Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black hole: A new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due to test-particle spin and the gravitational self-force

The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. In the test-mass limit, well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes. In the finite-mass case, there are a large variety of ways to define an ISCO in a post-Newtonian (PN) context. Here I generalize the gauge-invariant ISCO condition of Blanchet and Iyer Classical Quantum Gravity 20, 755 (2003) to the case of spinning (nonprecessing) binaries. The Blanchet-Iyer ISCO condition has two desirable and unexpected properties: (1) it exactly reproduces the Schwarzschild ISCO in the test-mass limit, and (2) it accurately approximates the recently calculated shift in the Schwarzschild ISCO frequency due to the conservative-piece of the gravitational self-force L. Barack and N. Sago, Phys. Rev. Lett. 102, 191101 (2009). The generalization of this ISCO condition to spinning binaries has the property that it also exactly reproduces the Kerr ISCO in the test-mass limit (up to the order at which PN spin corrections are currently known). The shift in the ISCO due to the spin of the test-particle is also calculated. Remarkably, the gauge-invariant PN ISCO condition exactly reproduces the ISCO shift predicted by the Papapetrou equations for a fully relativistic spinning particle. It is surprising that an analysis of the stability of the standard PN equations of motion is able (without any form of ``resummation'') to accurately describe strong-field effects of the Kerr spacetime. The ISCO frequency shift due to the conservative self-force in Kerr is also calculated from this new ISCO condition, as well as from the effective-one-body Hamiltonian of Barausse and Buonanno Phys. Rev. D 81, 084024 (2010). These results serve as a useful point of comparison for future gravitational self-force calculations in the Kerr spacetime.