Questions: Electromagnetic Field Tensor and Covariance

Question 1

In frame S, there is a purely magnetic field B pointing in the z-direction and no electric field (E = 0). An observer in frame S' moves relative to S along the x-axis. What does the observer in S' measure?

• A. The same purely magnetic field B, since magnetic fields are Lorentz invariant
• B. Both an electric field and a magnetic field, because E and B mix under Lorentz boosts
• C. No fields at all, since moving observers see fields Doppler-shifted to zero
• D. Only an electric field, since the magnetic field is fully converted to electric in the moving frame

Answer: (B)

E and B are not separately Lorentz invariant — they are components of a single antisymmetric tensor Fμν and mix under boosts. Under a boost along x, the transverse components transform as E'_y = γ(E_y − vB_z) and B'_y = γ(B_y + vE_z/c²), with similar expressions for z-components. Starting with E = 0 and B in the z-direction, a boost along x produces nonzero transverse E fields in S'. This mixing is the physical motivation for packaging E and B into a single tensor rather than treating them as independently meaningful fields.

Question 2

In one inertial frame, a plane electromagnetic wave has E and B fields that are perpendicular to each other (E·B = 0). A physicist claims this orthogonality might not hold in other frames. Is this correct?

• A. Yes — orthogonality of E and B is frame-dependent and can change under boosts
• B. No — E·B is a Lorentz invariant; if it equals zero in one frame, it equals zero in all frames
• C. Only if the wave is circularly polarized; linear polarization does not preserve orthogonality
• D. Yes — the invariant is |B|² − |E|²/c², not E·B, so orthogonality can vary

Answer: (B)

The pseudoscalar F_μν F̃^μν ∝ E·B is a Lorentz invariant: every inertial observer computes the same value. If E·B = 0 in one frame, it is zero in all frames. The orthogonality of E and B for plane electromagnetic waves is a frame-independent fact, not an artifact of the particular frame used to describe them. Similarly, B² − E²/c² is the other independent invariant — if this is positive in one frame, it is positive in all frames.

Question 3

The four Maxwell equations (two inhomogeneous, two homogeneous) can be expressed as exactly two tensor equations using the electromagnetic field tensor Fμν.

• T. True
• F. False

Answer: True

In tensor form: the two inhomogeneous equations (Gauss's law and Ampère's law) become ∂_ν F^μν = μ₀ J^μ, and the two homogeneous equations (Faraday's law and ∇·B = 0) become ∂_[λ F_μν] = 0, equivalently ∂_ν F̃^μν = 0 with the dual tensor. Both are tensor equations that transform covariantly under Lorentz transformations — they hold in the same form in every inertial frame. This compactness is not merely aesthetic; it makes Lorentz covariance manifest and demonstrates that Maxwell's equations are already fully consistent with special relativity.

Question 4

The electric field E and magnetic field B are independently Lorentz invariant: their individual magnitudes may change between frames, but the physical distinction between electric and magnetic is frame-independent.

• T. True
• F. False

Answer: False

E and B are not independently invariant — they are observer-dependent projections of the single antisymmetric tensor Fμν and mix freely under Lorentz boosts. A purely electric field in one frame has both electric and magnetic components in a moving frame, and vice versa. There is no frame-independent division of Fμν into 'the electric part' and 'the magnetic part.' What are frame-independent are the two Lorentz scalar invariants: F_μν F^μν ∝ B² − E²/c² and F_μν F̃^μν ∝ E·B.

Question 5

Why does the fact that E and B mix under Lorentz transformations motivate packaging them into a single tensor Fμν? What does the tensor formalism make possible that component-by-component transformation rules do not?

Model answer: The mixing under boosts reveals that E and B are not independently meaningful objects — they are observer-dependent projections of a single underlying structure. Using separate transformation rules for E and B components is correct but opaque: it requires memorizing six transformation equations and gives no immediate insight into what is frame-independent. The tensor Fμν contains all six independent components and transforms via the single law F'^μν = Λ^μ_ρ Λ^ν_σ F^ρσ, which is manifestly Lorentz covariant. This makes several things possible: Maxwell's equations reduce to two compact tensor equations obviously covariant by form; Lorentz invariants like E·B and B² − E²/c² arise naturally as tensor contractions; and the field's intrinsic character can be classified frame-independently.

The tensor formalism is not just notation — it reveals the deep structure of electromagnetism as a relativistic field theory and is the direct foundation for the Lagrangian formulation and extension to quantum field theory.