In a differential geometry textbook I'm reading it sometimes mentions a given vector field along a hypersurface of a Lorentzian Manifold being "unit normal" but it doesn't actually define the term anywhere. I tried googling the definition but it didn't come up with anything useful. So I was wondering if anyone here could tell me what it means to be "unit normal" in this context.
Any help is appreciated, cheers.
Best Answer
For a (Pseudo-)Riemannian manifold $(M,g)$ with a hypersurface $S\subset M$ (i.e. an embedded submanifold of codimension $1$), a vector field $N\in\mathfrak{X}M$ is said to be a unit normal of $S$ if it has unit magnitude on $S$ and is everywhere orthogonal to $TS$. In other words, $g(N(p),N(p))=\pm 1$ for all $p\in S$ and $g(N(p),v)=0$ for any $p\in S,v\in T_pS$. Such vector fields exist provided $S$ is orientable and the orthogonal compliment of $TS$ is nowhere null, but are not unique, since $-N$ is also a unit normal, and the behavior away from $S$ is not constrained.