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Carter constant

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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. Carter's constant along with the energy , axial angular momentum , 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).

Formulation

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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.[1] The Carter constant can be written as follows:

,

where is the latitudinal component of the particle's angular momentum, is the conserved energy of the particle, is the particle's conserved axial angular momentum, is the rest mass of the particle, and is the spin parameter of the black hole.[2] Note that here 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 using its four-velocity as where is the four-momentum and is the Kerr metric. 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 . The angular momentum component along is which coincides with .

Because functions of conserved quantities are also conserved, any function of and the three other constants of the motion can be used as a fourth constant in place of . This results in some confusion as to the form of Carter's constant. For example it is sometimes more convenient to use:

in place of . The quantity is useful because it is always non-negative. In general any fourth conserved quantity for motion in the Kerr family of spacetimes may be referred to as "Carter's constant". In the limit, and , where is the norm of the angular momentum vector, see Schwarzschild limit below.

As generated by a Killing tensor

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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 (different than used above). In component form:

,

where is the four-velocity of the particle in motion. The components of the Killing tensor in Boyer–Lindquist coordinates are:

,

where are the components of the metric tensor and and are the components of the principal null vectors:

with

.

The parentheses in are notation for symmetrization:

Schwarzschild limit

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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. In this case one only needs , , and 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 in spherical coordinates where and , where and and where and similarly for , we have

.

Since and represent an orthonormal basis, the Hodge dual of in an orthonormal basis is

consistent with although here and are with respect to proper time. Its norm is

.

Further since and , upon substitution we get

.

In the Schwarzschild case, all components of the angular momentum vector are conserved, so both and are conserved, hence is clearly conserved. For Kerr, is conserved but and are not, nevertheless is conserved.

The other form of Carter's constant is

since here . This is also clearly conserved. In the Schwarzschild case both and , where are radial orbits and with corresponds to orbits confined to the equatorial plane of the coordinate system, i.e. for all times.

See also

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References

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  1. ^ Carter, Brandon (1968). "Global structure of the Kerr family of gravitational fields". Physical Review. 174 (5): 1559–1571. Bibcode:1968PhRv..174.1559C. doi:10.1103/PhysRev.174.1559.
  2. ^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. New York: W. H. Freeman and Co. p. 899. ISBN 0-7167-0334-3.