An Extension of Lorentz Transformations

1. An Extension of Lorentz Transformations DometriousGordine Virginia Union University Howard University REU Program

2. Maxwell’s Equations • Lorentz transformations • (symmetry of Maxwell’s equations) Q:Can we extendto non-constant v? v is a constant Introduction (matrix format)

3. 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

4. Compute the extended Lorentz matrix • …and in matrix form: • Now need the transformation on the EM fields… The Quest

5. Use the definition • …to which weapplythe modified Lorentzmatrix twice (because it is a rank-2 tensor) • For example: The Quest red-underlinedfields vanishidentically Use thatFμν= –Fνμ

6. This simplifies—a little—to, e.g.: With the above-calculated partial derivatives: The Quest This is very clearly exceedingly unwieldy.We need a better approach.

7. Use the formal tensor calculus • Maxwell’s equations: • General coordinate transformations: The Quest note: opposite derivatives • 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