The Carter constant is a conserved quantity for motion around black holes in the general relativistic formulation of gravity.
Its SI base units are kg2⋅m4⋅s−2.
Carter's constant was derived for a spinning, charged black hole by Australian theoretical physicist Brandon Carter in 1968.
, and particle rest mass
provide the four conserved quantities necessary to uniquely determine all orbits in the Kerr–Newman spacetime (even those of charged particles).
Carter noticed that the Hamiltonian for motion in Kerr spacetime was separable in Boyer–Lindquist coordinates, allowing the constants of such motion to be easily identified using Hamilton–Jacobi theory.
is the latitudinal component of the particle's angular momentum,
is the particle's conserved axial angular momentum,
is the rest mass of the particle, and
is the spin parameter of the black hole.
denotes the covariant components of the four-momentum in Boyer-Lindquist coordinates which may be calculated from the particle's position
parameterized by the particle's proper time
Thus, the conserved energy constant and angular momentum constant are not to be confused with the energy
measured by an observer and the angular momentum
This results in some confusion as to the form of Carter's constant.
In general any fourth conserved quantity for motion in the Kerr family of spacetimes may be referred to as "Carter's constant".
is the norm of the angular momentum vector, see Schwarzschild limit below.
Noether's theorem states that each conserved quantity of a system generates a continuous symmetry of that system.
Carter's constant is related to a higher order symmetry of the Kerr metric generated by a second order Killing tensor field
The components of the Killing tensor in Boyer–Lindquist coordinates are: where
are the components of the metric tensor and
are the components of the principal null vectors: with The parentheses in
are notation for symmetrization: The spherical symmetry of the Schwarzschild metric for non-spinning black holes allows one to reduce the problem of finding the trajectories of particles to three dimensions.
to determine the motion; however, the symmetry leading to Carter's constant still exists.
Carter's constant for Schwarzschild space is: To see how this is related to the angular momentum two-form
represent an orthonormal basis, the Hodge dual of
in an orthonormal basis is consistent with
are with respect to proper time.
, upon substitution we get In the Schwarzschild case, all components of the angular momentum vector are conserved, so both
corresponds to orbits confined to the equatorial plane of the coordinate system, i.e.