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.