the OP was getting at, why does this number have pi in it? I have to admit that I have no idea...... any physicists out there?

Because we observe the magnetic field, one metre away from a wire carrying 1 Amp of current to be 2 x 10^-7 Tesla. This number itself is not special, it is simply the number that comes out when we define units like the Ampere and Tesla. We can use different, equally valid units to get a different number. (however the value that this number represents will always be the same.)

The permeability of free space takes the value that it does for the same reason, because of the system of units we have adopted.

For example I could define a new unit of magnetic field called an Iwnh, where 1 Iwnh = pi Tesla. In this system of units the magnetic permeability of free space would be 4 x 10^-7 with units of Iwnh.metres/Ampere (instead of the SI units of Tesla.metre/Ampere). Note too that I could have used a different unit of distance or current, rather than magnetic field to acheive exactly the same result.

In short the fundamentality (or fundamentalness? :rolleyes: ) of pi appearing in certain equations is no more fundamental than our system of units we have adopted.

Claude.

[Dbjergaard]

My point was "Why should we waste our time asking why a number is the way it is, when it is there because it is what is required to agree with what we observe in the universe?" The day after I posted this question I got in an argument with my chemistry teacher about why some constant was what it was (I HATE memorizing stuff, and chemistry is a lot of that) half way through the argument I realized that this motivation to argue about why the constant was what it was is the same thing that drives us to ask this question in science. As I was arguing I used the analogy of gravity.
We have known for a long time that the acceleration due to gravity is 9.81ms^-2 Why is it THIS value? We could measure it, and do calculations from it, but we did not know WHY it is this value. Then Einstein propounded his theory of general relativity and now we realize that the inverse square law, and 9.81ms^-2 all come about because of the curvature of space around the massive body earth. I may have missed a concept, but the demonstration of the drive to find the underlying truth behind phenomena can still be seen in this example. On the other hand, if you were taking a physical constant (ie the coupling constant 1/137....) and constructing it with pi and e and ones multiplied by threes added together and then wondering why it took 4 pi and 2 e to get the number, THEN your a fool because YOU put it there. There was no underlying physical observation that allowed you to put 4pi in there, you just did such that you got 1/137... out.

[rbj]

Well, I guess the reason pi is included in many equations is because the equation have something to do with a circle in some way.

or, more precisely, a sphere. because it is natural, in inverse-square laws, to imagine the inverse-square quantity as some sort of fixed flux distributed over a surface area or a sphere (so that it is all equidistant and perpedicular to the direction of flux) that is 4 \pi r^2 , then any inverse-square law that is expressing in terms of flux, will have a 4 \pi in the denominator. but then when you express the same inverse-square law with a big constant of proportionality that absorbs this 4 \pi factor, when you do anything like Gauss's law on it, that 4 \pi will come right back out. the extra factor of 2, when you see the Einstein equation is due to "the square-root appearing in the proper time formula":

dT = \sqrt{g_{\mu,\nu} dx^{\mu} dx^{\nu}}

(from http://groups.google.com/group/sci.physics.research/msg/a0ec7e6c84fc8499 )

at least that's how it is told to me.