Formulation
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. The Carter constant can be written as follows:
-
- ,
where is the latitudinal component of the particle's momentum, is the energy of the particle, is the particle's axial angular momentum, is the rest mass of the particle, and is the spin parameter of the black hole. 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".
Read more about this topic: Carter Constant
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The formulation was used as a motto by the English Nonconformist clergyman Richard Baxter (1615-1691)
“You do not mean by mystery what a Catholic does. You mean an interesting uncertainty: the uncertainty ceasing interest ceases also.... But a Catholic by mystery means an incomprehensible certainty: without certainty, without formulation there is no interest;... the clearer the formulation the greater the interest.”
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