Lapse, shift, induced metric and second fundamental form

PDF p. 4: Remark 1.3, decomposition

Let $(M,g,T)$ be stationary Lorentzian data as in Stationary Lorentzian data and tensor conventions, with signature $(-,+,+,+)$ and Levi–Civita connection $\mathrm{\nabla }$. Let ${\mathrm{\Sigma }}_0$ be a smooth embedded spacelike hypersurface, meaning that $g$ is positive definite on each three-dimensional tangent space, and let $n$ be its future-directed unit normal: $g(n,n)=-1$ and $g(n,U)=0$ for every tangent vector $U$. These hypersurface conventions are those of Spacelike sections, maximality and lapse–shift data.

The induced metric is $h(U,V)=g(U,V)$ for tangent vectors $U,V$, and the second fundamental form in this sign convention is $k(U,V)=-g({\mathrm{\nabla }}_{U}n,V)$. Define the lapse $N=-g(T,n)$ and the shift $X=T-Nn$ along ${\mathrm{\Sigma }}_0$. Thus $T=Nn+X$ with $X$ tangent. With $\mathrm{\mid }X{\mathrm{\mid }}_h^2=h(X,X)$ and $f=-g(T,T)$, the notation satisfies $f=N^2-\mathrm{\mid }X{\mathrm{\mid }}_h^2$. No sign condition on $N$ is imposed by these definitions.

Given: The stationary data and spacelike hypersurface in the Definition.

Goal: Justify the defined fields, the decomposition and the identity for $f$.

1.1 The hypersurface supplier gives the smooth future unit normal, positive-definite induced metric and well-defined second fundamental form with the displayed sign. It also gives the lapse and the tangent shift $X=T-Nn$, so $T=Nn+X$ is the unique decomposition of that form. [F1, F2, given]

2.1 Since $g(n,n)=-1$ and $g(n,X)=0$, bilinearity gives $g(T,T)=-N^2+g(X,X)$. The definition of $h$ makes $g(X,X)=\mathrm{\mid }X{\mathrm{\mid }}_h^2$, and hence $f=-g(T,T)=N^2-\mathrm{\mid }X{\mathrm{\mid }}_h^2$. This calculation requires no positivity assumption on $N$. [F1, F2, step 1.1] ∎