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Special relativity · Electrodynamics §9

Charge density is frame-dependent,
while total charge is invariant.

How do charge density and current density transform between inertial frames, and why is a neutral current-carrying wire charged in a moving frame?

Read §9 of the 1905 relativity paper →

SR-12 · Special Relativity §9

Charge and current density in moving frames

Ideal relativistic four-current model, host calculation

Relativistic Four-Current Visualization

Mode: neutral-conductor (v = 0.60c, γ = 1.2500)
Stationary Frame K (Laboratory)ρ = 0 C/m³Equal ion & electron linear density → Neutral wire (ρ = 0)Moving Frame k (Speed v = 0.60c)ρ' = -2.5017e-9 C/m³, J'x = 1 A/m²Differential Lorentz contraction → Net charge density ρ' ≠ 0
Set 0 for neutral conductor
Conduction current along x
QuantityStationary Frame (K)Moving Frame (k)UnitLorentz Transformation Law
Charge Density ρ0-2.5017e-9C/m³ρ' = γ (ρ - vJx/c²)
Current Density JxvaluevalueA/m²J'x = γ (Jx - vρ)
Lorentz Factor γ1.2511 / √(1 - v²/c²)
Four-Current Invariant (cρ)² - |J|²-1A²/m⁴Exact scalar invariant across all frames

Predict: Is a Neutral Wire Still Neutral in a Moving Frame?

A neutral wire in the laboratory carries a current in the +x direction. Described from a frame moving in the +x direction at 0.6c, is the wire still electrically neutral?

Overview: Electric charge density and current density transform together under a boost; a wire that is electrically neutral in one frame carries a net charge density in another.

Four-Current Invariant: The four-current (cρ, J) transforms as a Lorentz four-vector. The combination (cρ)² - |J|² is an exact relativistic invariant in all inertial frames.

Current Loops & Total Charge: For a current-carrying loop, Lorentz contraction shortens the wire segments while the charge per unit length shifts, ensuring total charge remains invariant while opposite legs carry equal and opposite static charges.

Continuity Invariance (§9): Section 9 shows that the continuity equation ∂ρ/∂t + ∇·J = 0 is invariant under Lorentz transformation: charge conservation holds identically in all inertial frames without modifying Maxwell's electrodynamics.

Not modeled: microscopic lattice dynamics and thermal vibrations, self-inductance and transient current startup, radiation reaction from accelerated charges, material resistance and Joule heating, finite wire thickness effects, gravitational fields and general relativistic curvature.

Worked case (readable without JavaScript)

Consider a neutral conductor in the stationary frame K with volumetric charge density ρ = 0 and current density Jx = 1 A/m², viewed from a frame k boosted along x at speed v = 0.6c (γ = 1.25).

ρ′=γ(ρ−vJxc2)=1.25(0−0.6c)=−0.75c≈−2.5017×10−9 C/m3\rho' = \gamma\left(\rho - \frac{v J_x}{c^2}\right) = 1.25\left(0 - \frac{0.6}{c}\right) = -\frac{0.75}{c} \approx -2.5017\times 10^{-9}\text{ C/m}^3
Jx′=γ(Jx−vρ)=1.25(1−0)=1.25 A/m2J'_x = \gamma\left(J_x - v\rho\right) = 1.25(1 - 0) = 1.25\text{ A/m}^2

Both coordinate frames agree exactly on the relativistic four-current invariant (cρ)² − |J|²:

(cρ)2−Jx2=0−1=−1 (A/m2)2,(cρ′)2−(Jx′)2=(−0.75)2−(1.25)2=0.5625−1.5625=−1 (A/m2)2(c\rho)^2 - J_x^2 = 0 - 1 = -1\text{ (A/m}^2)^2,\qquad (c\rho')^2 - (J'_x)^2 = (-0.75)^2 - (1.25)^2 = 0.5625 - 1.5625 = -1\text{ (A/m}^2)^2

For a rectangular current loop of length lx = 1 m carrying current I = 1 A at 0.6c, Lorentz contraction shortens the x-legs to l′x = lx/γ = 0.8 m. The top leg carries charge q′+ = −(v I lx)/c² ≈ −2.0014×10⁻⁹ C while the bottom leg carries q′− = +(v I lx)/c² ≈ +2.0014×10⁻⁹ C. The total charge remains identically zero, verifying that total charge is an exact Lorentz scalar.

Return to paper section 9 →SR-08: Field frame change