In a frame S, two events are observed: event 1: a pion is created at rest at the origin and event 2: the pion disintegrates after time \(\tau .\) Another observer in a frame \(\mathrm{S}^{\prime}\) is moving in the positive direction along the positive \(x\) -axis with a constant speed \(v\) and observes the same two events in his frame. The origins of the two frames coincide at \(t=t^{\prime}=0 .\) (a) Find the positions and timings of these two events in the frame \(S^{\prime}\) (a) according to the Galilean transformation, and (b) according to the Lorentz transformation.

For the Galilean transformation, the equations are given by: \[ x' = x - vt \] \[ t' = t \] For event 1, in frame S, we have x = 0 and t = 0. Applying the Galilean transformation: Event 1: \[ x' = 0 - v \times 0 = 0 \] \[ t' = 0 \] For event 2, in frame S, we have x = 0 and t = τ. Applying the Galilean transformation: Event 2: \[ x' = 0 - vτ \] \[ t' = τ \] According to the Galilean transformation, the position and timing of the two events in frame S' are: Event 1: \(x' = 0, t' = 0\) Event 2: \(x' = -vτ, t' = τ\)

For the Lorentz transformation, the equations are given by: \[ x' = \frac{x - vt}{\sqrt{1- \frac{v^2}{c^2}}} \] \[ t' = \frac{t - \frac{vx}{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}} \] Where c is the speed of light. For event 1, in frame S, we have x = 0 and t = 0. Applying the Lorentz transformation: Event 1: \[ x' = \frac{0 - v \times 0}{\sqrt{1- \frac{v^2}{c^2}}} = 0 \] \[ t' = \frac{0 - \frac{v \times 0}{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}} = 0 \] For event 2, in frame S, we have x = 0 and t = τ. Applying the Lorentz transformation: Event 2: \[ x' = \frac{0 - vτ}{\sqrt{1- \frac{v^2}{c^2}}} \] \[ t' = \frac{τ - \frac{v \times 0}{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{τ}{\sqrt{1 - \frac{v^2}{c^2}}} \] According to the Lorentz transformation, the position and timing of the two events in frame S' are: Event 1: \(x' = 0, t' = 0\) Event 2: \(x' = \frac{-vτ}{\sqrt{1 - \frac{v^2}{c^2}}}, t' = \frac{τ}{\sqrt{1 - \frac{v^2}{c^2}}}\)