# Surface gravity of bifurcate Killing horizon in static wedge

Setup: I am attempting to verify a claim from my class notes. The claim is that the surface gravity of a bifurcate Killing horizon whose geodesic generators $\chi^a$ are timelike and hypersurface orthogonal in wedge I (i.e., the spacetime is static in wedge I) is given by

$$\kappa = \frac{dN}{dl}|_{l=0} \quad , \tag{1}\label{eq1}$$

where $l$ is the proper length from the bifurcation surface and $N^2=-\chi_a\chi^a$. Presumably--although this wasn't stated explicitly--we are considering some convex normal neighborhood near the horizon where any point in this neighborhood is connected to the bifurcation surface via a unique (spacelike) geodesic.

Eq. \eqref{eq1} was then used to argue the metric near $l=0$ can be expressed

$$ds^2 = -\kappa^2 l^2 dt^2 + dl^2 + k_{ij}\,(l,x^i) dx^i dx^j,$$

where $t$ is the Killing parameter associated with $\chi^a$.

My Attempt: In a static spacetime, the metric takes the general form

$$ds^2 = -N^2(x^i)dt^2 + h_{ij}\,(x^i) dx^i dx^j,$$

where $N^2=-g_{ab}\,\chi^a\chi^b=-\chi_a \chi^a$. Using Eq. 12.5.16 in Wald's GR text for the surface gravity, we have (for the limit approaching the horizon)

\begin{align} \kappa^2 &= \lim \{-(\chi^b \nabla_b \chi^c)(\chi^a\nabla_a\chi_c)/\chi^d\chi_d\}\\ &=\frac{1}{4}\lim \{-\nabla^c(\chi^b\chi_b)\nabla_c(\chi^a\chi_a)/\chi^d\chi_d\}\\ &=\frac{1}{4}\lim \{\nabla^c N^2\nabla_c N^2/N^2\}\\ &=\lim \{ \nabla^c N \nabla_c N \},\\ \end{align}

where in going to the second line I have used Killing's equation ($\nabla_a \chi_b +\nabla_b \chi_a=0$) in the following way:

$$\nabla_a (\chi^b \chi_b) = 2\chi^b \nabla_a \chi_b = -2 \chi^b \nabla_b \chi_a \,.$$

It seems that it remains for me to argue two things: 1) that having chosen $l$ as one of my coordinates,

$$\lim_{l\to 0}\;\{\nabla_c N \nabla^c N\}=\left(\frac{dN}{dl}|_{l=0}\,\right)^2, \tag{2} \label{eq2}$$

And 2) that one can locally and covariantly expand the metric in some kind of Taylor series, where e.g.

\begin{align} N(l,x^i) &= N(0,x^i)+l\frac{dN}{dl}|_{l=0}\;+\mathcal{O}(l^2). \tag{3} \label{eq3} \\ &= l \sqrt{\kappa} + \mathcal{O}(l^2), \end{align} where I have used the fact that $\chi_a \chi^a = 0$ on the bifurcation surface.

My Question(s): It is not at all clear to me how \eqref{eq2} holds. In particular, I would have thought

$$\left(\frac{dN}{dl}\right)^2=\xi^a\xi^b \nabla_a N \nabla_b\, N \ne g^{ab} \,\nabla_a N\nabla_b N,$$

for $\xi^a = (\partial/\partial l)^a$. As far as \eqref{eq3} goes, I would suspect that one needs to make some sort of "Riemann normal coordinates argument", but I'm at least as confused on this issue as the other.