(i) Obtain Maxwell's equations in empty space from the action functional

$$S\left[A_{\mu}\right]=-\frac{1}{\mu_{0}} \int d^{4} x \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$

where $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$.

(ii) A modification of Maxwell's equations has the action functional

$$\tilde{S}\left[A_{\mu}\right]=-\frac{1}{\mu_{0}} \int d^{4} x\left\{\frac{1}{4} F_{\mu \nu} F^{\mu \nu}+\frac{1}{2 \lambda^{2}} A_{\mu} A^{\mu}\right\},$$

where again $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ and $\lambda$ is a constant. Obtain the equations of motion of this theory and show that they imply $\partial_{\mu} A^{\mu}=0$.

(iii) Show that the equations of motion derived from $\tilde{S}$ admit solutions of the form

$$A^{\mu}=A_{0}^{\mu} e^{i k_{\nu} x^{\nu}}$$

where $A_{0}^{\mu}$ is a constant 4-vector, and the 4 -vector $k_{\mu}$ satisfies $A_{0}^{\mu} k_{\mu}=0$ and $k_{\mu} k^{\mu}=-1 / \lambda^{2}$.

(iv) Show further that the tensor

$$T_{\mu \nu}=\frac{1}{\mu_{0}}\left\{F_{\mu \sigma} F_{\nu}^{\sigma}-\frac{1}{4} \eta_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta}-\frac{1}{2 \lambda^{2}}\left(\eta_{\mu \nu} A_{\alpha} A^{\alpha}-2 A_{\mu} A_{\nu}\right)\right\}$$

is conserved, that is $\partial^{\mu} T_{\mu \nu}=0$.