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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.

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closed as off-topic by ACuriousMind, John Rennie, Gert, HDE 226868, Sebastian Riese Nov 21 '15 at 2:42

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, John Rennie, Gert, HDE 226868, Sebastian Riese
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ 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