Show that Frank in the fixed system will also determine the time dilation result by having the sparkler be at rest in the system K’.
Strategy We should be able to proceed similarly to the derivation we did before when the sparkler was at rest in system K. In this case Mary lights the sparkler in the moving system K’. The time interval over which the sparkler is lit is given by T_{0}^{\prime}=t_{2}^{\prime}-t_{1}^{\prime} and the sparkler is placed at the position x_{1}^{\prime}=x_{2}^{\prime} \text { so that } x_{2}^{\prime}-x_{1}^{\prime}=0 . \text { In this case } T_{0}^{\prime} is the proper time. We use the Lorentz transformation from Equation (2.18) to determine the time difference T=t_{2}-t_{1} as measured by the clocks of Frank and his colleagues.
\begin{aligned}&x=\frac{x^{\prime}+v t^{\prime}}{\sqrt{1-\beta^{2}}} \\&y=y^{\prime} \\&z=z^{\prime} \\&t=\frac{t^{\prime}+\left(v x^{\prime} / c^{2}\right)}{\sqrt{1-\beta^{2}}}\end{aligned} (2.18)
