Integrability, Resonance, and Symmetry Breaking

Kerr geodesic motion is Liouville integrable, but stationarity and axisymmetry alone do not protect that property. A deformation may preserve energy and axial angular momentum while destroying the Killing tensor, turning resonant invariant tori into island chains and permitting chaotic layers. These dynamical signatures diagnose loss of integrability; they do not by themselves identify which field equation or matter source produced the deformation.

Required background. Killing Fields and Isometry Groups supplies manifest first integrals; Affine and Variational Geodesics supplies Hamiltonian geodesic flow and deviation.

Helpful background. Einstein’s Equation distinguishes arbitrary metric deformations from solutions; The Riemann Curvature Tensor supplies invariant geometry; Exact Solutions of Einstein’s Equations supplies the Kerr benchmark; Frames, Coframes, and Anholonomy prepares invariant frame checks.

Exact integrability in Kerr

On the cotangent bundle, geodesic motion has Hamiltonian

$$H=\frac12g^{ab}p_ap_b.$$

The four quantities

$$H, \qquad E=-p_t, \qquad L_z=p_\phi, \qquad K=K^{ab}p_ap_b$$

are functionally independent on a generic phase-space region and Poisson-commute. The Killing-vector equations give $\{H,E\}=\{H,L_z\}=0$; the Killing-tensor equation gives $\{H,K\}=0$; stationarity and axisymmetry of $K^{ab}$ give the remaining brackets. This is Liouville integrability for four configuration-space degrees of freedom. Restricting to a fixed mass shell and quotienting the cyclic coordinates leaves the familiar decoupled radial and polar motions.

Bound generic trajectories admit action–angle variables $(J_i,w^i)$ with $\dot w^i=\Omega_i(J)$. A resonance occurs when a nonzero integer vector satisfies $k_i\Omega_i=0$. Resonance is compatible with exact integrability: on Kerr the trajectory simply closes in the corresponding angle projection, while the invariant torus remains intact. The hidden-symmetry construction and involution checks are summarized in Frolov, Krtouš, and Kubizňák 2017, §§6.1–6.2, pp. 117–125, Open PDF.

What a geometric perturbation can break

Write a stationary-axisymmetric deformation as

$$g_{ab}(\varepsilon)=g^{\rm Kerr}_{ab}+\varepsilon h_{ab}+O(\varepsilon^2).$$

The two coordinate symmetries preserve $E$ and $L_z$. A deformed quadratic integral would require

$$K_{ab}(\varepsilon)=K^{(0)}_{ab}+\varepsilon K^{(1)}_{ab}+\cdots$$

to solve $\nabla_{(a}K_{bc)}=0$ order by order. At first order this is an overdetermined linear system for $K^{(1)}_{ab}$ with a source determined by $h_{ab}$ and $K^{(0)}_{ab}$. Failure of its integrability conditions obstructs that deformed quadratic integral. The stronger principal conformal Killing–Yano structure may also fail to deform, but these are not equivalent tests: a Killing tensor need not be the square of a Killing–Yano form, and loss of that factorization alone does not prove nonintegrability. Preserving the locations of two Killing vectors or making the metric look “Kerr-like” does not settle either equation.

When the exact tensor is lost, perturbation theory may still produce an adiabatic or finite-order approximate invariant. Such a quantity is useful only with its error estimate and timescale: $dK_{\rm app}/d\lambda=O(\varepsilon^{N+1})$ does not imply exact conservation over arbitrarily long motion. Nor does a formal vacuum multipole expansion guarantee a regular global spacetime. Brink discusses the relation between stationary-axisymmetric vacuum geometry, quadratic integrals, and separability in Brink 2010, §§II–IV, Open PDF.

Resonant islands and chaotic diagnostics

For a small generic Hamiltonian perturbation, nonresonant tori satisfying suitable nondegeneracy conditions can persist in deformed form, while resonant tori are the first places where the foliation can split into alternating stable and unstable periodic orbits. A Poincaré section then shows island chains rather than a single smooth invariant curve. Separatrix splitting can create stochastic layers, and larger perturbations may connect them.

No single numerical picture proves the geometric cause. A responsible diagnosis combines: convergence under integrator tolerance; constraint preservation; multiple initial conditions; a recurrence, rotation-number, or frequency-map test; and, where possible, direct failure of the Killing-tensor equation. A positive Lyapunov exponent depends on the time parameter and norm used, although bounded-versus-exponential separation under a fixed physical prescription remains informative. Coordinate changes can distort a section but cannot create an exact conserved tensor that does not exist.

This comparison is conditional. Special deformations can preserve a Killing tensor; some nonseparable systems show no detectable chaos in the sampled region; and apparent irregularity can come from poor coordinates or numerics. The discriminant is therefore a hierarchy: solve the tensor equation first, test involution second, and use phase-space diagnostics to map the consequences.

Exercise

Show that if $K^{ab}$ is stationary, then its quadratic integral Poisson-commutes with $E=-p_t$.

Solution

In canonical coordinates, $\{E,K^{ab}p_ap_b\}=-\{p_t,K^{ab}p_ap_b\}=\partial_t(K^{ab}p_ap_b)$. Canonical momenta are independent phase-space coordinates, so this equals $(\partial_tK^{ab})p_ap_b$. Stationarity of the tensor makes it zero. The same argument with $\partial_\phi K^{ab}=0$ gives $\{L_z,K\}=0$.

References

• Brink, Jeandrew. “Spacetime Encodings. III. Second Order Killing Tensors.” Physical Review D 81 (2010): 022001. DOI; Open PDF.
• 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.