Observables and Unconstrained Spin Tensor Dynamics in General Relativity from Scattering Amplitudes

TL;DR

This work extends the amplitude-based approach to spinning bodies in general relativity by incorporating an unconstrained spin tensor with a dynamical boost degree of freedom $\mathbf{K}$ in addition to the spin vector $\mathbf{S}$. Through field theory and worldline formalisms, it derives three- and four-point amplitudes, two-body amplitudes, the eikonal phase, and an effective long-range Hamiltonian that include both $\mathbf{S}$ and $\mathbf{K}$, and shows that $\mathbf{K}$ generally affects waveforms and conservative dynamics unless special Wilson coefficients decouple it to recover Kerr-like physics. The paper provides explicit toy models (Newtonian bound states) and an exotic black-hole solution (Rasheed–Larsen) to illustrate the physical content of $\mathbf{K}$ and connects these degrees of freedom to dynamical multipole moments and potential Goldstone-like interpretations. It also demonstrates the consistency of the eikonal construction with the Hamiltonian and highlights the necessity of including $\mathbf{K}$ for accurate modeling of both conventional and exotic compact objects in gravitational-wave physics. The framework lays the groundwork for unconstrained, high-order spin calculations and offers a path to identifying new light degrees of freedom in compact-object dynamics detectable by future observations.

Abstract

In a previous Letter, we showed that physical scattering observables for compact spinning objects in general relativity can depend on additional degrees of freedom in the spin tensor beyond those described by the spin vector alone. In this paper, we provide further details on the physics of these additional degrees of freedom, whose commutation relations and Poisson brackets are inherited from the underlying Lorentz symmetry, and on their consequence on observables. In particular, we give the waveform at leading order in Newton's constant and up to second order in the components of the spin tensor, and the conservative impulse, boost and spin kick, exhibiting spin magnitude change, through next-to-leading-order in Newton's constant and third order in the components of the spin tensor. We provide explicit examples -- a Newtonian two-particle bound system and a certain black-hole solution in an exotic matter-coupled gravitational theory -- that exhibit these degrees of freedom and are described by our four-dimensional and worldline field theories. We also discuss connections between these degrees of freedom and dynamical worldline multipole moments. We construct effective two-body Hamiltonians, we demonstrate explicitly that the extra degrees of freedom beyond the spin vector are necessary to describe the complete dynamics, and we explicitly remove certain unphysical singularities. Moreover, we show that the previously proposed eikonal (or radial action) formula correctly captures observables derived from the classical Hamiltonian. Finally, we comment on possible descriptions of the additional degrees of freedom from the perspective of Goldstone's theorem.

Figures (7)

• Figure 1: Scattering of a particle of momentum ${\bm p}_2$ off a two-body Newtonian bound state of internal angular momentum $\bm S$ and Laplace-Runge-Lenz vector $\bm A$ pointing along the $\hat{\bm x}$ axis. Both the particle and the bound state are moving along the $\hat{\bm z}$ direction.
• Figure 2: The three- and four-point Compton amplitudes. All momenta are outgoing.
• Figure 3: Diagrammatic representation of Eq. \ref{['eq:sewing']}. The dashed lines across the graviton propagators indicate on-shell conditions and summation over physical states.
• Figure 4: Plots showing various components of $32\pi \times 10^6\times {\hat{h}}_+$ at location given by the angles $(\theta, \phi)=(7, 4)\pi/10$, for $\bm S = (\cos\pi/4, \sin\pi/4, 0)$ and unit $\bm K$ vector written in terms of $c\psi_k\equiv\cos\psi_K$ and $s\psi_K\equiv\sin\psi_K$ for particles of equal masses $m_1=m_2=1$ with $u_i = \frac{1}{\sqrt{24}}(5, 0, 0, \pm 1)$. We use the covariant impact parameter defined in Ref. Bern:2020buy and choose it to be $\bm b_{\text{cov}} = (50, 0, 0)$ in units of the inverse particles' mass. The retarded-time axis is parameterized by $\hat{\tau} = \tau / |\bm b_{\text{cov}}|$.
• Figure 5: Plots showing various components of $32\pi \times 10^6\times {\hat{h}}_\times$ at location given by the angles $(\theta, \phi)=(7, 4)\pi/10$, for $\bm S = (\cos\pi/4, \sin\pi/4, 0)$ and unit $\bm K$ vector written in terms of $c\psi_k\equiv\cos\psi_K$ and $s\psi_K\equiv\sin\psi_K$ for particles of equal masses $m_1=m_2=1$ with $u_i = \frac{1}{\sqrt{24}}(5, 0, 0, \pm 1)$. We use the covariant impact parameter defined in Ref. Bern:2020buy and choose it to be $\bm b_{\text{cov}} = (50, 0, 0)$ in units of the inverse particles' mass. The retarded-time axis is parameterized by $\hat{\tau} = \tau / |\bm b_{\text{cov}}|$.