In an extreme binary black hole system, an orbit will increase its angle of inclination (ι) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits; and develop an analysis that is independent of and complements radiation reaction models. For a Schwarzschild black hole, the polar orbits represent the abutment between the prograde and retrograde orbits at which Q is at its maximum value for given values of latus rectum (l) and eccentricity (e). The introduction of spin (S = |J|/M 2 ) to the massive black hole causes this boundary, or abutment, to be moved towards greater orbital inclination; thus it no longer cleanly separates prograde and retrograde orbits.To characterise the abutment of a Kerr black hole (KBH), we first investigated the last stable orbit (LSO) of a test-particle about a KBH, and then extended this work to general orbits. To develop a better understanding of the evolution of Q we developed analytical formulae for Q in terms ofl, e, andS to describe elliptical orbits at the abutment, polar orbits, and last stable orbits (LSO). By knowing the analytical form of ∂Q/∂l at the abutment, we were able to test a 2PN flux equation for Q. We also used these formulae to numerically calculate the ∂ι/∂l of hypothetical circular orbits that evolve along the abutment. From these values we have determined that ∂ι/∂l = − 122.7S − 36S 3 l −11/2 − 63/2S + 35/4S 3 l −9/2 − 15/2Sl −7/2 − 9/2Sl −5/2 . By taking the limit of this equation forl → ∞, and comparing it with the published result for the weak-field radiation-reaction, we found the upper limit on ∂ι/∂l for the full range ofl up to the LSO. Although we know the value of ∂Q/∂l at the abutment, we find that the second and higher derivatives of Q with respect tol exert an influence on ∂ι/∂l. Thus the abutment becomes an important analytical and numerical laboratory for studying the evolution 1 arXiv:1007.4189v4 [gr-qc] 19 Jan 2011of Q and ι in Kerr spacetime and for testing current and future radiation back-reaction models for near-polar retrograde orbits.

No abstract

No abstract

No abstract

We argue that there is nothing puzzling in the fact that the Hamiltonian formulation of a covariant theory, General Relativity, after a non-covariant change of field variables is not canonically related to the formulation based on the original variable, the metric tensor. Were such a puzzle to be "solved" it would lead to the conclusion that a covariant theory can be converted into a non-covariant one in many different ways and without consequence. The non-canonicity of transformations from covariant to non-covariant variables shows the need to work in the original variables so as to be able to restore the covariant gauge transformations in the Hamiltonian approach. Any modification of Dirac's procedure for the constrained Hamiltonian with the aim to prove the legitimacy of non-covariant changes of field variables, or rejection of Dirac's procedure as "not fundamental and undoubted" because it does not allow such changes (as recently suggested by Shestakova [CGQ, 28 (2011) 055009] ), is equivalent to the rejection of the covariance of General Relativity, and to surrender to such temptation is truly puzzling.