18 | March | 2023 | Physics Town

A First Course in String Theory

(a) Show that for time-independent fields, the Maxwell equation $\displaystyle{T_{0ij}=0}$ implies that $\displaystyle{\partial_i E_j - \partial_j E_i = 0}$ . Explain why this condition is satisfied by the ansatz $\displaystyle{\vec E = - \nabla \Phi}$ .

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Eq. (3.23):

$\displaystyle{ \begin{aligned} T_{\mu \lambda \nu} &= \partial_\lambda F_{\mu \nu} + \partial_\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} \\ \end{aligned}}$