= Solution
\b[The two identities.] Set $F_{\beta\gamma}=\nabla_\beta K_\gamma$. The <Killing equation> says $F_{\beta\gamma}=-F_{\gamma\beta}$. The <Levi-Civita connection> has symmetric lower connection indices, so they cancel in the antisymmetric difference:
$$
\boxed{2\nabla_\beta K_\gamma
=\nabla_\beta K_\gamma-\nabla_\gamma K_\beta
=\partial_\beta K_\gamma-\partial_\gamma K_\beta.}
$$
Use normalized antisymmetrization, so
$$
3K_{[\alpha}F_{\beta\gamma]}
=K_\alpha F_{\beta\gamma}
+K_\beta F_{\gamma\alpha}
+K_\gamma F_{\alpha\beta}.
$$
Put $a^\sigma=K^\tau\nabla_\tau K^\sigma$. Contracting with $F^{\beta\gamma}$, the second term is $a^\gamma F_{\gamma\alpha}$ and the third is the same after using antisymmetry. Thus
$$
3F^{\beta\gamma}K_{[\alpha}F_{\beta\gamma]}
=K_\alpha F^{\beta\gamma}F_{\beta\gamma}
+2F_{\sigma\alpha}a^\sigma.
$$
Rearranging proves the <Killing derivative contraction identity>
$$
\boxed{
K_\alpha(\nabla^\beta K^\gamma)(\nabla_\beta K_\gamma)
=3(\nabla^\beta K^\gamma)K_{[\alpha}\nabla_\beta K_{\gamma]}
-2(\nabla_\sigma K_\alpha)(K^\tau\nabla_\tau K^\sigma).}
$$
\b[Restriction to a Killing horizon.] On a <Killing horizon>, the generator $K$ is normal to the horizon as well as tangent to its null generators. Consequently $K_{[\alpha}\nabla_\beta K_{\gamma]}=0$ there. One can see this without assuming <hypersurface orthogonality> away from the horizon: if the horizon is locally $\Phi=0$, write its dual one-form as $K=f\,d\Phi+\Phi q$ near it; then $K\wedge dK=0$ on $\Phi=0$. By the definition of <surface gravity>, $a^\sigma=\kappa K^\sigma$ on the horizon, and lowering that relation gives $K^\sigma\nabla_\sigma K_\alpha=\kappa K_\alpha$. The contraction identity becomes
$$
K_\alpha F^{\beta\gamma}F_{\beta\gamma}
=-2\kappa^2K_\alpha.
$$
Away from points where $K$ vanishes, cancel a nonzero component of $K_\alpha$; at a regular <bifurcation surface> extend by continuity. Therefore
$$
\boxed{\kappa^2=-\frac12(\nabla^\beta K^\gamma)(\nabla_\beta K_\gamma).}
$$
This is <surface gravity from the Killing derivative>; it concerns the horizon, not every point in the exterior.
\b[Static spherical metric.] Choose the <Killing vector> $K=\partial_t$. Its dual one-form is $-A\,dt$, and the first identity immediately gives
$$
F_{rt}=-\frac{A'}2,\qquad F_{tr}=\frac{A'}2,
$$
with all other components zero. Hence
$$
F^{\beta\gamma}F_{\beta\gamma}
=2g^{rr}g^{tt}F_{rt}^2
=-\frac{BA'^2}{2A}.
$$
Using the horizon identity and the simultaneous simple zeros,
$$
\boxed{\kappa^2=\lim_{r\to r_+}\frac{BA'^2}{4A}
=\frac{A'(r_+)B'(r_+)}4,\qquad
\kappa=\frac12\sqrt{A'(r_+)B'(r_+)}.}
$$
Here the positive value is chosen for a regular outer <Killing horizon> with $A,B>0$ on the exterior side. This is <surface gravity of a static spherical horizon>. A different normalization $K\mapsto cK$ rescales <surface gravity> by $c$; when an asymptotically flat normalization is wanted, the time coordinate is chosen so $A\to1$ at infinity.
\b[Euclidean regularity and temperature.] Let $\delta=r-r_+$, $a=A'(r_+)$ and $b=B'(r_+)$. On the exterior side, define the proper radial coordinate $R=2\sqrt{\delta/b}$. After <Wick rotation>, the near-horizon metric is
$$
ds_E^2=dR^2+\kappa^2R^2d\tau^2+r_+^2d\Omega^2
+\text{terms smooth in }R^2.
$$
The radial-time plane is a polar plane with angular coordinate $\vartheta=\kappa\tau$. Its circumference-to-radius ratio is $\kappa\,\Delta\tau$; the <conical singularity> disappears precisely when
$$
\boxed{\Delta\tau=\frac{2\pi}{\kappa}.}
$$
This is the <Euclidean black-hole regularity condition>. If a signed <surface gravity> convention is used, the period uses $|\kappa|$.
In a thermal <quantum field theory>, imaginary-time periodicity is the inverse-temperature condition expressed by the <KMS condition>. Therefore smoothness identifies the <Hawking temperature>
$$
\boxed{T_H=\frac{\kappa}{2\pi}}
$$
in units $\hbar=k_B=1$, or $T_H=\hbar\kappa/(2\pi k_B)$ when $\kappa$ is measured as an inverse time. It provides the thermal interpretation of <surface gravity> in <black-hole thermodynamics>. The regular Euclidean construction describes an equilibrium thermal state; the collapse calculation of <Hawking radiation> in the last solution supplies the outgoing spectrum. The simple-zero hypothesis excludes an <extremal black hole>, for which this polar-plane argument changes.
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