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) Gauss & Ampere Gauss & Faraday 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? • Generalize v → v0 + aμxμ = v0 + a0x0 + a1x1 + a2x2 + a3x3 • Expand all functions of v, but treat the aμ as small… … … is messy • For example, expanding • …turnsthestandardLorentz-boostmatrix • …into: …so that The Quest

4. Even just one specific electric and magnetic field component: This is clearly exceedingly unwieldy.We needa better approach. The Quest

5. 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.

6. The Gauss-Ampere equations: • …transform as Transform Maxwell’s Eq’s: Should vanish, = “condition X” The original equation, = 0

7. This produces “Condition X”: • where we needthis “X” tovanish. • Similarly, transforming the second half: • produces “Condition Y”: • where we needthis “Y” tovanish. • These conditions, “X=0” and “Y=0” insure that the particular coordinate transformation is a symmetry of Maxwell equations • They comprise 2·24 = 48 equations, for only 4 parameters aμ! Transform Maxwell’s Eq’s:

8. The Conditions are “reciprocal” • …in the former, new coordinates are functions of the old, • …in the second, old coordinates are functions of the new. • Introduce “small” deviations from linearity, • …so that the inverse transformation is, to lowest order: • Insert these into “X” and “Y” above; keep only 1st order terms. Evaluating Conditions:

9. For example, • Multiply out, compute derivatives, while keeping 1st order terms: • Now contract with the inverse-transformation: • …which expands (to 1st order) to: • …and simplifies upon transforming ν (“nu”) to the new system Evaluating Conditions:

10. Writing out the small A’s for every choice of every free index: Evaluating Conditions: Most of these vanish to 1st order.

11. Writing out the conditions, for every choice of the free index… • For example, Evaluating Conditions: • Similarly, we obtain a0 = 0. • However, there appear no restrictions on a2 and a3. • Since the initial coordinate system was chosen so the Lorentz-boost is in the x- (i.e., 1st) direction, the “X = 0” conditions allow Lorentz-boosts with

12. The “Y = 0” conditions are evaluated in the same fashion • A little surprisingly, they turn out to produce no restriction on the remaining extension parameters, a2 and a3. • Summarizing: • This permits velocities that are: • homogeneous (same direction everywhere) • constant in time (no acceleration/deceleration) • but the magnitude of which may vary (linearly, slowly)in directionsperpendicular to the boost velocity Conclusions:

13. Open questions: • Second and higher order effects (some conditions on a2 and a3 ?) • Combination of Lorentz-boosts with rotations • Consequences on the relativistic mechanics of moving bodies • …and especially, moving charged bodies. • Acknowledgments: • Funding from the REU grant PHY-1358727 Thank You! This leaves: