Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p S$ is degenerate).

If $K$ is a null vector field of $S$, show that the integral curves of $K$ are null geodesics of $S$.

Everyone has a hint?

ERRATA: I would like understand why that problem is equivalent to show that $$\nabla_K K=\lambda K,$$

where $\lambda\in C^\infty(S)$ is a smooth function. Cause, by $\nabla_K K=\lambda K$, if $\alpha$ is a integral curve of $K$, $\frac{d\alpha}{dt}=K(\alpha(t))$, then

$$\frac{D}{dt}\Big(\frac{d\alpha}{dt}\Big)=K(\alpha (t)),$$

and I don't understand why the right-hand side is zero.

Maybe I found something like that in the book General Relativity: With Applications to Astrophysics, By Norbert Straumann, p. 273.

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It's sufficient to consider $\nabla \psi = K$, I think. But I don't understand the final line, in short the notation.

Thanks again.


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