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.

enter image description here

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

Thanks again.

  • 2
    $\begingroup$ Hi and welcome to the Physics SE! Please note that this is not a homework help site. Please see this Meta post on asking homework questions and this Meta post for "check my work" problems. $\endgroup$ – John Rennie Nov 20 '15 at 15:38
  • $\begingroup$ @JohnRennie, Thanks for the advise. I will add more informations about my question. $\endgroup$ – Irddo Nov 20 '15 at 15:43
  • $\begingroup$ Adding more information will not make it on-topic. You have to show work and find the conceptual issue -- that's what the question should be. $\endgroup$ – Ryan Unger Nov 20 '15 at 15:53
  • $\begingroup$ In any case, the proof is given on page 684 of N. Straumann, General Relativity. 2013, Springer. $\endgroup$ – Ryan Unger Nov 20 '15 at 15:57
  • $\begingroup$ Thanks, @0celo7. I just needed look a little more the book. $\endgroup$ – Irddo Nov 20 '15 at 16:08