Comparison with the <3+1 decomposition of spacetime> gives the diagonal spatial metric
$$
\boxed{
\gamma_{ij}
=\operatorname{diag}\left(
\frac{(r+M)^2}{r^2},\,
(r+M)^2,\,
(r+M)^2\sin^2\theta
\right).
}
$$
The mixed metric coefficient is $\beta_r=g_{tr}=M/r$. Raising its index with $\gamma^{rr}=r^2/(r+M)^2$ yields the <shift vector>
$$
\boxed{
\beta^r=\frac{Mr}{(r+M)^2},
\qquad \beta^\theta=\beta^\phi=0.
}
$$
Now
$$
\beta_i\beta^i=\frac{M^2}{(r+M)^2}
$$
and $g_{tt}=-\alpha^2+\beta_i\beta^i=-(r-M)/(r+M)$. The positive <lapse function> is consequently
$$
\boxed{\alpha=\frac{r}{r+M}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 357 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-357/2/c/solution