*
Dometrious Gordine
Virginia Union University
Howard University REU Program
*
* Maxwell’s Equations
Gauss & Ampere
* Lorentz transformations
(symmetry of Maxwell’s equations)
Gauss & Faraday
Q: Can we extend
to non-constant v?
v is a constant
(matrix format)
*
* 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
* …turns the
standard
Lorentz-boost
matrix
* …into:
…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
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Transform the Maxwell’s equations:
Use that the equations in old coordinates hold.
Compute the transformation-dependent difference.
Derive conditions on the aμ parameters.
*
* The Gauss-Ampere equations:
* …transform as
Should vanish, = “condition X”
The original equation, = 0
*
* 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 aμ !
*
* 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.
*
* 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
*
* Writing out the small A’s for every choice of every free index:
Most of these vanish to 1st order.
*
* Writing out the conditions, for every choice of the free index…
* For example,
* Similarly, we obtain a0 = 0.
* However, there appear no restrictions on a2 and a3.
* Since the initial coordinate system was chosen so the Lorentzboost is in the x- (i.e., 1st) 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, 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 directions perpendicular to the boost velocity
*
* 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