= Komar mass of the magnetic dilaton black hole
{c}
{title2=$M_K(r)=r_+/2-P^2/r$}
For the <magnetically charged dilaton black hole>, the metric-dual one-form of $k=\partial_t$ is $k^\flat=-Vdt$. With orientation $dt\wedge dr\wedge d\theta\wedge d\phi$, its <Hodge star operator> gives $\star dk^\flat=-V'R^2\sin\theta\,d\theta\wedge d\phi$. Therefore the <Komar mass> is $R^2V'/2=r_+/2-P^2/r$. Its radial dependence is allowed because the <Ricci tensor> is nonzero; the divergence of $dk^\flat$ is sourced by $R_{ab}k^b$. At the horizon $M_K(r_+)=(r_+-r_-)/2=\kappa A_H/(4\pi)$.
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