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An Extension of Lorentz Transformations. 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. Introduction. (matrix format). - PowerPoint PPT Presentation
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Dometrious Gordine
Virginia Union UniversityHoward University REU Program
*An Extension of Lorentz Transformations
*Introduction*Maxwell’s Equations
*Lorentz transformations*(symmetry of Maxwell’s equations)
(matrix format)
v is a constant
Q: Can we extendto non-constant v?
*The Quest*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
*The Quest*Compute the extended Lorentz matrix
*…and in matrix form:
*Now need the transformation on the EM fields…
*The Quest*Use the definition*…to which we apply
the modified Lorentzmatrix twice (because it is a rank-2 tensor)*For example:
red-underlinedfields vanish
identically
Use thatFμν= –Fνμ
*The Quest*This simplifies—a little—to, e.g.:
This is very clearly exceedingly unwieldy.We need a better approach.
With the above-calculated partial derivatives:
*The Quest*Use the formal tensor calculus*Maxwell’s equations:
*General coordinate transformations:
note: opposite derivatives
…to be continued
Transform the Maxwell’s equations: Use that the equations in old coordinates hold. Compute the transformation-dependent difference. Derive conditions on the aμ parameters.