Chapter 16

Show that under the Galilean transformation $$ \bar{x}=x, \quad \bar{y}=y, \quad \bar{z}=z-v t, \quad \bar{t}=t $$ the equation $$ \nabla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=0 $$ becomes $$ \frac{\partial^{2} \phi}{\partial \bar{x}^{2}}+\frac{\partial^{2} \phi}{\partial \bar{y}^{2}}+\left(1-\frac{v^{2}}{c^{2}}\right) \frac{\partial^{2} \phi}{\partial \bar{z}^{2}}+\frac{2 v}{c^{2}} \frac{\partial^{2} \phi}{\partial \bar{t} \partial \bar{z}}-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial \bar{t}^{2}}=0 $$

Show that the Lorentz transformation may be written vectorially thus: $$ \begin{aligned} \overline{\mathbf{r}} &=\gamma[(\mathrm{n} \cdot \mathrm{r}) \mathrm{n}-\mathrm{v} t]+[\mathrm{r}-(\mathrm{n} \cdot \mathrm{r}) \mathrm{n}] \\\ \bar{t} &=\gamma\left(t-\frac{\mathrm{v} \cdot \mathrm{r}}{c^{2}}\right) \end{aligned} $$ where \(\mathrm{n}\) is a unit vector in the direction of \(\mathrm{v}\).