Hamilton-Jacobi Separation in Kerr

Kerr geodesics are exceptional because their Hamilton–Jacobi equation separates into one radial and one polar ordinary differential equation. The separation is not a lucky algebraic trick: it reflects the metric’s principal hidden symmetry and supplies the fourth first integral needed for complete geodesic integrability. Here $m=G M_{\mathrm{phys}}$ and $a=J_{\mathrm{phys}}/M_{\mathrm{phys}}$ (with $c=1$) are length parameters, and the treatment concerns test-particle and null-ray kinematics, not self-force or orbital stability.

Required background. Killing Fields and Isometry Groups supplies the manifest energy and axial-angular-momentum integrals; Affine and Variational Geodesics supplies the Hamilton–Jacobi formulation of geodesic motion.

Helpful background. Einstein’s Equation fixes the background field equation; The Riemann Curvature Tensor supplies curvature conventions; Exact Solutions of Einstein’s Equations locates Kerr in the family atlas; Frames, Coframes, and Anholonomy prepares the principal-frame interpretation.

The separable inverse metric

In Boyer–Lindquist coordinates define

$$\Sigma=r^2+a^2\cos^2\theta, \qquad \Delta=r^2-2mr+a^2.$$

The decisive property is that multiplying the inverse metric by $\Sigma$ makes its $r$ and $\theta$ dependence additive. Equivalently, for any covector $p_a$,

$$\begin{aligned} \Sigma g^{ab}p_ap_b={}& \Delta p_r^2+p_\theta^2 -\frac{\big[(r^2+a^2)p_t+a p_\phi\big]^2}{\Delta}\\ &+\frac{\big[p_\phi+a\sin^2\theta\,p_t\big]^2}{\sin^2\theta}. \end{aligned}$$

This identity is the coordinate-level answer to why additive separation works. Stationarity and axisymmetry make $p_t=-E$ and $p_\phi=L_z$ constant, but those two Killing symmetries alone do not explain the split between $r$ and $\theta$.

For a particle of rest mass $\mu$, take

$$S=+\frac12\mu^2\lambda-Et+L_z\phi+S_r(r)+S_\theta(\theta).$$

The plus sign is fixed by the parameter-dependent Hamilton–Jacobi equation $\partial_\lambda S+H=0$: on the mass shell, $H=\tfrac12g^{ab}p_ap_b=-\mu^2/2$, so $\partial_\lambda S=+\mu^2/2$. If one uses only the mass-shell equation below, the additive $\lambda$ term drops out, but writing it with the wrong sign would make the two Hamilton–Jacobi formulations inconsistent.

Substitution into $g^{ab}\partial_aS\partial_bS=-\mu^2$ leaves one function of $r$ equal to minus one function of $\theta$. Both must therefore equal a constant. This derivation and its geometric generalization are developed in Frolov, Krtouš, and Kubizňák 2017, §§6.1–6.2, printed pp. 117–125, Open PDF.

Radial and polar potentials

Choose the separated constant $Q$ so that the first-order equations read

$$\Sigma^2\left(\frac{dr}{d\lambda}\right)^2=R(r), \qquad \Sigma^2\left(\frac{d\theta}{d\lambda}\right)^2=\Theta(\theta),$$

with

$$R=\big[E(r^2+a^2)-aL_z\big]^2 -\Delta\big[\mu^2r^2+(L_z-aE)^2+Q\big],$$ $$\Theta=Q-\cos^2\theta \left[a^2(\mu^2-E^2)+\frac{L_z^2}{\sin^2\theta}\right].$$

Only regions with $R\geq0$ and $\Theta\geq0$ are accessible. Zeros of either potential are turning points unless the zero is multiple; a double radial root instead identifies a constant-radius orbit. For null rays set $\mu=0$. The affine parameter $\lambda$ above is the physical affine parameter; introducing Mino time by $d\lambda_{\rm M}=d\lambda/\Sigma$ removes the common $\Sigma$ and decouples the radial and polar motions.

The cyclic coordinates follow from the same separated action. With $P(r)=E(r^2+a^2)-aL_z$,

$$\Sigma\frac{dt}{d\lambda} =a(L_z-aE\sin^2\theta) +\frac{(r^2+a^2)P}{\Delta},$$ $$\Sigma\frac{d\phi}{d\lambda} =\left(\frac{L_z}{\sin^2\theta}-aE\right) +\frac{aP}{\Delta}.$$

Their $1/\Delta$ behavior at a horizon is a Boyer–Lindquist coordinate effect; a horizon-penetrating chart combines the divergent pieces into regular ingoing quantities.

There are two widespread constant conventions. On this page $Q$ is the quantity displayed in $\Theta$, while

$$K=Q+(L_z-aE)^2$$

is the direct quadratic contraction with a suitably normalized rank-two Killing tensor. Papers may call either $Q$ or $K$ “the Carter constant.” Stating the radial potential removes the ambiguity. Carter’s original separation result is Carter 1968, pp. 1559–1571.

Checks and scope

In the Schwarzschild limit $a\to0$,

$$\Theta=Q-L_z^2\cot^2\theta, \qquad L^2=Q+L_z^2.$$

Thus the hidden constant becomes the familiar total angular momentum squared after spherical symmetry is restored. For an equatorial orbit, $\theta=\pi/2$ and $d\theta/d\lambda=0$ require $Q=0$; reflection symmetry then keeps the orbit in the plane. These two limits independently check the signs in $\Theta$.

The potentials also expose parameter redundancies and dimensions. For null geodesics with $E\neq0$, rescaling the affine parameter rescales $(E,L_z,\sqrt Q)$ together, so the unparametrized trajectory depends on the impact parameters $\xi=L_z/E$ and $\eta=Q/E^2$. A radial spherical photon orbit at $r=r_0$ must satisfy both $R(r_0)=0$ and $R'(r_0)=0$; the first condition alone identifies a turning point, not a constant-radius ray. For massive geodesics, $E$ is specific energy when the momentum is divided by $\mu$, and that normalization should not be mixed with the null impact-parameter convention.

Boyer–Lindquist separation is valid on any coordinate patch where the displayed inverse metric applies, including regions where $\Delta$ changes sign, but the interpretation of $r$ as a spatial variable and $t$ as a time coordinate then changes. The separated first integrals remain tensorial statements; the chart-based signs of individual coordinate velocities do not.

Separation gives quadratures, not automatically elementary closed-form trajectories. It also does not decide whether a permitted orbit is stable, bound, future-complete, or physically realizable under radiation reaction. Those questions require analysis of the potentials or dynamical methods beyond the exact background geometry.

Exercise

Show that a null geodesic with constant latitude away from the axis must obey both $\Theta(\theta_0)=0$ and $\Theta'(\theta_0)=0$.

Solution

In Mino time, $(d\theta/d\lambda_{\rm M})^2=\Theta(\theta)$. A constant solution has zero polar velocity, so $\Theta(\theta_0)=0$. Its differentiated equation is $2\theta'\theta''=\Theta'\theta'$, which loses information at $\theta'=0$; using the second-order geodesic equation, or taking the continuous limit from nearby trajectories, additionally gives $\theta''=\Theta'(\theta_0)/2=0$. Hence the root must be stationary. The equatorial case satisfies this automatically by reflection symmetry when $Q=0$.

References

• Carter, Brandon. “Global Structure of the Kerr Family of Gravitational Fields.” Physical Review 174 (1968): 1559–1571. DOI.
• Frolov, Valeri P., Pavel Krtouš, and David Kubizňák. “Black Holes, Hidden Symmetries, and Complete Integrability.” Living Reviews in Relativity 20 (2017): 6. DOI; Open PDF.