Electricity and Magnetism · Problem 5.12

The key insight is that Gauss's law must be frame-independent—the total electric flux through any closed surface should give the same enclosed charge regardless of which inertial frame you measure from. When you boost to a frame moving parallel to a charged sheet, both the electric field and the area element perpendicular to it transform by factors involving $\gamma$, but these transformations must conspire so that $\oint \vec{E} \cdot d\vec{A}$ remains proportional to $Q_{\text{enc}}$. Working through the algebra with arbitrary $\gamma$ (rather than a specific velocity) directly demonstrates this cancellation and confirms that the fundamental form of Gauss's law survives relativistic transformations.

The key insight is that Gauss's law must be frame-independent—the total electric flux through any closed surface should give the same enclosed charge regardless of which inertial frame you measure from. When you boost to a frame moving parallel to a charged sheet, both the electric field and the area element perpendicular to it transform by factors involving $\gamma$, but these transformations must conspire so that $\oint \vec{E} \cdot d\vec{A}$ remains proportional to $Q_{\text{enc}}$. Working through the algebra with arbitrary $\gamma$ (rather than a specific velocity) directly demonstrates this cancellation and confirms that the fundamental form of Gauss's law survives relativistic transformations.