Dometrious Gordine

Virginia Union University

Howard 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 applythe modified Lorentzmatrix twice (because it is a rank-2 tensor)

*For example:

red-underlined

fields vanish

identically

Use that

Fμν= –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.