Questions: Lorenz Gauge

Question 1

In the Lorenz gauge, the scalar potential φ at a field point depends on the charge distribution at what time?

• A. The current time t — the potential responds instantaneously
• B. The retarded time t − r/c, where r is the source-to-field distance
• C. The advanced time t + r/c — the potential anticipates future sources
• D. An averaged time, since the gauge condition mixes past and present

Answer: (B)

In the Lorenz gauge, both φ and A satisfy wave equations whose solutions are the retarded potentials: the field at point r and time t depends on sources at the retarded time t − r/c, meaning the information takes exactly r/c (light travel time) to propagate from source to field point. This makes causal structure manifest — cause precedes effect by the light-travel delay. Option A describes the Coulomb gauge, where φ satisfies an instantaneous Poisson equation (∇²φ = −ρ/ε₀) — an apparent instantaneous action that is nonetheless physically harmless because it cancels in the final E and B fields.

Question 2

What is the key mathematical simplification that imposing the Lorenz gauge condition achieves in Maxwell's equations?

• A. It eliminates the vector potential A, leaving only a scalar equation for φ
• B. It reduces Maxwell's four equations to two
• C. It decouples the equations for φ and A, each satisfying its own wave equation driven by its source
• D. It transforms the equations to static form, valid only for slowly varying fields

Answer: (C)

Without a gauge condition, the equations for φ and A are coupled — changes in one affect the other. Substituting the Lorenz condition ∇·A + (1/c²)∂φ/∂t = 0 into Maxwell's equations yields two independent wave equations: □²φ = −ρ/ε₀ and □²A = −μ₀J. Each potential is driven only by its own source (charge density or current density), with no cross-coupling. This is a dramatic simplification that makes the equations amenable to the retarded Green's function technique and to quantization. By contrast, in the Coulomb gauge φ satisfies an instantaneous equation, and the equation for A contains a coupling term involving φ.

Question 3

The Lorenz gauge condition ∂_μ A^μ = 0 takes the same form in all inertial reference frames — it is a Lorentz-covariant equation.

• T. True
• F. False

Answer: True

This is the defining virtue of the Lorenz gauge. Written in four-vector notation as ∂_μ A^μ = 0, the condition is the four-divergence of the four-potential, which is a Lorentz scalar equation — it transforms properly under Lorentz boosts and has the same form in all inertial frames. This is in sharp contrast to the Coulomb gauge ∇·A = 0, which is a condition on only the spatial components of A and does not maintain the same form after a Lorentz boost (a boosted observer generally sees a non-zero ∇·A). The covariance of the Lorenz gauge makes it the natural choice for relativistic electrodynamics and quantum field theory.

Question 4

In the Coulomb gauge, the scalar potential φ propagates causally at the speed of light, which makes the Coulomb gauge more suitable for problems involving radiation than the Lorenz gauge.

• T. True
• F. False

Answer: False

This is backwards. In the Coulomb gauge, φ satisfies the instantaneous Poisson equation ∇²φ = −ρ/ε₀, meaning φ responds to charge density changes instantaneously across all space — an apparent violation of causality. In reality, causality is preserved because the instantaneous φ is exactly canceled by compensating terms in A when computing the physical fields E and B, but this cancellation is non-obvious and requires careful treatment. The Lorenz gauge is far superior for radiation and relativistic problems precisely because both φ and A satisfy wave equations that propagate at c, making the causal structure manifest rather than hidden.

Question 5

Why does the Lorenz gauge make the causal structure of electrodynamics transparent, while the Coulomb gauge obscures it, even though both gauges describe identical physical fields?

Model answer: In the Lorenz gauge, both φ and A satisfy wave equations and their solutions are retarded potentials — each depends on sources at the retarded time t − r/c, so the light-travel delay is explicit. In the Coulomb gauge, φ satisfies an instantaneous Poisson equation (φ responds everywhere at once to charge changes), but A contains compensating terms that, together with φ, always produce retarded E and B. The causality is real in both cases, but in the Coulomb gauge it is hidden in a cancellation between φ and A, whereas in the Lorenz gauge it is directly encoded in each potential individually.

Gauge freedom means many (φ, A) pairs describe the same E and B. Physical observables (E, B) are always causal — they propagate at c. But the potentials themselves are not directly observable, so they can behave non-causally (like Coulomb-gauge φ) without violating physics, as long as the final fields come out causal. The Lorenz gauge is the 'honest' representation: each potential individually reflects the causal structure. The Coulomb gauge is valid but involves a sort of accounting trick — instantaneous φ plus compensating A conspire to give causal fields. For this reason, the Lorenz gauge is preferred in relativistic and quantum-field-theory contexts where manifest covariance simplifies the analysis greatly.