Carter constant

Remarks

Example: for $a = 0$ the Kerr metric is the Schwarzschild metric, and $$\mathcal{Q} = p_\theta^2 + L_z^2 \cot^2\theta = \left(p_\theta^2 + \frac{p_\phi^2}{\sin^2\theta}\right) - p_\phi^2 = L^2 - L_z^2,$$ where $L^2$ is the squared total angular momentum. Both $L^2$ and $L_z$ are conserved by spherical symmetry, so $\mathcal{Q}$ is conserved; an equatorial orbit has $\mathcal{Q} = 0$.

Non-example: $p_\theta$ alone is not conserved, even for $a = 0$. On an inclined circular orbit around a Schwarzschild black hole, $p_\theta$ vanishes at the highest and lowest points and is nonzero in between.

Equivalent form: many texts use $K = \mathcal{Q} + (L_z - aE)^2$ instead. Then $K = K^{\mu\nu} p_\mu p_\nu$ for a symmetric Killing tensor $K^{\mu\nu}$, and the conservation of $K$ along geodesics follows from the Killing tensor equation $\nabla_{(\alpha} K_{\beta\gamma)} = 0$. The constants $\mu$, $E$, $L_z$ come from the metric and the Killing vector fields $\partial_t$ and $\partial_\phi$.