Classical potential for general spinning bodies

TL;DR

The paper develops an on-shell, amplitude-based framework to derive the spin-dependent classical gravitational potential for general spinning bodies at leading order in $G$ and to all orders in spin. By mapping a one-particle EFT worldline action to three-point amplitudes and employing Hilbert-space matching, the authors compute the leading PN, $1$PM potential and verify Kerr black hole limits via unity Wilson coefficients. A key result is that finite-spin corrections cancel between Wilson-coefficient deformations and Hilbert-space boosts, so minimal coupling reproduces the Kerr potential even at finite spin; in the chiral basis, the minimal coupling amplitude factorizes into universal spin-independent and spin-structure parts, enabling a straightforward classical-spin limit. The work also establishes universality at tree level and, up to $s=2$, at one loop, while discussing how to maintain universality for higher spins with local contact terms. Overall, the approach provides a coherent, general framework to understand spin effects in gravitational scattering and offers a clean benchmark against Kerr black hole physics.

Abstract

In this paper we compute the spin-dependent terms of the gravitational potential for general spinning bodies at the leading Newton's constant $G$ and to all orders in spin. We utilize the on-shell approach, which extracts the classical potential directly from the scattering amplitude. For spinning particles, extra care is required due to the fact that the spin space of each particle is independent. Once the appropriate matching procedures are applied, taking the classical-spin limit we obtain the potential for general spinning bodies. When the Wilson coefficients are set to unity, we successfully reproduced the potential for the Kerr black hole. Interestingly, for finite spins, we find that the finite-spin deviations from Kerr Wilson coefficients cancel with that in the matching procedure, reproducing the Kerr potential without the need for taking the classical-spin limit. Finally, we find that when cast into the chiral basis, the spin-dependence of minimal coupling exhibits factorization, allowing us to take the classical-spin limit straight forwardly.

Figures (2)

• Figure 1: The graviton exchange diagram between source $a$ and $b$ that yields the leading $\frac{1}{q^2}$ singularity, which is responsible for the classical potential.
• Figure 2: The triangle cut used to compute the 2 PM potential.