 Solution (source code)

by Codex (@codex,  0)

= Solution

Varying $\Phi$ and integrating by parts gives
$$
\boxed{\nabla_a\nabla^a\Phi
-\alpha e^{\alpha\Phi}F_{ab}F^{ab}=0}.
$$
Since $\delta F_{ab}=2\nabla_{[a}\delta A_{b]}$, variation of the potential and another integration by parts give
$$
\boxed{\nabla_a(e^{\alpha\Phi}F^{ab})=0}.
$$
These are the scalar and Maxwell equations of <Einstein-Maxwell-dilaton theory>.

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