"Bill Hobba" wrote in message ...
"OC" wrote in message
om...
The first postulate of Special Relativity says that the laws of
electromagnetism and optics are valid in the same frames of reference
where the laws of mechanics are valid.
Maybe Einstein did not spell it out for Androcles, but physicists know
that the laws of mechanics, as expressed by Newton's laws, are valid
in inertial frames of reference.

He did. From his 1905 paper on relativity:

'They suggest rather that, as has already been shown to the first order of
small quantities, the same laws of electrodynamics and optics will be valid
for all frames of reference for which the equations of mechanics hold good.1
We will raise this conjecture (the purport of which will hereafter be called
the ``Principle of Relativity'') to the status of a postulate'


The second postulate says that the speed of light in empty space does
not depend on the state of motion of the source.
This is based on experimental results (in the same way as, for
example, Coulomb's law was based on experiments).

Yes. But the connection is a little subtler. From the POR alone we know
such a velocity must exist - that its value is the speed of light follows
from experiment and theoretical considerations. However it is useful to
note that the evidence we have is compatible with the photon having a very
small mass, in which case the speed of light is not constant - but an
invariant speed c still exists.


Doesn't this invariant speed c show up once we try to find the
transfomations that keeps Maxwell's equation invariant (i.e., once we
find the Lorentz transformations)?



Complaints by crackpots notwithstanding, the first postulate is quite
reasonable.

It is both reasonable had has experimental and theoretical support.

The second postulate can be challenged only by proper experiments.


I consider it is even more reasonable than the first postulate because it is
predicted by it - only its numerical value is not. But all sorts of
considerations lead is the its value. To a very good experimental accuracy
(basically how well we experimentally know the photon has exactly zero mass
or equivalently charge is conserved).


It was my understanding that an invariant speed shows up once we have
the Lorentz transformations.
And these are found, for example, by keeping the Maxwell'sequations
invariant for inertial frames of reference.

But if we have the first postulate, and we do not look at the specific
form of the laws of mechanics, electromagnetism and optics, the second
postulate should still yield the Lorentz transformations.



By the way, I do not agree with Bill Hobba's interpretation of
Newton's laws.
The first law introduces the concept of inertia, and allows to identfy
in principle inertial frames of references.

That is the exact point I was making. On the surface it looks as though it
follows from Newton's second law which strictly spiking is a definition
anyway. But digging a little deeper shows such an analysis is superficial -
What Newton's first law is saying, at least conceptually, inertial frames
exist - exactly as you suggest. What Newton's second law is saying is look
to the forces - it is a 'half law' as Feynman would say. All this is at
length in his famous lectures under the chapter the Characteristics of Force
in volume 1 (I can not be bothered getting the page).


The problem with Newton's laws is that, when they are introduced,
there is no satisfactory definition of force from a formal point of
view.
There is usually an operative definition or a semi-empirical
definition (with experiments that show that forces add like vectors).

And, the first law gives us a way to identify inertial frame of
references.
Once we can identify "real2 forces (by some definition), the frames of
reference where the first law is valid are inertial.
And in these frames, the second law gives us the link between forces
and their effect on the motion of a body.


The second links force to acceleration, via a quantification of
inertia (mass).

On the surface it is merely a definition of force. What one needs to add is
other things like, classically, for fundamental forces, it is always
derivable for a potential ie is conservative. This is what Feynman means by
it is 'half a law'.


But, in order to use the second law as definition of force, one needs
a definition of mass.

Physicists prefer using the second law as definition of mass (as
quantitative measure of inertia).


These two laws require that frames of references and kinematics
(velocity, acceleration) are already defined (this the easy part), and
need force also to be already defined (which is the most questionable
part).

Well I am actually a proponent of not basing classical mechanics on Newton's
laws but rather on the PLA as espoused by Landau in Mechanics. None of
these issues arise in that approach - the PLA and the POR are seen as the
basis. I am not the only one - Taylor believes that is the best approach
even for beginning student - see http://www.eftaylor.com/leastaction.html.

"PLA" as in "Principle of Least Action"?

OC

(Bilge) wrote in message ...
shevek:
(Bilge) wrote:

Well, ``gauge'' is the proper term, however, if you prefer, you can
use the term, ``phase''. Make the following substitution in the
lagrangian for the fields (or if you wish, wavefunctions):


\Psi - \Psi' = \Psi\exp(-ia(x)) and simillarly for its h.c.

a(x) is an arbitrary function of the spacetime variables.

Ah, thanks.. that helps.

Sure. I can give you a specific example, if you wish. Note that the
function a(x) in the substitution above is only the simplest way to
write the phase and it corresponds to a U(1) symmetry. What it does,
in effect, is tell you that the free particle lagrangian is not going
to be covariant under a (local) change of phase unless you add a field
to the lagrangian and the field itself transforms properly. In the case
of E&M, the function a(x) is what corresponds to the gauge transform,
A'_u = A_u + (1/e) d_u a(x), where e is the electric charge.

What is also interesting, is that you can obtain basically the
same result by taking spacetime to be five dimensional with the
fifth dimension being intrinsically circul ar. What you then have
is your basic kaluza-klein model.


Ok.. I like the idea of guage symmetry as a "phase" symmetry, as I am
used to absolute phases being unobservable, only relative ones are
observable. Of course in the usual kind of guage transformation, as
you put it above, 1/e d_u a(x), the relation to a phase is hard to
see.. it appears more as a phase once you write down the
wavefunctions, or if you include a higher circular dimension..


If you write the argument to the exponent assuming the symmetry is
SU(2) instead of U(1), you end up with the weak interaction. For
the group SU(3), you get the qcd. If you extend the idea of adding
more dimensions to accomodate these interactions, what you have is
called string theory.


Wow.. I like the way you put that, certainly much easier to follow
than what I am used to seeing for these theories. So, each of these
explanations relies on the existence of a particle (or field)
wavefunction Psi, presumably with certain properties that give it the
form of the exponential.. or can it be made even more general?


[...]

I'll stay tuned. What with neutrino oscillations however I'm not sure
how continuous the symmetry will be.

Neutrino oscillations, in and of themselves, are not so mysterious,
since any time you have a hamiltonian of the form H_0 + V1 + V2,
where the states which diagonalize V1, don't diagonalize V2, the
states for which V1 is diagonal will get mixed by V2 and vice versa.
So, if the weak eigenstates are, \nu_e, \nu_u, \nu_tau, and those
are not the mass eigenstates, the mass eigenstates will be some
linear combination of those three and vice versa. What that means
in the overall cosmological picture us anybody's guess. The same is
true for the kobayashi-moskawa matrix which describes quark mixing.
It obviously occurs, but so far nature has apparently not made the
point of why it occurs, very clear.

OK, I'm a little out of my territory here (as you probably noticed)
but thanks for your help.. in lectures on neutrino oscillation I
hadn't heard the separation between weak eigenstates and mass
eigenstates noted; it is usually presented as neutrino wavefunctions
being superpositions of e.g. muon neutrino and electron neutrino
eigenstates. Maybe that's why I have been having trouble
understanding neutrino oscillation in the context of lepton number
conservation?

And what kind of weak reaction will give rise to a neutrino in a
superposition of lepton states? For a neutron decay for example,
doesn't that require that some of the other reaction products
(electron and proton) are also in mixed states? After all electrons
are massive; shouldn't some change flavor as well?

Thanks - shevek

(Bilge) wrote in message ...
shevek:

Basically SR is about being very careful what you mean by time and
distance, as real readings of clocks and meter sticks, as defined by
electromagnetic forces. Right?

Actually, it probably makes more sense and enables you avoid potentially
circular arguments if you say that special relativity defines the rela-
tionship between inertial frames. That requires special relativity to
hold regardless of whether you create a ``clock'' or ``meter stick'' using
electromagnetic, strong or weak interactions. A priori, there is no
reason to expect such a theory is possible, so the fact that we have
such a theory is a pretty good indication that special relativity applies
to all phenomena (excluding gravity which changes the metric itself).

One could say that clocks and meter sticks are defined by electro-
magnetic forces, but if you impose the further constraint that the
photon defines a massless particle, you still have a relativistic
theory, although possibly not the correct theory. However, there are
lots of constraints imposed by experiments on the photon mass that
have to be consistent with what the rest of the theories say. Once
your theory is self-consistent, that's all you can say. Then, your
inertial coordinates are whatever your theory says they are. Attempting
to define time and distance beyond any measurement of time and
distance that can be performed using the physics responsible for
the universe we inhabit is mental masturbation. Even worse, it
introduces additional physics which correspond to no phenomena.


OK, well I'm familiar with guage invariance in terms of freedom of
magnetic vector potential and electric potential (A_mu for you
4-vector fanatics) but I was looking for something more fundamental,
like what really -is- the magnetic vector potential? Weyl's idea was
that it corresponded to a fundamental freedom in choice of metric or
coordinate system, but Einstein thought that would cause problems
elsewhere.. anyway, just fishing for ideas, sorry if I'm being
unclear.

Maybe this will help. I have several different angles for you to
try and picture A, which are rather different, but are related in
a rather fundamental way. First of all, just using maxwell's equations,
the vector potential (as well as the scalar potential) are nothing
but some mathematical artifacts of some vector identities applied
to some differential equations. So, we should establish the vector
potential as something other than an artifact directly from an
experiment with electrons which is the equivalent to the double slit
experiment for light using quantum theory. Consider am experiment
in which ne arranges to have a beam of electrons pass on either
side of a solenoid, such that the magnetic field is zero in the
region of interest (one can always make the field small enough
that it clearly cannot be responsble for the effect shown here).
In the ascii art version below, ``O'' is the solenoid, as seen looking
down its axis and the .'s are the electrons going from left to
right.

. . At point P the wavefunction is \Psi(x,t)
P . . P' is split into \Psi_1(x,t) and \Psi_2(x,t),
. O . corresponding to the two possible paths.
. .
. . Even though B is zero iutside the solenoid,
the vector potential os _not_ zero there.

The solution of the schroedinger equation with the vector potential
included is given by the original wavefintion x a phase factor, i.e.,

\Psi(x,t) - \Psi(x,t)\exp(iS)

with S = (e/hbar) \integral A . dx, where dx is taken over the
path of the electron. If the beam is recombined at point point P',
then the wavefunction there will be the sum of the wave functions
taken over the two paths,

\Psi(x,t) = \Psi_1(x,t)\exp(iS_1) + \Psi_2(x,t)\exp(iS_2)

Taking the mod squared, gives a term that is proportional to
\exp(i(S_2 - S_1)), which is just,

S_2 - S1 = (e/hbar) [\integral A.dx_2 - \integral A.dx_1]

= (e/hbar) [\integral A.dx]

Where the last integral is taken over the closed path defined by
the difference between the two paths. The upshot is that, despite
the absence of any electromagnetic forces on the electrons, you
get an interference pattern that shifts as you change the field
inside the solenoid. That establishes the vector potential A as
an electromagnetic effect beyond that which can be described in
terms of E and B and consequently, beyond that which maxwell's
theory can describe.


The relativistic extension is to replace the vector potential, A,
with A^u giving the integral, \integral A^u d_u.

For picture number two, we start by noting that maxwell's equations
may be written in the form of a wave equation as,

box A^u = j^u

and blindly take A^u to be the photon to start. This is just a
compact way of writing the equations for curl B and div E
in terms of vector and scalar potentials as a single equation.

In free space, we don't have any sources, so our wave equation
is box A^u = 0, which has solutions,

A^u = e^u \exp(-ip.x)

Where e^u is the four-vector polarization and p.x mean p^u x_u or what
might be written in more familiar terms, i(k.x - wt). This equation has
four polarization states yet we know the photon is a spin 1, which has
only three possible polarizations and we know that the physical photon has
only two, so what we have for A^u contains two superfluous degrees of
freedom, since the number of real degrees of freedom can't depend upon how
you write an equation. First we note that in writing the equation above,
we assumed that d_u A^u = 0, which is known as the lorentz condition. We
can use that to eliminate one of the four polarization states, e^u. If we
eliminate the time polarization, we are left with the vector potential, A,
i.e., A^u = (0, A^1, A^2, A^3). You can now interpret your vector
potential as the photon. However, we still need to eliminate one of those
degrees of freedom. We do that by noting that the photon is always
polarized transverse to it's momentum, so that if p is along the z
direction so that p = (0, 0, p_z) and the most general polarization is
given by A = (A_x, A_y, A_z), then we can't have a polarization along z.

Note that if we write p = -i\hbar\grad, what that really means is
that A_z can be a constant, since the derivative of a constant is
zero. That is an example of gauge transformation. So long as a change
to A does not intrduce any change which affects any observable result,
the change is simply an artifact of the mathematics. More generally,
we can change A_z by any scalar function d^u\Phi, such that its
derivative, d_u d^u\Phi = 0, i.e., A'^u = A^u + d^u\Phi leaves the
physics unchanged if d_u A'^u = d_u A^u + d_u d^u \Phi.

I can still make a further generalization and deal with virtual
photons, but I'm going to stop here.


Thank you Bilge!

And the experiment was first carried out by Aharnov-Bohm.

I'll leave you with some peace and quiet for now..

Thanks again -

OC:
(Bilge) wrote:
OC:

By the way, I do not agree with Bill Hobba's interpretation of
Newton's laws.
The first law introduces the concept of inertia, and allows to identfy
in principle inertial frames of references.

In what way does it do that? Newton defines inertial motion as motion
which is force free. Force free means inertial motion. It's a circular
definition.

I did say that "force" needs to defined somehow before Newton's laws.

Once there is a definition for force (which does not require the
concept of inertia), it is possible to introduce inertia as that
property of a physical object such that the state of motion does not
change if there are no (net) forces acting on the object (Newton's
first law).

Unfortunately, that doesn't fix anything. It just adds another layer
of circularity. What precisely, tells you if a force is acting on
an object?

"OC" wrote in message
om...
"Bill Hobba" wrote in message
...
"OC" wrote in message
om...
The first postulate of Special Relativity says that the laws of
electromagnetism and optics are valid in the same frames of reference
where the laws of mechanics are valid.
Maybe Einstein did not spell it out for Androcles, but physicists know
that the laws of mechanics, as expressed by Newton's laws, are valid
in inertial frames of reference.

He did. From his 1905 paper on relativity:

'They suggest rather that, as has already been shown to the first order
of
small quantities, the same laws of electrodynamics and optics will be
valid
for all frames of reference for which the equations of mechanics hold
good.1
We will raise this conjecture (the purport of which will hereafter be
called
the ``Principle of Relativity'') to the status of a postulate'


The second postulate says that the speed of light in empty space does
not depend on the state of motion of the source.
This is based on experimental results (in the same way as, for
example, Coulomb's law was based on experiments).

Yes. But the connection is a little subtler. From the POR alone we
know
such a velocity must exist - that its value is the speed of light
follows
from experiment and theoretical considerations. However it is useful to
note that the evidence we have is compatible with the photon having a
very
small mass, in which case the speed of light is not constant - but an
invariant speed c still exists.


Doesn't this invariant speed c show up once we try to find the
transfomations that keeps Maxwell's equation invariant (i.e., once we
find the Lorentz transformations)?

That fixes the numerical value - that such a value exists follows from the
POR alone. A very important point to note is that Maxwell's equations is
only one way to fix the value - many other experiments can also be used - eg
particle decay times.




Complaints by crackpots notwithstanding, the first postulate is quite
reasonable.

It is both reasonable had has experimental and theoretical support.

The second postulate can be challenged only by proper experiments.


I consider it is even more reasonable than the first postulate because
it is
predicted by it - only its numerical value is not. But all sorts of
considerations lead is the its value. To a very good experimental
accuracy
(basically how well we experimentally know the photon has exactly zero
mass
or equivalently charge is conserved).


It was my understanding that an invariant speed shows up once we have
the Lorentz transformations.
And these are found, for example, by keeping the Maxwell'sequations
invariant for inertial frames of reference.

But if we have the first postulate, and we do not look at the specific
form of the laws of mechanics, electromagnetism and optics, the second
postulate should still yield the Lorentz transformations.

Please see, for example, the following for what is really happening -
http://arxiv.org/abs/physics/0110076.




By the way, I do not agree with Bill Hobba's interpretation of
Newton's laws.
The first law introduces the concept of inertia, and allows to identfy
in principle inertial frames of references.

That is the exact point I was making. On the surface it looks as though
it
follows from Newton's second law which strictly spiking is a definition
anyway. But digging a little deeper shows such an analysis is
superficial -
What Newton's first law is saying, at least conceptually, inertial
frames
exist - exactly as you suggest. What Newton's second law is saying is
look
to the forces - it is a 'half law' as Feynman would say. All this is at
length in his famous lectures under the chapter the Characteristics of
Force
in volume 1 (I can not be bothered getting the page).


The problem with Newton's laws is that, when they are introduced,
there is no satisfactory definition of force from a formal point of
view.
There is usually an operative definition or a semi-empirical
definition (with experiments that show that forces add like vectors).

And, the first law gives us a way to identify inertial frame of
references.

Does it? Please operationally define that method taking into account how to
define a coordinate system and sync clocks to determine if a particle is
moving at constant velocity and how you determine no force is acting on it -
in particular I think you might find syncing clocks a bit difficult without
the assumption of isotropy. I think you will find Landaus definition, based
on symmetry, as found on page 5 of Mechanics (Landau - Mechanics) much more
satisfactory. But its real resolution requires GR. I have seen
presentations that define it by means of accelerometers but in the final
analysis they must assume, for example, the speed of light is isotropic so
they can ensure the Einstein sync procedure works - or what they do is
assume it works anyway as a definition.

Once we can identify "real2 forces (by some definition),

Give me that definition please.

the frames of
reference where the first law is valid are inertial.
And in these frames, the second law gives us the link between forces
and their effect on the motion of a body.


The second links force to acceleration, via a quantification of
inertia (mass).

On the surface it is merely a definition of force. What one needs to
add is
other things like, classically, for fundamental forces, it is always
derivable for a potential ie is conservative. This is what Feynman
means by
it is 'half a law'.


But, in order to use the second law as definition of force, one needs
a definition of mass.

Which in no way detracts from my point. BTW one can use conservation of
momentum to define mass without any reference to the concept of force.


Physicists prefer using the second law as definition of mass (as
quantitative measure of inertia).

Does that apply to all physicists? In particular does it apply to Landau
when he derived its existence from the PLA on page 7 of Mechanics and showed
it must be positive? Indeed it is possible to present classical mechanicals
without any reference at all to force. Also you might like to check out
chapter 10 of the Feynman Lectures where he connects mass to conservation of
momentum rather than a definition of force.



These two laws require that frames of references and kinematics
(velocity, acceleration) are already defined (this the easy part), and
need force also to be already defined (which is the most questionable
part).

Well I am actually a proponent of not basing classical mechanics on
Newton's
laws but rather on the PLA as espoused by Landau in Mechanics. None of
these issues arise in that approach - the PLA and the POR are seen as
the
basis. I am not the only one - Taylor believes that is the best
approach
even for beginning student - see
http://www.eftaylor.com/leastaction.html.

"PLA" as in "Principle of Least Action"?

Yes. Please check out the link - I think it is rather good especially
http://www.eftaylor.com/pub/FmaAJPguest5.pdf.

Thanks
Bill


OC

(Bilge) wrote in message ...
OC:
(Bilge) wrote:
OC:

By the way, I do not agree with Bill Hobba's interpretation of
Newton's laws.
The first law introduces the concept of inertia, and allows to identfy
in principle inertial frames of references.

In what way does it do that? Newton defines inertial motion as motion
which is force free. Force free means inertial motion. It's a circular
definition.

I did say that "force" needs to defined somehow before Newton's laws.

Once there is a definition for force (which does not require the
concept of inertia), it is possible to introduce inertia as that
property of a physical object such that the state of motion does not
change if there are no (net) forces acting on the object (Newton's
first law).

Unfortunately, that doesn't fix anything. It just adds another layer
of circularity. What precisely, tells you if a force is acting on
an object?


I am not saying that it fixes the problem.
But it shifts the problem to the definition of force.

The definition of force is supposed to be able to tell us when a force
is acting on a physical body, in a way that does not require knowledge
about the motion of the object.
It is not an enitrely satisfactory way to deal with the problem.
But, provided we have such a definition of force, there is no
"circularity" in Newton's laws.

OC

OC:
(Bilge) wrote in message news:

Unfortunately, that doesn't fix anything. It just adds another layer
of circularity. What precisely, tells you if a force is acting on
an object?


I am not saying that it fixes the problem.
But it shifts the problem to the definition of force.

The definition of force is supposed to be able to tell us when a force
is acting on a physical body, in a way that does not require knowledge
about the motion of the object.

But, we don't have such a definition without invoking something
like the principle of equivalence, in which case we have general
relativity.

It is not an enitrely satisfactory way to deal with the problem.
But, provided we have such a definition of force, there is no
"circularity" in Newton's laws.

Since newton did not provide such a definition, newton's laws
are circular. You did precisely what I said you did in the
last post, You simply added another layer that doesn't resolve
anything.

OC:

I am not saying that it fixes the problem.
But it shifts the problem to the definition of force.

The definition of force is supposed to be able to tell us when a force
is acting on a physical body, in a way that does not require knowledge
about the motion of the object.
It is not an enitrely satisfactory way to deal with the problem.
But, provided we have such a definition of force, there is no
"circularity" in Newton's laws.

That's rather specious, since newton's laws were intended to define
forces. Since we don't have such a definition, nor do I say any way
to come up with one withing newton's laws, what you are saying is
equivalent to saying ``if there was a santa clause, he could climb
down the chimney''.