**General
Relativity For Teletubbies**

**Sir Kevin Aylward B.Sc., Warden
of the King’s Ale**

**Galilean
Transformation**

**Of**

**Maxwell’s
Equations**

__Back
to the __**Contents**__ section__

__Abstract__

*There are often
claims that Maxell’s Equations predict that the speed of light is invariant
irrespective of source or observer motion.
The argument pretty much, invariably, states that the EM wave equation
has a velocity given by:*

* ** *

*With the claim
that, because there are no other velocity terms in the expression, the velocity
is not with respect to anything. These claims are usually the result of the
claimers simply copying from those that copied, that copied, down a long chain,
with no evaluation as to whether such a claim is even rational. The claim is
false, and it is certainly not controversial that such a claim is false. Indeed
the considerable effort in engaging in the Michaelson-Morley Experiment is
testament to this. The MMX was specifically designed to detect a change in
light’s velocity in relation to the observers velocity via way of the Earth’s
motion through the Solar system, principally because that is what Maxwell’s
Equations indicated would be the case. *

*However, if as
an independent assumption, Maxwell’s Equations are subject to the Lorentz
Transformations, then they do “predict” an invariant velocity of light. *

*The following is
a non-original summary derivation of the expected change in lights’ velocity
according to the Galilean Transformation applied to Maxwell’s Equations.*

__Galilean
Transformation of Maxwell’s Equation__

In
order to actually determine what Maxwell’s Equations predict about observers
moving relative to EM fields, one has to *actually
calculate* what the wave equation will be from Maxwell’s equations under a
Galilean Transformation.

The core of this analysis was taken from:

For
two reference frames** S, S ^{’}**,
with one moving with relative with respect to another, the Galilean
Transformation is given by:

** **

The full Lorentz transformation of the Electromagnetic Fields and Sources, as provided by the Wikipedia article:

https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity

are:

The
Galilean Transform of the fields and sources are thus obtained by taking the
limit of *c*→∞
so that *γ*→1,
which results in:

** **

** **

After
swapping **v**→−**v**

In
the frame **S**, Maxwell’s Equations
are:

Thus,
in the transformed frame **S ^{’}, **Maxwell’s
Equations are:

With the derivatives also transformed by:

And the primes throughout, simply for convenience, now swapped to non primes, results in the transformed Maxwell’s Equations of:

Taking
the curl of the last equation, and substituting for **E** from the 1^{st} and 3^{rd} equations gives the
modified wave equation for **B**:

** **

Whence a solution of the form

is obtainable with

** **

resulting in a group velocity for the wave of:

which results in a change in velocity :

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