An Extension of Lorentz Transformations

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.