Introduction to Electrodynamics · Problem 12.12

The Lorentz transformation relates coordinates in two inertial frames, and you're now inverting it—finding the "reverse" transformation that expresses the unbarred frame in terms of the barred one. The key insight is that if frame $\bar{S}$ moves with velocity $v$ relative to frame $S$, then frame $S$ moves with velocity $-v$ relative to $\bar{S}$, so you simply swap the roles of the two frames and flip the sign of $v$. Working through the algebra carefully and verifying that your result matches the given equations builds confidence in the symmetry of relativity.

The Lorentz transformation relates coordinates in two inertial frames, and you're now inverting it—finding the "reverse" transformation that expresses the unbarred frame in terms of the barred one. The key insight is that if frame $\bar{S}$ moves with velocity $v$ relative to frame $S$, then frame $S$ moves with velocity $-v$ relative to $\bar{S}$, so you simply swap the roles of the two frames and flip the sign of $v$. Working through the algebra carefully and verifying that your result matches the given equations builds confidence in the symmetry of relativity.