Maurer-Cartan perturbation theory and scattering amplitudes in general relativity
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2023
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Doctoral Thesis
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Abstract
We study formal perturbations of Minkowski spacetime in general relativity using techniques from algebraic deformation theory. This is based on a formulation of the Einstein equations as a Maurer-Cartan equation of a differential graded Lie algebra. We extend this algebraic structure to the conformal compactification of Minkowski spacetime, obtaining a sheaf of differential graded Lie algebras on the Einstein cylinder. This extension is regular in the sense that the Maurer-Cartan equation is symmetric hyperbolic including across the boundary of Minkowski spacetime. The differential describes the Minkowski background, it exhibits an explicit de Rham piece which simplifies homology calculations. With this differential graded Lie algebra at hand we can apply standard Maurer-Cartan perturbation theory. This provides a basic iteration scheme which we use to construct formal power series solutions of the Einstein equations, with precise control over the asymptotics of the solutions. Maurer-Cartan perturbation theory is organized through homology calculations, in particular the gauge character of the Einstein equations is naturally built into the setup. The homology in degree two is the obstruction space, it is finite-dimensional and can be identified with the physical mass and angular momentum charges. The Lie bracket induces a map from the space of linearized solutions to this obstruction space, measuring the charges generated by the nonlinear self-interaction. This map encodes a perturbative positive mass theorem, which we prove using the spinor formalism. The obstructions force us to adapt the basic Maurer-Cartan iteration scheme by a renormalization scheme, which is based on a parametrization of the obstruction space in terms of a family of Kerr-tails at spacelike infinity. Our setup allows us to make rigorous contact with the physics literature on scattering amplitudes. We show rigorously, in low order formal perturbation theory, that the gauge independent amplitudes describe the radiative null asymptotics of the formal solutions. Our approach is to introduce gauge independent, bounded multilinear operators, that are given by integrals over null hypersurfaces in Minkowski spacetime. When applied to the formal solutions of the Einstein equations, these operators measure the radiative null asymptotics of the solutions. We then prove that these operators have a momentum space kernel, and that this kernel is given by the physical scattering amplitudes.
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ETH Zurich
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09577 - Willwacher, Thomas / Willwacher, Thomas