First Law of Mechanics for Spinning Compact Binaries: Dipolar Order

TL;DR

This paper derives a first law of mechanics for spinning compact binaries at dipolar order using a variational approach, linking it to Hamiltonian formulations and setting the stage for including quadrupolar effects.

Contribution

It introduces a new variational formula for spacetimes with Killing vectors and applies it to derive a first law for spinning binary systems at dipolar order.

Findings

Derived a variational formula for spacetimes with Killing vectors.

Established the equivalence of the new first law with previous Hamiltonian results.

Set the groundwork for extending the law to quadrupolar order.

Abstract

Building upon the Noether charge formalism of Iyer and Wald, we derive a variational formula for spacetimes admitting a Killing vector field, for a generic energy-momentum distribution with compact support. Applying this general result to the particular case of a binary system of spinning compact objects moving along an exactly circular orbit, modelled using the multipolar gravitational skeleton formalism, we derive a first law of compact binary mechanics at dipolar order. We prove the equivalence of this new result with the canonical Hamiltonian first law previously derived for binary systems of spinning compact objects, for spins colinear with the orbital angular momentum. This paper paves the way to an extension of the first law of binary mechanics to the next quadrupolar order, thereby accounting for the spin-induced and tidally-induced deformability of the compact bodies.