Kerr Black Hole Dynamics from an Extended Polyakov Action

TL;DR

The paper introduces a covariant hypersurface model for spinning black holes based on an extended Polyakov action on a rigid worldvolume $\mathbb{S}^2\times \mathbb{R}$, coupling to the background metric and Riemann tensor. At leading order, the $(1,1)$ mode of the worldvolume dynamics reproduces the Kerr three-point amplitude with a spin-dependent structure $A_3^{(1,1)}(v,a,k) = -i\kappa\big( (mv\cdot \varepsilon)^2 \cosh(k\cdot a) + i (mv\cdot \varepsilon)(k\cdot S\cdot \varepsilon) \frac{\sinh(k\cdot a)}{k\cdot a} \big)$, and fixing the couplings $\xi_{1,2,4}$ to Kerr values yields a direct landfall on the known Kerr amplitude. The framework further generates a family of generalized worldline theories via higher modes, with amplitudes expressed as entire functions; it yields spin-resummed bending angles and a stationary metric consistent with the Einstein equations at first order, and reveals a ring singularity structure tied to specific basis elements. The work establishes a novel action principle for spin in gravity that can be leveraged to compute higher-point amplitudes (e.g., gravitational Compton) and to study the Mathisson-Papapetrou-Dixon dynamics in a post-Minkowskian expansion, with potential extensions to higher dimensions and strong-coupling regimes. Overall, the approach provides a covariant, non-multipole-based route to characterize black hole spin and interactions, offering new tools for precision gravitational scattering and binary-merger phenomenology.

Abstract

We examine a hypersurface model for the classical dynamics of spinning black holes. Under specific rigid geometric constraints, it reveals an intriguing solution resembling expectations for the Kerr Black three-point amplitude. We explore various generalizations of this formalism and outline potential avenues for employing it to analyze spinning black hole attraction.

Figures (2)

• Figure 1: Spinning hypersurface from worldvolume (left) to the disc in target spacetime (right), with coordinate axes labeled. The radius of the black hole is $a_{t(l,j)}$.
• Figure 2: The related vectors are $v_{2}^\mu=(1,0,0,0)$, $v_{1}^\mu=(y,\sqrt{y^2-1},0,0)$, $b^\mu=(0,0,-|b|,0)$, $a^\mu=(0,0,0,|a|)$ .