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Now that we have this in hand, we can easily see how to transform the electric
and magnetic fields when we boost a frame. Of course, that does not guarantee
that the result will be simple.
To convert
from to , we must contract its indices
with the transformation tensors,
|
(16.165) |
Note that since is a linear transformation:
|
(16.166) |
(where I have deliberately inserted a space to differentiate the first
index from the second) we can write this in terms of the components of
as:
or (in a compressed notation):
|
(16.168) |
This is just a specific case of the general rule that can be used in
general to transform any nth rank tensor by contracting it appropriately
with each index.
As we saw in our discussion of Thomas precession, we will have occasion
to use this result for the particular case of a pure boost in an
arbitrary direction that we can without loss of generality pick to be
the 1 direction. Let's see how this goes. Recall that for a pure
boost in the one direction is the matrix formed with a lower right
quadrant identity and an upper left quadrant with
on the diagonal and on the corners). Thus:
so:
Note that we have extracted the ordinary cartesian components of
and
from after transforming it. I leave the rest of them to
work out yourself. You should be able to show that:
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(16.170) |
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(16.171) |
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(16.172) |
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(16.173) |
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(16.174) |
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(16.175) |
The component of the fields in the direction of the boost is unchanged,
the perpendicular components of the field are mixed (almost as if they
were space-time pieces) by the boost. If you use instead the general
form of for a boost and express the components in terms of dot
products, you should also show that the general transformation is
given by:
A purely electric or magnetic field in one frame will thus be a mixture of
electric and magnetic fields in another. We see that truly, there is little
reason to distinguish them. We have to be a little careful, of course. If
there is a monopolar (static) electric field in any frame, we cannot transform
it completely into a magnetostatic field in another, for example. Why?
Because the equations above will lead to some mixture for all , and
in nature as a constraint.
I encourage you to review the example given in Jackson and meditate upon
the remarks therein. We will not spend valuable class time on this,
however. Instead we will end this, after all, purely
mathematical/geometrical kinematical interlude (no Hamiltonians or
Lagrangians = no physics) and do some physics. Let us deduce the
covariant dynamics of relativistic particles in (assumed fixed)
electromagnetic fields.
Next: Relativistic Dynamics
Up: The Lorentz Group
Previous: Covariant Formulation of Electrodynamics
Contents
Robert G. Brown
2007-12-28