TITLE: Normal to the Hypersurfaces
QUESTION [1 upvotes]: I am trying to understand the derivation of the Hilbert-Einstein action. However it requires a knowledge about hyper-surfaces for the boundaries of the integrals and also about the normal to the hyper-surface $n^{a}$ for the derivations.
Could anyone refer me to a good source or try explain to me the normal to the hyper-surface and also why $n^{a}n^{b}n^{c}$ = 0?
In the derivation it says that:
If we define $g^{ab}$ as $h^{ab}+t n^{a}n^{b}$, where $t=n_a n^a$ then
$n^{c}(h^{ab}+t n^{a}n^{b})({\partial}_{c}{\delta g}_{ab}-{\partial}_{a}{\delta g}_{cb})=-n^{c}h^{ab}{\partial}_{a}{\delta g}_{cb}$ on hypersurface.
See page 31 on this manuscript for the derivation

REPLY [1 votes]: The problem is not proving 
$$n^a n^b n^c =0 $$
But in proving 
$$n^a n^b n^c (\partial_{c}\delta g_{ab} - \partial_{a}\delta g_{cb}) = 0$$
Observe the parenthetic term, we will write it as
$$f_{abc} = (\partial_{c}\delta g_{ab} - \partial_{a}\delta g_{cb})$$
Then observe
$$ f_{abc} = -f_{cba}$$
We get the result immediately, because index gymnastics gives us:
$$n^b (n^an^c f_{abc}) = 0$$
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