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| Paul Stowe... |
 Posted: Tue Sep 15, 2009 8:30 pm |
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If I wanted to write the wave equation for a frame of reference where
a source is moving with respect to me would it be
Del'^2E - (1/c^2)(d^2E/dt^2) - [2z/Sqrt(1 - z^2)](d^2E/dx'^2) = 0
Del'^2B - (1/c^2)(d^2B/dt^2) - [2z/Sqrt(1 - z^2)](d^2B/dx'^2) = 0
Where
Del' = d/dx'^2 + d/dy^2 + d/dz^2
And
x' = x/Sqrt(1 - z^2)
And
z = v/c
This expression describe elipsoidal field distributions contracted in
the moving x direction by Sqrt(1 -z^2) In the observer's frame v = 0
and it reduces to the tradional wave equation.
Is it the right wave equations?
Thanks |
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| Tom Roberts... |
| Posted: Fri Sep 18, 2009 5:13 am |
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Paul Stowe wrote:
[quote:289e22da93]If I wanted to write the wave equation for a frame of reference where
a source is moving with respect to me would it be
Del'^2E - (1/c^2)(d^2E/dt^2) - [2z/Sqrt(1 - z^2)](d^2E/dx'^2) = 0
Del'^2B - (1/c^2)(d^2B/dt^2) - [2z/Sqrt(1 - z^2)](d^2B/dx'^2) = 0
Where
Del' = d/dx'^2 + d/dy^2 + d/dz^2
x' = x/Sqrt(1 - z^2)
z = v/c
[/quote:289e22da93]
Why do you want to write an equation in terms of such mixed-frame quantities?
Equations only really make sense when written in terms of quantities ALL
measured in a single frame. In your equation, it appears that c, y, z, t, E, and
B are measured in one frame, and x'is measured in another. Plus you use the
symbol z for two COMPLETELY DIFFERENT things. To modern eyes this equation looks
VERY strange (and even downright wrong).
This last brings us to the fact that you must specify the theoretical context
you want to use. Three possibilities:
1. In the theory known as classical electrodynamics, it is clear how to write
the wave equation for the electric field in the primed frame:
Del'^2 E' - (1/c^2)(d^2E'/dt'^2) = 0
This is, of course, the same as in the unprimed frame, with primes added to
every quantity except constants like c and 0 -- that's what "invariance" means.
In classical electrodynamics, Maxwell's equations are invariant under such
coordinate transforms (Lorentz transforms), and hence so is this wave equation.
Note also that your transform for x' is WRONG in this theoretical context, and
that transforms for t', E', and B' must be included.
[I ignore the fact that these wave equations in terms of E
and B are of limited use; and also the fact that in classical
electrodynamics both E and B are 3-VECTORS. The wave equation
in terms of the 4-vector potential A is much more general.]
2. In an ether theory in which the speed of light is isotropically c in the
ether frame, and Galilean relativity applies, your transform for x' is also
WRONG, and with the correct equations one finds that the wave speed is no longer
isotropic in the moving frame, so one must split c into 3 components. The result
is more complicated than in classical electrodynamics, and is also refuted
experimentally.
3. If you are trying to invent some new theory in which your transform for x' is
correct, I doubt it can be self-consistent for the simple reason that it
purports to describe an object that is at rest in BOTH the primed and the
unprimed frames, which is absurd (I assume the "z" in your t' equation is v/c,
not the z coordinate, but the absurdity remains with the other interpretation).
[quote:289e22da93]This expression describe elipsoidal field distributions contracted in
the moving x direction by Sqrt(1 -z^2)
[/quote:289e22da93]
To interpret your equations and statements I must assume:
* that all quantities without primes in your equations are the same in
the primed frame as in the unprimed frame (i.e. not typos on your part)
* that you meant E and B to be vectors without bothering to mention it
* we are careful to only apply these equations in a way in which they
could be valid (i.e. for 3-vectors E and B perpendicular to each other
and the propagation direction of a wave, with E and B representing
their norms)
* we ignore the inconsistency with the hairy-ball theorem (which implies
that monopole EM radiation from a point source is impossible)
[These last 3 seem inherent in your description.]
It seems highly unlikely to me that the single symbol c can express the speed of
waves in this moving frame; but then I don't know what theoretical context you
have in mind, nor do I know how the coordinates transform in that context (your
stated transforms are self-inconsistent). Your claim is true at most for a
specific physical configuration which you did not describe (presumably
outward-going waves from a point source). As for all differential equations,
boundary conditions are important, and different ones can completely change the
character of the solutions.
[quote:289e22da93]Is it the right wave equations?
[/quote:289e22da93]
Not for any sensible theory of electrodynamics I know of. I think these
equations are refuted by the fact that Maxwell's equations apply accurately in
our labs (which are always moving).
Tom Roberts |
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| Paul Stowe... |
| Posted: Fri Sep 25, 2009 1:21 pm |
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On Sep 18, 8:13 am, Tom Roberts <tjrob... at (no spam) sbcglobal.net> wrote:
[quote:65cbc35472]Paul Stowe wrote:
If I wanted to write the wave equation for a frame of reference where
a source is moving with respect to me would it be
Del'^2E - (1/c^2)(d^2E/dt^2) - [2z/Sqrt(1 - z^2)](d^2E/dx'^2) = 0
Del'^2B - (1/c^2)(d^2B/dt^2) - [2z/Sqrt(1 - z^2)](d^2B/dx'^2) = 0
Where
Del' = d/dx'^2 + d/dy^2 + d/dz^2
x' = x/Sqrt(1 - z^2)
z = v/c
Why do you want to write an equation in terms of such mixed-frame quantities?
Equations only really make sense when written in terms of quantities ALL
measured in a single frame. In your equation, it appears that c, y, z, t, E, and
B are measured in one frame, and x'is measured in another. Plus you use the
symbol z for two COMPLETELY DIFFERENT things. To modern eyes this equation looks
VERY strange (and even downright wrong).
[/quote:65cbc35472]
I'm sorry, z does not represent two different things... As fo r x' it
is used because we're seeing the profile of a moving field wrt to
ours, and that is what we want to define. If we wanted to describe
the field profile in the FOR where the source is not moving and we
are, the we'd just use the standard equation with x', t'.
[quote:65cbc35472]This last brings us to the fact that you must specify the theoretical context
you want to use. Three possibilities:
1. In the theory known as classical electrodynamics, it is clear how to write
the wave equation for the electric field in the primed frame:
Del'^2 E' - (1/c^2)(d^2E'/dt'^2) = 0
[/quote:65cbc35472]
Yes, for a source in the another frame, switching to it (where it's
not moving but the whole FOR is)
[quote:65cbc35472]This is, of course, the same as in the unprimed frame, with primes added to
every quantity except constants like c and 0 -- that's what "invariance" means.
In classical electrodynamics, Maxwell's equations are invariant under such
coordinate transforms (Lorentz transforms), and hence so is this wave equation.
Note also that your transform for x' is WRONG in this theoretical context, and
that transforms for t', E', and B' must be included.
[I ignore the fact that these wave equations in terms of E
and B are of limited use; and also the fact that in classical
electrodynamics both E and B are 3-VECTORS. The wave equation
in terms of the 4-vector potential A is much more general.]
[/quote:65cbc35472]
But not relevant to this particular discussion.
[quote:65cbc35472]2. In an ether theory in which the speed of light is isotropically c in the
ether frame, and Galilean relativity applies, your transform for x' is also
WRONG, and with the correct equations one finds that the wave speed is no longer
isotropic in the moving frame, so one must split c into 3 components. The result
is more complicated than in classical electrodynamics, and is also refuted
experimentally.
[/quote:65cbc35472]
Nope, not Galilean relativity...
[quote:65cbc35472]3. If you are trying to invent some new theory in which your transform for x' is
correct, I doubt it can be self-consistent for the simple reason that it
purports to describe an object that is at rest in BOTH the primed and the
unprimed frames, which is absurd (I assume the "z" in your t' equation is v/c,
not the z coordinate, but the absurdity remains with the other interpretation).
[/quote:65cbc35472]
Nope, just trying to get the proper expression of a source's profile
as seen in one frame when that frame is moving wrt to the source. Not
interested is switching to the other's FOR.
[quote:65cbc35472]This expression describe elipsoidal field distributions contracted in
the moving x direction by Sqrt(1 -z^2)
To interpret your equations and statements I must assume:
* that all quantities without primes in your equations are the same in
the primed frame as in the unprimed frame (i.e. not typos on your part)
* that you meant E and B to be vectors without bothering to mention it
* we are careful to only apply these equations in a way in which they
could be valid (i.e. for 3-vectors E and B perpendicular to each other
and the propagation direction of a wave, with E and B representing
their norms)
* we ignore the inconsistency with the hairy-ball theorem (which implies
that monopole EM radiation from a point source is impossible)
[/quote:65cbc35472]
Think light bulb...
[quote:65cbc35472][These last 3 seem inherent in your description.]
It seems highly unlikely to me that the single symbol c can express the speed of
waves in this moving frame; but then I don't know what theoretical context you
have in mind, nor do I know how the coordinates transform in that context (your
stated transforms are self-inconsistent). Your claim is true at most for a
specific physical configuration which you did not describe (presumably
outward-going waves from a point source). As for all differential equations,
boundary conditions are important, and different ones can completely change the
character of the solutions.
[/quote:65cbc35472]
There's nothing radical here.
[quote:65cbc35472]Is it the right wave equations?
Not for any sensible theory of electrodynamics I know of. I think these
equations are refuted by the fact that Maxwell's equations apply accurately in
our labs (which are always moving).
[/quote:65cbc35472]
Are you claiming that the form of my expression has no basis in
science? If so, how does one describe the electron's field profile
moving at 0.95c for example? Feynman indicates that it is elipsoidal. |
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| Dono.... |
| Posted: Tue Oct 13, 2009 6:53 am |
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On Sep 15, 11:30 pm, Paul Stowe <theaether... at (no spam) gmail.com> wrote:
[quote:cbf3fd0191]If I wanted to write the wave equation for a frame of reference where
a source is moving with respect to me would it be
Del'^2E - (1/c^2)(d^2E/dt^2) - [2z/Sqrt(1 - z^2)](d^2E/dx'^2) = 0
Del'^2B - (1/c^2)(d^2B/dt^2) - [2z/Sqrt(1 - z^2)](d^2B/dx'^2) = 0
Where
Del' = d/dx'^2 + d/dy^2 + d/dz^2
And
x' = x/Sqrt(1 - z^2)
And
z = v/c
This expression describe elipsoidal field distributions contracted in
the moving x direction by Sqrt(1 -z^2) In the observer's frame v = 0
and it reduces to the tradional wave equation.
Is it the right wave equations?
Thanks
[/quote:cbf3fd0191]
Not even close, the correct equation is :
Del'^2E - (1/c^2)( at (no spam) ^2E/ at (no spam) t^2)=0
Del'^2B - (1/c^2)( at (no spam) ^2B/ at (no spam) t^2)=0
See here:
http://en.wikipedia.org/wiki/Electromagnetic_wave_equation
[quote:cbf3fd0191]This expression describe elipsoidal field distributions contracted in
the moving x direction by Sqrt(1 -z^2) In the observer's frame v = 0
and it reduces to the tradional wave equation.
Is it the right wave equations?
[/quote:cbf3fd0191]
Not even close. You need to apply the Lorentz transforms:
x'=\gamma(x-vt)
y'=y
z'=z
t'=\gamma(t-vx/c^2)
and you will get:
Del'^2E' - (1/c^2)( at (no spam) ^2E'/ at (no spam) t'^2)=0
Del'^2B' - (1/c^2)( at (no spam) ^2B'/ at (no spam) t^2)=0
where
Del' = d/dx'^2 + d/dy^2 + d/dz^2 |
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