Is the Horizon r=2GM a Null Surface?

[quasar987]

Homework Statement

In my GR final, there was a question that asked to show that the horizon r=2GM is a null surface, i.e. $d\tau^2=0$. How does that work, since just plugging r=2GM and dr=0 in the Schwarzschild metric yields

$$d\tau^2=-(2GM)^2d\Omega^2$$

which is obviously not zero.

 

[Dick]

A null surface has the property that its normal vector is null. What's a normal covector to the surface defined by r-2GM=0? Is it null?