# Constants of motion and Killing symmetries

A constant of motion in general relativity is a quantity built from a particle's four-momentum that keeps the same value all along a geodesic, and the systematic source of such quantities is spacetime symmetry: every Killing vector of the geometry yields one conserved scalar along every geodesic.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> This article covers how Killing vectors and their higher-rank generalizations, Killing tensors, generate conserved quantities; how the super-Hamiltonian turns those constants into an effective-potential picture for radial and polar motion; and how separability of the [Hamilton–Jacobi equation](https://www.edgepedia.org/hamilton-jacobi-equation) is tied to complete integrability. It stops short of solving the orbits in any specific metric.

| Key fact | Statement |
|---|---|
| Killing-vector theorem | A Killing vector \(\xi^{a}\) makes \(p_{a} \cdot \xi^{a}\) constant along a geodesic, by Noether's theorem.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> |
| Kerr integrals | Kerr geodesic motion has four independent, mutually Poisson-commuting first integrals: the Hamiltonian, \(E = -t^{a} p_{a}\), \(L_{z} = \phi^{a} p_{a}\), and the Carter constant \(Q = K^{ab} p_{a} \cdot p_{b}\).<sup>[2](https://ar5iv.labs.arxiv.org/html/1311.3836)</sup> |
| Carter constant | Quadratic in momentum; identified by Walker and Penrose (1970) with a rank-2 Killing tensor of the Kerr geometry.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> |
| Killing tensor | A symmetric tensor whose totally symmetrized covariant derivative vanishes, \(\nabla_{(c} k_{ab)} = 0\) for rank 2.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> |
| Integrability | Two Killing-vector constants plus the mass plus one Killing-tensor constant make a stationary axisymmetric system completely integrable.<sup>[3](https://link.springer.com/article/10.1140/epjp/s13360-025-07201-3)</sup> |
| Separability | A completely integrable geodesic equation has a fully separable Hamilton–Jacobi equation iff the Lagrangian is a composite of the involutive first integrals.<sup>[4](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/021A05353D8FA4AE9851F1599A39B6F0/S033427000001119Xa.pdf/separation_of_the_hamiltonjacobi_equation_for_the_kerr_metric.pdf)</sup> |

## Killing vectors and the conserved-quantity theorem

A Killing vector \(\xi^{a}\) is a vector field whose flow is an isometry of the spacetime, meaning the metric does not change along it. The conservation statement is precise: if \(p^{b}\) is the geodesic tangent (proportional to the four-velocity) and \(\xi^{b}\) is any Killing vector, then the scalar \(p_{b} \cdot \xi^{b}\) keeps a constant value along the geodesic.<sup>[5](https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/07%3A_Symmetries/7.01%3A_Killing_Vectors)</sup> Equivalently, the explicit spacetime symmetry generated by \(\xi^{a}\) implies, through [Noether's theorem](https://www.edgepedia.org/noethers-theorem) applied to the geodesic Lagrangian, that \(p_{a} \cdot \xi^{a}\) remains constant along the particle's worldline.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup>

**Timelike and axial Killing vectors.** In a [stationary spacetime](https://www.edgepedia.org/stationary-spacetime) the timelike Killing vector \(\partial_t\) gives a conserved energy; in Schwarzschild the conserved \(p_t\) along the \(\partial_t\) direction is interpreted as the mass-energy of the test particle.<sup>[5](https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/07%3A_Symmetries/7.01%3A_Killing_Vectors)</sup> Concretely, for equatorial timelike geodesics in Schwarzschild the energy is \(E = (1 - 2M/r)\, dt/d\tau\) and the rotational Killing vector gives angular momentum \(L = r^2\, d\varphi/d\tau\); both are conserved.<sup>[6](https://arxiv.org/html/2604.16644)</sup> In a general stationary axisymmetric spacetime, explicit symmetries are characterized by Killing vector fields, and these generate conserved quantities that simplify both geodesic motion and field equations.<sup>[7](https://utf.mff.cuni.cz/~krtous/papers/LDKVOIKT-JHEP07(2025)098.pdf)</sup>

## The super-Hamiltonian, integrability, and the effective-potential idea

Geodesic motion is naturally Hamiltonian, and any symmetric tensor K contracted with the momenta produces a quantity C = K_{αβ} ẋ^α ẋ^β (or K^{ab}p_a p_b) that is conserved when K satisfies the Killing tensor equation.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup><sup> • </sup><sup>[3](https://link.springer.com/article/10.1140/epjp/s13360-025-07201-3)</sup> A 2025 result establishes the clean correspondence: incorporating Killing vectors as the p = 1 case, Killing tensors are in one-to-one correspondence with monomial constants of geodesic motion C_p = K_{μ1...μp} p^{μ1}…p^{μp} along curves satisfying the geodesic equation p^ν ∇_ν p^μ = 0.<sup>[7](https://utf.mff.cuni.cz/~krtous/papers/LDKVOIKT-JHEP07(2025)098.pdf)</sup>

In Kerr, the eight-dimensional first-order system has exactly four such first integrals: the Hamiltonian \(H = -\tfrac{1}{2}\mu^{2}\) (fixing the rest mass), the energy \(E = -t^{a}p_{a}\), the axial angular momentum \(L_{z} = \phi^{a}p_{a}\), and the [Carter constant](https://www.edgepedia.org/carter-constant) \(Q = K^{ab}p_{a}p_{b}\). These are independent for non-degenerate orbits and have vanishing Poisson brackets, so the system is completely integrable in the sense of Liouville.<sup>[2](https://ar5iv.labs.arxiv.org/html/1311.3836)</sup>

The same integrability underlies practical orbit computation. Using action-angle variables from the Hamilton–Jacobi solution in [Boyer–Lindquist coordinates](https://www.edgepedia.org/boyer-lindquist-coordinates), the canonical equations take a simple form with constant frequencies, and the energy of any bound orbit around a Kerr black hole splits as \(e = \omega_{r} \cdot j_{r} + \omega_{\theta} \cdot j_{\theta} + \omega_{\phi} \cdot l_{z} + \langle z \rangle\).<sup>[2](https://ar5iv.labs.arxiv.org/html/1311.3836)</sup>

## Hidden symmetries: Killing tensors and the Carter constant

Familiar Killing vectors, associated with explicit spacetime symmetry, are Killing tensors of rank 1; the geometric structure encoded in Killing tensors of rank 2 and higher is called a hidden symmetry, because it produces conserved quantities of higher order in the momentum without any corresponding visible isometry.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> A rank-2 Killing tensor \(k_{ab}\) is a symmetric tensor whose symmetrized covariant derivative vanishes, \(\nabla_{(c} k_{ab)} = 0\); the rank-s version of the equation was given by Stackel in 1895.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup>

The history is concrete. [Brandon Carter](https://www.edgepedia.org/brandon-carter) (whose 1968 papers proved the separability results discussed below) obtained an additional quadratic integral of geodesic motion in Kerr by separating variables in the Hamilton–Jacobi equation.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> Two years later, Walker and Penrose showed that this constant corresponds one-to-one to a rank-2 Killing tensor of the Kerr geometry, making the hidden symmetry explicit.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> The same 1968 papers also showed that the [Klein–Gordon equation](https://www.edgepedia.org/klein-gordon-equation), not just the Hamilton–Jacobi equation, allows complete separation of variables in Kerr.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup>

 The result is not exclusive to Kerr. For any axisymmetric spacetime, the most general Killing tensor satisfies \(\xi_{\alpha\beta;\mu} + \xi_{\mu\alpha;\beta} + \xi_{\beta\mu;\alpha} = 0\) and yields a conserved quantity \(Q = \xi_{\alpha\beta} Z^\alpha Z^\beta\), with the Kerr Carter constant \(Q_{\mathrm{c}}\) recovered as a special case; the Killing vectors of the same spacetime give \(E = \tau_\alpha Z^\alpha\) and \(l = \xi_\alpha Z^\alpha\).<sup>[8](https://doi.org/10.1140/epjc/s10052-022-10544-1)</sup> More generally, in a stationary axisymmetric metric the two Killing vectors \(\partial_\tau\) and \(\partial_\phi\) supply two constants of motion, the test-particle mass is a third, and one Killing tensor supplying a fourth constant makes the system completely integrable.<sup>[3](https://link.springer.com/article/10.1140/epjp/s13360-025-07201-3)</sup>

## Separability and the Hamilton–Jacobi equation

The connection between constants of motion and separability runs through the Hamilton–Jacobi equation. A recent theorem states that a completely integrable geodesic equation has a fully separable Hamilton–Jacobi equation if and only if the Lagrangian is a composite of the involutive first integrals.<sup>[4](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/021A05353D8FA4AE9851F1599A39B6F0/S033427000001119Xa.pdf/separation_of_the_hamiltonjacobi_equation_for_the_kerr_metric.pdf)</sup> In a stationary axisymmetric metric, the two Killing-vector constants, the mass, and a Killing-tensor constant together supply the four motion constants needed for complete integrability.<sup>[3](https://link.springer.com/article/10.1140/epjp/s13360-025-07201-3)</sup> For Kerr this condition is met: Carter's separation-constant method is equivalent to solving the first-order Kerr geodesic equations with r- and θ-separated functions containing the Carter constant \(Q_{\mathrm{c}}\).<sup>[8](https://doi.org/10.1140/epjc/s10052-022-10544-1)</sup>

## Comparison with flat-space and Newtonian conservation laws

In flat spacetime there are enough Killing vectors to guarantee conservation of energy-momentum and angular momentum.<sup>[5](https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/07%3A_Symmetries/7.01%3A_Killing_Vectors)</sup>

In Schwarzschild, spherical symmetry yields four independent, mutually Poisson-commuting integrals of geodesic motion, satisfying Liouville's theorem.<sup>[1](https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf)</sup> Under [symplectic reduction](https://www.edgepedia.org/symplectic-reduction) of the [Schwarzschild metric](https://www.edgepedia.org/schwarzschild-metric) via the rotation group SO(3), Carter's Killing tensor quantity can be interpreted as the remnant of the square of angular momentum.<sup>[4](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/021A05353D8FA4AE9851F1599A39B6F0/S033427000001119Xa.pdf/separation_of_the_hamiltonjacobi_equation_for_the_kerr_metric.pdf)</sup>

## Recent developments

Three post-2023 results sharpen the picture. First, a 2025 JHEP paper established the one-to-one correspondence between Killing tensors of any rank (with Killing vectors as rank 1) and monomial constants of geodesic motion \(C_p\), formalizing exactly which polynomial quantities can be conserved and in what geometry.<sup>[7](https://utf.mff.cuni.cz/~krtous/papers/LDKVOIKT-JHEP07(2025)098.pdf)</sup> Second, 2025 work extended the framework to sourced Plebański spacetimes, showing the same four-constant structure (two Killing-vector constants, mass, and a Killing-tensor constant) at work beyond vacuum Kerr.<sup>[3](https://link.springer.com/article/10.1140/epjp/s13360-025-07201-3)</sup> Third, a 2026 analysis found that Schwarzschild timelike geodesics possess three hidden conserved quantities, analogs of the Newtonian LRL vector (an LRL angle, an LRL Killing-vector time, and an LRL proper-time), each conserved for all timelike geodesics, and obtained their generating symmetry transformations by applying Noether's theorem in reverse to the geodesic Lagrangian.<sup>[6](https://arxiv.org/html/2604.16644)</sup>

## References

1. Frolov, Krtouš, Kubizňák, *Black holes, hidden symmetries, and complete integrability*, https://utf.mff.cuni.cz/~krtous/papers/BHHSCI-arxiv.pdf
2. *A Note on Celestial Mechanics in Kerr Spacetime*, https://ar5iv.labs.arxiv.org/html/1311.3836
3. *Four-function generalization and separable structures of the Plebański spacetime with sources*, Eur. Phys. J. Plus (2025), https://link.springer.com/article/10.1140/epjp/s13360-025-07201-3
4. *The separation of the Hamilton-Jacobi equation for the Kerr metric*, https://www.cambridge.org/core/services/aop-cambridge-core/content/view/021A05353D8FA4AE9851F1599A39B6F0/S033427000001119Xa.pdf/separation_of_the_hamiltonjacobi_equation_for_the_kerr_metric.pdf
5. Crowell, *Killing Vectors*, General Relativity, Physics LibreTexts, https://phys.libretexts.org/Bookshelves/Relativity/General_Relativity_(Crowell)/07%3A_Symmetries/7.01%3A_Killing_Vectors
6. *Hidden symmetry group for particle orbits (timelike geodesics) in Schwarzschild spacetime*, arXiv:2604.16644, https://arxiv.org/html/2604.16644
7. *On a lower-dimensional Killing vector origin of irreducible Killing tensors*, JHEP07(2025)098, https://utf.mff.cuni.cz/~krtous/papers/LDKVOIKT-JHEP07(2025)098.pdf
8. *All analytic solutions for geodesic motion in axially symmetric space-times*, Eur. Phys. J. C (2022), https://doi.org/10.1140/epjc/s10052-022-10544-1

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Foundations and field equations › Geodesic motion › Constants of motion and Killing symmetries*

*Initially written Sep 17, 2026 · Edited: Oct 11, 2026; Oct 15, 2026*

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