Re: Help from 'honest' SR-cult cretins needed!

From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 01/03/05 

Date: 3 Jan 2005 09:31:22 GMT

JP <jp@nospam.com> writes:

>On Sun, 02 Jan 2005 22:59:53 -0600, eleaticus wrote:

>> Yep, I'm writing a piece about one pillar of the SR-cult's fraudulent claim
>> that the Newtonian/Galilean transformations of various equations are not
>> invariant.
>>
>It sounds like one of your pillars has been objectively rebutted.

The title of the thread sounds quite ridiculous. It is the same as "Help
from 'honest' idiots and morons needed, so that we can see in what way
they are idiots and morons". The fact is that anybody, even somebody as
stupid as Eleaticus, should know that nobody is going to offer to help
anybody who uses such terminology to describe them. Eleaticus, if you
*really* want help from a person who knows about relativity, then it is
the height of stupidity to insult them at the same time that you are
asking them for help.

Eleaticus does not like the form for the Galilean Transformation, given
by

        t' = t,

        x' = x - vt,

        y' = y,

        z' = z.

This is because of his refusal to acknowledge the introduction of t' as a
coordinate in the primed frame. He claims that the fact that t' and t are
equal makes the introduction of t' as a new symbol invalid. He never
justifies his claim of invalidity - we are just expected to accept his
unsupported view that variables represented by different variables cannot
be equal (or cannot be identically equal). The fact that y' and y are
always equal, and that z' and z are always equal, does not seem to inspire
his ire, in spite of the fact that y' and y have different symbols, and z'
and z have different symbols.

Of course, it is possible to get past Eleaticus's unsupported objection
by regarding the Galilean Transformation as a time-dependent
transformation between (x,y,z) and (x',y',z'), so that

        x' = x - vt,

        y' = y,

        z' = z.

Eleaticus's other big problem with comprehension is not so easily dealt
with.

In a coordinate system with t' for the primed system, the differential
operators transform as

        d/dt' = d/dt + v d/dx,

        d/dx' = d/dx,

        d/dy' = d/dy,

        d/dz' = d/dz,

and in Eleaticus's preferred approach,

        (d/dt)_primed = (d/dt)_unprimed + v d/dx,

        d/dx' = d/dx,

        d/dy' = d/dy,

        d/dz' = d/dz,

where (d/dt)_unprimed is the derivative with respect to t with respect to
the unprimed coordinate system, and (d/dt)_primed is the derivative with
respect to t with respect to the primed coordinate system.

Eleaticus's difficulty in comprehension as far as this is concerned is
demonstrated by his conviction that the presence of the term involving
d/dx in the expression for d/dt' is indicative of t' depending on x
(t, y, z, being held constant). Eleaticus has been very explicit that
this is his interpretation. The fact that this is his interpretation
demonstrates beyond a shadow of a doubt that Eleaticus is completely
ignorant of multivariable calculus.

In contrast to Eleaticus's own personal interpretation, the presence of
the term involving d/dx in the expression for d/dt' is actually indicative
of the fact that x depends on t' (x', y', z', being held constant), and
NOT the other way around as Eleaticus would have it.

This cam be most easily seen by taking the Chain Rule,

d/dt' = dt/dt' d/dt + dx/dt' d/dx + dy/dt' d/dy + dz/dt' d/dz,

d/dx' = dt/dx' d/dt + dx/dx' d/dx + dy/dx' d/dy + dz/dx' d/dz,

d/dy' = dt/dy' d/dt + dx/dy' d/dx + dy/dy' d/dy + dz/dy' d/dz,

d/dz' = dt/dz' d/dt + dx/dz' d/dx + dy/dz' d/dy + dz/dz' d/dz,

and noting that the coefficient of d/dx in the expression for d/dt' is
dx/dt', i.e. the rate at which x changes relative to t' (x', y', z' being
held constant).

Maxwell's homogeneous equations are invariant under the Galilean
Transformation, with transformation laws:

        E_x' = E_x,

        E_y' = E_y - v B_z,

        E_z' = E_z + v B_y,

        B_x' = B_x,

        B_y' = B_y,

        B_z' = B_z.

Maxwell's inhomogeneous equations are also invariant under the Galilean
transformation, with transformation laws:

        E_x' = E_x,

        E_y' = E_y,

        E_z' = E_z,

        B_x' = B_x,

        B_y' = B_y + v/c^2 E_z,

        B_z' = B_z - v/c^2 E_y,

        \rho' = \rho,

        J_x' = J_x - v \rho,

        J_y' = J_y,

        J_z' = J_z.

Note the the transformation laws for the charge density and current
density are as they should be under the Galilean transformation.

So we now have that the homogeneous equations are invariant under the
Galilean Transformation, and the inhomogeneous equations are invariant
under the Galilean Transformation, but Maxwell's Equations as a whole are
NOT invariant under the Galilean Transformation, since the transformation
laws required for the EM field for the two cases are inconsistent with
each other.

The transformation law for the EM field which makes the homogeneous
equations invariant will not also make the inhomogeneous equations
invariant. The transformation law for the EM field which makes the
inhomogeneous equations invariant will not also make the homogeneous
equations invariant.

On the other hand, all of Maxwell's equations are invariant under the
Lorentz Transformation, with transformation laws:

        E_x' = E_x,

        E_y' = \gamma (E_y - v B_z),

        E_z' = \gamma (E_z + v B_y),

        B_x' = B_x,

        B_y' = \gamma (B_y + v/c^2 E_z),

        B_z' = \gamma (B_z - v/c^2 E_y),

        \rho' = \gamma (\rho - v/c^2 J_x),

        J_x' = \gamma (J_x - v \rho),

        J_y' = J_y,

        J_z' = J_z,

where \gamma = 1/sqrt(1 - v^2/c^2).

David

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