In the <spherical coordinate system>, $|d\mathbf x|^2=dr^2+r^2d\Omega_2^2$ and $\mathbf x\mathbin\cdot d\mathbf x=r\,dr$. Hence
$$
|d\mathbf x|^2-\frac{(\mathbf x\mathbin\cdot d\mathbf x)^2}{1+r^2}
=\left(1-\frac{r^2}{1+r^2}\right)dr^2+r^2d\Omega_2^2
=\frac{dr^2}{1+r^2}+r^2d\Omega_2^2.
$$
Thus the <static coordinates on anti-de Sitter spacetime> have
$$
\boxed{w(r)=1+r^2},
$$
which is positive for every $r\geq0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-309/1/a/solution