# Is the surface gravity on a null surface trivial? (Wald eq. 12.5.2)

In Ch 12.5 of Robert Wald's General Relativity text, he considers a null hypersurface with normal vector $\chi^a$. By definition, on the null hypersurface, we have $\chi^a\chi_a=0$. i.e., $\chi^a\chi_a$ is constant on the hypersurface, so we trivially find that $\nabla^a(\chi^b\chi_b)$ must also be normal to the hypersurface. Wald then argues that because both $\nabla^a(\chi^b\chi_b)$ and $\chi^a$ are normal to the hypersurface, they must be proportional--that is, there must be a function $\kappa$ such that

$$\nabla^a(\chi^b\chi_b)=-2\kappa\chi^a. \tag{Wald 12.5.2}\label{Wald}$$

However, Wald later claims (below equation 12.5.15) that \eqref{Wald} implies $\nabla^a(\chi^b\chi_b)$ is not zero.

Question: Is Wald not contradicting himself by essentially using $\nabla^a(\chi^b\chi_b)=0$ to imply $\nabla^a(\chi^b\chi_b)\ne 0$ on the null surface?

The quantity $\chi^2$ is constant along the horizon, which means that $v(\chi^2)=0$ for any vector $v$ in the tangent space to the horizon. This can also be written as $v^a\nabla_a\chi^2=0$, and the vector $\nabla^a\chi^2$ points along the normal direction.