Definition of surface gravity via the non-affine geodesic equation

I have found a discrepancy in the way different sources define surface gravity (or derive) via the non-affine geodesic equation satisfied by the a Killing vector $$\xi$$ on a Killing Horizon (KH), up to a sign. That is, on the one hand Ref. 1, Ref.2, Ref. 3 (and many others) claim that on the KH $$\xi^{\mu}\xi_{\nu;\mu}=\kappa\xi_\nu,\tag{1a}\label{1a}$$ while Ref. 4, Ref. 5 claim that on the KH $$\xi^{\mu}\xi_{\nu;\mu}=-\kappa\xi_\nu.\tag{1b}\label{1b}$$

As harmless as it may seem, I don't believe this is just convention, as in that case different authors should get answers differing by a sign but all of the quoted references agree that e.g. for the Schwarzschild BH, $$\kappa=1/4M$$, not $$k=\pm1/4M$$, the upper sign referring to \eqref{1a} and the lower sign to \eqref{1b}.

Furthermore, upon affine reparametrizazion of geodesics along e.g. the future Schwarzschild horizon (parametrized by $$u=t-r^\ast$$), since $$\kappa$$ is constant one gets in the two cases that the affine parameter $$\lambda$$ is $$\lambda\propto e^{\pm u}$$ where again the sign is according to \eqref{1a} and \eqref{1b}; in the second case this means that $$\lambda\propto U$$, where $$U$$ is a Kruskal null coordinate, but the same is not true in the first case (or possibly not in the same Kruskal patch?). What is the subtetly going on here?

References

1. A relativist's toolkit. The Mathematics of black hole mechanics, E. Poisson. CUP, 2008. Eq. $$(5.40)-(5.41)$$;
2. General Relativity, R. M. Wald. University of Chicago Press, 1984. Eq. $$(12.5.2)-(12.5.5)$$;
3. Townsend's review of BH, eq. $$(2.82)$$;
4. Spacetime and geometry, S. Carroll. CUP, 2019. Eq. $$(6.8)$$
5. Particle creation by Black Holes, SWH. Just after Eq. $$(2.16)$$

I don't have the Carroll book in front of me, but in Hawking's paper he was describing in the paragraph above (2.16) that $$\lambda$$ was a parameter on the past horizon of the global Schwarzschild spacetime. On the past horizon, the surface gravity is defined as $$\xi^a\nabla_a \xi^b = - \kappa \xi^b$$, and with this definition $$\kappa$$ is positive. This equation is saying that $$\xi^a$$ is getting shorter as you move up the past horizon toward the bifurcation surface. The analogous equation on the future horizon is $$\xi^a\nabla_a \xi^b = \kappa \xi^b$$, with the same (positive) value of $$\kappa$$.
• I was just thinking it might be about past/future horizons (mentioned that in the chat). I checked the lecture notes on which the book is based and unfortunately that part is missing, so I can't provide Carroll's paragraph here. It's possible that it's an error as well, since that equation is never used again and $\kappa=1/4M$ is derived by using the formula for $\kappa^2$, which is clearly the same. Then, I wonder: do you think that most books use the second one without mentioning anything because in typical applications one deals with future horizons? Commented Feb 9 at 16:52