Lemma: Geometric normalization

For (eq:metric), surface gravity is

The coordinate $U$ in (eq:U) is affine on the past Kruskal horizon, and its exterior restriction satisfies

Proof. In ingoing Eddington–Finkelstein coordinates,

The relevant connection coefficients are

At $r=r_{\mathrm{h}}$, therefore,

Since $F^{\prime }(r)=r_{\mathrm{h}}/r^2$, the stated value follows. The normalization of $\chi$ at infinity removes a rescaling ambiguity.

Set

Their product is

and $2\kappa =1/r_{\mathrm{h}}$. Hence the radial metric extends across the horizons in the form

with $A$ smooth and nonzero at $UV=0$. On the past horizon $V=0$, the coefficient is constant along $U$, so the horizon generators have vanishing affine acceleration in coordinate $U$. Finally,

which gives $\chi (U)=-\kappa U$.