Introduction to Lorentz Invariant Quantities

In special relativity, quantities invariant under Lorentz transformation remain identical across inertial frames moving at constant velocity. This ensures that physical laws like Maxwell’s equations hold universally. For CSIR NET aspirants, understanding which quantities are invariant — charge density, charge, current, or electric field — is crucial for solving electrodynamics questions.

Question Analysis

The question asks which physical quantity remains unchanged between inertial frames connected by Lorentz transformations in special relativity. Lorentz invariance means a physical quantity has the same value regardless of the relative motion between observers.

Option Breakdown

• a. Charge density: Changes due to Lorentz contraction of volume. A uniform charge density \( \rho_0 \) in one frame appears as \( \gamma \rho_0 \) in a boosted frame, where \( \gamma = 1 / \sqrt{1 – v^2 / c^2} \).
• b. Charge: The total charge \( Q = \int \rho \, d^3x \) stays the same across frames. Charge conservation (\( \partial_\mu j^\mu = 0 \)) and integration over simultaneity surfaces prove \( Q’ = Q \) using Lorentz transformations.
• c. Current: The current \( I = dQ/dt \) varies because time dilates and charge motion differs between frames. Current density transforms as part of the four-current \( j^\mu \).
• d. Electric field: Fields mix under boosts; parallel components may stay similar, but perpendicular ones transform as \( E’_\perp = \gamma (E + v \times B)_\perp \), hence not invariant.

Correct answer: b. Charge

Why Charge is Lorentz Invariant

The total electric charge \( Q \) is invariant under Lorentz transformation. This can be shown using the four-current \( j^\mu = (c\rho, \mathbf{j}) \), where \( \partial_\mu j^\mu = 0 \) ensures charge conservation. Derivations show that:

\( Q’ = \int \rho’ \, d^3x’ = Q \)

after accounting for simultaneity adjustments with the Lorentz transformations:

\( t = \gamma (v x’ / c^2 + t’) \), \( x = \gamma (x’ + v t’) \).

Although the charge density \( \rho \) transforms to \( \gamma \rho \), the contracted volume changes by \( 1 / \gamma \), preserving the total charge \( Q \).

Transformations of Other Quantities

Other quantities change under Lorentz transformations:

• • Charge density and current: Form components of the four-current vector, transforming as:

ρ' = γ(ρ - v·j / c²)
j'x = γ(j_x - vρ)

• • Current: Depends on the frame because both motion and time measurement vary.
  • Electric field: Components mix with the magnetic field:

E'∥ = E∥
E'⊥ = γ(E⊥ + v × B)_⊥

This mixing means the electric field is not Lorentz invariant.

Lorentz Transformation Behavior Comparison