Question

Show that the potential representation (Eq. 12.133) automatically satisfies [Suggestion: Use Prob. 12.54.]

Short answer

The potential representation is automatically satisfied$\frac{\partial {\mathrm{G}}^{\mathrm{\mu \nu }}}{\partial x^v}=0$

Step-by-step solution

Expression for the field tensor equation:

Using equation 12.130, write the equation for the field tensor.

${\mathit{F}}^{\mathbf{\mu }\mathbf{\nu }}\mathbf{=}\mathbf{}\frac{\mathbf{\partial }{\mathbf{F}}_{\mathbf{\mu }\mathbf{\nu }}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{\lambda }}}\mathbf{+}\frac{\mathbf{\partial }{\mathbf{F}}_{\mathbf{\nu }\mathbf{\lambda }}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{\mu }}}\mathbf{+}\frac{\mathbf{\partial }{\mathbf{F}}_{\mathbf{\lambda }\mathbf{\mu }}}{\mathbf{\partial }{\mathbf{x}}^{\mathbf{\nu }}}$ .........(1)

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{also}\mathrm{know}\mathrm{that}: \\ \frac{\partial }{\partial {\mathrm{x}}^2}={\partial }_{\mathrm{\lambda }} \\ \frac{\partial }{\partial {\mathrm{x}}^{\mathrm{\mu }}}={\partial }_{\mathrm{\mu }} \\ \frac{\partial }{\partial {\mathrm{x}}^{\mathrm{\nu }}}={\partial }_{\mathrm{\nu }} \\ \mathrm{Also},\mathrm{the}\mathrm{value}\mathrm{of}{\mathrm{F}}_{\mathrm{\mu \nu },}{\mathrm{F}}_{\mathrm{\mu \lambda }\mathrm{and}}{\mathrm{F}}_{\mathrm{\nu \lambda }}\mathrm{is}\mathrm{given}\mathrm{by}: \\ {\mathrm{F}}_{\mathrm{\mu \nu }}={\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\nu }}-{\partial }_{\mathrm{\nu }}{\mathrm{A}}_{\mathrm{\mu }} \\ {\mathrm{F}}_{\mathrm{\mu \lambda }}={\partial }_{\mathrm{\nu }}{\mathrm{A}}_{\mathrm{\lambda }}-{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\nu }} \\ {\mathrm{F}}_{\mathrm{\nu \mu }}={\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\mu }}-{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\lambda }} \end{gathered}$$

Show that :

${\mathbf{\partial }}_{\mathbf{\lambda }}{\mathbf{\partial }}_{\mathbf{\mu }\mathbf{}}{\mathbf{A}}_{\mathbf{V}\mathbf{}}\mathbf{=}{\mathbf{\partial }}_{\mathbf{\mu }}\mathbf{}{\mathbf{\lambda }}_{\mathbf{\lambda }\mathbf{}}{\mathbf{A}}_{\mathbf{V}\mathbf{}}$

$$\begin{gathered} \mathrm{Substitute}{\partial }_{\mathrm{\lambda }}\mathrm{for}\frac{\partial }{\partial {\mathrm{X}}^{\mathrm{\lambda }}},{\partial }_{\mathrm{\mu }}\mathrm{for}\frac{\partial }{\partial {\mathrm{X}}^{\mathrm{\mu }}},{\partial }_{\mathrm{V}}\mathrm{for}\frac{\partial }{\partial {\mathrm{X}}^{\mathrm{V}}},{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{V}}-{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\mu }}\mathrm{for}{\mathrm{F}}_{\mathrm{\mu v}},{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\lambda }}-{\partial }_{\mathrm{\lambda }}-{\mathrm{A}}_{\mathrm{V}} \\ \mathrm{for}{\mathrm{F}}_{\mathrm{v\lambda }}\mathrm{and}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\mu }}-{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\lambda }}{\mathrm{F}}_{\mathrm{i}\mathrm{\mu }}\mathrm{in}\mathrm{equation}\left(1\right) \\ \frac{\partial {\mathrm{F}}_{\mathrm{\mu V}}}{\partial {\mathrm{X}}^2}+\frac{\partial {\mathrm{F}}_{\mathrm{V\lambda }}}{\partial {\mathrm{X}}^{\mathrm{\mu }}}+\frac{\partial {\mathrm{F}}_{\mathrm{\lambda \mu }}}{\partial {\mathrm{X}}^{\mathrm{V}}}={\partial }_{\mathrm{\lambda }}{\mathrm{F}}_{\mathrm{\mu V}}+{\partial }_{\mathrm{\mu }}{\mathrm{F}}_{\mathrm{V\lambda }}+{\partial }_{\mathrm{V}}{\mathrm{F}}_{\mathrm{\lambda \mu }} \\ \frac{\partial {\mathrm{F}}_{\mathrm{\mu V}}}{\partial {\mathrm{X}}^2}+\frac{\partial {\mathrm{F}}_{\mathrm{V\lambda }}}{\partial {\mathrm{X}}^{\mathrm{\mu }}}+\frac{\partial {\mathrm{F}}_{\mathrm{\lambda \mu }}}{\partial {\mathrm{X}}^{\mathrm{V}}}={\partial }_{\mathrm{\lambda }}\left({\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{V}}-{\partial }_{\mathrm{V}}\mathrm{A\mu }\right)+{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\lambda }}-{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{V}}\right)+{\partial }_{\mathrm{V}}\left({\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\mu }}-{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\lambda }}\right) \\ \frac{\partial {\mathrm{F}}_{\mathrm{\mu V}}}{\partial {\mathrm{X}}^2}+\frac{\partial {\mathrm{F}}_{\mathrm{V\lambda }}}{\partial {\mathrm{X}}^{\mathrm{\mu }}}+\frac{\partial {\mathrm{F}}_{\mathrm{\lambda \mu }}}{\partial {\mathrm{X}}^{\mathrm{V}}}=\left({\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{V}}-{\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\mu }}\right)+\left({\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\lambda }}-{\partial }_{\mathrm{\mu }}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{V}}\right)+\left({\partial }_{\mathrm{V}}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\mu }}-{\partial }_{\mathrm{V}}{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\lambda }}\right) \\ \mathrm{On}\mathrm{further}\mathrm{solving}, \\ \frac{\partial {\mathrm{F}}_{\mathrm{\mu V}}}{\partial {\mathrm{X}}^2}+\frac{\partial {\mathrm{F}}_{\mathrm{V\lambda }}}{\partial {\mathrm{X}}^{\mathrm{\mu }}}+\frac{\partial {\mathrm{F}}_{\mathrm{\lambda \mu }}}{\partial {\mathrm{X}}^{\mathrm{V}}}=\left({\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{V}}-{\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\mu }}\right)+\left({\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\lambda }}-{\partial }_{\mathrm{\mu }}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{V}}\right)+\left({\partial }_{\mathrm{V}}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\mu }}-{\partial }_{\mathrm{V}}{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\lambda }}\right) \\ \frac{\partial {\mathrm{F}}_{\mathrm{\mu V}}}{\partial {\mathrm{X}}^2}+\frac{\partial {\mathrm{F}}_{\mathrm{V\lambda }}}{\partial {\mathrm{X}}^{\mathrm{\mu }}}+\frac{\partial {\mathrm{F}}_{\mathrm{\lambda \mu }}}{\partial {\mathrm{X}}^{\mathrm{V}}}=\left({\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\nu }}-{\partial }_{\mathrm{\mu }}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\nu }}\right)+\left(-{\partial }_{\mathrm{\lambda }}{\partial }_{\mathrm{V}}{\mathrm{A}}_{\mathrm{\mu }}+{\partial }_{\mathrm{\nu }}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\mu }}\right)+\left({\partial }_{\mathrm{\mu }}{\partial }_{\mathrm{\lambda }}{\mathrm{A}}_{\mathrm{\lambda }}-{\partial }_{\mathrm{\nu }}{\partial }_{\mathrm{\mu }}{\mathrm{A}}_{\mathrm{\lambda }}\right) \\ \frac{\partial {\mathrm{F}}_{\mathrm{\mu V}}}{\partial {\mathrm{X}}^2}+\frac{\partial {\mathrm{F}}_{\mathrm{V\lambda }}}{\partial {\mathrm{X}}^{\mathrm{\mu }}}+\frac{\partial {\mathrm{F}}_{\mathrm{\lambda \mu }}}{\partial {\mathrm{X}}^{\mathrm{V}}}=0 \\ \mathrm{Hence},\mathrm{the}\mathrm{above}\mathrm{equation}\mathrm{satisfies}\frac{\partial {\mathrm{G}}^{\mathrm{\mu \nu }}}{\partial {\mathrm{x}}^{\mathrm{\nu }}}=0 \\ \mathrm{Therefore},\mathrm{the}\mathrm{potential}\mathrm{representattion}\mathrm{is}\mathrm{automatically}\mathrm{satisfied}\frac{\partial {\mathrm{G}}^{\mathrm{\mu \nu }}}{\partial {\mathrm{x}}^{\mathrm{\nu }}}=0 \end{gathered}$$