The principle of least action/time, and geodesics of spacetime

[Alexander]

Originally posted by Hurkyl
If you want. One would probably call it the friction (force) field if you did.

How can you cook friction field? Say, here is a book sitting on a table at place (x,y). What is then the direction of "friction field" at this (x,y) point?

 

[pmb]

Consequence # 2: all fundamental constants are velocity-invariant (i.e., moving observer sees same G, h, c, e as non moving one).

That c is invariant does not follow since it's not *defined* as a fundamental constant. c is defined as the speed of the propagation of an EM wave. And if Maxwell's equations prove to be invalid and its the Proca Lagrangian which becomes the Lagrangian used to derive the EM equations then "c" will not be invariant.

Pete

 

[Alexander]

c is a fundamental constant along with h and G.

e is likely not (it seems to be a derivative from h and c).

 

[Hurkyl]

Doggone! Schwarzschild has found a magnetic monopole but he won't show it to us! Where do you keep yours? I keep mine in my sock drawer.

It's only a magnetic monopole if the magnetic field is not divergence free. Maxwell's equations certainly permit electromagnetic fields where the electric field is everywhere zero, but there is a static nonzero magnetic field. One such solution is:

E(x, y, z) = (0, 0, 0)
B(x, y, z) = (-y, x, 0) / (x² + y²)

Now, take Coulomb law and apply Lorents transformations to it - you'll get Maxwell equations as a mathematical consequence of transform electric field from stationary into moving reference system.

Care to try it? Coulomb's law is a law of electrostatics, not electrodynamics, the Maxwell equations cannot fall out of this manipulation.

Again you got it backward. Magnetic force is a consequence of magnetic field. Magnetic force may be absent even if magnetic field is there (say, probe charge is not moving). So, magnetic field is more fundamental than magnetic force.

Whether or not any of that is true, that does not change how magnetic force and magnetic field are DEFINED; magnetic force is the force that acts on currents and magnetic field is a force field associated with magnetic force.

How can you cook friction field? Say, here is a book sitting on a table at place (x,y). What is then the direction of "friction field" at this (x,y) point?

You're the one who suggested we lift a force field from frictional force, not me. I don't see why a force field has to be a vector field as opposed to a scalar field anyways; for example the "coefficient of kinetic friction field" &mu is a scalar field for a surface, and the kinetic friction force law is F = &mu N. Even better, we could define &mu to be &mu times the normal vector to the surface, yielding a more general kinetic frictional force law F = -&mu.G where G is the net force acting on the object. (this is well-defined because frictional force will always be orthogonal to &mu)

 

[marcus]

I don't want to get into the discussion and am not finding fault either---everybody sounds right so I suspect there must be some semantic trouble explaining the possible disagreements. But I do have a kind of footnote (which may be old news to everybody).

I just looked in the ancient and highly-revered sophomore physics text by Edward Purcell called Electricity and Magnetism---the Berkeley Phyics Series volume 2 (1965)
It does static charge. Then, before any discussion of magnetism, it introduces just enough special relativity to derive the electric field of a moving charge. Then it derives "magnetic" effects as a relativistic corrections to electric fields.

Eg on page 185 "...We have derived the magnetic field of a straight current by analyzing only the *electric* field of moving charges..." (Purcell's emphasis)

Then after some more discussion he says on page 186 "...We can now see plainly that the magnetic interaction between moving charges is a relativistic effect...In a world of moving electric charges, magnetism would vanish if c were infinite..."

Someone who learned EM from Purcell might be tempted to consider the electric field "realer" than the magnetic field, tho both are presumably convenient idealizations so the point may be moot. It may even involve a question of taste about which people may honorably hold different opinions.

 

[Alexander]

Purcell is indeed correct. Electric charge is "something" which produces what we label as "electric field". Then, moving observer sees this field as having "magnetcic" term [vE]/c² just due to his motion (hense v) and due to Lorents transforms (from which c comes).

Return to absolute space (=make c--->oo) and magnetic field vanishes.

 

[Hurkyl]

Eg on page 185 "...We have derived the magnetic field of a straight current by analyzing only the *electric* field of moving charges..." (Purcell's emphasis)

(boldface mine)

Even here, Purcell does not define the magnetic field as a the relativistic portion of an electric field; he derives it. The meaning of "magnetic field" exists independantly of electricy and relativity.

Alexander is making a common engineer's mistake; he has confused "definition" with "law" or "theorem" or something similar. This is a very insidious mistake because one you internalize it, it is extremely difficult to comprehend anything where the incorrect internalized definition may actually be false.

In this case, he has internalized some facts about electrostatics and magnetostatics, and does not realize how specialized those hypotheses are.


I don't entirely doubt the possibility that the magnetic field may be redundant information; the Maxwell equations allow one to solve for the magnetic field at all times given the electric field at all times and the boundary condition of knowing the entire magnetic field at one particular time... I don't doubt it may be possible to develop an intelligent way to select the boundary condition to yield a unique solution for the magnetic field. The anomalous magnetic moment of various particles could possibly be explained away with current loops in LQG or ST as opposed to being an inherent property of a particle as demanded by the standard model.


But Alexander is blinded by very special cases of electromagnetism to allow him to recognize this. Coulomb's law only applies to electrostatic fields. The Biot-Savart law only applies to magnetostatic fields. The only situation where a static electromagnetic field remains static after applying a Lorentz boost is in the special case where all currents involved are parallel to the boost. (such as in Purcell's example with a straight current). Alexanders claims fall apart for dynamic electromagnetic fields.

In particular, a magnetodynamic field is divided into two parts, an induction field and a radiation field, while an electrodynamic field is divided into a retarded Coulomb field, two intermediate fields, and a radiation field.

Reference: Electromagnetic Field Theory by Bo Thide, sections 7.1 and 7.2
http://www.plasma.uu.se/CED/Book/



In particular, the magnetic field at a point x in space is (where &rho, j, and a are evaluated at a retarded time allowing for light speed propagation from y to x, and letting r = x - y, and integration is over all of space)

B = (&mu₀ / 4 &pi) &int j * r / r³ d³y + (&mu₀ / 4 &pi c) &int j' * r / r² d³y

Whereas the electric field is:

E = (1 / 4 &pi &epsilon₀) &int &rho r / r³ d³y + (1 / 4 &pi &epsilon₀ c) &int (j.r) r / r⁴ d³y
+ (1 / 4 &pi &epsilon₀ c) &int (j * r) * r / r⁴ d³y + (1 / 4 &pi &epsilon₀ c²) &int (j' * r) * r / r³ d³y


As you can see, the "law" B = v*E/c² for the fields generated by volume elements only holds for the first term in each field (if we define v as j/&rho); the retarded coulomb field and the induction field. This law does not hold for the entire field! One piece of interest is that of the radiation field (the final terms). The radiative electric field of a volume element is best expressed in terms of the radiative magnetic field:

E = c B * r / r

This is far from the results predicted by extrapolating from the special case of boosting along a straight current.

 

[Alexander]

Originally posted by Hurkyl


Even here, Purcell does not define the magnetic field as a the relativistic portion of an electric field; he derives it. The meaning of "magnetic field" exists independantly of electricy and relativity.

Nope. Remove either electricity or relativity and magnetic field vanishes.

When Purcell (following Einstein and Feynmann) applies Lorents transformation to electric charge he gets the term [vE]/c² as a mathematical product. Note that magnetic field does not exist anywhere yet. Then he says: this term IS what we usually label as "magnetic field", so let's follow Einstein, Feynmann, Landay, etc and label this term by letter "B".

Labeling (=definitions) is just a nick-naming mathematical objects consisting of many symbols by just one - to shorten mathematical paperwork. Thus instead of more bulky relativistic equation(s) we end up with more elegant one(s) which we call "Maxwell equations" when we replace bulky term [vE]/c² by just one symbol "B".

Voila - magnetic field is "created".

So clearly cross product of electric field with velocity of observer is what we define as magnetic field (or magnetic component of e/m field).

Absense of "magnetic" charges follows as a mathematical by-product of definition of magnetic field.

 

[Hurkyl]

Then he says: this term IS what we usually label as "magnetic field", so let's follow Einstein, Feynmann, Landay, etc and label this term by letter "B".

Yes, that's exactly my point. He says:

this term IS what we usually label as "magnetic field"

He does not say:

We shall define "magnetic field" to be this term.


IOW, he says that this term satisfies the definition of a magnetic field, so we shall denote it with B.



And I repeat: B = vE/c^2 only works for static electromagnetic fields. It fails for dynamic fields.

 

[Alexander]

Originally posted by Hurkyl


And I repeat: B = vE/c^2 only works for static electromagnetic fields. It fails for dynamic fields.

How so? If an electric field E is moving with a speed c, then we have: B=[cxE]/c². This vector product, as you can easily see, is equal to E/c by magnitude and orthogonal to both c and E vectoirs.

That is INDEED the correct relationship between electric field and "magnetic" component (in SI system) in a moving with speed c electric field (=photon, e/m wave).

So the definition of magnetic field holds correct for moving with c electric field too.

 

[Hurkyl]

How so? If an electric field E is moving with a speed c, then we have: B=[cxE]/c2. This vector product, as you can easily see, is equal to E/c by magnitude and orthogonal to both c and E vectoirs.

(a) You changed your formula. In your previous usage, you said

B = v*E/c^2

where v was the relative velocity between the reference frame of the charge and the reference frame of the observer.

Now you're saying v is the velocity of an electromagnetic wave (which is circular, but not the main criticism).

Which is it? Or do you have a better definition of v that covers both cases?


(b) This new formula only holds for waves where the magnetic field is exactly perpindicular to the electric field and has an amplitude of a fixed constant times that of the electric field. Consider the counterexample:

E(x, y, z, t) = -C₁ sin x cos t j
B(x, y, z, t) = C₂ cos x sin t k


(c) Coulomb's Law itself fails in dynamical fields.

It is impossible to derive electromagnetic radiation from Coulomb's Law + Relativity; Coulomb's law requires that the electric field point (more or less) at an oscillating charge, meaning it cannot point roughly perpendicular to the velocity vector of the resulting wave.

 

[pmb]

Alexander wrote

you can't refer to Maxwell equation(s) trying to prove constancy of speed of light, because Maxwell equations are CONSEQUENCE of relativity, because they are DERIVED from relativity (namely, from existence of electric charge, Lorents transformations of coordinates (which gives rize to magnetic component), and 3-dimensionality of space).

From "Elementary Derivation of the Equivalence of Mass and Energy," A. Einstein, Bulletin of the American Mathematical Society 41, 223-230 (1935)

The very first two sentances of this paper reads as follows

The special theory of relativity grew out of the Maxwell electromagnetic equations. So it came about that even in the derivation of the mechanical concepts and their relations the consideration of those of the electromagnetic field has played an essential role.

Pete

 

[Alexander]

Originally posted by Hurkyl
(a) You changed your formula. In your previous usage, you said

B = v*E/c^2

where v was the relative velocity between the reference frame of the charge and the reference frame of the observer.

Now you're saying v is the velocity of an electromagnetic wave (which is circular, but not the main criticism).

Which is it? Or do you have a better definition of v that covers both cases?

The formula is correct. You use B=[vE]/c² if by E you mean the electric field measured in the system of observer and B=[vE']gamma/c² if by E' you mean the electric field measured in the system of electric charge. Gamma = (1-(v/c)²)^-1/2

 

[Alexander]

Originally posted by pmb
From "Elementary Derivation of the Equivalence of Mass and Energy," A. Einstein, Bulletin of the American Mathematical Society 41, 223-230 (1935)

The very first two sentances of this paper reads as follows:

The special theory of relativity grew out of the Maxwell electromagnetic equations. So it came about that even in the derivation of the mechanical concepts and their relations the consideration of those of the electromagnetic field has played an essential role.


Pete

Historically yes, mathematically no. Nature does not care much what we discovered first - skeleton of dino or skeleton of humanoid.

 

[pmb]

Originally posted by Alexander
Historically yes, mathematically no. Nature does not care much what we discovered first - skeleton of dino or skeleton of humanoid.

More than historically. In any case = Maxwell's equations are not what you seem to think they are. E.g. this comment

Now, take Coulomb law and apply Lorents transformations to it - you'll get Maxwell equations as a mathematical consequence of transform electric field from stationary into moving reference system. Instead of B in your equations will be term [vE]/c2.

indicates that you seem to think that the equations which relate the E and B field in one frame to an E and B field in another frame are Maxwell's equations. They are not. The equation you called "Coulomb's Law" is one of Maxwell's equations. Coulombs law cannot be derived from the principle of relativity nor can it be derived from the principle of the constancy of light.

That relation between the magnetic field and the electric field field follows from relativity.

For a list of Maxwell's equations see
www.geocities.com/physics_world/maxwell.htm

Pete

 

[Alexander]

Again incorrect. Coulomb law is more fundamental than Maxwell equations. It follows from Heizenberg uncertainty principle, translational symmetry of space and 3-dimensionality of space.

 

[pmb]

Originally posted by Alexander
Again incorrect. Coulomb law is more fundamental than Maxwell equations. It follows from Heizenberg uncertainty principle, translational symmetry of space and 3-dimensionality of space.

You missed my point. Coulomb's law **IS** one of Maxwell's equations!

Please provide a proof, or a referance to a proof, of this claim -- "It follows from Heizenberg uncertainty principle, translational symmetry of space and 3-dimensionality of space."

Pmb

 

[Alexander]

Coulomb law is NOT one of Maxwell equations. Gauss law is, which is derived from Coulomb law and from definition of electric field.

Inverse square law for some of natural forces (like gravity and electric force) follows mathematically from interaction of massles virtual particles in 3-d space, providing that space is translationary symmetric (=momentum conservation). See QED texts for derivation of inverse square law.

Existence of virtual particles follows from Heizenberg uncertainty principle (which in turns follows from wave nature of all particles).

 

[pmb]

Alexander wrote

Coulomb law is NOT one of Maxwell equations. Gauss law is, which is derived from Coulomb law and from definition of electric field.

Thank you. Yes. That is correct. I meant to say that Coulomb's law is equivalent to one of Maxwell's equations namely Gauss's law.

Anyway - Your claim was "Coulomb law is more fundamental than Maxwell equations" which is pretty far from being the truth.

Let's forget about this claim that Coulomb's law is derivable from quantum principles as you suggest - that is not relavent to your comment to which I was originally referring to. Nameley your claim that Maxwell's equations are derived from relativity. Or to quote you directly

you can't refer to Maxwell equation(s) trying to prove constancy of speed of light, because Maxwell equations are CONSEQUENCE of relativity, because they are DERIVED from relativity (namely, from existence of electric charge, Lorents transformations of coordinates (which gives rize to magnetic component), and 3-dimensionality of space).

Please derive Coulomb's law from
(1) existence of electric charge
(2) Lorents transformations of coordinates
(3) 3-dimensionality of space

I question the derivation from quantum principle since they axioms of QED might be based on Maxwell's equations and since I'm not formally familiar with QED I have no wish to discuss it here.

Did you really mean that it Maxwell's equations are derivable from quantum principles *plus* special relativity?

Pete

 

[Alexander]

Yes. Maxwell equations are classical limit (h--->0) of QED equations of interactions of charged particles with virtual massles bosons (virtual photons in QED). The result of this interaction is exchange by momentum (in translationary symmetric space momentum shall conserve).

There is no "force" in quantum interactions - there is only exchange by momentum, energy, spin, charge and other conserved quantities. Force is a classic concept standing for "average of momentum change rate", F = <dp/dt>. The momentum of virtual photon is what we call "force", so to speak - in this case "Coulomb force", and a bunch of virtual bosons is what we call "field" ("electric field" if those bosons are virtual photons). Due to Heizenberg uncertainty principle (HUP)if two particles (say, electrons) are sharing same virtual boson (that is where interaction of electrons or other distant from each other particles comes from), then the momentum of this boson is inversely proportional to the distance between particles.

Without going into QED the Coulomb law can not be derived. Here is a rough sketch of how inverse square "forces" and inverse square "fields" originate from HUP of exchange by virtual massles boson. If two electrons are separated by distance r, then to "share" or "exchange" by same virtual photon (which is moving with speed c) the photon must exist for at least t~r/c time. The longer the virtual photon is around, the less energy it can have according to HUP: E~h/t, thus the less momentum it can carry p=E/c~h/ct. Recalling that classic "force" is nothing else but the avarage rate of momentum exchange F~p/t, you get inverse square law: F~p/t=(h/ct)/t =h/ct²=h/(c(r/c)²)=hc/r²

By the way, hc is indeed close to e²/4piepsilon (which is a factor in classic Coulomb law of interaction of two electrons), and the difference (called fine constant factor) arises in transition grom QED to classic limit due to screening of actual charge of electron (which is about sqrt(137) more than its classic limit 1.6x10^-19) by polarized virtual pairs around it.

 

[pmb]

Alexander

I'm not sure why you went into all that. When I said that I wasn't formally familiar with QED that only meant that I couldn't go into a grad level quantum field theory class and pass a test on the subject. I didn't mean don't understand/know it. I'm just not interested in it at the present time as I suggested.

Thanks for the effort though.

Pete

 

[Integral]

Seems like it is time for some moderator intervention in this thread. It really seems that it is starting to circle with no hope of final resolution.

It is clear that historically relativity was NOT necessary to derive the constancy of the speed of light. It is also clear that Einstein was able to POSTULATE the constancy of the speed of light due to the work of Maxwell. That fact that Maxwell’s derivation of the speed of light as a constant created a 30 year period of turmoil in Classical Physics is sufficient proof for me that relativity is not NECESSARY to discuss the constancy of the speed of light.

This does not mean that given a new and greater understanding of the universe that the constancy can be seen to play a fundamental role which can be viewed independently of Maxwell’s Equations. Since I am not familiar with a development of relativity that does not rely on the constancy of the speed of light, it is not clear how one can use relativity to prove c constant.

If Alexander or pmb wishes to start a thread dedicated to that topic, the discussion can continue.