"Edward Green" wrote in message
ups.com...
Timo A. Nieminen wrote:

On Sun, 8 Jul 2006, Edward Green wrote:

The formal simularity of the Coriolis force and the Lorentz force law

-2w x v vs. -qB x v

suggests that the magnetic field may correspond to a local rotation of
space (inertial coordinate system) as seen by charge vs. that seen by
mass.

Very Maxwellian. Have you read Maxwell's papers on this kind of thing?
(Although he would have said "local rotation of the (a)ether", not
space.)

No I haven't. As regards rotation of the "aether" vs. "space", that is
of course merely (rather emotionally loaded, for some) semantics;
though we could say that if one of the watermarks of the aether is
"preferred rest frame", then the rotational aether, so to speak, has
never gone away.

For some insight: one can obtain Maxwell's equations from Coulomb's law
and special relativity.

The whole package? I knew that we could get a magnetic effect from SR
+ electric field (well, so I've heard), but I didn't know we could get
the whole deal.

Yes you can - check out
http://www.cse.secs.oakland.edu/hask...Relativity.htm

However some other assumptions are also used eg charge is not dependant on
velocity and forces add linearly (it is a very interesting exercise to go
through the derivation and see exactly what the assumptions are). Bilge
correctly points out that Jackson notes it is not quite possible to do it
from SR and Coulombs law alone. However with the extra non stated
assumptions added it is derivable. It is interesting to see, for example,
exactly what breaks down in gravity. Here, while rest mass certainly is
invariant, E=MC2 strongly suggests that for moving mass not only should we
include mass as the source of gravitation but energy as well so the source
of gravity being the invariant rest mass may not be true. The linear adding
of forces looks doubtful with gravity as well.

Thanks
Bill

Actually, I've heard mixed comments about this: the
derivations I've seen centered on length contraction, and charge
consequently appearing bunched up, and I've seen those dismissed as at
best heuristic (which I guess means plausibility results we don't
happen to like) and at worst nonsensical -- i.e., as an example of a
failed oolie, like the infamous "flow above and below the wing must
meet up".

I suppose a proper relativistic derivation doesn't care how the distant
sources look to us, but focuses on the local properties of the field
and the requirements of Lorentz invariance. Something like?

What happens if you start from Newton's law of
universal gravitation and SR? Surely you must get a gravito-magnetic
term.
Heaviside did this, and it's a nice exercise. It isn't necessarily easier
to follow Heaviside than to do it from scratch.

Consider that E and B are defined in terms of the Lorentz force,
F=q(E+vxB), while D and H are defined in terms of source densities
(charge
and current densities) and are thus in different units. So, even in free
space, you need constitutive relations for unit conversion (unless you
choose a perverted set of units). What are the gravitational constitutive
relations?

When I set this as a P/F open-book exam, one student was cunning enough
to
find it on www, so you can search for it rather than doing it, but I
recommend trying it for an hour or so first.

Thanks for the constructive comments. One knows one should not be
sucked into profitless "no, what I said wasn't totally stupid"
arguments, but in the absence of positive feedback, one sometimes
falters. (Was it Pauli who said "You know, what professor Einstein says
is not so stupid...")?

I was just about prepared to stubbornly stand my ground: if we adopt as
the operational _meaning_ of "in the presence of a magnetic field
charged particles seem to see themselves at the center of a different
rotational rest frame than neutral massive particles" as "they obey an
appropriate Coriolis law analogue", then my musuing is a tautology.
And if one thinks about the geometric meaning of the cross product --
rotating the force relative to the velocity around a fixed axis in a
fixed sense, and proprotional in magnitude to the projection of the
velocity in a plane perpendicular to that axis -- the sense of having
something to do with rotation is inescapable. I also venture that
every static magnetic source (even an infinitely long straight wires)
involves effective circulation of charge.

Frame dragging?

"dda1" wrote in message
ups.com...

Igor wrote:
Tom Roberts wrote:
Igor wrote:
Here's a good link:
http://abacus.bates.edu/~msemon/Noteon.pdf

The authors forgot to mention that the putative E field:
E = (m/q) w x (w x r) (eq. 5)
does not satisfy Maxwell's equations, because div E != 0 yet there are
no charges present.


Tom Roberts

Yeah, I can see your point. Well, nobody called it a perfect analogy.
I think the paper that was referenced and the original paper by Coisson
from 1973 were just to point out the similarities. They were both
published in the American Journal of Physics, which tends to deal with
new ways of looking at old physics more often than not. Unfortunately,
the Coisson paper does not appear to be available online, although I
know I have a copy of it somewhere.


American Journal of Physics is run by an imbecile (Jan Tobochnick) and
has as charter the publication of "no new reserach" (see the web page).

That characterization is silly. He is a well respected legitimate scientist
and teacher.


AmJPhys publishes regurgitations (as you well pointed out) of old
stuff, basically reinterpretations of older papers. Another junk
journal.


Isn't it the Journal of the American Association of Physics Teachers? That
its focus is not on research but understanding known physics looks quite
reasonable to me. In fact whenever I get to a library the AMJP is one
publication I always browse and learn a lot.

Bill

Bilge wrote:

Edward Green:
The formal simularity of the Coriolis force and the Lorentz force law

-2w x v vs. -qB x v

Actually, you mean -2m w x v.

Yes.

suggests that the magnetic field may correspond to a local rotation of
space (inertial coordinate system) as seen by charge vs. that seen by
mass.

I'm not sure what you mean by the ``formal similarity...''

I think you meant "I'm not sure what you mean by 'formal simularity'.
That's similarity happening at the very same time. ;-)

I meant something like "term by term identity, after changing the
labels".

However, note that for a neutral particle, changing coordinates
to a rotating frame does not give it a charge and two different
particles with the same charge but different masses have different
radii of curvature in the same magnetic field.

True. As I mentioned, this seems to suggest that the rotational rest
frame seen by charge and that seen by mass are different. No don't ask
me to quantify this -- but it doesn't seem like such a very weird idea.
After all, cannot EM and gravity be put on the same footing by a
geometric theory called Kaluza-Klein? Possibly the extra dimensions
are exactly what is required to give this statement meaning.

What we call spacetime coordinates are numbers we can use to describe
all of the objects we observe in experiments in the same way. The
only reason that gravity can be described as spacetime curvature
(and hence transformed away locally by a suitable change of coordinates)
is that the equivalence principle, in which gravitational and inertial
masses are postulated to e equivalent, holds to the precision experiments
can so far test.

I'm not sure what you are getting at, but I rather think I just
answered you. Yes, this idea would imply that not all particles
behaved similarly under "geometry", but then, I think we are by
implication talking about a more complicated theory than one involving
gravity and mass alone. Different aspects of the particle may sample
different aspects of the environment -- like an ice skater feeling the
wind.

"Edward Green" wrote in message
oups.com...
| Bilge wrote:
|
| Edward Green:
| The formal simularity of the Coriolis force and the Lorentz force law
|
| -2w x v vs. -qB x v
|
| Actually, you mean -2m w x v.
|
| Yes.
|
| suggests that the magnetic field may correspond to a local rotation of
| space (inertial coordinate system) as seen by charge vs. that seen by
| mass.
|
| I'm not sure what you mean by the ``formal similarity...''
|
| I think you meant "I'm not sure what you mean by 'formal simularity'.
| That's similarity happening at the very same time. ;-)
|
| I meant something like "term by term identity, after changing the
| labels".
|
| However, note that for a neutral particle, changing coordinates
| to a rotating frame does not give it a charge and two different
| particles with the same charge but different masses have different
| radii of curvature in the same magnetic field.
|
| True. As I mentioned, this seems to suggest that the rotational rest
| frame seen by charge and that seen by mass are different. No don't ask
| me to quantify this -- but it doesn't seem like such a very weird idea.
| After all, cannot EM and gravity be put on the same footing by a
| geometric theory called Kaluza-Klein? Possibly the extra dimensions
| are exactly what is required to give this statement meaning.
|
| What we call spacetime coordinates are numbers we can use to describe
| all of the objects we observe in experiments in the same way. The
| only reason that gravity can be described as spacetime curvature
| (and hence transformed away locally by a suitable change of coordinates)
| is that the equivalence principle, in which gravitational and inertial
| masses are postulated to e equivalent, holds to the precision
experiments
| can so far test.
|
| I'm not sure what you are getting at, but I rather think I just
| answered you. Yes, this idea would imply that not all particles
| behaved similarly under "geometry", but then, I think we are by
| implication talking about a more complicated theory than one involving
| gravity and mass alone. Different aspects of the particle may sample
| different aspects of the environment -- like an ice skater feeling the
| wind.

Not all particles behave similarly when viewed from different frames of
reference. This ball is disobeying Newton's first law, which it cannot do
and it doesn't:
http://ww2010.atmos.uiuc.edu/(Gh)/gu...s/coriolis.mov
Androcles.

Edward Green:
Bilge wrote:

However, note that for a neutral particle, changing coordinates
to a rotating frame does not give it a charge and two different
particles with the same charge but different masses have different
radii of curvature in the same magnetic field.

True. As I mentioned, this seems to suggest that the rotational rest
frame seen by charge and that seen by mass are different. No don't ask
me to quantify this -- but it doesn't seem like such a very weird idea.

Depends upon one's notion of weird... But, I digress.

After all, cannot EM and gravity be put on the same footing by a
geometric theory called Kaluza-Klein?

``Seductively similar footing'' might be more accurate. Kaluza-Klein
theory died due to some problems which were not well understood. However,
the idea was resurrected, improved, expanded and lives on in what is
now known as string theory.

Possibly the extra dimensions are exactly what is required to give
this statement meaning.

If you allow another dimension, then it's not only relatively
straight-forward to create a (somewhat naive, but suggestive) theory,
but also to see how it relates to traditional E&M. In the kaluza-klein
theory, the fifth dimension was postulated to be intrinsically circular,
so that the metric had the form,

ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - (Rdw)^2

A typical wavefunction for a charged particle would then be of the form,

\Psi(t,x,y,z,w) = \Psi(0)\exp[-i(Et - p.x - Rp_w w)]

where p.x means p_x x - p_y y - p_z z and p_w is the momentum in the
fifth dimension. So, we can write that as,

\Psi(t,x,y,z,w) = \Psi(t,x,y,z)\exp[-iR p_w w]

with \Psi(t,x,y,z) defined by \Psi(0)\exp[-i(Et - p.x)]

Now compare that with the standard theory in which we have some initial
wavefunction, \Psi(t,x,y,z) = \Psi(0)\exp[-i(Et - p.x)]. The standard
theory requires that the physics remain invariant under a change of
_phase_, i.e., a gauge transformation,

\Psi - \Psi' = \Psi\exp(-iS)

must not result in any change of physics. (This requirement alone leads
to the existence of a globally conserved charge).

If you then make the identification: S = Rp_w w, you have some
basis for your claim. You can make that stronger by allowing S to
be a function of the coordinates (i.e., a local rather than
global gauge transformation). The goals of the kaluza-klein theory
were even more ambitious. Note that the mass-energy-momementum relation
(from which the earlier wave equation was obtained) now becomes,

E^2 - p.p - (p^4)^2 = 0

which suggests identifying p^4 with the usual mass, so that if you
redefine the mass as a five dimensional quantity, the electron is
massless. From there one can try to quantize the mass based on the
fact that the 5th dimension is intrinsically circular, so that one
might hope to explain the ratios of the charged particle masses
via a winding number, since \Psi(w) = \Psi(w + 2n\pi) must hold.
While that is suggestive, I don't think that sort of idea has lead
anywhere.

But, there are still more similarities. Note that in general
relativity, one defines a covariant derivative, such that when
operating on a four-vector, V^a, D_u V^a = d_u V^a + C^a_ub V^b,
where the C^a_ub are the connection coefficients (christoffel
symbols). The riemann (curvature) tensor is then obtained from
the commutator of the covariant derivatives:

[D_u, D_v] V^a = R^a_buv V^b

In qed, one obtains a gauge covariant derivative,

D_u == d_u + ieA_u

which suggests identifying iA_u as the ``electromagnetic connection
coeficients.'' (But note the factor of `i'.) By taking the commutator
of the covariant derivatives, we get:

[D_u, D_v] = (1/ie)F_vu

where F_vu is just the faraday tensor from classical E&M. Now, we
can take the partial to recover maxwell's (inhomogeneous)
equations, d_v F_vu = j^u. One then views the faraday tensor as
the electromagnetic ``curvature.''

As a final analogy, the homogeneous maxwell's equations
d^a F^bc + permutations = 0, are (in this language) a purely
geometric result, analogous to the bianchi identities.
(Look under ``fiber bundles,'' for more information on this
approach. This also analogizes to the weak and strong interactions,
and yang-mills theories, in general.)

However, the bottom line is that if you want to treat E&M
as some sort of space(time) rotation, you can't do it in
4-d. In 4-d, E&M corresponds to invariance under a change of
phase.

What we call spacetime coordinates are numbers we can use to describe
all of the objects we observe in experiments in the same way. The
only reason that gravity can be described as spacetime curvature
(and hence transformed away locally by a suitable change of coordinates)
is that the equivalence principle, in which gravitational and inertial
masses are postulated to e equivalent, holds to the precision experiments
can so far test.

I'm not sure what you are getting at, but I rather think I just
answered you.

Essentially, my point is that the traditional concept of space and
time applies to everything in the universe, so any adaptation of
the geometry to account for forces must apply in the _same_ way to
everything we can measure. That idea begat general relativity. By
simply eliminating one's preconceptions of how geometry has to be,
one find that gravity has a geometric origin and is not a real force,
in that it can be transformed away.

That is impossible for E&M (at least in 4-dimensions). To do what
you propose is equivalent to finding a coordinate transformation that
transforms away the electric charge.

Yes, this idea would imply that not all particles
behaved similarly under "geometry", but then, I think we are by
implication talking about a more complicated theory than one involving
gravity and mass alone. Different aspects of the particle may sample
different aspects of the environment -- like an ice skater feeling the
wind.

Well, it certainly is more complicated theory - it's called string
theiry (or M-theory), it requires 11 dimensions and it is so complicated
that nobody understands it. To the extent that some physicists understand
something about it, none have been able to suggest a realistic experiment
to test it. (This is not to say that it's wrong, but merely a fact). Once
you try to include E&M as a geometric artifact, you are stuck with having
to include the strong and weak interactions as well and start using
phrases like ``Calabi-Yau manifold'' when you speak of geometry.
Orthogonal rotations don't cut it.

In article .com, Edward Green
says...

The formal simularity of the Coriolis force and the Lorentz force law

-2w x v vs. -qB x v

suggests that the magnetic field may correspond to a local rotation of
space (inertial coordinate system) as seen by charge vs. that seen by
mass.

I'm not sure about that, but there is a sense in which Lorentz-like
forces should be expected, in the low-velocity limit.

Let L be an arbitrary lagrangian for a point-mass written in terms
of spatial coordinates and spatial velocities. Assume that the velocities
are not too large, so that L can be expanded as a power series in velocity.
Taking the first few terms, we have

L = A + B_i V^i + C_ij V^i V^j + ...

where A, B_i and C_ij are functions of the coordinates (but are
independent of velocity).

Whatever the origin of the terms A, B_i, and C_ij, we can make
the following interpretations

A = scalar potential, which gives rise to gradient forces
proportional to d/dx^i A

B_i = vector potential, which gives rise to Lorentz forces
(as well as Coriolis forces), which are proportional to
curl(B) x V

C_ij = tensor potential, which gives rise to centrifugal force
(as well as the usual kinetic energy term 1/2 mv^2).

Two contributions to B_i are the following:

1. If the particle is charged and there is an electromagnetic
vector potential A_i, then there is a contribution to B_i of
the form q A_i. This is what gives rise to the Lorentz force.

2. If you are using noninertial coordinates, then there is
a contribution to B_i of the form m g_0i, where g_uv is the
metric tensor. This is what gives rise to the Coriolis force.

--
Daryl McCullough
Ithaca, NY

Bill Hobba wrote:

"Edward Green" wrote in message...

Timo A. Nieminen wrote:

For some insight: one can obtain Maxwell's equations from Coulomb's law
and special relativity.

The whole package? I knew that we could get a magnetic effect from SR
+ electric field (well, so I've heard), but I didn't know we could get
the whole deal.

Yes you can - check out
http://www.cse.secs.oakland.edu/hask...Relativity.htm

Thanks very much for the reference.

The problem with discussing interesting topics with knowledgable people
is that they invariably come up with challenging follow on reading. ;-)

However some other assumptions are also used eg charge is not dependant on
velocity and forces add linearly (it is a very interesting exercise to go
through the derivation and see exactly what the assumptions are). Bilge
correctly points out that Jackson notes it is not quite possible to do it
from SR and Coulombs law alone. However with the extra non stated
assumptions added it is derivable. It is interesting to see, for example,
exactly what breaks down in gravity. Here, while rest mass certainly is
invariant, E=MC2 strongly suggests that for moving mass not only should we
include mass as the source of gravitation but energy as well so the source
of gravity being the invariant rest mass may not be true. The linear adding
of forces looks doubtful with gravity as well.

I've started to think about how one would approach such a derivation.
I think I would have assumed the invariance and velocity independence
of charge without a qualm, also the linear addition of forces. Special
relativity has "linear" written all over it, anyway. I assume
classical EM is a linear limit of a more general classical theory of
the electromagnetic phenomenon.

Tom Roberts wrote:

Edward Green wrote:

Tom Roberts wrote:

Snip profound argument that if we express the Coriolis force in such a
way that there is no Coriolis force, then there is no Coriolis force

Not at all! You ignored the fact that my equations used
_physical_quantities_.


I'm well aware that the Coriolis force is a so-called fictious force.

Then you should abide by the consequences.

You entirely missed the point.

Given that the Coriolis force is a fictious force -- one in particular
arising in a rotating reference frame -- and given that the Lorentz
force is formally identical to it -- changing labels but keeping
velocity as itself -- then the suggestion arises that in the presence
of a magnetic field charged particles, vis. a vis. their charge,
effectively see themselves in a frame rotating with respect to whatever
frame we would otherwise consider not to be rotating, when we use
massive neutral particles to establish the latter.

Whether these comments are deep or shallow, they do _not_ suffer from
ignorance of the meaning of the Coriolis force, nor its "fictitious"
origin. On the contrary, this awareness is at the heart of the thing.

...

"Edward Green" wrote in message
ups.com...

Bill Hobba wrote:

"Edward Green" wrote in message...

Timo A. Nieminen wrote:

For some insight: one can obtain Maxwell's equations from Coulomb's
law
and special relativity.

The whole package? I knew that we could get a magnetic effect from SR
+ electric field (well, so I've heard), but I didn't know we could get
the whole deal.

Yes you can - check out
http://www.cse.secs.oakland.edu/hask...Relativity.htm

Thanks very much for the reference.

The problem with discussing interesting topics with knowledgable people
is that they invariably come up with challenging follow on reading. ;-)

However some other assumptions are also used eg charge is not dependant
on
velocity and forces add linearly (it is a very interesting exercise to go
through the derivation and see exactly what the assumptions are). Bilge
correctly points out that Jackson notes it is not quite possible to do it
from SR and Coulombs law alone. However with the extra non stated
assumptions added it is derivable. It is interesting to see, for
example,
exactly what breaks down in gravity. Here, while rest mass certainly is
invariant, E=MC2 strongly suggests that for moving mass not only should
we
include mass as the source of gravitation but energy as well so the
source
of gravity being the invariant rest mass may not be true. The linear
adding
of forces looks doubtful with gravity as well.

I've started to think about how one would approach such a derivation.
I think I would have assumed the invariance and velocity independence
of charge without a qualm, also the linear addition of forces. Special
relativity has "linear" written all over it, anyway. I assume
classical EM is a linear limit of a more general classical theory of
the electromagnetic phenomenon.

Excepting quantum corrections (ie classically) EM is fully in accord with
all experimental evidence ie is linear, charge is invariant, and the source
of EM fields, Coulombs law holds etc. That is why Maxwell's equations are
so great - they are correct as far as we know.

Thanks
Bill

Tom Roberts says...

Edward Green wrote:
Tom Roberts wrote:
Not really.

Snip profound argument that if we express the Coriolis force in such a
way that there is no Coriolis force, then there is no Coriolis force

Not at all! You ignored the fact that my equations used
_physical_quantities_.

Well, the Kaluza-Klein approach to the unification of gravity
and electromagnetism interprets electromagnetic forces as a
manifestation of general relativity of 5-dimensional spacetime.
What this means is that the supposedly *physical* force of
electromagnetism can be explained in terms of *fictitious*
forces. So the distinction between "physical" and "fictitious"
may not be readily observable (the Kaluza-Klein
theory can be distinguished from E&M in 4-D spacetime, but
only if probed at high enough energies to detect spatial
variations of fields along the extra, curled-up dimension).

In a certain sense, it's the *theory* that tells you what is
physical and what is fictitious.

--
Daryl McCullough
Ithaca, NY