Reimagining Gravity: Generalized Symmetries, Double Copy, and Spinning Black Holes

Author: Kim, Joon-Hwi

Year: 2027

Degree: Dissertation (Ph.D.)

Advisor: Cheung, Clifford W.

Committee Members: Simmons-Duffin, David; Teukolsky, Saul A.; Parra-Martinez, Julio; Cheung, Clifford W.

Option: Physics

DOI: 10.7907/zx2g-0673

Abstract

General relativity is a century-old subject. Yet modern explorations through generalized symmetries, scattering amplitudes, and effective field theory have raised open problems, motivating a reassessment of conventional views on gravitation, spacetime, and spin. First, do generalized symmetries exist in dynamical gravity as a low-energy effective field theory? Second, is there a field-theoretic explanation for the tree-level double copy relationship between general relativity and Yang-Mills theory? Third, what are the exact equations of motion or Lagrangians of four-dimensional spinning black holes in external fields, in their point-particle effective theory?

In this dissertation, three different perspectives on gravity are developed to shed light on each of these puzzles.

First, we view gravity as a gauge theory of the Lorentz group. This refers to the vielbein formulation of gravity. It implies that the effective field theory of gravity, at the fully nonlinear level, exhibits a one-form symmetry valued in the center of the local Lorentz group. The charged objects are the Wilson loops of the spin connection, i.e., spin holonomies in various representations. The symmetry operator is a topological operator associated with a certain area measured in Planck units. Their linking admits an elegant physical interpretation in classical gravitation: the symmetry operator materializes a tetradic cousin of cosmic string that induces quantized spin precession angles as a gravitational Aharonov-Bohm effect. Notably, our construction identifies a new symmetry of the standard model at scales below the lightest neutrino mass. The absence of global symmetries in quantum gravity suggests that this gravitational one-form symmetry is either gauged or explicitly broken, the latter of which mandates the existence of fermions.

Second, we investigate how far gravity can be viewed as a gauge theory of diffeomorphisms. Based on an established result on Born-Infeld theory, we point out that color-kinematics duality at the equations of motion level mandates treating the diffeomorphism algebra formally like a gauge algebra in the internal, fiberwise sense of Yang-Mills theory. We show that this seemingly radical prescription admits a concrete construction within a mathematically consistent framework: field theory of a dynamical frame field. We demonstrate that color-kinematics duality systematically defines diffeomorphism gauge connection, diffeomorphism covariant derivative, and even diffeomorphism Wilson line. In this context, we revisit the teleparallel formulation of gravity and evaluate its relevance to double copy in three or four dimensions. The Misner string is reinterpreted as a topological classical solution in any diffeomorphism gauge theory, endable on Newman-Unti-Tamburino charges.

Third, we view four-dimensional gravity as a nonlinear interaction between self-dual and anti-self-dual parts. We begin by reviewing the simplicity of self-dual gravity in the context of double copy, Pleba'nski's second heavenly equation, and Mason-Newman or Lax pair formulations. This motivates the program of understanding full gravity by perturbing away from the simpler self-dual sector, an insight often advocated by twistor theorists. Chiral formulations of gravity are discussed in this context, via both Lorentz and diffeomorphism gauge theory perspectives. In the former, we review Pleba'nski gravity. In the latter, we derive a Chalmers-Siegel version of teleparallel gravity that perturbs away from self-dual gravity in the Mason-Newman characterization.

We then discuss how this view on four-dimensional gravity can be applied to the derivation of four-dimensional black hole solutions and their effective dynamics, eventually leading to an identification of an integrable subsector of the general-relativistic spinning black hole binary problem.

We begin by a pedagogical demonstration of the simplicity of self-dual backgrounds. The motion of a relativistic charged particle in the background of a self-dual dyon is shown to be maximally superintegrable by being isomorphic to the hydrogen atom. Similarly, the motion of a relativistic particle in a self-dual black hole background is maximally superintegrable by being isomorphic to the Kepler problem.

A precise relation between these two backgrounds is established by the Kerr-Schild double copy. We derive a Kerr-Schild metric for the self-dual Taub-Newman-Unti-Tamburino solution by double copying a gauge potential of the self-dual dyon. A simple complexified coordinate transformation is constructed to explicitly show that this new metric is diffeomorphic to the well-known Gibbons-Hawking instanton metric.

By taking this result as a crucial lemma, it is then shown that the Kerr metric represents the nonlinear superposition of self-dual and anti-self-dual Taub-Newman-Unti-Tamburino solutions. This elevates the Newman-Janis algorithm to a rigorous derivation of the Kerr metric and explicates its origin. More generally, the five-parameter family of solutions including Kerr-Newman and Kerr-Taub-Newman-Unti-Tamburino black holes represents systems of Taub-Newman-Unti-Tamburino instantons and chiral dyons.

Based on this realization, we propose a probe counterpart of the Newman-Janis algorithm, which uniquely constrains the effective equations of motion of Kerr and Kerr-Newman black holes in external gravitational and electromagnetic fields within the self-dual sector. It is shown that these effective equations of motion are maximally superintegrable in the backgrounds of self-dual black holes, via dynamical Newman-Janis shifts of conserved charges. This identifies a hidden symmetry as an infrared principle that can pinpoint black holes among all massive spinning objects in the point-particle effective theory, in the self-dual sector.

In the Lagrangian formulation, we construct a unique class of effective worldline actions for Kerr and Kerr-Newman black holes in generic, non-self-dual external fields. We provide their explicit all-orders formulae in terms of expansions in curvature and spin length. This is facilitated by developing an "in-in" formalism for all-orders geodesic deviation as an alternative to the Synge framework.

A universal framework arises for spinning-particle mechanics in which spin is added to spacetime as the imaginary part while chirality and holomorphy are inherently linked. We demonstrate how exact spinning black hole equations of motion are derived beyond the self-dual sector.

Finally, we derive the classical Compton scattering amplitudes of spinning black holes, directly from the explicit effective point-particle actions by implementing a chiral worldline perturbation theory in which the plus-helicity spin exponentiation is Lagrangian-level manifest to all orders. Our amplitudes are free of spurious poles and exhibit correct residues on physical factorization channels, for both same-helicity and mixed-helicity configurations. Physically, this computation views the Kerr black hole as a dynamical Taub-Newman-Unti-Tamburino pair and systematically understands it by perturbing away from the self-dual Taub-Newman-Unti-Tamburino worldline, while providing an invariant characterization of its dynamics.

These explorations spark a new approach to spinning black hole dynamics that makes maximal use of the simplicity and the manifest Newman-Janis property of the self-dual sector, finding applications to post-Minkowskian gravity.

Files