Canonical Formulation of Spin in General Relativity

TL;DR

The work addresses the challenge of incorporating spin into the canonical ADM formulation of general relativity to enable accurate conservative dynamics in the post-Newtonian regime. It develops an action-based approach that yields a clean, canonical spin structure at linear order and validates it through an order-by-order PN construction, confirming consistency with Noetherian conserved quantities and the Poincaré algebra. The thesis then extends the formalism to include spin-induced quadrupole deformations and provides explicit next-to-leading order spin Hamiltonians (spin–orbit, spin(1)–spin(2), and spin(1)–spin(1)) up to $3$PN for maximal spins, with generalization to neutron stars via the quadrupole parameter $C_Q$. These results significantly advance analytic modeling of spinning compact binaries, supporting gravitational wave predictions and integration into effective-one-body frameworks, while outlining directions for future dissipative and higher-multipole extensions.

Abstract

The present thesis aims at an extension of the canonical formalism of Arnowitt, Deser, and Misner from self-gravitating point-masses to objects with spin. This would allow interesting applications, e.g., within the post-Newtonian (PN) approximation. The extension succeeded via an action approach to linear order in the single spins of the objects without restriction to any further approximation. An order-by-order construction within the PN approximation is possible and performed to the formal 3.5PN order as a verification. In principle both approaches are applicable to higher orders in spin. The PN next-to-leading order spin(1)-spin(1) level was tackled, modeling the spin-induced quadrupole deformation by a single parameter. All spin-dependent Hamiltonians for rapidly rotating bodies up to and including 3PN are calculated.