= First-order Maxwell action
{title2=$S=\frac1{16\pi}\int\sqrt{-g}\,[F^{ab}F_{ab}-2F^{ab}(\nabla_aA_b-\nabla_bA_a)]$}
Vary the antisymmetric auxiliary <electromagnetic field tensor> and the <electromagnetic four-potential> independently. The first variation sets $F_{ab}=\nabla_aA_b-\nabla_bA_a$ algebraically, and variation of the potential gives $\nabla_bF^{ab}=0$. Substitution of the algebraic equation reduces the first-order action to the usual negative quadratic <Maxwell action>. Metric variation gives the same stress tensor after eliminating the auxiliary field, because the chain-rule term multiplying its variational equation then vanishes.
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