= Solution
Use the intended positive-mass family: write $b=r_+>0$ and $a=r_-=2P^2/b\ge0$, so $R^2=r(r-a)$, $V=1-b/r$ and $ab=2P^2$. Lowering the index of the <Killing vector field> gives $k^\flat=-Vdt$, hence
$$
dk^\flat=V'\,dt\wedge dr.
$$
For the stipulated orientation and Lorentzian signature, $\sqrt{-g}=R^2\sin\theta$ and the <Hodge star operator> obeys
$$
\star(dt\wedge dr)=-R^2\sin\theta\,d\theta\wedge d\varphi.
$$
The minus sign follows from $g^{tt}g^{rr}=-1$. Thus the <Komar mass of the magnetic dilaton black hole> is
$$
\boxed{M_{\rm Komar}(r)=\frac12R^2V'=\frac b2\left(1-\frac ar\right)=\frac{r_+}{2}-\frac{P^2}{r}.}
$$
Its asymptotic mass is $M=r_+/2$.
For any <Killing vector field>, its antisymmetric derivative satisfies $\nabla_a(dk^\flat)^{ab}=-2R^b{}_ck^c$. Applying <Stokes theorem> between two spheres relates the difference of their <Komar mass> to this Ricci source. It vanishes in vacuum but need not vanish for the nonzero <Ricci tensor> of <Einstein-Maxwell-dilaton theory>. Here, explicitly, $dM_{\rm Komar}/dr=P^2/r^2$. The potential $A=P\cos\theta\,d\varphi$ is only a local gauge potential near the polar axes; the mass computation uses its globally defined field strength and the metric, so does not depend on this gauge-patch issue.
Back to article page
