First Law of Mechanics for Black Hole Binaries with Spins

TL;DR

The paper extends the first law of binary mechanics to spinning point-particle binaries by deriving a direct link between the redshift observable and the (Fokker-type) Hamiltonian, valid to linear order in spin and including spin–spin couplings of arbitrary order. It develops a canonical ADM Hamiltonian framework for spinning bodies, constructs the Fokker Lagrangian/Hamiltonian by eliminating field degrees of freedom, and proves a first-law relation relating $M$, $J$, $z_A$, $m_A$, and $S_A$ for circular orbits. The authors then compute spin–orbit contributions to the redshift through 2.5PN order, confirm consistency with near-zone PN results, and specialize to corotating binaries by introducing Kerr-like constitutive relations, yielding explicit expressions for proper rotation frequencies up to 2PN. The work provides a versatile foundation for cross-checks with gravitational self-force calculations, numerical relativity, and effective-one-body models, and outlines future extensions to higher spins, second-order self-force, and broader quasi-equilibrium analyses.

Abstract

We use the canonical Hamiltonian formalism to generalize to spinning point particles the first law of mechanics established for binary systems of non-spinning point masses moving on circular orbits [Le Tiec, Blanchet, and Whiting, Phys. Rev. D 85, 064039 (2012)]. We find that the redshift observable of each particle is related in a very simple manner to the canonical Hamiltonian and, more generally, to a class of Fokker-type Hamiltonians. Our results are valid through linear order in the spin of each particle, but hold also for quadratic couplings between the spins of different particles. The knowledge of spin effects in the Hamiltonian allows us to compute spin-orbit terms in the redshift variable through 2.5PN order, for circular orbits and spins aligned or anti-aligned with the orbital angular momentum. To describe extended bodies such as black holes, we supplement the first law for spinning point-particle binaries with some "constitutive relations" that can be used for diagnosis of spin measurements in quasi-equilibrium initial data.