2
$\begingroup$

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?

$\endgroup$
0

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.