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Dometrious Gordine
Virginia Union University
Howard University REU Program
*
* Maxwell’s Equations
* Lorentz transformations
* (symmetry of Maxwell’s equations)
Q: Can we extend
to non-constant v?
v is a constant
(matrix format)
*
* Q: Can we extend Lorentz transformations, but so as to still be a
symmetry of Maxwell’s equations?
* Standard: boost-speed (v) is constant.
* Make v → v0 + aμxμ = v0 + a0x0 + a1x1 + a2x2 + a3x3
* Expand all functions of v, but treat the aμ as small
* …that is, keep only linear (1st order) terms
and
*
* Compute the extended Lorentz matrix
* …and in matrix form:
* Now need the transformation on the EM fields…
*
* Use the definition
* …to which we apply
the modified Lorentz
matrix twice (because it is a rank-2 tensor)
* For example:
*
* This simplifies—a little—to, e.g.:
With the above-calculated partial derivatives:
This is very clearly exceedingly
unwieldy.
We need a better approach.
*
* Use the formal tensor calculus
* Maxwell’s equations:
* General coordinate transformations:
note: opposite derivatives
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Transform the Maxwell’s equations:
Use that the equations in old coordinates hold.
Compute the transformation-dependent difference.
Derive conditions on the aμ parameters.
…to be continued