Questions: Lorentz Covariance of Maxwell's Equations

Question 1

An observer at rest sees a stationary point charge producing a pure electric field with no magnetic field. A second observer moves at constant velocity relative to the first. What does the moving observer measure?

• A. The same pure electric field — the field is a physical object that cannot change based on observer motion
• B. No fields at all — in the moving frame, the charge is at a different position so its effects cancel
• C. A combination of electric and magnetic fields — the moving charge appears as a current, and E and B mix under Lorentz boosts
• D. A pure magnetic field — velocity converts electric fields to magnetic fields completely

Answer: (C)

This is the central physical insight of Lorentz covariance: E⃗ and B⃗ are not separately invariant quantities; they mix under Lorentz boosts just as space and time coordinates mix. The moving observer sees the charge as moving — a moving charge is a current — so a magnetic field appears. Neither observer is wrong; they are both making correct measurements in their own inertial frames. This is precisely why the electromagnetic field tensor F^μν is necessary: it packages all six field components together so the transformation law is clean.

Question 2

What does it mean to say that Maxwell's equations are 'Lorentz covariant'?

• A. The speed of light is the same in all inertial frames, so Maxwell's equations must involve c
• B. Maxwell's equations take the same mathematical form in every inertial reference frame, so no frame is privileged
• C. Maxwell's equations were derived assuming a stationary ether, and Lorentz showed how to correct them for moving frames
• D. The electromagnetic fields E⃗ and B⃗ are unchanged (invariant) under Lorentz transformations

Answer: (B)

Covariance means the equations preserve their form under Lorentz transformations — not that the fields themselves are unchanged (they are not invariant; they transform). Option C is historically backwards: Maxwell's equations needed no correction, unlike Newtonian mechanics. Option D confuses covariance (same form) with invariance (same value). The equations ∂_μ F^μν = J^ν and ∂_[μ F_νλ] = 0 are manifestly covariant because every index is contracted according to consistent tensor rules.

Question 3

Maxwell's equations required modification when special relativity was developed, just as Newton's laws required modification.

• T. True
• F. False

Answer: False

This is historically and physically backwards. Maxwell's equations were already exactly relativistic — they did not need modification. It was *Newtonian mechanics* that required modification (giving way to relativistic mechanics). Einstein's 1905 paper started from Maxwell's equations as correct and reconstructed kinematics around them. The Lorentz transformations were originally derived to find the symmetry group of Maxwell's equations, not the other way around. This is why the paper was titled 'On the Electrodynamics of Moving Bodies.'

Question 4

Under a Lorentz boost, a pure electric field in one inertial frame becomes a mixture of electric and magnetic field components in another frame.

• T. True
• F. False

Answer: True

This is one of the key physical consequences of Lorentz covariance. E⃗ and B⃗ are not independently Lorentz-invariant; they transform into each other under boosts. This is why the electromagnetic field tensor F^μν, which packages all six field components together, is the natural object in relativistic electrodynamics. Two relativistic invariants — E⃗·B⃗ and E²−c²B² — remain unchanged across frames, but the individual E⃗ and B⃗ vectors generally do not.

Question 5

Why does packaging the electric and magnetic fields into the electromagnetic field tensor F^μν make Lorentz covariance manifest, while treating them as separate 3-vectors obscures it?

Model answer: In 3D notation, E⃗ and B⃗ appear as separate objects with separate transformation laws that only mix correctly if you apply the right formulas. In 4D tensor notation, F^μν is a single rank-2 tensor that transforms automatically under Lorentz transformations via the standard tensor rule: F'^μν = Λ^μ_α Λ^ν_β F^αβ. Maxwell's equations written as ∂_μ F^μν = μ₀J^ν have all indices contracted, which by the rules of tensor calculus guarantees the equation holds in all frames simultaneously. No separate frame-by-frame verification is needed. The covariance is 'manifest' because it is built into the notation: a properly contracted tensor equation is automatically Lorentz covariant.

This is the payoff of the 4D formalism. The 3D equations hide the relativistic structure; the 4D equations reveal it. The same logic applies throughout modern physics: gauge theories, general relativity, and quantum field theory all use tensor/spinor notation precisely because it makes symmetry properties manifest rather than requiring case-by-case verification.