Are there conserved quantities on the black hole horizons?

I am facing a conceptual difficulty while reading about black-hole(BH) horizons. Consider for simplicity a Schwarzchild(SW) BH, with a null hypersurface $$\zeta$$ denoting its horizon at $$r=2M$$ and let $$g_{\alpha\beta}$$ denote the metric. Finally suppose $$K^{\alpha}$$ is a Killing vector for this space. Then, we know that $$C = g_{\alpha\beta}\dot{x}^\alpha K^\beta$$ is a conserved charge/quantity of this space.

MY QUESTION: Since $$g_{\alpha\beta}$$ (in the $$(t,r,\theta,\phi)$$ coordinate) blows up at $$r=2M$$, I am having trouble understanding whether the quantity $$C$$ is defined on $$\zeta$$ i.e. is $$C$$ still conserved at the hypersurface $$\zeta$$?.

A mathematical approach would be better. If anyone can find relevant papers please point them out (I couldn't find any that discuss this issue at the horizon $$\zeta$$). Also it would be better if someone can generalise for a non-SW BH and answer the same question.

• So as long as the horizon is not a physical singularity, I can still use the Killing fields to calculate $C$. Is this correct ? @JohnRennie Jul 14 '20 at 7:51

The quantity $$C = g_{\alpha\beta}\dot{x}^\alpha K^\beta$$ is a scalar invariant and is therefore independent of the coordinate system used.
Since coordinate systems exist that are not singular at the event horizon (for example Kruskal Szekeres) you could calculate $$C$$ at the horizon using these coordinates. The quantity is not affected by a change of coordinates.