Abstract
So far we have the field equations, the Maxwell equations, and using the Lorentz force law have constructed equations describing the conservation of energy and momentum between the fields and charges. Our next task is to cast these equations in a form that is consistent with special relativity. Special relativity has two fundamental predictions:
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Frame Invariance, that is, the laws of physics are the same in all inertial frames of reference.
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The speed of light is the same in all inertial frames.
The second of these yields the Lorentz transformation between one frame moving at constant velocity w.r.t. another.
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Notes
- 1.
This experiment was actually performed by Michelson and Morley in 1887 to attempt to measure our relative velocity w.r.t. the aether which was believed to fill the vacuum to allow the propagation of electromagnetic waves.
- 2.
This is known as a spacelike interval, if s is imaginary then it describes a timelike interval, that is, an object travelling faster than c.
- 3.
In everyday experience this effect is small, the electrons in a copper wire of 1 mm\(^2\) cross sectional area carrying 1 A have an average drift speed of \({\sim } 10^{-4}\,\mathrm{ms}^{-1}\) which is why household wiring doesn’t charge up.
- 4.
In electrostatics this reduces to the Coulomb gauge \(\nabla \cdot \mathbf{A}=0\) which gives Poisson equations for both \(\phi \) and \(\mathbf{A}\)
$$-\nabla ^2 \phi =\frac{\rho }{\epsilon _0}$$and
$$-\nabla ^2 \mathbf{A}=\mu _0 \mathbf{J}.$$ - 5.
In general if we have a well defined transformation that gives \(x'^{\alpha }= x'^{\alpha }(x^0,x^1,x^2,x^3)\) a covariant vector transforms as:
$$\begin{aligned} a'_\alpha = \frac{\partial x^0}{\partial x'^\alpha }a_0 +\frac{\partial x^1}{\partial x'^\alpha }a_1 \frac{\partial x^2}{\partial x'^\alpha }a_2 \frac{\partial x^3}{\partial x'^\alpha }a_3= \frac{\partial x^\beta }{\partial x'^\alpha }a_\beta \end{aligned}$$(3.45)and a contravariant vector transforms as
$$\begin{aligned} a'^{\alpha } =\frac{\partial x'^{\alpha }}{\partial x^\beta }a^\beta \end{aligned}$$(3.46)See the appendix on tensors for details.
- 6.
This is a member of the Lorentz Group of transformations, see e.g., Classical Electrodynamics 2nd Ed., J. D. Jackson, (Wiley 1975) for details.
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Chapman, S. (2021). A Frame Invariant Electromagnetism. In: Core Electrodynamics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-66818-1_3
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