Write $f(r)=1-r_+^2/r^2$ and let a dot denote differentiation with respect to an <affine parameter> $\tau$. The time-translation and $z$-translation <Killing vector field>[Killing vector fields], together with rotational symmetry of the unit $S^3$, give the <geodesic conserved quantity from a Killing vector>[conserved quantities]
$$
E=f\dot t,
\qquad
P=\dot z,
\qquad
J^2=r^4\lVert\dot\Omega_3\rVert^2.
$$
Define $\varepsilon=-g_{ab}\dot x^a\dot x^b$, so $\varepsilon=1,0,-1$ for timelike, null and spacelike geodesics respectively. The normalization equation becomes
$$
-\frac{E^2}{f}+\frac{\dot r^2}{f}+\frac{J^2}{r^2}+P^2=-\varepsilon.
$$
It therefore has the <effective potential> form
$$
\frac12\dot r^2+\widetilde V(r)=0,
\qquad
\boxed{\widetilde V(r)=\frac12\left[f(r)\left(\varepsilon+P^2+\frac{J^2}{r^2}\right)-E^2\right]}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-311/1/a/solution