Recently, I came across a strange definition of Hawking quasi-local mass, which states that given a surface $S$ in the spacetime, the Hawking mass of $S$ is defined as $$m(S)=\sqrt{\frac{\mathrm{Area}(S)}{16\pi}}\left(1-\frac{1}{4\pi}\int_S\rho\mu\right),\tag{1}$$ where $\rho$ is the convergence of the outer null normal and $-\mu$ is the convergence of the inner null normal. The definition is included in Some remarks on the quasi-local mass by Christodoulou and Yau. I call it strange because I used to learn about Hawking mass by defining $$m(S)=\sqrt{\frac{\mathrm{Area}(S)}{16\pi}}\left(1-\frac{1}{16\pi}\int_S H^2\right)\tag{2}$$ with $H$ as the mean curvature of $S$. This latter definition requires only knowledge of Riemannian geometry, and I'm thinking, is there possibly a way to show that it is in fact equivalent to the former definition, the more relativistic one?
I have to admit that in the first definition, there are many technical terms with which I have little acquaintance, so I grabbed a book titled Semi-Riemannian geometry with applications to relativity and authored by Barrett O'Neill. Does this book help to bridge the gap between these two definitions? On its 56th page, I'm told that a tangent vector $v$ is null if and only if $\langle v,v\rangle=0$ and $v\neq 0$. If this is what the null means, then what does it mean as to the convergence of the outer/inner null normal? Is there something that can be said to converge in the sense of (mathematical) limits?
Can anyone help me out? Thank you.
Edit. I hope it adds more information. According to Wikipedia, a null hypersurface is a hypersurface on which the normal vector takes on a null value. But I'm not sure if the surface $S$ in question is a null hypersurface because Christodoulou and Yau did not explicitly define $S$ in their remarks.