Dometrious Gordine Virginia Union University Howard University REU Program

* Maxwell’s Equations * Lorentz transformations (symmetry of Maxwell’s equations) (matrix format) v is a constant Gauss & AmpereGauss & Faraday

…so that

* Even just one specific electric and magnetic field component: This is clearly exceedingly unwieldy. We need a better approach.

* Use the formal tensor calculus * Maxwell’s equations: * General coordinate transformations: 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 a μ parameters.

* The Gauss-Ampere equations: * …transform as The original equation, = 0Should vanish, = “condition X”

* This produces “Condition X”: * where we need this “X” to vanish. * Similarly, transforming the second half: * produces “Condition Y”: * where we need this “Y” to vanish. * 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 a μ !

* 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 1 st order terms.

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

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

* Writing out the conditions, for every choice of the free index… * For example, * Similarly, we obtain a 0 = 0. * However, there appear no restrictions on a 2 and a 3. * Since the initial coordinate system was chosen so the Lorentz- boost is in the x- (i.e., 1 st ) direction, the “X = 0” conditions allow Lorentz-boosts with

* 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, a 2 and a 3. * 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 directions perpendicular to the boost velocity

* Open questions: * Second and higher order effects (some conditions on a 2 and a 3 ?) * 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