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

[vshiro]

Hi all,
I am trying to reformulate the axioms of Special Relativity. It seems intuitively true that all inertial frams should be equivalent (*), but there seems to be no philosophical justification that light should travel at constant velocity to all inertial frames (+).
Could someone show me, without sacrificing too much detail, the proofs for the following:
1. the principle of least time for light, which states that light always travels in paths of least time. Does the proof need the premise of condition (+)?
2. given the principle of least action/time, derive that light always travel in geodesics of the space to which it is confined.
Thanks, all.

--Shiro

 

[Hurkyl]

but there seems to be no philosophical justification that light should travel at constant velocity to all inertial frames

Light is an electromagnetic wave.
The wave equation for electromagnetic waves indicates that they should propagate at speed 1/sqrt(&mu₀&epsilon₀)
&mu₀ and &epsilon₀ are constants of the vacuum, and are the same for all observers.
Therefore, the speed of light = c = 1/sqrt(&mu₀&epsilon₀) is constant for all observers.[/size]

 

[Alexander]

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).

Better approach is to postulate that ALL fundamental constants (G,h,c,e) are velocity-invariant (=independent of observer's motion). In essense this means that all physics is velocity-invariant as usually postulated to be separate from constancy of c postulate.

 

[Alexander]

actually fundamental constants are not only velocity invariant, but also acceleration invariant, position invariant, time invariant (as far as we know). So they are invariant in any and all reference systems and for any and all observers no matter where/when he/they are or how weird they are moving.

 

[Hurkyl]

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

er... [?]

http://scienceworld.wolfram.com/biography/Maxwell.html
http://scienceworld.wolfram.com/biography/Einstein.html

Maxwell's theory was published in 1873. Special Relativity was published in 1905.

Heck, Maxwell died in the year Einstein was born (1879), how could his equations possibly be derived from Special Relativity?


The equations comprising Maxwell's Equations date back even further.

http://scienceworld.wolfram.com/physics/MaxwellEquations.html

The equations composing Maxwell's laws are:

Gauss's Law. (Gauss died in 1855)
Absence of magnetic monopoles
Faraday's Law. (Faraday died in 1867)
Ampere's Law. (Ampere died in 1836)

And Maxwell's modification of Ampere's law to include displacement current.


Even the first line of Einstein's paper On the Electrodynamics of Moving Bodies references Maxwell's electrodynamics:

It is known that Maxwell's electrodynamics--as usually understood at the present time--when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.

And Einstein even uses the equations in his paper. See the first sentence of section 6:

Let the Maxwell-Hertz equations for empty space hold good for the stationary system K, so that we have

[insert equations here]

where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that of the magnetic force.

And from there, Einstein goes on to derive the transformation law for the EM field based on the hypothesis that the Maxwell equations are valid in all inertial frames of reference.



It's true that relativity added to Maxwell's theory by explaining what happens when one changes their frame of reference, and corrects the EM force law (which is independant of the Maxwell equations), but by no means imaginable were the Maxwell equations derived from Special Relativity.

 

[Alexander]

Do you care about historic way of who and in which sequence stubmles upon observed facts, or do you care how these facts are related to each other (which one is causal consequence of which other one)?

Prime cause af it all (of e/dynamics) is an "electric charge" property of some particles. So far we don't know what it is (what causes it) - this is where present knowledge blurrs into unknown. Correct me if I am wrong and electric charge is finally derived from more fundamental entities.

Electric charge + 3D space --> Electric field --> Coulomb inverse square electric field law (square only if to use 3D space, 1st power if to use 2D, etc) ---> Gauss law (First Maxwell equation).

Electric field + motion of observer --> magnetic field. (Here Lorents transformations of coordinates are vital. There is no magnetic field if to use Galilean ones. Magnetic field is simply relativistic component of moving electric field. Absense of magnetic monopole as a consequence.) --> Magnetic Gauss law (second Maxwell equation).


Magnetic field (better say, component) + motion of observer ---> Faraday's law of electric induction (3rd Maxwell equation) and equation for displacement currents (last Maxwell equation).

Finally, motion of charge with acceleration ----> both electric and magnetic components changing in such mathematical manner which we label "e/m waves (or light, depending on rate of acceleration change)".


All arrows here stand fot "mathematically results in".

 

[Hurkyl]

(which one is causal consequence of which other one)

It's somewhat imprecise to talk about causality when the objects involved don't occur at points in time. One can talk about logical implication, but the arrows of implication generally do not have a unique direction.

The "historic way" demonstrates that Maxwell's laws can be derived without the assumption of a constant speed of light.

 

[Alexander]

If you assume that speed of light is not constant, then there is no Lorents transformations. Then magnetic field vanishes (as a mathematical result - there is NO magnetic field if you transform coordinates using Galilean transformations instead of Lorents). As a consequence of absense of magnetic field last 3 Maxwell equations vanish. Only Gauss law survives.

Instead others you'll have, for example, just ordinary wave equation for ELECTRIC wave (notice here: NOT for e/m wave, because there is NO magnetic field). Speed of this wave would depend on observer and on source motion - like sound wave for instance.

 

[Hurkyl]

If you assume that speed of light is not constant, then there is no Lorents transformations. Then magnetic field vanishes (as a mathematical result - there is NO magnetic field if you transform coordinates using Galilean transformations instead of Lorents). As a consequence of absense of magnetic field last 3 Maxwell equations vanish. Only Gauss law survives.

Who cares about galilean transformations?

The magnetic field is the force field that deflects currents. The inverse square magnetostatic force between stationary current patterns is

F = (&mu₀/4&pi) I₂ (I₁ r) / d²

from which we conclude an inverse square magnetic field

B = (&mu₀/4&pi) (I r) / d²


which is in direct analogy with electrostatics. Stationary charge patterns generate inverse square electric fields, stationary current patterns generate inverse square magnetic fields.

Varying magnetic field introduces an emf via Faraday's law. Varying electric fields emulate a current. Combine with Gauss's law to yield a wave equation. Assume the vacuum looks the same to all observers and you have constant speed of light.

[/size]

 

[DrChinese]

Originally posted by vshiro
...there seems to be no philosophical justification that light should travel at constant velocity to all inertial frames (+).
Could someone show me, without sacrificing too much detail, the proofs for the following:
1. the principle of least time for light, which states that light always travels in paths of least time. Does the proof need the premise of condition (+)?
2. given the principle of least action/time, derive that light always travel in geodesics of the space to which it is confined.
--Shiro

Unless I misunderstand what you are asking... why would there be a philosophical basis for the measurement of c yielding a constant value? This is a consequence of the theory of relativity, which was put forth to explain the observed facts. You cannot derive special relativity from other theory, ergo it cannot be proven.

I think it is interesting that the least action principle for light applies equally well when light moves from one medium to another. I am not sure if this could be called a geodesic. However, a photon changes its path in response to a change in medium such that its path is still the optimal fastest path.

 

[Alexander]

Originally posted by Hurkyl




Who cares about galilean transformations?

The magnetic field is the force field that deflects currents. The inverse square magnetostatic force between stationary current patterns is

F = (&mu₀/4&pi) I₂ (I₁ r) / d²

from which we conclude an inverse square magnetic field

B = (&mu₀/4&pi) (I r) / d²


which is in direct analogy with electrostatics. Stationary charge patterns generate inverse square electric fields, stationary current patterns generate inverse square magnetic fields.

Varying magnetic field introduces an emf via Faraday's law. Varying electric fields emulate a current. Combine with Gauss's law to yield a wave equation. Assume the vacuum looks the same to all observers and you have constant speed of light.

[/size]

Looks like you got it all backward. Magnetic field is NOT some kind of new field. It simply is relativistic part of electric field. Move by electric field - and you'll get magnetic. All 3 Maxwell equations mathematically follow from the definition of magnetic field B=[v x E]gamma/c², which in turn follows from Lorents transformations of electric field into moving reference system of observer.

 

[Kirk Gaulden]

Formulation of space-time mechanics

Hello,

Space-time mechanics are derived from relativity and parodoxia of the manner relative to the fundamental forces of nature. Using {x,y,z,t},
in association of manifold energies from one part of the universe to the other to balance motion of subspace entities as superpartner
particles that support subspace and hyperspace. Subspace: being a manifold of gravity space-times associated to protect us from hyperspace radiation.Hyperspace: being bosonic and fermion dynamics
that are the equivalancy principal of the universe. The parodoxia:
involves repeating associations to relativity and the basic structure
of the four fundamental forces of nature. Using this information toward that end involves membranes and p-branes where {t}does not equal zero but is a step toward dimensional exploration. A step up from gravity. The exo-gravitational universe, we cannot see it because it to heavy to be supported by non-dimensional space. It is what causes space to expand.

 

[Hurkyl]

Looks like you got it all backward. Magnetic field is NOT some kind of new field.[/quoet]

Nah, you have it backward! B=[v x E]gamma/c^2 is a derived relationship between the magnetic and electric fields (subject to suitable restructions), where the electric and magnetic fields each have their own definitions.

Hurkyl

 

[Kirk Gaulden]

A magnetic field is not a constant, it is in transition to the next energy level of the infinite parodoxia determined by quark plasma densities of bosonic-fermion dynamics, change to form an equivalancy
principal across the board energy level of the four fundamental
forces seeing gravity is time, therefore, subspace is space-time
membrane in gravity and p-branes are condensed forms of dimensional space-times in gravity at {t}, the promotion of lensing potiential
gravity produces manifolds to transfer energy and motion at a different level, this can be done through every fundamental force.
This creates the interaction chromodynamics and fermion dynamics in the same way.

 

[Alexander]

OK. Then where is it derived from, and what is the definition of B ?

 

[Kirk Gaulden]

It is derived from dt^2 and Einsteins equivalancy principal,
and the parodoxia, as for B, I think your talking about hyperspace: the equivalent of a space based on pure bosonic-fermion space which exchanges and absorbs energies creating hyperluminal particles called
a subuniverse. These items are applied to the above statements.
All of the events mentioned herein are above present human mathematical abilities. This is why I sent the information to the
jet propulsion lab, a manifold is the first part of a wormhole as
I understand it, dualaity takes care of the second half.

 

[Hurkyl]

The definition of B is the force field associated with magnetic force, which is defined as the force which deflects currents. I.E. B is the vector field such that:

F = I x B

Where I is a current element and F is the force acting on that current element.


The definition of a force field is always the field associated with a particular force.


The field of electrostatics showed that the electric field at point x generated by a point charge q at y is (neglecting the subscripts and boldface):

E = (1 / 4 &pi &epsilon) q (y - x) / |y - x|³

The field of magnetostatics showed that the magnetic field at point x generated by a point current I at y is:

B = (&mu / 4 &pi) I * (y - x) / |y - x|³

Mix in the relation I = q v, and your "definition" falls out of the mathematics as being a relationship between the electric field generated by a point charge and the magnetic field generated by a point charge.


Another way is to take the Maxwell equations as the fundamental law as opposed to the laws of electrostatics and magnetostatics (but the definition of the terms involved is still the same!) Generate the wave equation and find the integral describing the solution, and your relationship pops out, again not as a true in general equation (it's obviously not true in general), but as a relationship between the fields generated by a point charge at a retarted time.


Another way is to start with your relationship (where it makes sense) as the fundamental law. Since we're assuming it, it doesn't need to be derived, but it's still not the definition; magnetic field is always defined as the force field for the magnetic force, which is always defined as the force that acts on currents.

 

[Kirk Gaulden]

magnetic field scattering

See, the problem is your not looking a the ability for mass/energy
to be linked, where energy, force and matter are all the samething
just at different levels associated by electron and chromodynamics scattering due to space-time fields regulated by gravity waves
at one level or the other fundamental forces placed in levels according to distribution of the enhancement of fields transitioning
magnetic field of the electron so a state of transition takes place
that create interia or microgravity. Normal space is potential gravity, knowing this gravity controls the forces abilities or inabilities.

 

[Alexander]

Originally posted by Hurkyl
The definition of B is the force field associated with magnetic force, which is defined as the force which deflects currents. I.E. B is the vector field such that:

F = I x B

Where I is a current element and F is the force acting on that current element.


The definition of a force field is always the field associated with a particular force.

No, this is not the DEFINITION of magnetic field. This is the DESCRIPTION of magnetic field of a current (=moving charge) which usually is given in introductory physics classes. In these classes when students did not take relativity yet, magnetic field is NOT defined, but only described. Yet there they mention that magnetic field is not field of its own, but a mere relativistic part of electric field of moving charge.


In electrodynamics magnetic field is defined as a term [vxE] arizing from Lorents coordinate transformation of components of electric force F (thus, of the components of electric field E=F/q) from stationary coordinate system of electric charge into the moving with velocity v reference system of observer.

Basicly, if you are STATIONARY versus electric charge, your force has only ONE direction (away or toward electric charge), but when you move, then Lorents transformations give you TWO vectors: one still directed radially toward (or away) charge, and another (delayed component) directed as if there is a new field in PERPENDICULAR to radial direction (and perpendicular to system's velocity vector direction). Namely this vector product [vxE] (with gamma/c² factor) is what we nick name as "magnetic field" and label as B to simplify relativcistic math in calculations.

 

[Hurkyl]

Definition of force field = field associated with a given force. Nothing more, nothing less.


And when we want to simplify relativistic equations, we use the electromagnetic potential 4-vector, or the electromagnetic field strength tensor.

 

[Alexander]

Say, you have a friction force (book on a table). Shall we introduce friction field then? And what field is that indeed?

 

[Kirk Gaulden]

reply to hurkyl

Interia.

 

[Hurkyl]

Say, you have a friction force (book on a table). Shall we introduce friction field then?

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

 

[Alexander]

Hurkul, analyse. What causes friction? Interaction of atoms. So, no new "field" is behind - just e/magnetic one.


Same with magnetic "field" - it is just electric field in motion.

 

[Kirk Gaulden]

Potiential static gravity under a hyperconductive electromagnetic field moment, where {x,y,z,t} is onto the four fundamental forces of nature. Gravity has been proven to be able to move at the speed of light. Angular momentum at 45'on a 360'sphere applied to quark field plasmas densities at wavelengths of active gravity by laser light should produce a manifold,electromagnetically. Gravity should absorb the hyper magnetic field to activate it.

 

[Hurkyl]

Hurkul, analyse. What causes friction? Interaction of atoms. So, no new "field" is behind - just e/ magnetic one.

So?

Friction field is still defined as the field associated with frictional force. Frictional force happens to be a macroscopic approximation to electromagnetic interactions, and thus frictional force can be computed in terms of electromagnetic force, and thus the frictional field can be computed in terms of the electromagnetic field, but the frictional field would not be defined in terms of the electromagnetic field. (and, of course the frictional force is not defined in terms of electromagnetic forces; it is a deduction that electromagnetic forces are the root cause of contact forces)

 

[Alexander]

So, what is the root cause of magnetic field?

 

[Hurkyl]

Magnetic force.

 

[schwarzchildradius]

If so there should be an example of a magnetic field that is not generated by a moving electric field. I've got one but I will not show you it.

 

[pmb]

Originally posted by Alexander
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

That's incorrect. First off Maxwell's equations exist seperately and independant of special relativity. The Principle of relativity states that all laws of physics are the same in all inertial frames. That means that Maxwell's equations hold in all frames of referance.

In fact it was Einstein himself who, in the very same issue of that journal SR appeared in, stated quite clearly and explicitly that the constancy of light "is contained in Maxwell's equations."

re - "(namely, from existence of electric charge, Lorents transformations of coordinates (which gives rize to magnetic component), and 3-dimensionality of space)." - Actually the Lorentz transformation is derived from the Principle of Relativity and Maxwell's equations don't depend on them per se - they demand them.

Pete

 

[marcus]

Originally posted by schwarzchildradius
If so there should be an example of a magnetic field that is not generated by a moving electric field. I've got one but I will not show you it.

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.

 

[marcus]

Originally posted by Kirk Gaulden
See, the problem is your not looking a the ability for mass/energy
to be linked, where energy, force and matter are all the samething
just at different levels associated by electron and chromodynamics scattering due to space-time fields regulated by gravity waves
at one level or the other fundamental forces placed in levels according to distribution of the enhancement of fields transitioning
magnetic field of the electron so a state of transition takes place
that create interia or microgravity. Normal space is potential gravity, knowing this gravity controls the forces abilities or inabilities.

Gentlemen and (possibly) ladies, I suspect that this is a spoof.

No real crackpot could make up gobbledegook this silly.

the made-up word "interia" (if it is made up by Auld Girken) is
fine coinage. Like any poet he wants to make sure we heard, and so he he repeats it in a one word message further down. I say he is brilliant at playing the loonie and does not believe a word of what he says.

 

[Alexander]

Originally posted by Hurkyl
Magnetic force.

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.

Did not you study electrodynamics? Magnetic field is derived there from Lorents transforms of electric field in the very beginning.

 

[Alexander]

Originally posted by schwarzchildradius
If so there should be an example of a magnetic field that is not generated by a moving electric field. I've got one but I will not show you it.

You don't have to. There is non.

 

[Alexander]

Originally posted by pmb
That's incorrect. First off Maxwell's equations exist seperately and independant of special relativity. The Principle of relativity states that all laws of physics are the same in all inertial frames. That means that Maxwell's equations hold in all frames of referance.

In fact it was Einstein himself who, in the very same issue of that journal SR appeared in, stated quite clearly and explicitly that the constancy of light "is contained in Maxwell's equations."

re - "(namely, from existence of electric charge, Lorents transformations of coordinates (which gives rize to magnetic component), and 3-dimensionality of space)." - Actually the Lorentz transformation is derived from the Principle of Relativity and Maxwell's equations don't depend on them per se - they demand them.

Pete

Correct casual sequence of postulates and mathematical consequences:

Postulate: all laws of Nature are independent of velocity of of observer (who tests them).

Consequence # 1: all fundamental constants (G, h, c, e) are independent on observer's velocity.

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

Consequence # 3: Coordinates shall transform by Lorents transform equations (not by Galileo), whenever you measure space and time by electromagnetic tools, say using any e/m clock (i.e., not a pendulum one) and any atom-based meter stick.

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. Feel free to nick name this cross product as "magnetic field"

 

[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.
[/size]

 

[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