= Solution
Let $f(r)=1-2M/r+Q^2/r^2$. A positive-radius horizon requires a root of $r^2-2Mr+Q^2=0$, giving
$$
r_\pm=M\pm\sqrt{M^2-Q^2}.
$$
For the usual asymptotically flat <Reissner-Nordstrom spacetime>, the black-hole conditions are
$$
\boxed{M>0,\qquad M\geq |Q|.}
$$
The strict inequality gives a nonextremal outer <event horizon> at $r_+$; equality gives an <extremal black hole>. If $M<|Q|$ with $M>0$, there is no horizon and the central <curvature singularity> is naked. Nonpositive $M$ cannot supply a positive-radius black-hole horizon; $M=Q=0$ is <Minkowski spacetime>. For $Q=0$, the zero root $r_-=0$ is the singularity, not an additional regular horizon.
In the static coordinates the vector $k=\partial_t$ has constant components and every metric component is independent of $t$. Its <Lie derivative> therefore satisfies
$$
(\mathcal L_k g)_{ab}=k^c\partial_cg_{ab}+g_{cb}\partial_ak^c+g_{ac}\partial_bk^c=0.
$$
For the torsion-free <Levi-Civita connection>, $\mathcal L_k g=2\nabla_{(a}k_{b)}$, proving the <Killing equation> directly. This also normalizes the <Killing vector field> to unit time translation at infinity.
At $r=r_+$, $k^2=-f=0$. Use the regular advanced chart $v=t+r_*$ with $dr_*/dr=f^{-1}$, in which
$$
ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_2^2,\qquad k=\partial_v.
$$
On the horizon, $k_a=(dr)_a$. Thus the <Killing vector field> is both null and normal to the horizon, and its orbits generate that <Killing horizon>. The normal lines of a <null hypersurface> are <null geodesics>, possibly with nonaffine parameter. To see this without assuming a separate geodesic property, the <Killing equation> gives
$$
k^a\nabla_a k_b=-k^a\nabla_b k_a=-\frac12\nabla_b(k^2)=\frac12f'(r)(dr)_b.
$$
Restriction to the future horizon therefore gives
$$
k^a\nabla_a k_b=\frac12f'(r_+)k_b.
$$
The acceleration is proportional to the tangent, which is exactly the <geodesic equation> with a nonaffine parameter. An affinely parametrized tangent can be obtained by rescaling $k$ along each generator; that rescaled vector need not itself extend to a spacetime <Killing vector field>. At a nonextremal <bifurcation surface>, $k$ vanishes, so a nonzero generator tangent must be described in a regular affine parametrization there.
On the past branch, use the retarded chart with radial-time cross term $-2\,du\,dr$. Now $k_a=-(dr)_a$ at the horizon, so the same calculation yields the opposite signed nonaffinity. Since $r_+$ is constant and the spacetime is spherically symmetric, $f'(r_+)/2$ is constant over the future horizon. This gives the two signs in the horizon relation without assuming their value beforehand.
Factor $f=(r-r_+)(r-r_-)/r^2$. Its derivative at the outer horizon gives the <Reissner-Nordstrom horizon surface gravity>
$$
\boxed{\kappa=\frac12 f'(r_+)=\frac{r_+-r_-}{2r_+^2}
=\frac{\sqrt{M^2-Q^2}}{\left(M+\sqrt{M^2-Q^2}\right)^2}.}
$$
It is positive for $M>|Q|$, and at $Q=0$ reduces to the <Schwarzschild surface gravity> $1/(4M)$. For $M=|Q|>0$, the root is double and $\kappa=0$. Thus \b[the horizon remains present when its surface gravity vanishes]: it is a <degenerate Killing horizon>, and $k$ itself has affine parametrization along its nonzero horizon generators. The strictly positive-constant description applies only to the nonextremal case; the zero value is the legitimate extremal limit.
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